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Guzm´an +,1, 2 and Ke Zhang +5 +1Instituto de Astrof´ısica, Pontificia Universidad Cat´olica de Chile, Av. Vicu˜na Mackenna 4860, 782-0436 Macul, Santiago, Chile +2N´ucleo Milenio de Formaci´on Planetaria (NPF), Chile +3Millennium Institute for Astrophysics, Chile +4naXys, Department of Mathematics, University of Namur, Rue de Bruxelles 61, 5000 Namur, Belgium +5Department of Astronomy, University of Wisconsin-Madison, 475 N. Charter Street, Madison, WI 53706, USA +ABSTRACT +ALMA observations of the disk around HD 163296 have resolved a crescent-shape substructure at +around 55 au, inside and off-center from a gap in the dust that extends from 38 au to 62 au. In this +work we propose that both the crescent and the dust rings are caused by a compact pair (period ratio +≃ 4 : 3) of sub-Saturn-mass planets inside the gap, with the crescent corresponding to dust trapped +at the L5 Lagrange point of the outer planet. This interpretation also reproduces well the gap in the +gas recently measured from the CO observations, which is shallower than what is expected in a model +where the gap is carved by a single planet. Building on previous works arguing for outer planets at +≈ 86 and ≈ 137 au, we provide with a global model of the disk that best reproduces the data and +show that all four planets may fall into a long resonant chain, with the outer three planets in a 1:2:4 +Laplace resonance. We show that this configuration is not only an expected outcome from disk-planet +interaction in this system, but it can also help constraining the radial and angular position of the +planet candidates using three-body resonances. +Keywords: protoplanetary disks — planet–disk interactions — hydrodynamics — planets and satellites: +dynamical evolution and stability — radiative transfer +1. INTRODUCTION +Substructures are ubiquitous in protoplanetary disks, +particularly in the dust density distribution exhibited +by high angular resolution observations (Andrews 2020; +Bae et al. 2022). The Atacama Large Millimeter Ar- +ray (ALMA) has revealed a variety of substructures, +whereas a large population of rings and gaps are shown +in continuum observations, to a lesser extent, in molec- +ular line emissions (e.g., van der Marel et al. 2019). +The advances in spatial resolution have been able to +resolve non-axisymmetric substructures within gaps, in- +cluding systems such as PDS 70 (Benisty et al. 2021), +HD 163296 (Isella et al. 2018), HD 100546 (P´erez et al. +2020), HD 97048 (Pinte et al. 2019), and LkCa 15 (Long +et al. 2022). These substructures may be due to embed- +ded planets induced by gravitational interactions (e.g., +Bae et al. 2022), which vary depending on the emis- +sion shape. +Point-like emissions are generally associ- +ated with an accreting planet surrounded by a circum- +planetary disk (CPD) (Perez et al. 2015; Szul´agyi et al. +2018), while crescent shapes may be related to the sta- +ble Lagrange points L4 and L5 of a star-planet system +(Rodenkirch et al. 2021; Long et al. 2022) or vortices. +These hypotheses frequently assume that a single sub- +structure is caused by a single planet, often in a Jo- +vian mass regime. +However, we have recently shown +in Garrido-Deutelmoser et al. (2022), that a pair of +lower-mass and gap-sharing planets can sculpt compact +and/or elongated vortices within the gap that last for +several thousands orbits. +The disk surrounding HD 163296 contains ringed +structures in the mm-continuum (Isella et al. 2018) and +several molecular tracers (Law et al. 2021; Zhang et al. +2021). In particular, inside the dust density gap that ex- +tends from 38 to 62 au, a crescent-shaped substructure +resides at around 55 au (Teague et al. 2021). Recently, +it was suggested that the emission comes from the dust +trapped around the stable point L5 of a Jupiter mass +planet orbiting at 48 au (Rodenkirch et al. 2021). Even +though this method seems to reproduce the broad fea- +tures of the dust continuum distribution, two aspects +remain unclear: +arXiv:2301.13260v1 [astro-ph.EP] 30 Jan 2023 + +ID2 +Garrido-Deutelmoser, et al. +1. the crescent feature resides at 55 au, off-center +from the dust gap, while the L5 point is co-orbital +to the planet at 48 au. Varying the planet’s eccen- +tricity to account for this shift is unlikely to help +as the stability of the crescent is damaged, leading +to its prompt disruption. +2. the Jupiter-mass planet needed to retain observ- +able amounts of dust at L5 and open a wide enough +gap in the dust, is expected to open a deep gap1 +(Duffell 2020). This prediction disagrees with the +recent results provided by Zhang et al. (2021), who +found that the dust density gap has a correspond- +ing CO gap ∼ 10 times shallower than the predic- +tions involving a Jupiter. The local gas depletion +depends on the planetary mass (∝ m−2 +p ) (Kana- +gawa et al. 2015), whereby opting for a lower mass +planet to carve a shallower gap, may not produce +a sufficient gravitational interaction to enforce the +dust trapping at L5. +In this work, we propose a that a compact pair of +sub-Saturn-mass planets can solve these issues, simul- +taneously accounting for the dust emission (the shifted +crescent and dust rings) and shallow gap in the CO. +This scenario is largely motivated by our recent work in +Garrido-Deutelmoser et al. (2022) where we showed that +a compact pair of gap-sharing planets generally lead to +nonaxisymmetric substructures like that observed in HD +163296. +2. SETUP +The hydrodynamics simulations and radiative trans- +fer calculations in this work largely follow the scheme +in Rodenkirch et al. (2021) and are only briefly sum- +marized here. We carried out 2D hydrodynamic simu- +lations using the FARGO3D multifluid code (Ben´ıtez- +Llambay et al. 2019; Masset 2000; Weber et al. 2019) +to produce gas and dust density distribution for a fidu- +cial disk model. The resulting density maps are read +into the RADMC3D code (Dullemond et al. 2012) to +calculate the radiative transfer image at λ ∼ 1.25 mm. +We use this image and the HD 163296 template (that +contain all the technical properties of the observation) +in the SIMIO2 package to achieve the synthetic ALMA +observation comparable to the dust continuum observa- +tion provided by Isella et al. (2018). +1 An increase in the local viscosity to produce a much shallower gap +comes at the expense of reducing the lifetime of the L5 crescent, +or even prevent its formation in the first place (Rodenkirch et al. +2021). +2 https://www.nicolaskurtovic.com/simio +2.1. Hydrodynamic Simulations +The initial surface density profiles for the gas (sub- +index g) and dust (sub-index d) are given by +Σg/d = Σg/d,0 +� r +r0 +�−0.8 +exp +� +− +� +r +rc,g/d +�γg/d� +, +(1) +where we set r0 = 48 au, the initial surface density +Σg,0 = 37.4 gr cm−2, the cut-off radius rc,g = 165 au, +and the exponent γg = 1. +Similarly, for the dust we +set Σd,0 = Σg,0/100 = 0.374 gr cm−2, rc,d = 90 au, +and γd = 2. We include the evolution for five indepen- +dent dust species. These grains have sizes in cm of 0.02, +0.071, 0.13, 0.26, and 1.92. We use an aspect ratio of +h(r) = h0(r/r0)f with h0 = 0.05 and a flaring index +f = 0.25, which implies a mid-plane temperature profile +described by T = 25(r/r0)−0.5 K. This setup coincides +with that from Rodenkirch et al. (2021). +The disk extends from rin = 5 au to rout = 197 au, +implying an initial disk mass of ∼ 0.15 M⊙. The compu- +tational domain is composed of nr = 512 logarithmically +spaced radial cells and nθ = 768 equally spaced cells in +the azimuthal [0, 2π] domain. We include a radially vari- +able viscosity of the standard parameter α (Shakura & +Sunyaev 1973) as +α(r) = αin − αin − αout +2 +� +1 + tanh +�r − ξ +σr0 +�� +, +(2) +where the inner and outer viscosities are αin = 1 × 10−4 +and αout = 5 × 10−3, ξ = 144 au indicate the midpoint +of the transition and σ = 1.25 defines the slope. Similar +to the values provided by Liu et al. (2018). +A system of 4 planets was embedded. The location of +the two outer ones is indicated in Teague et al. (2018) +through kinematic detections. The third planet’s posi- +tion (i.e., the inner planet of the outer pair) is strongly +associated with the potential velocity kink reported by +(Pinte et al. 2020). The two inner planets are tightly +packed and their parameters (masses and orbits) were +derived numerically guided by the results in Garrido- +Deutelmoser et al. (2022), where it was found that the +planets should be gap-sharing with forming vortices at +their Lagrange points, implying a condition in the plan- +etary separation3 of: +∆a ≲ 4.6H ≃ 11.5 au, +(3) +where H the scale high of the disk. In turn, the masses +are constrained by the width of the gap. After a few +3 This expression has been tested for planets with masses near the +thermal mass of Mth = M⋆h3 ∼ 0.25MJ ≃ 80M⊕. + +HD 163296: crescent and resonant chain +3 +dozen simulations attempting to match disk morphol- +ogy in continuum observations at the ∼ 48 au region, +we choose to place the planets at a1 = 46, a2 = 54, +a3 = 84.5, and a4 = 137 au with their respective masses +of M1 = 85M⊕, M2 = 60M⊕, M3 = 0.4MJup, and +M4 = 1MJup. The four bodies can gravitationally in- +teract between them, but they do not feel the disk. We +ran an extra model to compare against previous works, +substituting a Jupiter at 48 au instead of the inner pack- +age of planets. Both cases were evolved for 0.48 Myrs, +equivalent to 2000 orbits of the innermost planet. +2.2. Radiative Transfer +We convert the 2D dust surface density into a 3D vol- +ume density assuming the vertical approximation given +by +ρdj(r, φ, θ) = Σ(r, φ) +√ +2πHdj +exp +� +− z2 +2H2 +dj +� +, +(4) +where z = r cos(θ) and the dust settling follows the diffu- +sion model Hdj = +� +Dz/(Dz + Stj)H, with Dz = 0.6α +the vertical diffusion coefficient, and Stj the Stokes num- +ber of the species j (Weber et al. 2022). We assume an +intrinsic volume density for the particles ρs = 2 gr cm−3 +and a power law for the grain size distribution, such that +n(a) ∝ a−3.5. We assumed a dust composition of 20% +amorphous carbon, 20% water ices and 60% silicates, +where the corresponding dust opacities were computed +with the code provided by (Bohren & Huffman 1983). +The polar direction is distributed in 64 equally spaced +cells and extended in [80.6◦, 99.4◦] inclination domain. +We use nphot = 108 photon packages to calculate the +dust temperature, and nscatt = 107 photon packages to +trace the thermal emission. We use a full anisotropic +scattering with polarization treatment. The system is +assumed to be at a distance of 101 pc with a central star +of mass 1.9 M⊙ and effective temperature Teff = 9, 330 +K. The inclination is taken to be i = 46◦ and position +angle PA= 133◦. +2.3. Synthetic Observations +We use SIMIO that contains a suit of functions for +CASA 5.6.2. We select the template designed for HD +163296 to create images with the same uv-coverage as +observation from Isella et al. (2018). We set the rescale +flux option in 0.4 to get similar intensities. In addition, +we add simple thermal noise4 of level 12mJy to finally +get RMS noise of 0.022 mJy beam−1. +4 https://simio-continuum.readthedocs.io/en/main/tutorials/ +tutorial 3.html +3. LOCAL GAP DEPLETION +Figure 1 shows the surface density after ∼ 0.5 Myr +(2,000 orbits) for the single-Jovian case (panel a) and the +two sub-Saturns with masses 85M⊕ and 60M⊕ (panel +b). The corresponding azimuthally-averaged profiles in +panel (c) show that ∼ 95% of gas is depleted for the +single-Jovian, while only ∼ 55% is depleted for the two- +planet case. Despite of their lower masses the planets +pair creates the same gap width as the Jovian. +This +shallower gaps for fixed gap width are expected in com- +pact multi-planet systems due the planet lower masses +(depth Σgap/Σ0 ∝ M −2 +p ) and angular flux transferred by +the neighbouring planets (Duffell & Dong 2015; Garrido- +Deutelmoser et al. 2022). +As argued by Zhang et al. (2021), if a Jupiter-mass +planet had opened the corresponding CO gap, it would +be 10 times deeper than what is actually observed. In- +stead, by embedding two planets we can alleviate these +differences and largely reduce this discrepancy as shown +in panel (c). Therefore, the depletion values would be +closer to the results from observations. In order to bet- +ter quantify this, we compare the CO column density +gaps with the surface density from both models5. As +shown in Figure 2, this approach reproduces the gas gap +reasonably well in the two-planet case, largely matching +the depths and widths with a small offset of the peaks +by ∼ 4 au. In turn, the single-Jovian case is too deep +compared to the observations as expected. +Beyond the third planet at ∼ 85 au, neither of the two +models (single Jovian or compact pair) is able to repro- +duce the gas gaps. +This was already noted in Zhang +et al. (2021) when comparing with the models from +Teague et al. (2021) and it may partly be explained by +the presence of a CO snowline at 65 au. Outside the +mid-plane CO snowline, CO freezes out at the disk mid- +plane and therefore the CO gap properties (e.g., width +and depth) may deviate from that of gas gaps due to +vertical temperature and CO abundance variations. We +recall that our work mostly focuses on the gap and cres- +cent at ∼ 50 au where these issues can be more securely +avoided. +4. THE CRESCENT FEATURE +A closely-packed planet pair can directly affect the +gas around each other’s coorbital regions by deposit- +ing angular momentum from wave steepening and sub- +sequent shocks. These shocks may strengthen the vor- +tencity around the stable L4 and L5 Lagrange points, +5 The Σ profiles were processed under smooth function methodol- +ogy described described in Appendix A. + +4 +Garrido-Deutelmoser, et al. +100 +50 +0 +50 +100 +x [au] +100 +50 +0 +50 +100 +y [au] +(a) +100 +50 +0 +50 +100 +x [au] +(b) +0 +30 +60 +90 +120 +150 +180 +r [au] +0.1 +1.0 +� / +0 +� +(c) +single-planet gap +two-planets gap +0.2 +0.5 +0.8 +1.1 +1.4 +1.7 +2.0 +2.3 +log10( ) +[gr cm 2] +Figure 1. Panels show a time evolution after 2000 orbits at 48 au (∼ 4.8 × 105 yrs). (a) Surface density Σ maps in log-scale for +a single Jupiter planet at 48 au. (b) The same as (a), but for a two-planet system with 85M⊕ and 60M⊕ instead the Jupiter. +The crosses denote the position of the planets and the white lines indicate their orbits. The disk rotates in a clockwise direction. +(c) Azimuthally averaged Σ/Σ0 profiles for both models. +thus enabling the effective gas and dust trapping for at +least thousands of orbital periods (Garrido-Deutelmoser +et al. 2022). +The overdensity around either L4 or L5 for the inner +and outer planet is a highly dynamic problem, with the +most prominent structure chaotically alternating loca- +tion. This said, we observe that for several combina- +tions of parameters (surface density, aspect ratio, plan- +etary masses, and so on), the outer planet often retains +large amounts of material around L5. The depicted be- +havior would leave an off-center substructure inside the +gap that greatly reproduces the distinctive emission in +the disk as shown in Figure 3. More quantitatively, our +models matches the observed azimuthal extent of ∼ 45◦ +and the radial intensity profiles passing through the cres- +cent (peaks and troughs inside the ring at ∼ 68 au; see +Appendix B for more details). +4.1. Dynamical behavior +Figure 4 compares the gas and dust density distribu- +tion for different dust grain sizes. All the dust species +show a clear shared gap between the two inner planets. +In the largest size, the co-orbital regions of each one +display an overdensity at L5, but the more prominent +substructure belongs to the second planet. +This out- +put has taken into account two conditions. The choice +of planetary masses in the inner system must be lower +for the body orbiting the substructure, and the pres- +ence of the two outer planets. If we neglect either, the +Lagrange points can still trap dust, but the mass dis- +tribution may change to make some L4 or the inner L5 +the most prominent. As shown in Garrido-Deutelmoser +et al. (2022), this evolution is highly dynamic and ir- +regular so that the overdensities around L4 and L5 con- +0 +30 +60 +90 +120 +150 +180 +210 +240 +r [au] +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +Normalized gap depth +CO gaps in Ke Zhang 2021 +single-planet gas gaps +two-planets gas gaps +Figure 2. Comparison of column density gap in C18O (2−1) +line observation found by Zhang et al. (2021) with gas gaps +derived from surface density profiles in this work. +stantly change. However, the final morphology in our +configuration comes from the early stages of evolution. +Theoretically, the stable Lagrange point L5 is located +at 60◦ from the planet at its trailing position. +How- +ever, our model shows that the center of the crescent is +slightly shifted and can vary between 65◦ and 85◦ for +reason we still do not understand and deserve further +investigation. Accordingly, the position of the proposed +planet will be also slightly shifted from the Lagrange +point. Finally, we observe that azimuthal extent of the +crescent remains roughly constant and equal to ∼ 45◦ +similar to the observations. +4.2. Behavior for different dust species + +HD 163296: crescent and resonant chain +5 +In our fiducial model, the vortensity for L5 of the outer +planet is stronger than that of the inner one (not shown). +Therefore, the dust accumulation is generally expected +to be greater around the orbit of the outer planet for all +sizes. This is especially true for small sizes that are well +coupled to the gas and librate with larger amplitudes +around L5 (panels b and c in Fig. 4). As the size of the +grains increases (Stokes numbers approach unity, panels +d to f), the dust distribution becomes compact toward +the center of the Lagrange point (Montesinos et al. 2020) +and we can even see some tenuous accumulation around +the inner planet’s L5 point at 1.9 cm (panel f). +5. A LAPLACE RESONANCE CHAIN +Our fiducial simulation has 4 planets with period ra- +tios P2/P1 = (54/46)3/2 = 1.27, (84.5/54)3/2 = 1.96, +and (137/84.5)3/2 = 2.06. +Therefore, the three outer +planets lie near a 1 : 2 : 4 commensurability, which be- +comes nearly exact (within 1%) if planet 3 changes from +84.5 au to 86 au. This fact begs the question of whether +disk-driven migration may have placed the planets in +their current, near resonant, orbits6. +From Figure 1, the planets carved relatively shallow +gaps, so we may estimate the rate of orbital migration +following Kanagawa & Szuszkiewicz 2020 as: +τa ≡ +���a +˙a +��� ≃ +�M⋆ +Mp +� � +M⋆ +Σmina2p +� h2 +p +ΩK,p +, +≃ 0.4 Myr +�10 gr cm−2 +Σmin +� �100 au +ap +�1/2 +�2M⊙ +M⋆ +� �1MJ +Mp +� � hp +0.1 +�2 +. +(5) +where Σmin corresponds to the local density at the base +of the density gap. Using the fiducial planetary parame- +ters and M⋆ = 1.9 M⊙, a fix aspect ratio h = 0.1 and the +surface density constraints from Zhang et al. (2021, Ta- +ble 5 therein), we observe that the migration timescales +are all comparable to the age of the system making mi- +gration a plausible scenario. +As a proof of concept, in Figure 5 we show an N-body +integration using REBOUND (Rein & Liu 2012) and pre- +scribing the damping timescales τa and τe ≡ e/| ˙e| for +each planet in order to mimic planet-disk interactions +in the REBOUNDx library (Tamayo et al. 2019). We set +τa/τe = 100 with τa computed using Eq. 5, see values +for the orbital decay timescales in Table 1. We begin +6 We note that a disk-driven migration may also lead to offsets +from the exact commensurabilities by either wake-planet interac- +tions (Baruteau & Papaloizou 2013), disk-driven precession (e.g., +Tamayo et al. 2015) or resonant repulsion (e.g., Papaloizou 2011). +the simulations with the planets further away from their +current positions and let them migrate due to their in- +teraction with the gaseous component of the disk, where +kinematic evidence has been detected in the gas at ∼ 260 +au (Pinte et al. 2020; Teague et al. 2021), and extensions +in CO (2-1) up to ∼ 500 au (Zhang et al. 2021). The +evolution shows that all planet pairs are captured into a +long resonant chain after ∼ 0.5 × 106 yrs. These corre- +spond to a two-body 4:3 mean-motion resonance (MMR) +for the innermost planets, and a double 2:1 - 2:1 MMR +for the two outer pairs, finally leading to the libration +of the following three-body angles: +φ123 = 3λ1 − 5λ2 + 2λ3, and +(6) +φ234 = 2λ2 − 6λ3 + 4λ4. +(7) +Both φ123 and φ234 have small-amplitude libration +(∼ 2◦) and their libration centers are 187◦ and 218◦, re- +spectively. Note that when the four planets reach the re- +ported semi-major axis at approximately the same time +∼ 106 yrs (gray vertical line in panel b), they are al- +ready captured in the two- and three-planet resonances, +showing that the proposed configuration with our hydro- +dynamical simulations presented in the previous sections +is possible. +5.1. Predicting the position angle (PA) of the planet +candidates using 3-body resonances +Because the orbits of planets i are coplanar and nearly +circular (ei ≲ 0.07), the mean longitudes λi are close to +the true longitudes ϖi +fi and will likely librate around +the same angles. Defining an arbitrary reference frame, +rotated by PA0 we write the position angles (PAs) as +PAi = ϖi + fi + PA0, and define the following three- +body angle, frame-independent7, combinations: +PA123 = 3PA1 − 5PA2 + 2PA3, +and +(8) +PA234 = 2PA2 − 6PA3 + 4PA4. +(9) +From panel (e) in Figure 5, we observe that these an- +gles librate around PA123 ∼ 185◦ and PA234 ∼ 210◦ sim- +ilar to φ123 and φ234, but with larger amplitudes (near +100◦ in both cases). This is expected due to the non-zero +eccentricities. +Assuming that we know two angles, say PA2 due to +the crescent and PA3 due to a velocity kink, we can use +the above relations to constrain PA1 and PA4 as: +PA1 ∼ 1 +3PA123 + 5 +3PA2 − 2 +3PA3, +and +(10) +PA4 ∼ 1 +4PA234 − 1 +2PA2 + 3 +2PA3. +(11) +7 Since the critical angles satisfy the D’Alembert property, these +combinations are independent of the reference frame. + +6 +Garrido-Deutelmoser, et al. +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +RA [arcsec] +1.5 +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +Dec [arcsec] +180◦ +90◦ +0◦ +270◦ +ALMA Band 6 +1.5 +1.0 +0.5 +0.0 +-0.5 +-1.0 +-1.5 +RA [arcsec] +Synthetic Observation +0.01 +0.1 +0.4 +1.0 +2.0 +Intensity [mJy beam 1] +Figure 3. Band 6 (λ ∼1.25 mm) comparison between dust continuum image from ALMA observation (Isella et al. 2018) and +our synthetic observation after ≈ 4.8 × 105 yrs. The synthesized beam are the same for both images (0.038′′ × 0.048′′, 82.5◦), +represented by white ellipse at the bottom left corner for each image. The synthetic image is projected with an inclination +i = 46◦ and a position angle PA= 133◦. The enumerated white dots indicate the position of potential planets associated with: +(1) and (4) resonance angles in Laplace chains (see §5.1), (2) Lagrange point L5 from the simulation, (3) velocity kink reported +by Pinte et al. (2020). The white dashed lines denote the orbits of planets in the simulation. The right bottom Cartesian +coordinate describe the prescription to estimate the azimuthal angles. The disk rotates in a clockwise direction. +Our model indicates that planet 2 (54 au) is ∼ 75◦ ± +10◦ ahead of the crescent center8, which corresponds to +PA2 ≃ 32◦. +In addition, as reported by Pinte et al. +(2020), the planet 3 at 86 au has PA3 ≃ 357◦ associated +with a velocity kink. Thus, our model predicts that the +planets at 46 au and 137 au should have position angles +of PA1 ∼ 237◦ and PA4 ∼ 212◦ respectively (see the left +panel in Figure 3). +Quite recently9, Alarc´on et al. (2022) localized strong +kinematic deviation in C I line emission. The position +of this structure lies inside the gap at 48 au, which +azimuthally coincides with our predicted planet 1 at +PA1 ∼ 237◦. We note that this predicted planet differs +from the one proposed by Isella et al. (2018) and in- +correctly quoted by Alarc´on et al. (2022) as coincident +with the outflow. The reason is that the disk rotates +in a clockwise direction so the proposed planet invoked +to explain the crescent as a L5 feature, similar to Ro- +denkirch et al. (2021), will actually show ahead of the +crescent. In this way, the C I deviation cannot be ex- +plained by a co-rotational planet that is also responsible +8 Due to the clockwise rotation of the disk, our angle convention +PA is given the coordinate axes at the bottom of Figure 3. +9 After the submission of our manuscript to the journal. +Table 1. Migration rate estimates +— +— +— +— +ap +Mp +Σmin [gr/cm/cm] +τa [Myr] +46 au +85 M⊕ +12 +1.8 +54 au +60 M⊕ +19 +1.5 +84.5 au +127 M⊕ +9.3 +1.1 +137 au +317 M⊕ +4.2 +0.8 +Note—The values of Σmin are taken from Table 5 in +Zhang et al. (2021), except for the innermost one +provided by our model. +for the crescent, unless the dust accumulation around +L4 becomes more prominent than that of L5, which is +unlikely (Rodenkirch et al. 2021; Garrido-Deutelmoser +et al. 2022). +5.2. Resonances in other systems +We remark that resonances may be a common out- +come in these young systems, including the embedded +planets in PDS 70 (Bae et al. 2019), as well as young, +but disk-free systems, like HR 8799 also in a long reso- + +2 +3 +1 +4HD 163296: crescent and resonant chain +7 +100 +50 +0 +50 +100 +y [au] +Gas +(a) +0.2 mm +(b) +100 +50 +0 +50 +100 +y [au] +0.7 mm +(c) +1.3 mm +(d) +100 +50 +0 +50 +100 +x [au] +100 +50 +0 +50 +100 +y [au] +2.6 mm +(e) +100 +50 +0 +50 +100 +x [au] +1.9 cm +(f) +0 +50 +100 +150 +0.0 +0.5 +1.0 +1.5 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +0 +1 +2 +3 +4 +5 +0 +10 +20 +30 +Figure 4. Face-on gas and dust surface density Σ from the +hydrodynamic model after ≈ 4.8×105 yrs (2000 orbits at 48 +au). The panels correspond to different fluids. The crosses +denote the position of the planets and the white dashed lines +indicate their orbits. The disk rotates in a clockwise direc- +tion. +nance chain involving four planets (Go´zdziewski & Mi- +gaszewski 2020). +Similar to our work, a compact multi-planet system +has been proposed using the axisymmetric dust gaps +and rings of HL Tau (ALMA Partnership et al. 2015), +where a resonant configuration may promote the sys- +tem’s dynamical stability (Tamayo et al. 2015). In our +case, we use not only the system’s migration history and +dust rings and gaps, but also add the constraints from +the crescent shape structure and the CO gas emission. +6. CONCLUSIONS +We have provided a global model for HD 163296 with +four planets (semi-major axes in the range of 40 − 140 +au) that can reproduce the rings and gaps in the dust +continuum and the shallow gaps in the gas constrained +by the CO emission. A key ingredient in our model is +the presence of two sub-Saturn-mass planets near the +4:3 resonance opening the gap at ∼ 48 au, where the +Time [Myr] +1.3 +1.6 +1.9 +2.2 +2.5 +period ratio +2 : 1 +4 : 3 +(a) +P4/P3 +P3/P2 +P2/P1 +Time [Myr] +0 +100 +200 +300 +400 +a [au] +(b) +a4 +a3 +a2 +a1 +Time [Myr] +0.00 +0.03 +0.06 +0.09 +e +(c) +e4 +e3 +e2 +e1 +Time [Myr] +0 +90 +180 +270 +360 +3pl [deg] +187 +218 +(d) +123 +234 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +Time [Myr] +0 +90 +180 +270 +360 +PA3pl [deg] +185 +210 +(e) +PA123 +PA234 +Figure 5. +Potential migratory history of the four-planet +system locking the planets in a long orbital resonance chain +leaving the outer three planets near a consecutive 2:1 com- +mensurability and the innermost pair near 4:3 (panels a and +b). The eccentricities remain small after the capture (panel +c) and the three-body resonant angles φ123 = 3λ1−5λ2+2λ3 +and φ234 = 2λ2 − 6λ3 + 4λ4 undergo small-amplitude libra- +tion (panel d). The bottom panel exhibits the corresponding +combinations of the position angles in the system: PA123 = +3PA1 − 5PA2 + 2PA3 and PA234 = 2PA2 − 6PA3 + 4PA4. +crescent corresponds to the L5 Lagrange point of the +outer planet at 54 au. +We show that the four-planet system may be part of +a long resonance chain with the inner two in a 4:3 MMR +and the outer three in a 1:2:4 Laplace resonance chain, +consistent with a history of convergent migration within +the disk. Our proposed three-body resonances allow to +relate the planetary radial and angular positions, and +based on the crescent location at 55 au and the proposed +location by Pinte et al. (2020) for the planet at ≃ 86 au, +our model predicts two planets: i) a sub-Saturn at 46 +au and PA ∼ 237◦; ii) a Jovian at 137 au and PA ∼ +212◦(Figure 3). +Overall, our work shows that tightly-spaced planetary +systems, often found at small orbital distances in tran- + +XXXXXX8 +Garrido-Deutelmoser, et al. +siting surveys, may leave detectable imprints in proto- +planetary disks at much larger separations. +Acknowledgements The authors would like to thank +Andrew Youdin, Kaitlin Kratter, Diego Mu˜noz, Matt +Russo, Pablo Ben´ıtez-Llambay, Sim´on Cassasus, and Xi- +mena S. Ramos for helpful discussions that improved the +quality of this work and Juan Veliz for his support with +the cluster logistics. Finally we thank the anonymous re- +viewer for the thorough and useful report. J.G. acknowl- +edge support by ANID, – Millennium Science Initiative +Program – NCN19 171 and FONDECYT Regular grant +1210425. The Geryon cluster at the Centro de Astro- +Ingenieria UC was extensively used for the calculations +performed in this paper. BASAL CATA PFB-06, the +Anillo ACT-86, FONDEQUIP AIC-57, and QUIMAL +130008 provided funding for several improvements to +the Geryon cluster. +C.P. acknowledges support from +ANID Millennium Science Initiative-ICN12 009, CATA- +Basal AFB-170002, ANID BASAL project FB210003, +FONDECYT Regular grant 1210425, CASSACA grant +CCJRF2105, and ANID+REC Convocatoria Nacional +subvencion a la instalacion en la Academia convocatoria +2020 PAI77200076. C.C. acknowledges FNRS Grant No. +F.4523.20 (DYNAMITE MIS-project). V.V.G. acknowl- +edges support from FONDECYT Regular 1221352, +ANID project Basal AFB-170002, and ANID, – Mil- +lennium Science Initiative Program – NCN19 171. 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J., et al. 2021, ApJS, 257, +5, doi: 10.3847/1538-4365/ac1580 + +10 +Garrido-Deutelmoser, et al. +APPENDIX +A. GAS GAP CALCULATION +r [au] +101 +102 +[gr cm 2] +(a) +6 +10 +20 +30 +60 +100 +200 +300 +0 +30 +60 +90 +120 +150 +180 +r [au] +101 +102 +[gr cm 2] +(b) +6 +10 +20 +30 +60 +100 +200 +300 +� +convolved +� +regions {r} +smooth fit +Figure 6. Black line denote the convolved surface density +profile of models and grey dashed line the respective smooth +function. Panel (a) and (b) represent the single-planet case +and two-planet case respectively. The crosses indicate the +position of the planets. +In Zhang et al. (2021) a smooth function was sub- +tracted from NCO column density profiles to better char- +acterize substructures in the residual values. To com- +pare these results with our models, we follow the same +procedure. First, the Σ maps from hydrodynamic simu- +lations were convolved with a circular Gaussian beam of +0.15′′, which has the same size as MAPS CO (2-1) line +observations. Then, their azimuthally averaged profiles +were interpolated every 2 au. In addition, the radial re- +gion {r [au] : 0 < r0 < 35, 59 < r1 < 72, 98 < r2 < +110, r3 > 170} was selected to describe the gaps. Both +were taken as input for the smoothfit10 module. The +Figure 6 shows the outputs of smoothed profile repre- +10 https://pypi.org/project/smoothfit/ +30 +60 +90 +120 +150 +180 +r [au] +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +�Intensity +� +[mJy beam 1] +180◦ +90◦ +0◦ +270◦ +350◦ +300◦ +ALMA Band 6 +Synthetic Obs. +Figure 7. Azimuthally averaged intensity profiles for syn- +thetic and ALMA observations after ≈ 4.8 × 105 yrs. The +crosses denote the semi-major axes of the planets. The insert +shows the ALMA observation in Band 6 with contours that +reproduce the crescent and rings as well as the angular slice +used for the azimuthal average denoted by ˆφ. +sented by the grey dashed line and the convolved sur- +face density profile in black lines. The cyan dots denote +the regions in which the function acts. The Figure 2 +shows the residual between lines to provide a reasonable +comparison with CO gaps observations. +B. RADIAL INTENSITY +We quantify the intensity around the substructure re- +gion of our synthetic model with the ALMA observation. +First, we deproject the images obtaining a face-on view +to convert them to polar coordinates and then generate a +radial profile by taking the azimuthal average between +PA of 300◦ and 350◦. This extension fully covers the +emission from the crescent. The results are shown in +the Figure 7, which is accompanied by a diagram show- +ing the angular slice. +Figure 7 show that radial intensity through the cres- +cent region reaches amplitudes higher than those ob- +served by a factor of 1.2 at 55 au. The emission from +the substructure is clearly off-centered on the gap and +resolved in spatial resolution, showing a gap in intensity +between it and the ring. The first ring reproduces the +intensities in a good way, while the second is noticeably +1.7 times fainter. + +HD 163296: crescent and resonant chain +11 +2 /3 +/3 +0 +/3 +2 /3 + [rad] +0.02 cm +Feel Disk = YES +0.02 cm +Feel Disk = NO +30 +40 +50 +60 +70 +r [au] +2 /3 +/3 +0 +/3 +2 /3 + [rad] +1.9 cm +30 +40 +50 +60 +70 +r [au] +1.9 cm +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 + [gr cm +2] +0.02 +0.04 +0.06 +0.08 +0.10 + [gr cm +2] +Figure 8. Dust surface density Σ for 0.02 cm and 1.9 cm +grain sizes. The crosses denote the position of the planets. +The upper and bottom white rectangles, indicate the La- +grange points L4 and L5 respect to the outer planet. +C. EFFECT FROM DISK GRAVITY ACTING ON +PLANETS +We briefly test whether turning on the full disk-planet +interaction may lead to morphological changes in the +structure of the crescent. We recall that in our fiducial +simulation (see §2.1), while the disk do feel the planets’ +gravity, the planets do not feel the disk. +We perform two-planet simulations considering only +the inner planet pair near the 4:3 commensurability (46 +au and 55 au) for up to 2000 orbits of the inner planet. +The initial density has been reduced by a factor of 100 +to avoid significant migration. In figure 8 we show the +density distribution for two dust fluids of 0.02 cm and +1.9 cm grain sizes in two cases: the full disk-planet in- +teraction is considered (left panels, displaying a slight +inward migration at the ∼ 10% level), and the disk grav- +ity acting on the planets is ignored (right panels, with +no migration). Despite of the slight orbital migration, +we do not observe any significant changes regarding the +amount and distribution of captured material at the L4 +and L5 Lagrange points. + diff --git a/-tFQT4oBgHgl3EQfKTWv/content/tmp_files/load_file.txt b/-tFQT4oBgHgl3EQfKTWv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8da473206f27df635b47e3d6707855a78f29d0fb --- /dev/null +++ b/-tFQT4oBgHgl3EQfKTWv/content/tmp_files/load_file.txt @@ -0,0 +1,752 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf,len=751 +page_content='Draft version February 3, 2023 Typeset using LATEX twocolumn style in AASTeX631 A gap-sharing planet pair shaping the crescent in HD 163296: a disk sculpted by a resonant chain Juan Garrido-Deutelmoser ,1, 2 Cristobal Petrovich ,1, 3 Carolina Charalambous ,4 Viviana V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Guzm´an ,1, 2 and Ke Zhang 5 1Instituto de Astrof´ısica, Pontificia Universidad Cat´olica de Chile, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Vicu˜na Mackenna 4860, 782-0436 Macul, Santiago, Chile 2N´ucleo Milenio de Formaci´on Planetaria (NPF), Chile 3Millennium Institute for Astrophysics, Chile 4naXys, Department of Mathematics, University of Namur, Rue de Bruxelles 61, 5000 Namur, Belgium 5Department of Astronomy, University of Wisconsin-Madison, 475 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Charter Street, Madison, WI 53706, USA ABSTRACT ALMA observations of the disk around HD 163296 have resolved a crescent-shape substructure at around 55 au, inside and off-center from a gap in the dust that extends from 38 au to 62 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In this work we propose that both the crescent and the dust rings are caused by a compact pair (period ratio ≃ 4 : 3) of sub-Saturn-mass planets inside the gap, with the crescent corresponding to dust trapped at the L5 Lagrange point of the outer planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This interpretation also reproduces well the gap in the gas recently measured from the CO observations, which is shallower than what is expected in a model where the gap is carved by a single planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Building on previous works arguing for outer planets at ≈ 86 and ≈ 137 au, we provide with a global model of the disk that best reproduces the data and show that all four planets may fall into a long resonant chain, with the outer three planets in a 1:2:4 Laplace resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We show that this configuration is not only an expected outcome from disk-planet interaction in this system, but it can also help constraining the radial and angular position of the planet candidates using three-body resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Keywords: protoplanetary disks — planet–disk interactions — hydrodynamics — planets and satellites: dynamical evolution and stability — radiative transfer 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' INTRODUCTION Substructures are ubiquitous in protoplanetary disks, particularly in the dust density distribution exhibited by high angular resolution observations (Andrews 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The Atacama Large Millimeter Ar- ray (ALMA) has revealed a variety of substructures, whereas a large population of rings and gaps are shown in continuum observations, to a lesser extent, in molec- ular line emissions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', van der Marel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The advances in spatial resolution have been able to resolve non-axisymmetric substructures within gaps, in- cluding systems such as PDS 70 (Benisty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021), HD 163296 (Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2018), HD 100546 (P´erez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2020), HD 97048 (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2019), and LkCa 15 (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' These substructures may be due to embed- ded planets induced by gravitational interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2022), which vary depending on the emis- sion shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Point-like emissions are generally associ- ated with an accreting planet surrounded by a circum- planetary disk (CPD) (Perez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Szul´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2018), while crescent shapes may be related to the sta- ble Lagrange points L4 and L5 of a star-planet system (Rodenkirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2022) or vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' These hypotheses frequently assume that a single sub- structure is caused by a single planet, often in a Jo- vian mass regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' However, we have recently shown in Garrido-Deutelmoser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2022), that a pair of lower-mass and gap-sharing planets can sculpt compact and/or elongated vortices within the gap that last for several thousands orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The disk surrounding HD 163296 contains ringed structures in the mm-continuum (Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2018) and several molecular tracers (Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In particular, inside the dust density gap that ex- tends from 38 to 62 au, a crescent-shaped substructure resides at around 55 au (Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Recently, it was suggested that the emission comes from the dust trapped around the stable point L5 of a Jupiter mass planet orbiting at 48 au (Rodenkirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Even though this method seems to reproduce the broad fea- tures of the dust continuum distribution, two aspects remain unclear: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='13260v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='EP] 30 Jan 2023 ID2 Garrido-Deutelmoser, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' the crescent feature resides at 55 au, off-center from the dust gap, while the L5 point is co-orbital to the planet at 48 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Varying the planet’s eccen- tricity to account for this shift is unlikely to help as the stability of the crescent is damaged, leading to its prompt disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' the Jupiter-mass planet needed to retain observ- able amounts of dust at L5 and open a wide enough gap in the dust, is expected to open a deep gap1 (Duffell 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This prediction disagrees with the recent results provided by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021), who found that the dust density gap has a correspond- ing CO gap ∼ 10 times shallower than the predic- tions involving a Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The local gas depletion depends on the planetary mass (∝ m−2 p ) (Kana- gawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2015), whereby opting for a lower mass planet to carve a shallower gap, may not produce a sufficient gravitational interaction to enforce the dust trapping at L5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In this work, we propose a that a compact pair of sub-Saturn-mass planets can solve these issues, simul- taneously accounting for the dust emission (the shifted crescent and dust rings) and shallow gap in the CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This scenario is largely motivated by our recent work in Garrido-Deutelmoser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2022) where we showed that a compact pair of gap-sharing planets generally lead to nonaxisymmetric substructures like that observed in HD 163296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' SETUP The hydrodynamics simulations and radiative trans- fer calculations in this work largely follow the scheme in Rodenkirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021) and are only briefly sum- marized here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We carried out 2D hydrodynamic simu- lations using the FARGO3D multifluid code (Ben´ıtez- Llambay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Masset 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2019) to produce gas and dust density distribution for a fidu- cial disk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The resulting density maps are read into the RADMC3D code (Dullemond et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2012) to calculate the radiative transfer image at λ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='25 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We use this image and the HD 163296 template (that contain all the technical properties of the observation) in the SIMIO2 package to achieve the synthetic ALMA observation comparable to the dust continuum observa- tion provided by Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 1 An increase in the local viscosity to produce a much shallower gap comes at the expense of reducing the lifetime of the L5 crescent, or even prevent its formation in the first place (Rodenkirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='nicolaskurtovic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='com/simio 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Hydrodynamic Simulations The initial surface density profiles for the gas (sub- index g) and dust (sub-index d) are given by Σg/d = Σg/d,0 � r r0 �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 exp � − � r rc,g/d �γg/d� , (1) where we set r0 = 48 au, the initial surface density Σg,0 = 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 gr cm−2, the cut-off radius rc,g = 165 au, and the exponent γg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Similarly, for the dust we set Σd,0 = Σg,0/100 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='374 gr cm−2, rc,d = 90 au, and γd = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We include the evolution for five indepen- dent dust species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' These grains have sizes in cm of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='071, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='13, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='26, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We use an aspect ratio of h(r) = h0(r/r0)f with h0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='05 and a flaring index f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='25, which implies a mid-plane temperature profile described by T = 25(r/r0)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This setup coincides with that from Rodenkirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The disk extends from rin = 5 au to rout = 197 au, implying an initial disk mass of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='15 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The compu- tational domain is composed of nr = 512 logarithmically spaced radial cells and nθ = 768 equally spaced cells in the azimuthal [0, 2π] domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We include a radially vari- able viscosity of the standard parameter α (Shakura & Sunyaev 1973) as α(r) = αin − αin − αout 2 � 1 + tanh �r − ξ σr0 �� , (2) where the inner and outer viscosities are αin = 1 × 10−4 and αout = 5 × 10−3, ξ = 144 au indicate the midpoint of the transition and σ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='25 defines the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Similar to the values provided by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' A system of 4 planets was embedded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The location of the two outer ones is indicated in Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2018) through kinematic detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The third planet’s posi- tion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', the inner planet of the outer pair) is strongly associated with the potential velocity kink reported by (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The two inner planets are tightly packed and their parameters (masses and orbits) were derived numerically guided by the results in Garrido- Deutelmoser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2022), where it was found that the planets should be gap-sharing with forming vortices at their Lagrange points, implying a condition in the plan- etary separation3 of: ∆a ≲ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6H ≃ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 au, (3) where H the scale high of the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In turn, the masses are constrained by the width of the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' After a few 3 This expression has been tested for planets with masses near the thermal mass of Mth = M⋆h3 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='25MJ ≃ 80M⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' HD 163296: crescent and resonant chain 3 dozen simulations attempting to match disk morphol- ogy in continuum observations at the ∼ 48 au region, we choose to place the planets at a1 = 46, a2 = 54, a3 = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5, and a4 = 137 au with their respective masses of M1 = 85M⊕, M2 = 60M⊕, M3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4MJup, and M4 = 1MJup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The four bodies can gravitationally in- teract between them, but they do not feel the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We ran an extra model to compare against previous works, substituting a Jupiter at 48 au instead of the inner pack- age of planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Both cases were evolved for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='48 Myrs, equivalent to 2000 orbits of the innermost planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Radiative Transfer We convert the 2D dust surface density into a 3D vol- ume density assuming the vertical approximation given by ρdj(r, φ, θ) = Σ(r, φ) √ 2πHdj exp � − z2 2H2 dj � , (4) where z = r cos(θ) and the dust settling follows the diffu- sion model Hdj = � Dz/(Dz + Stj)H, with Dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6α the vertical diffusion coefficient, and Stj the Stokes num- ber of the species j (Weber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We assume an intrinsic volume density for the particles ρs = 2 gr cm−3 and a power law for the grain size distribution, such that n(a) ∝ a−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We assumed a dust composition of 20% amorphous carbon, 20% water ices and 60% silicates, where the corresponding dust opacities were computed with the code provided by (Bohren & Huffman 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The polar direction is distributed in 64 equally spaced cells and extended in [80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6◦, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4◦] inclination domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We use nphot = 108 photon packages to calculate the dust temperature, and nscatt = 107 photon packages to trace the thermal emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We use a full anisotropic scattering with polarization treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The system is assumed to be at a distance of 101 pc with a central star of mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 M⊙ and effective temperature Teff = 9, 330 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The inclination is taken to be i = 46◦ and position angle PA= 133◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Synthetic Observations We use SIMIO that contains a suit of functions for CASA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We select the template designed for HD 163296 to create images with the same uv-coverage as observation from Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We set the rescale flux option in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 to get similar intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In addition, we add simple thermal noise4 of level 12mJy to finally get RMS noise of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='022 mJy beam−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 4 https://simio-continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='io/en/main/tutorials/ tutorial 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='html 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' LOCAL GAP DEPLETION Figure 1 shows the surface density after ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 Myr (2,000 orbits) for the single-Jovian case (panel a) and the two sub-Saturns with masses 85M⊕ and 60M⊕ (panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The corresponding azimuthally-averaged profiles in panel (c) show that ∼ 95% of gas is depleted for the single-Jovian, while only ∼ 55% is depleted for the two- planet case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Despite of their lower masses the planets pair creates the same gap width as the Jovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This shallower gaps for fixed gap width are expected in com- pact multi-planet systems due the planet lower masses (depth Σgap/Σ0 ∝ M −2 p ) and angular flux transferred by the neighbouring planets (Duffell & Dong 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Garrido- Deutelmoser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' As argued by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021), if a Jupiter-mass planet had opened the corresponding CO gap, it would be 10 times deeper than what is actually observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In- stead, by embedding two planets we can alleviate these differences and largely reduce this discrepancy as shown in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Therefore, the depletion values would be closer to the results from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In order to bet- ter quantify this, we compare the CO column density gaps with the surface density from both models5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' As shown in Figure 2, this approach reproduces the gas gap reasonably well in the two-planet case, largely matching the depths and widths with a small offset of the peaks by ∼ 4 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In turn, the single-Jovian case is too deep compared to the observations as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Beyond the third planet at ∼ 85 au, neither of the two models (single Jovian or compact pair) is able to repro- duce the gas gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This was already noted in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021) when comparing with the models from Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021) and it may partly be explained by the presence of a CO snowline at 65 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Outside the mid-plane CO snowline, CO freezes out at the disk mid- plane and therefore the CO gap properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', width and depth) may deviate from that of gas gaps due to vertical temperature and CO abundance variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We recall that our work mostly focuses on the gap and cres- cent at ∼ 50 au where these issues can be more securely avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' THE CRESCENT FEATURE A closely-packed planet pair can directly affect the gas around each other’s coorbital regions by deposit- ing angular momentum from wave steepening and sub- sequent shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' These shocks may strengthen the vor- tencity around the stable L4 and L5 Lagrange points, 5 The Σ profiles were processed under smooth function methodol- ogy described described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 4 Garrido-Deutelmoser, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 100 50 0 50 100 x [au] 100 50 0 50 100 y [au] (a) 100 50 0 50 100 x [au] (b) 0 30 60 90 120 150 180 r [au] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 � / 0 � (c) single-planet gap two-planets gap 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='3 log10( ) [gr cm 2] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Panels show a time evolution after 2000 orbits at 48 au (∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 × 105 yrs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (a) Surface density Σ maps in log-scale for a single Jupiter planet at 48 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (b) The same as (a), but for a two-planet system with 85M⊕ and 60M⊕ instead the Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The crosses denote the position of the planets and the white lines indicate their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The disk rotates in a clockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (c) Azimuthally averaged Σ/Σ0 profiles for both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' thus enabling the effective gas and dust trapping for at least thousands of orbital periods (Garrido-Deutelmoser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The overdensity around either L4 or L5 for the inner and outer planet is a highly dynamic problem, with the most prominent structure chaotically alternating loca- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This said, we observe that for several combina- tions of parameters (surface density, aspect ratio, plan- etary masses, and so on), the outer planet often retains large amounts of material around L5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The depicted be- havior would leave an off-center substructure inside the gap that greatly reproduces the distinctive emission in the disk as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' More quantitatively, our models matches the observed azimuthal extent of ∼ 45◦ and the radial intensity profiles passing through the cres- cent (peaks and troughs inside the ring at ∼ 68 au;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' see Appendix B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Dynamical behavior Figure 4 compares the gas and dust density distribu- tion for different dust grain sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' All the dust species show a clear shared gap between the two inner planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In the largest size, the co-orbital regions of each one display an overdensity at L5, but the more prominent substructure belongs to the second planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This out- put has taken into account two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The choice of planetary masses in the inner system must be lower for the body orbiting the substructure, and the pres- ence of the two outer planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' If we neglect either, the Lagrange points can still trap dust, but the mass dis- tribution may change to make some L4 or the inner L5 the most prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' As shown in Garrido-Deutelmoser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2022), this evolution is highly dynamic and ir- regular so that the overdensities around L4 and L5 con- 0 30 60 90 120 150 180 210 240 r [au] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 Normalized gap depth CO gaps in Ke Zhang 2021 single-planet gas gaps two-planets gas gaps Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Comparison of column density gap in C18O (2−1) line observation found by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021) with gas gaps derived from surface density profiles in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' stantly change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' However, the final morphology in our configuration comes from the early stages of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Theoretically, the stable Lagrange point L5 is located at 60◦ from the planet at its trailing position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' How- ever, our model shows that the center of the crescent is slightly shifted and can vary between 65◦ and 85◦ for reason we still do not understand and deserve further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Accordingly, the position of the proposed planet will be also slightly shifted from the Lagrange point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Finally, we observe that azimuthal extent of the crescent remains roughly constant and equal to ∼ 45◦ similar to the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Behavior for different dust species HD 163296: crescent and resonant chain 5 In our fiducial model, the vortensity for L5 of the outer planet is stronger than that of the inner one (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Therefore, the dust accumulation is generally expected to be greater around the orbit of the outer planet for all sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This is especially true for small sizes that are well coupled to the gas and librate with larger amplitudes around L5 (panels b and c in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' As the size of the grains increases (Stokes numbers approach unity, panels d to f), the dust distribution becomes compact toward the center of the Lagrange point (Montesinos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2020) and we can even see some tenuous accumulation around the inner planet’s L5 point at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 cm (panel f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' A LAPLACE RESONANCE CHAIN Our fiducial simulation has 4 planets with period ra- tios P2/P1 = (54/46)3/2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='27, (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5/54)3/2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='96, and (137/84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5)3/2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Therefore, the three outer planets lie near a 1 : 2 : 4 commensurability, which be- comes nearly exact (within 1%) if planet 3 changes from 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 au to 86 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This fact begs the question of whether disk-driven migration may have placed the planets in their current, near resonant, orbits6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' From Figure 1, the planets carved relatively shallow gaps, so we may estimate the rate of orbital migration following Kanagawa & Szuszkiewicz 2020 as: τa ≡ ���a ˙a ��� ≃ �M⋆ Mp � � M⋆ Σmina2p � h2 p ΩK,p , ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 Myr �10 gr cm−2 Σmin � �100 au ap �1/2 �2M⊙ M⋆ � �1MJ Mp � � hp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (5) where Σmin corresponds to the local density at the base of the density gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Using the fiducial planetary parame- ters and M⋆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 M⊙, a fix aspect ratio h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1 and the surface density constraints from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021, Ta- ble 5 therein), we observe that the migration timescales are all comparable to the age of the system making mi- gration a plausible scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' As a proof of concept, in Figure 5 we show an N-body integration using REBOUND (Rein & Liu 2012) and pre- scribing the damping timescales τa and τe ≡ e/| ˙e| for each planet in order to mimic planet-disk interactions in the REBOUNDx library (Tamayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We set τa/τe = 100 with τa computed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 5, see values for the orbital decay timescales in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We begin 6 We note that a disk-driven migration may also lead to offsets from the exact commensurabilities by either wake-planet interac- tions (Baruteau & Papaloizou 2013), disk-driven precession (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', Tamayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2015) or resonant repulsion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', Papaloizou 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' the simulations with the planets further away from their current positions and let them migrate due to their in- teraction with the gaseous component of the disk, where kinematic evidence has been detected in the gas at ∼ 260 au (Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Teague et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021), and extensions in CO (2-1) up to ∼ 500 au (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The evolution shows that all planet pairs are captured into a long resonant chain after ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 × 106 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' These corre- spond to a two-body 4:3 mean-motion resonance (MMR) for the innermost planets, and a double 2:1 - 2:1 MMR for the two outer pairs, finally leading to the libration of the following three-body angles: φ123 = 3λ1 − 5λ2 + 2λ3, and (6) φ234 = 2λ2 − 6λ3 + 4λ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (7) Both φ123 and φ234 have small-amplitude libration (∼ 2◦) and their libration centers are 187◦ and 218◦, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Note that when the four planets reach the re- ported semi-major axis at approximately the same time ∼ 106 yrs (gray vertical line in panel b), they are al- ready captured in the two- and three-planet resonances, showing that the proposed configuration with our hydro- dynamical simulations presented in the previous sections is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Predicting the position angle (PA) of the planet candidates using 3-body resonances Because the orbits of planets i are coplanar and nearly circular (ei ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='07), the mean longitudes λi are close to the true longitudes ϖi +fi and will likely librate around the same angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Defining an arbitrary reference frame, rotated by PA0 we write the position angles (PAs) as PAi = ϖi + fi + PA0, and define the following three- body angle, frame-independent7, combinations: PA123 = 3PA1 − 5PA2 + 2PA3, and (8) PA234 = 2PA2 − 6PA3 + 4PA4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (9) From panel (e) in Figure 5, we observe that these an- gles librate around PA123 ∼ 185◦ and PA234 ∼ 210◦ sim- ilar to φ123 and φ234, but with larger amplitudes (near 100◦ in both cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This is expected due to the non-zero eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Assuming that we know two angles, say PA2 due to the crescent and PA3 due to a velocity kink, we can use the above relations to constrain PA1 and PA4 as: PA1 ∼ 1 3PA123 + 5 3PA2 − 2 3PA3, and (10) PA4 ∼ 1 4PA234 − 1 2PA2 + 3 2PA3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (11) 7 Since the critical angles satisfy the D’Alembert property, these combinations are independent of the reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 6 Garrido-Deutelmoser, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 RA [arcsec] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 Dec [arcsec] 180◦ 90◦ 0◦ 270◦ ALMA Band 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 RA [arcsec] Synthetic Observation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 Intensity [mJy beam 1] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Band 6 (λ ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='25 mm) comparison between dust continuum image from ALMA observation (Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2018) and our synthetic observation after ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 × 105 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The synthesized beam are the same for both images (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='038′′ × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='048′′, 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5◦), represented by white ellipse at the bottom left corner for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The synthetic image is projected with an inclination i = 46◦ and a position angle PA= 133◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The enumerated white dots indicate the position of potential planets associated with: (1) and (4) resonance angles in Laplace chains (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1), (2) Lagrange point L5 from the simulation, (3) velocity kink reported by Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The white dashed lines denote the orbits of planets in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The right bottom Cartesian coordinate describe the prescription to estimate the azimuthal angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The disk rotates in a clockwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Our model indicates that planet 2 (54 au) is ∼ 75◦ ± 10◦ ahead of the crescent center8, which corresponds to PA2 ≃ 32◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In addition, as reported by Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2020), the planet 3 at 86 au has PA3 ≃ 357◦ associated with a velocity kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Thus, our model predicts that the planets at 46 au and 137 au should have position angles of PA1 ∼ 237◦ and PA4 ∼ 212◦ respectively (see the left panel in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Quite recently9, Alarc´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2022) localized strong kinematic deviation in C I line emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The position of this structure lies inside the gap at 48 au, which azimuthally coincides with our predicted planet 1 at PA1 ∼ 237◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We note that this predicted planet differs from the one proposed by Isella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2018) and in- correctly quoted by Alarc´on et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2022) as coincident with the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The reason is that the disk rotates in a clockwise direction so the proposed planet invoked to explain the crescent as a L5 feature, similar to Ro- denkirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021), will actually show ahead of the crescent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In this way, the C I deviation cannot be ex- plained by a co-rotational planet that is also responsible 8 Due to the clockwise rotation of the disk, our angle convention PA is given the coordinate axes at the bottom of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 9 After the submission of our manuscript to the journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Migration rate estimates — — — — ap Mp Σmin [gr/cm/cm] τa [Myr] 46 au 85 M⊕ 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 54 au 60 M⊕ 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 au 127 M⊕ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1 137 au 317 M⊕ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 Note—The values of Σmin are taken from Table 5 in Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021), except for the innermost one provided by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' for the crescent, unless the dust accumulation around L4 becomes more prominent than that of L5, which is unlikely (Rodenkirch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Garrido-Deutelmoser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Resonances in other systems We remark that resonances may be a common out- come in these young systems, including the embedded planets in PDS 70 (Bae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2019), as well as young, but disk-free systems, like HR 8799 also in a long reso- 2 3 1 4HD 163296: crescent and resonant chain 7 100 50 0 50 100 y [au] Gas (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 mm (b) 100 50 0 50 100 y [au] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='7 mm (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='3 mm (d) 100 50 0 50 100 x [au] 100 50 0 50 100 y [au] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6 mm (e) 100 50 0 50 100 x [au] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 cm (f) 0 50 100 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0 1 2 3 0 1 2 3 4 5 0 10 20 30 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Face-on gas and dust surface density Σ from the hydrodynamic model after ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8×105 yrs (2000 orbits at 48 au).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The panels correspond to different fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The crosses denote the position of the planets and the white dashed lines indicate their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The disk rotates in a clockwise direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' nance chain involving four planets (Go´zdziewski & Mi- gaszewski 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Similar to our work, a compact multi-planet system has been proposed using the axisymmetric dust gaps and rings of HL Tau (ALMA Partnership et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2015), where a resonant configuration may promote the sys- tem’s dynamical stability (Tamayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In our case, we use not only the system’s migration history and dust rings and gaps, but also add the constraints from the crescent shape structure and the CO gas emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' CONCLUSIONS We have provided a global model for HD 163296 with four planets (semi-major axes in the range of 40 − 140 au) that can reproduce the rings and gaps in the dust continuum and the shallow gaps in the gas constrained by the CO emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' A key ingredient in our model is the presence of two sub-Saturn-mass planets near the 4:3 resonance opening the gap at ∼ 48 au, where the Time [Myr] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 period ratio 2 : 1 4 : 3 (a) P4/P3 P3/P2 P2/P1 Time [Myr] 0 100 200 300 400 a [au] (b) a4 a3 a2 a1 Time [Myr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='09 e (c) e4 e3 e2 e1 Time [Myr] 0 90 180 270 360 3pl [deg] 187 218 (d) 123 234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 Time [Myr] 0 90 180 270 360 PA3pl [deg] 185 210 (e) PA123 PA234 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Potential migratory history of the four-planet system locking the planets in a long orbital resonance chain leaving the outer three planets near a consecutive 2:1 com- mensurability and the innermost pair near 4:3 (panels a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The eccentricities remain small after the capture (panel c) and the three-body resonant angles φ123 = 3λ1−5λ2+2λ3 and φ234 = 2λ2 − 6λ3 + 4λ4 undergo small-amplitude libra- tion (panel d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The bottom panel exhibits the corresponding combinations of the position angles in the system: PA123 = 3PA1 − 5PA2 + 2PA3 and PA234 = 2PA2 − 6PA3 + 4PA4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' crescent corresponds to the L5 Lagrange point of the outer planet at 54 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We show that the four-planet system may be part of a long resonance chain with the inner two in a 4:3 MMR and the outer three in a 1:2:4 Laplace resonance chain, consistent with a history of convergent migration within the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Our proposed three-body resonances allow to relate the planetary radial and angular positions, and based on the crescent location at 55 au and the proposed location by Pinte et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2020) for the planet at ≃ 86 au, our model predicts two planets: i) a sub-Saturn at 46 au and PA ∼ 237◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' ii) a Jovian at 137 au and PA ∼ 212◦(Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Overall, our work shows that tightly-spaced planetary systems, often found at small orbital distances in tran- XXXXXX8 Garrido-Deutelmoser, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' siting surveys, may leave detectable imprints in proto- planetary disks at much larger separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Acknowledgements The authors would like to thank Andrew Youdin, Kaitlin Kratter, Diego Mu˜noz, Matt Russo, Pablo Ben´ıtez-Llambay, Sim´on Cassasus, and Xi- mena S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Ramos for helpful discussions that improved the quality of this work and Juan Veliz for his support with the cluster logistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Finally we thank the anonymous re- viewer for the thorough and useful report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' acknowl- edge support by ANID, – Millennium Science Initiative Program – NCN19 171 and FONDECYT Regular grant 1210425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The Geryon cluster at the Centro de Astro- Ingenieria UC was extensively used for the calculations performed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' BASAL CATA PFB-06, the Anillo ACT-86, FONDEQUIP AIC-57, and QUIMAL 130008 provided funding for several improvements to the Geryon cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' acknowledges support from ANID Millennium Science Initiative-ICN12 009, CATA- Basal AFB-170002, ANID BASAL project FB210003, FONDECYT Regular grant 1210425, CASSACA grant CCJRF2105, and ANID+REC Convocatoria Nacional subvencion a la instalacion en la Academia convocatoria 2020 PAI77200076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' acknowledges FNRS Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='20 (DYNAMITE MIS-project).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' acknowl- edges support from FONDECYT Regular 1221352, ANID project Basal AFB-170002, and ANID, – Mil- lennium Science Initiative Program – NCN19 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' acknowledges the support of the Office of the Vice Chan- cellor for Research and Graduate Education at the Uni- versity of Wisconsin – Madison with funding from the Wisconsin Alumni Research Foundation.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', Ben´ıtez-Llambay, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2019, ApJ, 884, 178, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='3847/1538-4357/ab412f Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', Booth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', Law, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' 2021, ApJS, 257, 5, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='3847/1538-4365/ac1580 10 Garrido-Deutelmoser, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' GAS GAP CALCULATION r [au] 101 102 [gr cm 2] (a) 6 10 20 30 60 100 200 300 0 30 60 90 120 150 180 r [au] 101 102 [gr cm 2] (b) 6 10 20 30 60 100 200 300 � convolved � regions {r} smooth fit Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Black line denote the convolved surface density profile of models and grey dashed line the respective smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Panel (a) and (b) represent the single-planet case and two-planet case respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The crosses indicate the position of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' (2021) a smooth function was sub- tracted from NCO column density profiles to better char- acterize substructures in the residual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' To com- pare these results with our models, we follow the same procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' First, the Σ maps from hydrodynamic simu- lations were convolved with a circular Gaussian beam of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='15′′, which has the same size as MAPS CO (2-1) line observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Then, their azimuthally averaged profiles were interpolated every 2 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In addition, the radial re- gion {r [au] : 0 < r0 < 35, 59 < r1 < 72, 98 < r2 < 110, r3 > 170} was selected to describe the gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Both were taken as input for the smoothfit10 module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The Figure 6 shows the outputs of smoothed profile repre- 10 https://pypi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='org/project/smoothfit/ 30 60 90 120 150 180 r [au] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='5 �Intensity � [mJy beam 1] 180◦ 90◦ 0◦ 270◦ 350◦ 300◦ ALMA Band 6 Synthetic Obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Azimuthally averaged intensity profiles for syn- thetic and ALMA observations after ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 × 105 yrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The crosses denote the semi-major axes of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The insert shows the ALMA observation in Band 6 with contours that reproduce the crescent and rings as well as the angular slice used for the azimuthal average denoted by ˆφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' sented by the grey dashed line and the convolved sur- face density profile in black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The cyan dots denote the regions in which the function acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The Figure 2 shows the residual between lines to provide a reasonable comparison with CO gaps observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' RADIAL INTENSITY We quantify the intensity around the substructure re- gion of our synthetic model with the ALMA observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' First, we deproject the images obtaining a face-on view to convert them to polar coordinates and then generate a radial profile by taking the azimuthal average between PA of 300◦ and 350◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' This extension fully covers the emission from the crescent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The results are shown in the Figure 7, which is accompanied by a diagram show- ing the angular slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Figure 7 show that radial intensity through the cres- cent region reaches amplitudes higher than those ob- served by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 at 55 au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The emission from the substructure is clearly off-centered on the gap and resolved in spatial resolution, showing a gap in intensity between it and the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The first ring reproduces the intensities in a good way, while the second is noticeably 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='7 times fainter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' HD 163296: crescent and resonant chain 11 2 /3 /3 0 /3 2 /3 [rad] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='02 cm Feel Disk = YES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='02 cm Feel Disk = NO 30 40 50 60 70 r [au] 2 /3 /3 0 /3 2 /3 [rad] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 cm 30 40 50 60 70 r [au] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 cm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='6 [gr cm 2] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='10 [gr cm 2] Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Dust surface density Σ for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='02 cm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 cm grain sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The crosses denote the position of the planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The upper and bottom white rectangles, indicate the La- grange points L4 and L5 respect to the outer planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' EFFECT FROM DISK GRAVITY ACTING ON PLANETS We briefly test whether turning on the full disk-planet interaction may lead to morphological changes in the structure of the crescent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We recall that in our fiducial simulation (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='1), while the disk do feel the planets’ gravity, the planets do not feel the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' We perform two-planet simulations considering only the inner planet pair near the 4:3 commensurability (46 au and 55 au) for up to 2000 orbits of the inner planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' The initial density has been reduced by a factor of 100 to avoid significant migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' In figure 8 we show the density distribution for two dust fluids of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='02 cm and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content='9 cm grain sizes in two cases: the full disk-planet in- teraction is considered (left panels, displaying a slight inward migration at the ∼ 10% level), and the disk grav- ity acting on the planets is ignored (right panels, with no migration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} +page_content=' Despite of the slight orbital migration, we do not observe any significant changes regarding the amount and distribution of captured material at the L4 and L5 Lagrange points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFQT4oBgHgl3EQfKTWv/content/2301.13260v1.pdf'} diff --git a/.gitattributes b/.gitattributes index bad571fc51d5005e7d423c1d2872fb8d72eb4017..a76d658368b0d69d826d786e72be02ded47ae480 100644 --- a/.gitattributes +++ b/.gitattributes @@ -5847,3 +5847,73 @@ gtFLT4oBgHgl3EQfZS_m/content/2301.12069v1.pdf filter=lfs diff=lfs merge=lfs -tex 8dE2T4oBgHgl3EQf8Ait/content/2301.04215v1.pdf filter=lfs diff=lfs merge=lfs -text 6dE2T4oBgHgl3EQfkgfV/content/2301.03980v1.pdf filter=lfs diff=lfs merge=lfs -text a9E_T4oBgHgl3EQfzByO/content/2301.08321v1.pdf filter=lfs diff=lfs merge=lfs -text +PdE5T4oBgHgl3EQfYg_i/content/2301.05575v1.pdf filter=lfs diff=lfs merge=lfs -text +3tE4T4oBgHgl3EQf0g0b/content/2301.05282v1.pdf filter=lfs diff=lfs merge=lfs -text 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sha256:7acbf1a3a72dbf1e28bb49cd1f94aeb32596577bc48f376afe2e749c156b7152 +size 151387 diff --git a/4NFQT4oBgHgl3EQf4Dar/content/tmp_files/2301.13430v1.pdf.txt b/4NFQT4oBgHgl3EQf4Dar/content/tmp_files/2301.13430v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..39dca56a8c7be7e40b3c165012ef44b3ec2bbfe6 --- /dev/null +++ b/4NFQT4oBgHgl3EQf4Dar/content/tmp_files/2301.13430v1.pdf.txt @@ -0,0 +1,1001 @@ +Published as a conference paper at ICLR 2023 +GENEFACE: +GENERALIZED +AND +HIGH-FIDELITY +AUDIO-DRIVEN 3D TALKING FACE SYNTHESIS +Zhenhui Ye1∗, Ziyue Jiang1∗, Yi Ren2, Jinglin Liu1, JinZheng He1, Zhou Zhao1† +1Zhejiang University +{zhenhuiye,jiangziyue,jinglinliu,jinzhenghe,zhaozhou}@zju.edu.cn +2Bytedance +ren.yi@bytedance.com +ABSTRACT +Generating photo-realistic video portrait with arbitrary speech audio is a crucial +problem in film-making and virtual reality. Recently, several works explore the us- +age of neural radiance field in this task to improve 3D realness and image fidelity. +However, the generalizability of previous NeRF-based methods to out-of-domain +audio is limited by the small scale of training data. In this work, we propose Gene- +Face, a generalized and high-fidelity NeRF-based talking face generation method, +which can generate natural results corresponding to various out-of-domain audio. +Specifically, we learn a variaitional motion generator on a large lip-reading cor- +pus, and introduce a domain adaptative post-net to calibrate the result. Moreover, +we learn a NeRF-based renderer conditioned on the predicted facial motion. A +head-aware torso-NeRF is proposed to eliminate the head-torso separation prob- +lem. Extensive experiments show that our method achieves more generalized and +high-fidelity talking face generation compared to previous methods1 . +1 +INTRODUCTION +Audio-driven face video synthesis is an important and challenging problem with several applications +such as digital humans, virtual reality (VR), and online meetings. Over the past few years, the com- +munity has exploited Generative Adversarial Networks (GAN) as the neural renderer and promoted +the frontier from only predicting the lip movement Prajwal et al. (2020)Chen et al. (2019) to gener- +ating the whole face Zhou et al. (2021)Lu et al. (2021). However, GAN-based renderers suffer from +several limitations such as unstable training, mode collapse, difficulty in modelling delicate details +Suwajanakorn et al. (2017)Thies et al. (2020), and fixed static head pose Pham et al. (2017)Taylor +et al. (2017)Cudeiro et al. (2019). Recently, Neural Radiance Field (NeRF) Mildenhall et al. (2020) +has been explored in talking face generation. Compared with GAN-based rendering techniques, +NeRF renderers could preserve more details and provide better 3D naturalness since it models a +continuous 3D scene in the hidden space. +Recent NeRF-based works Guo et al. (2021)Liu et al. (2022)Yao et al. (2022) manage to learn an +end-to-end audio-driven talking face system with only a few-minutes-long video. However, the +current end-to-end framework is faced with two challenges. 1) The first challenge is the weak +generalizability due to the small scale of training data, which only consists of about thousands-many +audio-image pairs. This deficiency of training data makes the trained model not robust to out-of- +domain (OOD) audio in many applications (such as cross-lingual Guo et al. (2021)Liu et al. (2022) or +singing voice). 2) The second challenge is the so-called ”mean face” problem. Note that the audio +to its corresponding facial motion is a one-to-many mapping, which means the same audio input +may have several correct motion patterns. Learning such a mapping with a regression-based model +leads to over-smoothing and blurry results Ren et al. (2021); specifically, for some complicated +∗Authors contribute equally to this work. +†Corresponding author +1Video samples and source code are available at https://geneface.github.io +1 +arXiv:2301.13430v1 [cs.CV] 31 Jan 2023 + +Published as a conference paper at ICLR 2023 +audio with several potential outputs, it tends to generate an image with a half-opened and blurry +mouth, which leads to unsatisfying image quality and bad lip-synchronization. To summarize, the +current NeRF-based methods are challenged with the weak generalizability problem due to the lack +of audio-to-motion training data and the ”mean face” results due to the one-to-many mapping. +In this work, we develop a talking face generation system called GeneFace to address these two +challenges. To handle the weak generalizability problem, we devise an audio-to-motion model to +predict the 3D facial landmark given the input audio. We utilize hundreds of hours of audio-motion +pairs from a large-scale lip reading datasetAfouras et al. (2018) to learn a robust mapping. As for +the ”mean face” problem, instead of using the regression-based model, we adopt a variational auto- +encoder (VAE) with a flow-based prior as the architecture of the audio-to-motion model, which +helps generate accurate and expressive facial motions. However, due to the domain shift between +the generated landmarks (in the multi-speaker domain) and the training set of NeRF (in the target +person domain), we found that the NeRF-based renderer fails to generate high-fidelity frames given +the predicted landmarks. Therefore, a domain adaptation process is proposed to rig the predicted +landmarks into the target person’s distribution. To summarize, our system consists of three stages: +1 Audio-to-motion. We present a variational motion generator to generate accurate and expressive +facial landmark given the input audio. +2 Motion domain adaptation. To overcome the domain shift, we propose a semi-supervised ad- +versarial training pipeline to train a domain adaptative post-net, which refines the predicted 3D +landmark from the multi-speaker domain into the target person domain. +3 Motion-to-image. We design a NeRF-based renderer to render high-fidelity frames conditioned +on the predicted 3D landmark. +The main contributions of this paper are summarized as follows: +• We present a three-stage framework that enables the NeRF-based talking face system to enjoy +the large-scale lip-reading corpus and achieve high generalizability to various OOD audio. We +propose an adversarial domain adaptation pipeline to bridge the domain gap between the large +corpus and the target person video. +• We are the first work that analyzes the ”mean face” problem induced by the one-to-many audio- +to-motion mapping in the talking face generation task. To handle this problem, we design a varia- +tional motion generator to generate accurate facial landmarks with rich details and expressiveness. +• Experiments show that our GeneFace outperforms other state-of-the-art GAN-based and NeRF- +based baselines from the perspective of objective and subjective metrics. +2 +RELATED WORK +Our approach is a 3D talking face system that utilizes a generative model to predict the 3DMM- +based motion representation given the driving audio and employs a neural radiance field to render +the corresponding images of a human head. It is related to recent approaches to audio-driven talking +head generation methods and scene representation networks for the human portrait. +Audio-driven Talking Head Generation +Generating talking faces in line with input audio has +long attracted the attention of the computer vision community. Earlier works focus on synthesizing +the lip motions on a static facial imageJamaludin et al. (2019)Tony Ezzat & Poggio (2002)Vou- +gioukas et al. (2020)Wiles et al. (2018). Then the frontier is promoted to synthesize the full headYu +et al. (2020)Zhou et al. (2019)Zhou et al. (2020). However, free pose control is not feasible in these +methods due to the lack of 3D modeling. With the development of 3D face reconstruction tech- +niquesDeng et al. (2019), many works explore extracting 3D Morphable Model (3DMM)Paysan +et al. (2009) from the monocular video to represent the facial movementTero Karras & Lehtinen +(2017)Yi et al. (2020) in the talking face system, which is named as model-based methods. With +3DMM, a coarse 3D face mesh M can be represented as an affine model of facial expression and +identity code: +M = M + Bidi + Bexpe, +(1) +2 + +Published as a conference paper at ICLR 2023 +Driving +Audio +HuBERT +Features +Variational Motion Generator +𝑧~𝑁 0,1 +Enhanced +Latent +Generated +Landmark +Domain Adaptative Post-net +Conv 1D +BN +ReLU +x N ++ +Refined +Landmark +3DMM NeRF Renderer +Head-NeRF +Torso-NeRF +Rendered +Head +Rendered +Frame +WaveNet- +like Decoder +Flow-based +Prior +Figure 1: The inference process of GeneFace. BN denotes batch normalization. +where M is the average face shape; Bid and Bexp are the PCA bases of identity and expression; i +and e are known as identity and expression codes. +By modeling the 3D geometry with 3DMM, the model-based works manage to manipulate the head +pose and facial movement. However, 3DMM could only define a coarse 3D mesh of the human +head, and delicate details (such as hair, wrinkle, teeth, etc.) are ignored. It raises challenges for +GAN-based methods to obtain realistic results. Recent advances in neural rendering have created +a prospect: instead of refining the geometry of 3DMM or adding more personalized attributes as +auxiliary conditions for GAN-based renderers, we could leave these delicate details to be modeled +implicitly by the hidden space of the neural radiance field. +Neural Radiance Field for Rendering Face +The recent proposed neural radiance field +(NeRF)Mildenhall et al. (2020)Kellnhofer et al. (2021)Pumarola et al. (2021)Sitzmann et al. (2019) +has attracted much attention in the human portrait rendering field since it could render high-fidelity +images with rich details such as hair and wrinkles. For instance, Sitzmann et al. (2019) presents +a compositional NeRF for generating each part of the upper body. NerFaceGafni et al. (2021) and +Pumarola et al. (2021) propose pose-expression-conditioned dynamic NeRFs for modeling the dy- +namics of a human face. EG3DChan et al. (2022) proposes a hybrid explicit–implicit tri-plane +representation to achieve fast and geometry-aware human face rendering. HeadNeRFHong et al. +(2022) proposes a real-time NeRF-based parametric head model. +Several works have also applied NeRF in the audio-driven talking face generation task. Zhang et al. +(2021) devise an implicit pose code to modularize audio-visual representations. AD-NeRF Guo +et al. (2021) first presents an end-to-end audio-driven NeRF to generate face images conditioned on +Deepspeech Hannun et al. (2014) audio features. Recently, SSP-NeRF Liu et al. (2022) proposed +a semantic-aware dynamic ray sampling module to improve the sample efficiency and design a +torso deformation module to stabilize the large-scale non-rigid torso motions. DFA-NeRF Yao et al. +(2022) introduces two disentangled representations (eye and mouth) to provide improved conditions +for NeRF. To achieve few-shot training, DFRFShen et al. (2022) conditions the face radiance field +on 2D reference images to learn the face prior, thus greatly reducing the required data scale (tens +of seconds of video) and improve the convergence speed (about 40k iterations). However, all of the +previous NeRF-based work focuses on better image quality or reducing the training cost, while the +generalizability to out-of-domain audio is relatively an oversight. +Our GeneFace could be regarded as bridging the advantages of the aforementioned two types of +works. Compared with previous 3DMM-based methods, our work could enjoy good 3D naturalness +and high image quality brought by the NeRF-based renderer. Compared with previous end-to-end +NeRF-based methods, we improve the generalizabity to out-of-domain audio via introducing a gen- +erative audio-to-motion model trained on a large lip reading corpus. +3 +GENEFACE +In this section, we introduce our proposed GeneFace. As shown in Fig. 1, GeneFace is composed of +three parts: 1) a variational motion generator that transforms HuBERT features Hsu et al. (2021) into +3D facial landmarks; 2) a post-net to refine the generated motion into the target person domain; 3) +a NeRF-based renderer to synthesize high-fidelity frames. We describe the designs and the training +process of these three parts in detail in the following subsections. +3 + +Published as a conference paper at ICLR 2023 +Large Video Corpus +Deep 3D Recon +3DMM Landmark +HuBERT +Features +WaveNet-like +Encoder +WaveNet-like +Decoder +Flow-based +Prior +𝜇, 𝜎 +Variational Motion Generator +Generated +Landmark +Pretrained +SyncNet +Training of Audio2motion +Audio +Figure 2: The structure of variational motion generator. Dashed arrows means the process is only +performed during training; and only the dashed rectangle part is used during inference. +3.1 +VARIATIONAL MOTION GENERATOR +To achieve expressive and diverse 3D head motion generation, we introduce a variational auto- +encoder (VAE) to perform a generative and expressive audio-to-motion transform, namely the vari- +ational motion generator, as shown in Fig. 2. +Audio and motion representation +To better extract the semantic information, we utilize Hu- +BERT, a state-of-the-art ASR model, to obtain audio features from the input wave and use it as the +condition of the variational motion generator. As for the motion representation, to represent detailed +facial movement in Euclidean space, we select 68 key points from the reconstructed 3D head mesh +and use their position as the action representations. Specifically, +LM3D = {(M − M)i|i ∈ I}, +(2) +where LM3D ∈ R68×3, M and M are the 3DMM mesh and mean mesh defined in Equation (1), I +is the index of the key landmark in the mesh. In this paper, we name this action representation 3D +landmarks for abbreviation. +Dilated convolutional encoder and decoder +Inspired by WaveNet, to better extract features from +the audio sequence and construct long-term temporal relationships in the output sample, we design +the encoder and decoder as fully convolutional networks where the convolutional layers have incre- +mentally increased dilation factors that allow its receptive field to grow exponentially with depth. +In contrast to previous works, which typically divide the input audio sequence into sliding windows +to obtain a smooth result, we manage to synthesize the whole sequence of arbitrary length within +a single forward. To further improve the temporal stability of the predicted landmark sequence, a +Gaussian filter is performed to eliminate tiny fluctuations in the result. +Flow-based Prior +We also notice that the gaussian prior of vanilla VAE limits the performance of +our 3D landmark sequence generation process from two prospectives: 1) the datapoint of each time +index is independent of each other, which induces noise to the sequence generation task where there +is a solid temporal correlation among frames. 2) optimizing VAE prior push the posterior distribution +towards the mean, limiting diversity and hurting the generative power. To this end, following Ren +et al. (2021), we utilize a normalizing flow to provide a complex and time-related distribution as the +prior distribution of the VAE. Please refer to Appendix A.1 for more details. +Training Process +Due to the introduction of prior flow, the closed-form ELBO is not feasible, +hence we use the Monte-Carlo ELBO loss Ren et al. (2021) to train the VAE model. Besides, +inspired by Prajwal et al. (2020), we independently train a sync-expert Dsync that measures the +possibility that the input audio and 3D landmarks are in-sync, whose training process can be found in +Appendix A.2 . The trained sync-expert is then utilized to guide the training of VAE. To summarize, +the training loss of our variational motion generator (VG) is as follows: +LVG(φ, θ, ϵ) = −Eqφ(z|l,a)[log pθ(l|z, a)]+KL(qφ(z|l, a)|pϵ(z|a))−Eˆl∼pθ(l|z,c)[log Dsync(ˆl)] (3) +where φ, θ, ϵ denote the model parameters of the encoder, decoder and the prior, respectively. c +denotes the condition features of VAE. The ground truth and predicted 3D landmarks are represented +by l and ˆl, respectively. +4 + +::.DPublished as a conference paper at ICLR 2023 +HuBERT +Features +Large Video Corpus +Variational +Motion +Generator +HuBERT +Features +Target Person Video +Domain Adaptive +Post-net +Generated Landmark ++ ++ +Refined Landmark +MLP-based +Disc. +as neg. +sample +as neg. +sample +GT target person +3DMM Landmark +MSE +as positive sample +Training of PostNet +Pretrained +SyncNet +Figure 3: The training process of Domain Adaptative Post-net. +3.2 +DOMAIN ADAPTIVE POST-NET +As we train the variational motion generator on a large multi-speaker dataset, the model can general- +ize well with various audio inputs. However, as the scale of the target person video is relatively tiny +(about 4-5 minutes) compared with the multi-speaker lip reading dataset (about hundreds of hours), +there exists a domain shift between the predicted 3D landmarks and the target person domain. As a +consequence, the NeRF-based renderer cannot generalize well with the predicted landmark, which +results in blurry or unrealistic rendered images. To this end, A naive solution is to fine-tune the +variational generator in the target person dataset. The challenge is that we generally only have a +short personalized video, and the generalizability of the model may be lost after the fine-tuning. +Under such circumstances, we design a semi-supervised adversarial training pipeline to perform a +domain adaptation. To be specific, we learn a post-net to refine the VAE-predicted 3D landmarks into +the personalized domain. We consider two requirements for this mapping: 1) it should preserve the +temporal consistency and lip-synchronization of the input sequence; 2) it should correctly map each +frame into the target person’s domain. To fulfill the first point, we utilize 1D CNN as the structure +of post-net and adopt the sync-expert to supervise the lip-synchronization; for the second point, we +jointly train an MLP-structured frame-level discriminator that measures the identity similarity of +each landmark frame to the target person. The detailed structure of the post-net and discriminator +can be found in Appendix A.3. +Training Process +The training process of post-net is shown in Fig.3. During training, the MLP +discriminator tries to distinguish between the ground truth landmark l′ extracted from the target +person’s video and the refined samples Gω(ˆl) generated from the large-scale dataset. We use the +LSGAN loss to update the discriminator : +LD(δ) = Eˆl∼pθ(l|z,c)[(Dδ(PNω(ˆl)) − 0)2] + El′∼p′(l)[(Dδ(l′) − 1)2] +(4) +where ω and δ are the parameters of the post-net PN and discriminator D. l′ is the ground truth +3DMM landmark of the target person dataset, and ˆl is the 3D landmarks refined by the post-net. +As for the training of post-net, the post-net competes with the discriminator while being guided by +the pre-trained sync-expert to maintain lip synchronization. Besides, we utilize the target person +dataset to provide a weak supervised signal to help the adversarial training. Specifically, we extract +the audio c′ of the target person video for VAE to predict the landmarks ˆl′ ∼ pθ(l|z, c′) and en- +courage the refined landmarks PNω(ˆl′) to approximate the ground truth expression l′. Finally, the +training loss of post-net is: +LPN(ω) = Eˆe∼pθ(l|z,c)[(Dδ(PNω(ˆl)) − 1)2] + Eˆl∼pθ(l|z,c)[Dsync(ˆl)] ++Eˆl′∼pθ(l|z,c′)[((PNω(ˆl′) − l′)2] +(5) +3.3 +NERF-BASED RENDERER +We obtain a robust and diverse audio-to-motion mapping through the variational motion generator +and post-net. Next, we design a NeRF-based renderer to render high-fidelity frames conditioned on +the predicted 3D landmarks. +5 + +DPublished as a conference paper at ICLR 2023 +Training of NeRF +Target Person Video +Deep 3D Recon +3DMM Landmark +Landmark +Encoder +Head Color +Encoder +Head +Pose +Head +NeRF +Torso +NeRF +Rendered +Head +Rendered +Frame +3DMM NeRF Renderer +Figure 4: The training process of NeRF-based renderer. +3D landmark-conditioned NeRF +Inspired by Guo et al. (2021), we present a conditional NeRF +to represent the dynamic talking head. Apart from viewing direction d and 3D location x, the +3D landmarks l will act as the condition to manipulate the color and geometry of the implicitly +represented head. Specifically, the implicit function F can be formulated as follows: +Fθ : (x, d, l) → (c, σ) +(6) +where c and σ denote the color and density in the radiance field. To improve the continuity between +adjacent frames, we use the 3D landmarks from the three neighboring frames to represent the facial +shape, i.e., l ∈ R3×204. We notice that some facial landmarks only change in a small range, which +numerically raises challenges for NeRF to learn the high-frequency image details. Therefore, we +normalize the input 3D landmarks point-wisely, which is necessary to achieve better visual quality. +Following the setting of volume rendering, to render each pixel, we emit a camera ray r(t) = o+t·d +in the radiance field, with camera center o, viewing direction d. The final color C is calculated by +aggregating the color c along the ray: +C(r, l; θ) = +� tf +tn +σθ(r(t), l) · cθ(r(t), l, d) · T(t)dt +(7) +where tn and tr is the near bound and far bound of ray r; cθ and σθ are the output of the implicit +function Fθ, T(t) is the accumulated transmittance along the ray from tn to t, which is defined as: +T(t) = exp(− +� t +tn +σθ(r(τ))dτ) +(8) +Head-aware Torso-NeRF +To better model the head and torso movement, we train two NeRFs to +render the head and torso parts, respectively. As shown in Fig. 4, we first train a head-NeRF to +render the head part, then train a torso-NeRF to render the torso part with the rendering image of the +head-NeRF as background. Following Guo et al. (2021), we assume the torso part is in canonical +space and provide the head pose h to torso-NeRF as a signal to infer the torso movement. The +torso-NeRF implicitly learns to expect the location of the rendered head, then rigid the torso from +canonical space to render a natural result. +However, this cooperation between head-NeRF and torso-NeRF is fragile since the torso-NeRF +cannot observe the head-NeRF’s actual output. Consequently, several recent works report that the +torso-NeRF produces head-torso separation artifacts Liu et al. (2022)Yao et al. (2022) when the head +pose is relatively large. Based on the analysis above, we propose to provide the torso-NeRF with a +perception of the rendering result of the head-NeRF. Specifically, we use the output color Chead of +the head-NeRF as a pixel-wise condition of the torso-NeRF. The torso’s implicit function Ftorso is +expressed as: +Ftorso : (x, Chead; d0, Π, l) → (c, σ) +(9) +where d0 is view direction in the canonical space, Π ∈ R3×4 is the head pose that composed of a +rotation matrix and a transform vector. +Training Process +We extract 3D landmarks from the video frames and use these landmark-image +pairs to train our NeRF-based renderer. The optimization target of head-NeRF and torso-NeRF is +to reduce the photo-metric reconstruction error between rendered and ground-truth images. Specifi- +cally, the loss function can be formulated as: +LNeRF (θ) = +� +r∈R +||Cθ(r, l) − Cg||2 +2 +(10) +6 + +Published as a conference paper at ICLR 2023 +where R is the set of camera rays, Cg is the color of the ground image. +4 +EXPERIMENTS +4.1 +DATASET PREPARATION AND PREPROCESSING +Dataset preparation. +Our method aims to synthesize high-fidelity talking face images with great +generalizability to out-domain audio. To learn robust audio-to-motion mapping, a large-scale lip- +reading corpus is needed. Hence we use LRS3-TEDAfouras et al. (2018) to train our variational +generator and post-net 2. Additionally, a certain person’s speaking video of a few minutes in length +with an audio track is needed to learn a NeRF-based person portrait renderer. To be specific, in order +to compare with the state-of-the-art method, we utilize the data set of Lu et al. (2021) and Guo et al. +(2021), which consist of 5 videos of an average length of 6,000 frames in 25 fps. +Data preprocessing. +As for the audio track, we downsample the speech wave into the sampling +rate of 16000 and process it with a pretrained HuBERT model. For the video frames of LRS3 and +the target person videos, we resample them into 25 fps and use Deng et al. (2019) to extract the head +pose and 3D landmarks. As for the target person videos, they are cropped into 512x512 and each +frame is processed with the help of an automatic parsing method Lee et al. (2020) for segmenting +the head and torso part and extracting a clean background. +4.2 +EXPERIMENTAL SETTINGS +Comparison baselines. +We compare our GeneFace with several remarkable works: 1) Wav2Lip +Prajwal et al. (2020), which pretrain a sync-expert to improve the lip-synchronization performance; +2) MakeItTalk Zhou et al. (2020), which also utilize 3D landmark as the action representation; 3) +PC-AVS Zhou et al. (2021), which first modularize the audio-visual representation. 4) LiveSpeech- +Portriat Lu et al. (2021), which achieves photorealistic results at over 30fps; 5) AD-NeRF Guo et al. +(2021), which first utilize NeRF to achieve talking head generation. For Wav2Lip, PC-AVS, and +MakeItTalk, the LRS3-TED dataset is used to train the model, and a reference clip of the target +person video is used during the inference stage; for LSP, both of LRS3-TED dataset and the target +person video is used to train the model; for the NeRF-based method, AD-NeRF, only the target +person video is used to train an end-to-end audio-to-image renderer. +Implementation Details. +We train the GeneFace on 1 NVIDIA RTX 3090 GPU, and the detailed +training hyper-parameters of the variational generator, post-net, and NeRF-based are listed in Ap- +pendix B. For variational generator and post-net, it takes about 40k and 12k steps to converge (about +12 hours). For the NeRF-based renderer, we train each model for 800k iterations (400k for head and +400k for the torso, respectively), which takes about 72 hours. +4.3 +QUANTITATIVE EVALUATION +Evaluation Metrics +We employ the FID score Heusel et al. (2017) to measure image quality. We +utilize the landmark distance (LMD)Chen et al. (2018) and syncnet confidence score Prajwal et al. +(2020) to evaluate lip synchronization. Furthermore, to evaluate the generalizability, we additionally +test all methods with a set of out-of-domain (OOD) audio, which consists of cross-lingual, cross- +gender, and singing voice audios. +Evaluation Results +The results are shown in Table 1. We have the following observations. (1) Our +GeneFace achieves good lip-synchronization with high generalizability. Since Wav2Lip is jointly +trained with SyncNet, it achieves the highest sync score that is higher than the ground truth video. +Our method performs best in LMD and achieves a better sync score than other baselines. When +tested with out-of-domain audios, while the sync-score of person-specific methods (LSP and AD- +NeRF) significantly drops, GeneFace maintains good performance. (2) Our GeneFace achieves the +best visual quality. We observe that one-shot methods (Wav2Lip, MakeItTalk, and PC-AVS) perform +2we select samples of good quality in the LRS3-TED dataset, the selected subset contains 19,775 short +videos from 3,231 speakers and is about 120 hours-long. +7 + +Published as a conference paper at ICLR 2023 +Method +FID ↓ +LMD↓ +Sync ↑ +FID(OOD) ↓ +Sync(OOD) ↑ +Wav2Lip +71.40 +3.988 +9.212 +68.05 +9.645 +MakeitTalk +57.96 +4.848 +4.981 +53.33 +4.933 +PC-AVS +96.81 +5.812 +6.239 +98.31 +6.156 +LSP +29.30 +4.589 +6.119 +35.21 +4.320 +AD-NeRF +27.52 +4.199 +4.894 +35.69 +4.225 +Ground Truth +0.00 +0.000 +8.733 +N/A +N/A +GeneFace (ours) +22.88 +3.933 +6.987 +27.38 +6.212 +Table 1: Quantitative evaluation with different methods. Best results are in bold. +/ɔɪ/ +/um/ +/f/ +/um/ +/ʃ/ +GeneFace +AD-NeRF +/i/ +/w/ +/s/ +/ɒ/ +/ju:/ +Audio +Figure 5: The comparison of generated key frame results. We show the phonetic symbol of the +key frame and the corresponding synthesized talking heads of AD-NeRF and GeneFace. We mark +the head-torso separation artifact, blurry mouth, un-sync results with brown, blue, and red arrow, +respectively. Please zoom in for better visualization. More qualitative comparisons can be found +in demo video. +poorly on FID due to low image fidelity. Since we use 3D landmarks as the condition of the NeRF +renderer, it address the mean face problem and leads to better lip syncronization and visual quality +than AD-NeRF. +4.4 +QUALITATIVE EVALUATION +To compare the generated results of each method, we show the keyframes of two clips in Fig.5. Due +to space limitations, we only compare our GeneFace with AD-NeRF here and provide full results +with all baselines in Appendix C.1. We observe that although both methods manage to generate +high-fidelity results, GeneFace solves several problems that AD-NeRF has: 1) head-torso separation +(brown arrow) due to the separate generation pipeline of head and torso part; 2) blurry mouth images +due to the one-to-many audio-to-lip mapping; 3) unsynchronized lip due to the weak generalizability. +User Study +We conduct user studies to test the quality of audio-driven portraits. Specifically, we +sample 10 audio clips from English, Chinese, and German for all methods to generate the videos, +and then involve 20 attendees for user studies. We adopt the Mean Opinion score (MOS) rating +protocol for evaluation, which is scaled from 1 to 5. The attendees are required to rate the videos +based on three aspects: (1) lip-sync accuracy; (2) video realness; (3) image quality. +We compute the average score for each method, and the results are shown in Table 2. We have +the following observations: 1) Our GeneFace achieves comparatively high lip-sync accuracy with +Wav2LipPrajwal et al. (2020) since both of them learn a generalized audio-to-motion mapping on a +large dataset with guidance from a sync-expert. 2) As for the video realness and image quality, the +Person-specific methods (LSP, AD-NeRF, and GeneFace) outperform one-shot methods (Wav2Lip, +MakeItTalk, and PC-AVS). Although LSP has slightly better image quality than GeneFace, our +method achieves the highest video realness and lip-sync accuracy score. +4.5 +ABLATION STUDY +In this section, we perform ablation study to prove the necessity of each component in GeneFace. +8 + +Published as a conference paper at ICLR 2023 +Methods +Wav2Lip +MakeItTalk +PC-AVS +LSP +AD-NeRF +GeneFace (ours) +Lip-sync Accuracy +3.77±0.25 +2.86±0.33 +3.11±0.30 +3.65±0.20 +3.05±0.26 +3.82±0.24 +Image Quality +3.38±0.19 +2.84±0.20 +2.73±0.25 +3.92±0.13 +3.44±0.22 +3.87±0.16 +Video Realness +3.27±0.26 +2.52±0.30 +2.46±0.28 +3.62±0.24 +3.31±0.24 +3.87±0.16 +Table 2: User study with different methods. The error bars are 95% confidence interval. +Setting +FID↓ +LMD↓ +Sync↑ +FID(OOD)↓ +Sync(OOD)↑ +GeneFace +22.88 +3.933 +6.987 +27.38 +6.212 +w/o prior flow +24.71 +4.063 +6.404 +29.55 +5.831 +w/o sync-expert +24.02 +4.151 +5.972 +30.77 +5.549 +w/o post-net +30.26 +4.532 +5.085 +35.58 +5.248 +w. fine-tune +25.75 +4.227 +6.875 +29.30 +5.966 +w/o head-aware +26.34 +3.948 +6.899 +28.89 +6.167 +Table 3: Ablation study results. The ablation settings are described in Sec. 4.5. +Varaiational motion generator +We test two settings on the variational motion generator: (1) w/o +prior flow, where we replace the flow-based prior with a gaussian prior. The results are shown in +Table 3 (line 2), where the Sync score drops by a relatively large margin. This observation suggests +that the temporal enhanced latent variable contributes to the stability of the predicted landmark se- +quence. (2) w/o sync-expert (line 3), where the variational motion generator is no longer supervised +by a pretrained sync-expert. We observe that it leads to a significant degradation in Sync score. +Domain adaptative post-net +In the setting w/o post-net, we remove the domain adaptative post- +net, the results are shown in Table 3 (line 4). It can be seen that directly using the 3D landmarks +predicted by the variational motion generator leads to a significant performance drop in FID and +Sync scores. To further investigate the efficacy of post-net, we utilize T-SNE to visualize the land- +marks of different domains in Fig. 10. The visualization results prove that there exists a significant +domain gap between the LRS3 dataset and the target person video, and our post-net successfully +rigs the predicted landmarks from the LRS3 domain into the target person domain. We also try to +replace the post-net with directly fine-tuning on the target person video (line 5), although it achieves +a competitive sync score on in-domain audios, its performance in OOD audio is worse. +Head-aware torso-NeRF +In the w/o head-ware setting, we remove head image condition of the +torso-NeRF. The results are shown in Table 3 (line 6). Due to the unawareness of the head’s location, +the head-torso separation occurs occasionally, which results in a drop in the FID score. +5 +CONCLUSION +In this paper, we propose GeneFace for talking face generation, which aims to solve the weak gen- +eralizability and mean face problem faced by previous NeRF-based methods. A variational motion +generator is proposed to construct a generic audio-to-motion mapping based on a large corpus. We +then introduce a domain adaptative post-net with an adversarial training pipeline to rig the predicted +motion representation into the target person domain. Moreover, a head-aware torso-NeRF is present +to address the head-torso separation issue. Extensive experiments show that our method achieves +more generalized and high-fidelity talking face generation compared to previous methods. Due to +space limitations, we discuss the limitations and future work in Appendix D. +ACKNOWLEDGMENT +This work was supported in part by the National Natural Science Foundation of China Grant No. +62222211, Zhejiang Electric Power Co.,Ltd.Science and Technology Project No.5211YF22006 and +Yiwise. +9 + +Published as a conference paper at ICLR 2023 +ETHICS STATEMENT +GeneFace improves the lip synchronization and expressiveness of the synthesized talking head +video. With the development of talking face generation techniques, it is much easier for people +to synthesize fake videos of arbitrary persons. In most situations, they utilize these techniques to +facilitate the movie and entertainment industry and reduce the bandwidth of video streaming by +sending audio signals only. However, the talking face generation techniques can be misused. As it is +more difficult for people to distinguish synthesized videos, the algorithm may be utilized to spread +fake information or obtain illegal profits. Potential solutions like digital face forensics methods to +detect deepfakes must be considered. We also plan to include restrictions in the open-source license +of the GeneFace project to prevent ”deepfake”-related abuse. We hope the public is aware of the +potential risks of misusing new techniques. +REFERENCES +Triantafyllos Afouras, Joon Son Chung, and Andrew Zisserman. Lrs3-ted: a large-scale dataset for +visual speech recognition. arXiv preprint arXiv:1809.00496, 2018. +Eric R. Chan, Connor Z. Lin, Matthew A. Chan, Koki Nagano, Boxiao Pan, Shalini De Mello, +Orazio Gallo, Leonidas J. 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ACM Transactions on Graphics (TOG), +39(6):1–15, 2020. +A +DETAILS OF MODELS +A.1 +VARIATIONAL MOTION GENERATOR +Following PortaSpeech, our variational motion generator consists of an encoder, a decoder, and +a flow-based prior model. The encoder, as shown in Fig. 6a, is composed of a 1D-convolution +followed by ReLU activation and layer normalization, and a non-causal WaveNet. The decoder, as +shown in Fig. 6b, consists of a non-causal WaveNet and a 1D transposed convolution followed by +ReLU and layer normalization. The prior model, as shown in Fig. 6c, is a normalizing flow, which is +composed of a 1D-convolution coupling layer and a channel-wise flip operation. HuBERT features +are utilized as the audio condition of these three modules. +A.2 +SYNC-EXPERT +Our sync-expert inputs a window of Tl consecutive 3D landmark frames and an audio feature clip of +size Ta × D, where Tl and Ta are the lengths of the video and audio clip respectively, and D is the +dimension of HuBERT features. The sync-expert is trained to discriminate whether the input audio +and landmarks are synchronized. It consists of a landmark encoder and an audio encoder, as shown +in Fig. 7, both of which are comprised of a stack of 1D-convolutions followed by batch normal- +ization and ReLU. We use cosine-similarity with binary cross-entropy loss to train the sync-expert. +Specifically, we compute cosine-similarity for the landmark embedding l and audio embedding a +12 + +Published as a conference paper at ICLR 2023 +Conv1D+BN +ReLU +x N +ℎ𝑢𝑏𝑒𝑟𝑡 𝑐𝑙𝑖𝑝 +Conv1D+BN +ReLU +x N +𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 𝑐𝑙𝑖𝑝 +cosine +similarity +Pos/Neg? +Figure 7: The structure of sync-expert. +Conv1D + ReLU ++Batch Norm +𝑐𝑜𝑎𝑟𝑠𝑒 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 +𝑟𝑒𝑓𝑖𝑛𝑒𝑑 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 ++ +x N +Figure 8: The structure of post-net. +to represent the probability that the input audio-landmark pair is synchronized. The training loss of +sync-expert can be represented as: +Lsync = CE( +a · l +max(||a||2 · ||l||2, ϵ)) +(11) +A.3 +DOMAIN ADAPTATIVE POST-NET AND DISCRIMINATOR +The domain adaptative post-net, as shown in Fig. +8, is composed of a stack of residual 1D- +convolution followed by ReLU and batch normalization. The discriminator is a MLP composed +of a stack of fully connected layers followed with ReLU and dropout. +B +DETAILED EXPERIMENTAL SETTINGS +B.1 +MODEL CONFIGURATIONS +We list the hyper-parameters of GeneFace in Tab. 4. +C +ADDITIONAL EXPERIMENTS +C.1 +QUALITATIVE RESULTS WITH ALL BASELINES +To compare the generated results of each method, we show the keyframes of one in-domain audio +clip in Fig.9. We have the following observations: 1) Wav2Lip achieves competitive lip-sync perfor- +mance yet generates blurry mouth results; 2) MakeItTalk and PC-AVS fail to preserve the speaker’s +identity, leading to unrealistic generated results; 3) LSP generates unnatural lip movement during the +transition phase of different syllables. Please see our supplementary video for better visualization. +13 + +Published as a conference paper at ICLR 2023 +Table 4: Hyper-parameter list +Hyper-parameter +GeneFace +Variational Motion Generator +Encoder Layers +8 +Decoder Layers +4 +Encoder/Decoder Conv1D Kernel +5 +Encoder/Decoder Conv1D Channel Size +192 +Latent Size +16 +Prior Flow Layers +4 +Prior Flow Conv1D Kernel +3 +Prior Flow Conv1D Channel Size +64 +Sync-expert Layers +14 +Sync-expert Channel Size +512 +Post-net and Discriminator +Post-net Layers +8 +Post-net Conv1D Kernel +3 +Post-net Conv1D Channel Size +256 +Discrimnator Layers +5 +Discrimnator Linear Hidden Size +256 +Discrimnator Dropout Rate +0.25 +NeRF-based Renderer +Head/Torso-NeRF Layers +11 +Head/Torso-NeRF Hidden Size +256 +Landmark/Head Color Encoder Layers +3 +Landmark/Head Color Encoder Hidden Size +128 +Setting +L2 error on 3D landmark↓ +LMD↓ +GeneFace (VAE + Flow + landmark NeRF) +0.0371 +3.933 +vanilla VAE + landmark NeRF +0.0385 +4.063 +Regression Model + landmark NeRF +0.0424 +4.305 +AD-NeRF +N/A +4.199 +Table 5: Ablation study on 3D Landmark L2 error. +C.2 +EVALUATION ON 3D LANDMARK L2 ERROR +To evaluate the contribution of the variational generator to the quality of the predicted landmark, we +adopt L2 error on the predicted 3D landmarks as the metric. We compare our vairiaitonal generator +(VAE+Flow) against vanilla VAE and a simple regression model trained with MSE loss. The results +are listed in Table 5. It can be seen that removing the prior flow or using a regression-based model +leads to a performance drop. +C.3 +T-SNE VISUALIZATION FOR DOMAIN ADAPTATION +To further investigate the efficacy of post-net, we utilize T-SNE to visualize the landmarks of dif- +ferent domains in Fig. 10. The visualization results prove that there exists a significant domain +gap between the LRS3 dataset and the target person video, and our post-net successfully rigs the +predicted landmarks from the LRS3 domain into the target person domain. +D +LIMITATIONS AND FUTURE WORK +There are mainly two limitations of the proposed approach. Firstly, we found the landmark sequence +generated by variational motion generator and post-net occasionally has tiny fluctuations, which +results in some artifacts such as shaking hairs, etc. Currently, we utilize a heuristic post-processing +method (Gaussian filter) to alleviate this problem. In future work, we will explore better modeling +the temporal information in the network architecture to further improve the stability. Secondly, the +current NeRF-based renderer is majorly based on the setting of vanilla NeRF, which results in a long +training and inference time. In future work, we will try to enhance the performance of the NeRF +backend by combining recent progress in accelerated and light-weight NeRF. +14 + +Published as a conference paper at ICLR 2023 +GeneFace +AD-NeRF +GT +LSP +PC-AVS +MakeItTalk +Wav2Lip +/i/ +/w/ +/s/ +/ɒ/ +/ju:/ +Audio +Figure 9: The comparison of generated key frame results. We show the phonetic symbol of the +key frame and the corresponding synthesized talking heads of all baselines. Please zoom in for +better visualization. More qualitative comparisons can be found in demo video. +Figure 10: The T-SNE visualization of 3DMM landmarks in different datasets. The green and blue +points denote the ground truth landmarks in LRS3 dataset and the target person video; The red and +yellow points represent the predicted landmarks without/with the domain adaptation. +15 + +person_train +postnet +pred_Irs3 +vae \ No newline at end of file diff --git a/4NFQT4oBgHgl3EQf4Dar/content/tmp_files/load_file.txt b/4NFQT4oBgHgl3EQf4Dar/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91848597f2156cb442a812a3c62600e3aa10b948 --- /dev/null +++ b/4NFQT4oBgHgl3EQf4Dar/content/tmp_files/load_file.txt @@ -0,0 +1,695 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf,len=694 +page_content='Published as a conference paper at ICLR 2023 GENEFACE: GENERALIZED AND HIGH-FIDELITY AUDIO-DRIVEN 3D TALKING FACE SYNTHESIS Zhenhui Ye1∗, Ziyue Jiang1∗, Yi Ren2, Jinglin Liu1, JinZheng He1, Zhou Zhao1† 1Zhejiang University {zhenhuiye,jiangziyue,jinglinliu,jinzhenghe,zhaozhou}@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='cn 2Bytedance ren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='yi@bytedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='com ABSTRACT Generating photo-realistic video portrait with arbitrary speech audio is a crucial problem in film-making and virtual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Recently, several works explore the us- age of neural radiance field in this task to improve 3D realness and image fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, the generalizability of previous NeRF-based methods to out-of-domain audio is limited by the small scale of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In this work, we propose Gene- Face, a generalized and high-fidelity NeRF-based talking face generation method, which can generate natural results corresponding to various out-of-domain audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Specifically, we learn a variaitional motion generator on a large lip-reading cor- pus, and introduce a domain adaptative post-net to calibrate the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Moreover, we learn a NeRF-based renderer conditioned on the predicted facial motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' A head-aware torso-NeRF is proposed to eliminate the head-torso separation prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Extensive experiments show that our method achieves more generalized and high-fidelity talking face generation compared to previous methods1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 1 INTRODUCTION Audio-driven face video synthesis is an important and challenging problem with several applications such as digital humans, virtual reality (VR), and online meetings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Over the past few years, the com- munity has exploited Generative Adversarial Networks (GAN) as the neural renderer and promoted the frontier from only predicting the lip movement Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020)Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2019) to gener- ating the whole face Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021)Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, GAN-based renderers suffer from several limitations such as unstable training, mode collapse, difficulty in modelling delicate details Suwajanakorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2017)Thies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020), and fixed static head pose Pham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2017)Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2017)Cudeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Recently, Neural Radiance Field (NeRF) Mildenhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020) has been explored in talking face generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Compared with GAN-based rendering techniques, NeRF renderers could preserve more details and provide better 3D naturalness since it models a continuous 3D scene in the hidden space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Recent NeRF-based works Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021)Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022)Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022) manage to learn an end-to-end audio-driven talking face system with only a few-minutes-long video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, the current end-to-end framework is faced with two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 1) The first challenge is the weak generalizability due to the small scale of training data, which only consists of about thousands-many audio-image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' This deficiency of training data makes the trained model not robust to out-of- domain (OOD) audio in many applications (such as cross-lingual Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021)Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022) or singing voice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2) The second challenge is the so-called ”mean face” problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Note that the audio to its corresponding facial motion is a one-to-many mapping, which means the same audio input may have several correct motion patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Learning such a mapping with a regression-based model leads to over-smoothing and blurry results Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' specifically, for some complicated ∗Authors contribute equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' †Corresponding author 1Video samples and source code are available at https://geneface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='io 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='13430v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='CV] 31 Jan 2023 Published as a conference paper at ICLR 2023 audio with several potential outputs, it tends to generate an image with a half-opened and blurry mouth, which leads to unsatisfying image quality and bad lip-synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To summarize, the current NeRF-based methods are challenged with the weak generalizability problem due to the lack of audio-to-motion training data and the ”mean face” results due to the one-to-many mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In this work, we develop a talking face generation system called GeneFace to address these two challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To handle the weak generalizability problem, we devise an audio-to-motion model to predict the 3D facial landmark given the input audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We utilize hundreds of hours of audio-motion pairs from a large-scale lip reading datasetAfouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2018) to learn a robust mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As for the ”mean face” problem, instead of using the regression-based model, we adopt a variational auto- encoder (VAE) with a flow-based prior as the architecture of the audio-to-motion model, which helps generate accurate and expressive facial motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, due to the domain shift between the generated landmarks (in the multi-speaker domain) and the training set of NeRF (in the target person domain), we found that the NeRF-based renderer fails to generate high-fidelity frames given the predicted landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Therefore, a domain adaptation process is proposed to rig the predicted landmarks into the target person’s distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To summarize, our system consists of three stages: 1 Audio-to-motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We present a variational motion generator to generate accurate and expressive facial landmark given the input audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2 Motion domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To overcome the domain shift, we propose a semi-supervised ad- versarial training pipeline to train a domain adaptative post-net, which refines the predicted 3D landmark from the multi-speaker domain into the target person domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3 Motion-to-image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We design a NeRF-based renderer to render high-fidelity frames conditioned on the predicted 3D landmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows: We present a three-stage framework that enables the NeRF-based talking face system to enjoy the large-scale lip-reading corpus and achieve high generalizability to various OOD audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We propose an adversarial domain adaptation pipeline to bridge the domain gap between the large corpus and the target person video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We are the first work that analyzes the ”mean face” problem induced by the one-to-many audio- to-motion mapping in the talking face generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To handle this problem, we design a varia- tional motion generator to generate accurate facial landmarks with rich details and expressiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Experiments show that our GeneFace outperforms other state-of-the-art GAN-based and NeRF- based baselines from the perspective of objective and subjective metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2 RELATED WORK Our approach is a 3D talking face system that utilizes a generative model to predict the 3DMM- based motion representation given the driving audio and employs a neural radiance field to render the corresponding images of a human head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' It is related to recent approaches to audio-driven talking head generation methods and scene representation networks for the human portrait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Audio-driven Talking Head Generation Generating talking faces in line with input audio has long attracted the attention of the computer vision community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Earlier works focus on synthesizing the lip motions on a static facial imageJamaludin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2019)Tony Ezzat & Poggio (2002)Vou- gioukas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020)Wiles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Then the frontier is promoted to synthesize the full headYu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020)Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2019)Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, free pose control is not feasible in these methods due to the lack of 3D modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' With the development of 3D face reconstruction tech- niquesDeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2019), many works explore extracting 3D Morphable Model (3DMM)Paysan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2009) from the monocular video to represent the facial movementTero Karras & Lehtinen (2017)Yi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020) in the talking face system, which is named as model-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' With 3DMM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' a coarse 3D face mesh M can be represented as an affine model of facial expression and identity code: M = M + Bidi + Bexpe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (1) 2 Published as a conference paper at ICLR 2023 Driving Audio HuBERT Features Variational Motion Generator 𝑧~𝑁 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='1 Enhanced Latent Generated Landmark Domain Adaptative Post-net Conv 1D BN ReLU x N + Refined Landmark 3DMM NeRF Renderer Head-NeRF Torso-NeRF Rendered Head Rendered Frame WaveNet- like Decoder Flow-based Prior Figure 1: The inference process of GeneFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' BN denotes batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' where M is the average face shape;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Bid and Bexp are the PCA bases of identity and expression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' i and e are known as identity and expression codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' By modeling the 3D geometry with 3DMM, the model-based works manage to manipulate the head pose and facial movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, 3DMM could only define a coarse 3D mesh of the human head, and delicate details (such as hair, wrinkle, teeth, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=') are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' It raises challenges for GAN-based methods to obtain realistic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Recent advances in neural rendering have created a prospect: instead of refining the geometry of 3DMM or adding more personalized attributes as auxiliary conditions for GAN-based renderers, we could leave these delicate details to be modeled implicitly by the hidden space of the neural radiance field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Neural Radiance Field for Rendering Face The recent proposed neural radiance field (NeRF)Mildenhall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020)Kellnhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021)Pumarola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021)Sitzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2019) has attracted much attention in the human portrait rendering field since it could render high-fidelity images with rich details such as hair and wrinkles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' For instance, Sitzmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2019) presents a compositional NeRF for generating each part of the upper body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' NerFaceGafni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021) and Pumarola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021) propose pose-expression-conditioned dynamic NeRFs for modeling the dy- namics of a human face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' EG3DChan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022) proposes a hybrid explicit–implicit tri-plane representation to achieve fast and geometry-aware human face rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' HeadNeRFHong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022) proposes a real-time NeRF-based parametric head model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Several works have also applied NeRF in the audio-driven talking face generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021) devise an implicit pose code to modularize audio-visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' AD-NeRF Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021) first presents an end-to-end audio-driven NeRF to generate face images conditioned on Deepspeech Hannun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2014) audio features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Recently, SSP-NeRF Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022) proposed a semantic-aware dynamic ray sampling module to improve the sample efficiency and design a torso deformation module to stabilize the large-scale non-rigid torso motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' DFA-NeRF Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022) introduces two disentangled representations (eye and mouth) to provide improved conditions for NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To achieve few-shot training, DFRFShen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022) conditions the face radiance field on 2D reference images to learn the face prior, thus greatly reducing the required data scale (tens of seconds of video) and improve the convergence speed (about 40k iterations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, all of the previous NeRF-based work focuses on better image quality or reducing the training cost, while the generalizability to out-of-domain audio is relatively an oversight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Our GeneFace could be regarded as bridging the advantages of the aforementioned two types of works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Compared with previous 3DMM-based methods, our work could enjoy good 3D naturalness and high image quality brought by the NeRF-based renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Compared with previous end-to-end NeRF-based methods, we improve the generalizabity to out-of-domain audio via introducing a gen- erative audio-to-motion model trained on a large lip reading corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3 GENEFACE In this section, we introduce our proposed GeneFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 1, GeneFace is composed of three parts: 1) a variational motion generator that transforms HuBERT features Hsu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021) into 3D facial landmarks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2) a post-net to refine the generated motion into the target person domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3) a NeRF-based renderer to synthesize high-fidelity frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We describe the designs and the training process of these three parts in detail in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3 Published as a conference paper at ICLR 2023 Large Video Corpus Deep 3D Recon 3DMM Landmark HuBERT Features WaveNet-like Encoder WaveNet-like Decoder Flow-based Prior 𝜇, 𝜎 Variational Motion Generator Generated Landmark Pretrained SyncNet Training of Audio2motion Audio Figure 2: The structure of variational motion generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Dashed arrows means the process is only performed during training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' and only the dashed rectangle part is used during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='1 VARIATIONAL MOTION GENERATOR To achieve expressive and diverse 3D head motion generation, we introduce a variational auto- encoder (VAE) to perform a generative and expressive audio-to-motion transform, namely the vari- ational motion generator, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Audio and motion representation To better extract the semantic information, we utilize Hu- BERT, a state-of-the-art ASR model, to obtain audio features from the input wave and use it as the condition of the variational motion generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As for the motion representation, to represent detailed facial movement in Euclidean space, we select 68 key points from the reconstructed 3D head mesh and use their position as the action representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Specifically, LM3D = {(M − M)i|i ∈ I}, (2) where LM3D ∈ R68×3, M and M are the 3DMM mesh and mean mesh defined in Equation (1), I is the index of the key landmark in the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In this paper, we name this action representation 3D landmarks for abbreviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Dilated convolutional encoder and decoder Inspired by WaveNet, to better extract features from the audio sequence and construct long-term temporal relationships in the output sample, we design the encoder and decoder as fully convolutional networks where the convolutional layers have incre- mentally increased dilation factors that allow its receptive field to grow exponentially with depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In contrast to previous works, which typically divide the input audio sequence into sliding windows to obtain a smooth result, we manage to synthesize the whole sequence of arbitrary length within a single forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To further improve the temporal stability of the predicted landmark sequence, a Gaussian filter is performed to eliminate tiny fluctuations in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Flow-based Prior We also notice that the gaussian prior of vanilla VAE limits the performance of our 3D landmark sequence generation process from two prospectives: 1) the datapoint of each time index is independent of each other, which induces noise to the sequence generation task where there is a solid temporal correlation among frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2) optimizing VAE prior push the posterior distribution towards the mean, limiting diversity and hurting the generative power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To this end, following Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021), we utilize a normalizing flow to provide a complex and time-related distribution as the prior distribution of the VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Please refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Training Process Due to the introduction of prior flow, the closed-form ELBO is not feasible, hence we use the Monte-Carlo ELBO loss Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021) to train the VAE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Besides, inspired by Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020), we independently train a sync-expert Dsync that measures the possibility that the input audio and 3D landmarks are in-sync, whose training process can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The trained sync-expert is then utilized to guide the training of VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To summarize, the training loss of our variational motion generator (VG) is as follows: LVG(φ, θ, ϵ) = −Eqφ(z|l,a)[log pθ(l|z, a)]+KL(qφ(z|l, a)|pϵ(z|a))−Eˆl∼pθ(l|z,c)[log Dsync(ˆl)] (3) where φ, θ, ϵ denote the model parameters of the encoder, decoder and the prior, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' c denotes the condition features of VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The ground truth and predicted 3D landmarks are represented by l and ˆl, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4 ::.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='DPublished as a conference paper at ICLR 2023 HuBERT Features Large Video Corpus Variational Motion Generator HuBERT Features Target Person Video Domain Adaptive Post-net Generated Landmark + + Refined Landmark MLP-based Disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' as neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' sample as neg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' sample GT target person 3DMM Landmark MSE as positive sample Training of PostNet Pretrained SyncNet Figure 3: The training process of Domain Adaptative Post-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='2 DOMAIN ADAPTIVE POST-NET As we train the variational motion generator on a large multi-speaker dataset, the model can general- ize well with various audio inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, as the scale of the target person video is relatively tiny (about 4-5 minutes) compared with the multi-speaker lip reading dataset (about hundreds of hours), there exists a domain shift between the predicted 3D landmarks and the target person domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As a consequence, the NeRF-based renderer cannot generalize well with the predicted landmark, which results in blurry or unrealistic rendered images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To this end, A naive solution is to fine-tune the variational generator in the target person dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The challenge is that we generally only have a short personalized video, and the generalizability of the model may be lost after the fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Under such circumstances, we design a semi-supervised adversarial training pipeline to perform a domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To be specific, we learn a post-net to refine the VAE-predicted 3D landmarks into the personalized domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We consider two requirements for this mapping: 1) it should preserve the temporal consistency and lip-synchronization of the input sequence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2) it should correctly map each frame into the target person’s domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To fulfill the first point, we utilize 1D CNN as the structure of post-net and adopt the sync-expert to supervise the lip-synchronization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' for the second point, we jointly train an MLP-structured frame-level discriminator that measures the identity similarity of each landmark frame to the target person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The detailed structure of the post-net and discriminator can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Training Process The training process of post-net is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' During training, the MLP discriminator tries to distinguish between the ground truth landmark l′ extracted from the target person’s video and the refined samples Gω(ˆl) generated from the large-scale dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We use the LSGAN loss to update the discriminator : LD(δ) = Eˆl∼pθ(l|z,c)[(Dδ(PNω(ˆl)) − 0)2] + El′∼p′(l)[(Dδ(l′) − 1)2] (4) where ω and δ are the parameters of the post-net PN and discriminator D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' l′ is the ground truth 3DMM landmark of the target person dataset, and ˆl is the 3D landmarks refined by the post-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As for the training of post-net, the post-net competes with the discriminator while being guided by the pre-trained sync-expert to maintain lip synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Besides, we utilize the target person dataset to provide a weak supervised signal to help the adversarial training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Specifically, we extract the audio c′ of the target person video for VAE to predict the landmarks ˆl′ ∼ pθ(l|z, c′) and en- courage the refined landmarks PNω(ˆl′) to approximate the ground truth expression l′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Finally, the training loss of post-net is: LPN(ω) = Eˆe∼pθ(l|z,c)[(Dδ(PNω(ˆl)) − 1)2] + Eˆl∼pθ(l|z,c)[Dsync(ˆl)] +Eˆl′∼pθ(l|z,c′)[((PNω(ˆl′) − l′)2] (5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='3 NERF-BASED RENDERER We obtain a robust and diverse audio-to-motion mapping through the variational motion generator and post-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Next, we design a NeRF-based renderer to render high-fidelity frames conditioned on the predicted 3D landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 5 DPublished as a conference paper at ICLR 2023 Training of NeRF Target Person Video Deep 3D Recon 3DMM Landmark Landmark Encoder Head Color Encoder Head Pose Head NeRF Torso NeRF Rendered Head Rendered Frame 3DMM NeRF Renderer Figure 4: The training process of NeRF-based renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3D landmark-conditioned NeRF Inspired by Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021), we present a conditional NeRF to represent the dynamic talking head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Apart from viewing direction d and 3D location x, the 3D landmarks l will act as the condition to manipulate the color and geometry of the implicitly represented head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Specifically, the implicit function F can be formulated as follows: Fθ : (x, d, l) → (c, σ) (6) where c and σ denote the color and density in the radiance field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To improve the continuity between adjacent frames, we use the 3D landmarks from the three neighboring frames to represent the facial shape, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=', l ∈ R3×204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We notice that some facial landmarks only change in a small range, which numerically raises challenges for NeRF to learn the high-frequency image details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Therefore, we normalize the input 3D landmarks point-wisely, which is necessary to achieve better visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Following the setting of volume rendering, to render each pixel, we emit a camera ray r(t) = o+t·d in the radiance field, with camera center o, viewing direction d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The final color C is calculated by aggregating the color c along the ray: C(r, l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' θ) = � tf tn σθ(r(t), l) · cθ(r(t), l, d) · T(t)dt (7) where tn and tr is the near bound and far bound of ray r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' cθ and σθ are the output of the implicit function Fθ, T(t) is the accumulated transmittance along the ray from tn to t, which is defined as: T(t) = exp(− � t tn σθ(r(τ))dτ) (8) Head-aware Torso-NeRF To better model the head and torso movement, we train two NeRFs to render the head and torso parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4, we first train a head-NeRF to render the head part, then train a torso-NeRF to render the torso part with the rendering image of the head-NeRF as background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Following Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021), we assume the torso part is in canonical space and provide the head pose h to torso-NeRF as a signal to infer the torso movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The torso-NeRF implicitly learns to expect the location of the rendered head, then rigid the torso from canonical space to render a natural result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, this cooperation between head-NeRF and torso-NeRF is fragile since the torso-NeRF cannot observe the head-NeRF’s actual output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Consequently, several recent works report that the torso-NeRF produces head-torso separation artifacts Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022)Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2022) when the head pose is relatively large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Based on the analysis above, we propose to provide the torso-NeRF with a perception of the rendering result of the head-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Specifically, we use the output color Chead of the head-NeRF as a pixel-wise condition of the torso-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The torso’s implicit function Ftorso is expressed as: Ftorso : (x, Chead;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' d0, Π, l) → (c, σ) (9) where d0 is view direction in the canonical space, Π ∈ R3×4 is the head pose that composed of a rotation matrix and a transform vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Training Process We extract 3D landmarks from the video frames and use these landmark-image pairs to train our NeRF-based renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The optimization target of head-NeRF and torso-NeRF is to reduce the photo-metric reconstruction error between rendered and ground-truth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Specifi- cally, the loss function can be formulated as: LNeRF (θ) = � r∈R ||Cθ(r, l) − Cg||2 2 (10) 6 Published as a conference paper at ICLR 2023 where R is the set of camera rays, Cg is the color of the ground image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4 EXPERIMENTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='1 DATASET PREPARATION AND PREPROCESSING Dataset preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Our method aims to synthesize high-fidelity talking face images with great generalizability to out-domain audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To learn robust audio-to-motion mapping, a large-scale lip- reading corpus is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Hence we use LRS3-TEDAfouras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2018) to train our variational generator and post-net 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Additionally, a certain person’s speaking video of a few minutes in length with an audio track is needed to learn a NeRF-based person portrait renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To be specific, in order to compare with the state-of-the-art method, we utilize the data set of Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021) and Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021), which consist of 5 videos of an average length of 6,000 frames in 25 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Data preprocessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As for the audio track, we downsample the speech wave into the sampling rate of 16000 and process it with a pretrained HuBERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' For the video frames of LRS3 and the target person videos, we resample them into 25 fps and use Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2019) to extract the head pose and 3D landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As for the target person videos, they are cropped into 512x512 and each frame is processed with the help of an automatic parsing method Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020) for segmenting the head and torso part and extracting a clean background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='2 EXPERIMENTAL SETTINGS Comparison baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We compare our GeneFace with several remarkable works: 1) Wav2Lip Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020), which pretrain a sync-expert to improve the lip-synchronization performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2) MakeItTalk Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020), which also utilize 3D landmark as the action representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3) PC-AVS Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021), which first modularize the audio-visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4) LiveSpeech- Portriat Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021), which achieves photorealistic results at over 30fps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 5) AD-NeRF Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2021), which first utilize NeRF to achieve talking head generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' For Wav2Lip, PC-AVS, and MakeItTalk, the LRS3-TED dataset is used to train the model, and a reference clip of the target person video is used during the inference stage;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' for LSP, both of LRS3-TED dataset and the target person video is used to train the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' for the NeRF-based method, AD-NeRF, only the target person video is used to train an end-to-end audio-to-image renderer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We train the GeneFace on 1 NVIDIA RTX 3090 GPU, and the detailed training hyper-parameters of the variational generator, post-net, and NeRF-based are listed in Ap- pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' For variational generator and post-net, it takes about 40k and 12k steps to converge (about 12 hours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' For the NeRF-based renderer, we train each model for 800k iterations (400k for head and 400k for the torso, respectively), which takes about 72 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='3 QUANTITATIVE EVALUATION Evaluation Metrics We employ the FID score Heusel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2017) to measure image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We utilize the landmark distance (LMD)Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2018) and syncnet confidence score Prajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020) to evaluate lip synchronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Furthermore, to evaluate the generalizability, we additionally test all methods with a set of out-of-domain (OOD) audio, which consists of cross-lingual, cross- gender, and singing voice audios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Evaluation Results The results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We have the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (1) Our GeneFace achieves good lip-synchronization with high generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Since Wav2Lip is jointly trained with SyncNet, it achieves the highest sync score that is higher than the ground truth video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Our method performs best in LMD and achieves a better sync score than other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' When tested with out-of-domain audios, while the sync-score of person-specific methods (LSP and AD- NeRF) significantly drops, GeneFace maintains good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2) Our GeneFace achieves the best visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We observe that one-shot methods (Wav2Lip, MakeItTalk, and PC-AVS) perform 2we select samples of good quality in the LRS3-TED dataset, the selected subset contains 19,775 short videos from 3,231 speakers and is about 120 hours-long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 7 Published as a conference paper at ICLR 2023 Method FID ↓ LMD↓ Sync ↑ FID(OOD) ↓ Sync(OOD) ↑ Wav2Lip 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='988 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='212 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='645 MakeitTalk 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='96 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='848 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='981 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='933 PC-AVS 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='81 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='812 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='239 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='31 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='156 LSP 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='589 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='119 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='320 AD-NeRF 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='199 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='894 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='69 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='225 Ground Truth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='733 N/A N/A GeneFace (ours) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='933 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='987 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='212 Table 1: Quantitative evaluation with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Best results are in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' /ɔɪ/ /um/ /f/ /um/ /ʃ/ GeneFace AD-NeRF /i/ /w/ /s/ /ɒ/ /ju:/ Audio Figure 5: The comparison of generated key frame results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We show the phonetic symbol of the key frame and the corresponding synthesized talking heads of AD-NeRF and GeneFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We mark the head-torso separation artifact, blurry mouth, un-sync results with brown, blue, and red arrow, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Please zoom in for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' More qualitative comparisons can be found in demo video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' poorly on FID due to low image fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Since we use 3D landmarks as the condition of the NeRF renderer, it address the mean face problem and leads to better lip syncronization and visual quality than AD-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='4 QUALITATIVE EVALUATION To compare the generated results of each method, we show the keyframes of two clips in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Due to space limitations, we only compare our GeneFace with AD-NeRF here and provide full results with all baselines in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We observe that although both methods manage to generate high-fidelity results, GeneFace solves several problems that AD-NeRF has: 1) head-torso separation (brown arrow) due to the separate generation pipeline of head and torso part;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2) blurry mouth images due to the one-to-many audio-to-lip mapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3) unsynchronized lip due to the weak generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' User Study We conduct user studies to test the quality of audio-driven portraits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Specifically, we sample 10 audio clips from English, Chinese, and German for all methods to generate the videos, and then involve 20 attendees for user studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We adopt the Mean Opinion score (MOS) rating protocol for evaluation, which is scaled from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The attendees are required to rate the videos based on three aspects: (1) lip-sync accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2) video realness;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (3) image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We compute the average score for each method, and the results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We have the following observations: 1) Our GeneFace achieves comparatively high lip-sync accuracy with Wav2LipPrajwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2020) since both of them learn a generalized audio-to-motion mapping on a large dataset with guidance from a sync-expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2) As for the video realness and image quality, the Person-specific methods (LSP, AD-NeRF, and GeneFace) outperform one-shot methods (Wav2Lip, MakeItTalk, and PC-AVS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Although LSP has slightly better image quality than GeneFace, our method achieves the highest video realness and lip-sync accuracy score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='5 ABLATION STUDY In this section, we perform ablation study to prove the necessity of each component in GeneFace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 8 Published as a conference paper at ICLR 2023 Methods Wav2Lip MakeItTalk PC-AVS LSP AD-NeRF GeneFace (ours) Lip-sync Accuracy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='77±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='33 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='11±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='65±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='05±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='82±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='24 Image Quality 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='38±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='73±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='44±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='16 Video Realness 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='27±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='46±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='87±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='16 Table 2: User study with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The error bars are 95% confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Setting FID↓ LMD↓ Sync↑ FID(OOD)↓ Sync(OOD)↑ GeneFace 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='933 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='987 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='212 w/o prior flow 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='063 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='404 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='55 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='831 w/o sync-expert 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='02 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='151 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='972 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='77 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='549 w/o post-net 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='532 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='085 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='58 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='248 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' fine-tune 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='227 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='875 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='966 w/o head-aware 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='948 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='899 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='89 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='167 Table 3: Ablation study results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The ablation settings are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Varaiational motion generator We test two settings on the variational motion generator: (1) w/o prior flow, where we replace the flow-based prior with a gaussian prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The results are shown in Table 3 (line 2), where the Sync score drops by a relatively large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' This observation suggests that the temporal enhanced latent variable contributes to the stability of the predicted landmark se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' (2) w/o sync-expert (line 3), where the variational motion generator is no longer supervised by a pretrained sync-expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We observe that it leads to a significant degradation in Sync score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Domain adaptative post-net In the setting w/o post-net, we remove the domain adaptative post- net, the results are shown in Table 3 (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' It can be seen that directly using the 3D landmarks predicted by the variational motion generator leads to a significant performance drop in FID and Sync scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' To further investigate the efficacy of post-net, we utilize T-SNE to visualize the land- marks of different domains in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The visualization results prove that there exists a significant domain gap between the LRS3 dataset and the target person video, and our post-net successfully rigs the predicted landmarks from the LRS3 domain into the target person domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We also try to replace the post-net with directly fine-tuning on the target person video (line 5), although it achieves a competitive sync score on in-domain audios, its performance in OOD audio is worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Head-aware torso-NeRF In the w/o head-ware setting, we remove head image condition of the torso-NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The results are shown in Table 3 (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Due to the unawareness of the head’s location, the head-torso separation occurs occasionally, which results in a drop in the FID score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 5 CONCLUSION In this paper, we propose GeneFace for talking face generation, which aims to solve the weak gen- eralizability and mean face problem faced by previous NeRF-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' A variational motion generator is proposed to construct a generic audio-to-motion mapping based on a large corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We then introduce a domain adaptative post-net with an adversarial training pipeline to rig the predicted motion representation into the target person domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Moreover, a head-aware torso-NeRF is present to address the head-torso separation issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Extensive experiments show that our method achieves more generalized and high-fidelity talking face generation compared to previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Due to space limitations, we discuss the limitations and future work in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported in part by the National Natural Science Foundation of China Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 62222211, Zhejiang Electric Power Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=',Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Science and Technology Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='5211YF22006 and Yiwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 9 Published as a conference paper at ICLR 2023 ETHICS STATEMENT GeneFace improves the lip synchronization and expressiveness of the synthesized talking head video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' With the development of talking face generation techniques, it is much easier for people to synthesize fake videos of arbitrary persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In most situations, they utilize these techniques to facilitate the movie and entertainment industry and reduce the bandwidth of video streaming by sending audio signals only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' However, the talking face generation techniques can be misused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' As it is more difficult for people to distinguish synthesized videos, the algorithm may be utilized to spread fake information or obtain illegal profits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Potential solutions like digital face forensics methods to detect deepfakes must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We also plan to include restrictions in the open-source license of the GeneFace project to prevent ”deepfake”-related abuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We hope the public is aware of the 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3661–3670, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Hang Zhou, Yu Liu, Ziwei Liu, Ping Luo, and Xiaogang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Talking face generation by adver- sarially disentangled audio-visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In AAAI, volume 33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 9299–9306, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Hang Zhou, Yasheng Sun, Wayne Wu, Chen Change Loy, Xiaogang Wang, and Ziwei Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Pose- controllable talking face generation by implicitly modularized audio-visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In CVPR, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4176–4186, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Yang Zhou, Xintong Han, Eli Shechtman, Jose Echevarria, Evangelos Kalogerakis, and Dingzeyu Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Makelttalk: speaker-aware talking-head animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' ACM Transactions on Graphics (TOG), 39(6):1–15, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' A DETAILS OF MODELS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='1 VARIATIONAL MOTION GENERATOR Following PortaSpeech, our variational motion generator consists of an encoder, a decoder, and a flow-based prior model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The encoder, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 6a, is composed of a 1D-convolution followed by ReLU activation and layer normalization, and a non-causal WaveNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The decoder, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 6b, consists of a non-causal WaveNet and a 1D transposed convolution followed by ReLU and layer normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The prior model, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 6c, is a normalizing flow, which is composed of a 1D-convolution coupling layer and a channel-wise flip operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' HuBERT features are utilized as the audio condition of these three modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='2 SYNC-EXPERT Our sync-expert inputs a window of Tl consecutive 3D landmark frames and an audio feature clip of size Ta × D, where Tl and Ta are the lengths of the video and audio clip respectively, and D is the dimension of HuBERT features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The sync-expert is trained to discriminate whether the input audio and landmarks are synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' It consists of a landmark encoder and an audio encoder, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 7, both of which are comprised of a stack of 1D-convolutions followed by batch normal- ization and ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We use cosine-similarity with binary cross-entropy loss to train the sync-expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Specifically, we compute cosine-similarity for the landmark embedding l and audio embedding a 12 Published as a conference paper at ICLR 2023 Conv1D+BN ReLU x N ℎ𝑢𝑏𝑒𝑟𝑡 𝑐𝑙𝑖𝑝 Conv1D+BN ReLU x N 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 𝑐𝑙𝑖𝑝 cosine similarity Pos/Neg?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Figure 7: The structure of sync-expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Conv1D + ReLU +Batch Norm 𝑐𝑜𝑎𝑟𝑠𝑒 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 𝑟𝑒𝑓𝑖𝑛𝑒𝑑 𝑙𝑎𝑛𝑑𝑚𝑎𝑟𝑘 + x N Figure 8: The structure of post-net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' to represent the probability that the input audio-landmark pair is synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The training loss of sync-expert can be represented as: Lsync = CE( a · l max(||a||2 · ||l||2, ϵ)) (11) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='3 DOMAIN ADAPTATIVE POST-NET AND DISCRIMINATOR The domain adaptative post-net, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 8, is composed of a stack of residual 1D- convolution followed by ReLU and batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The discriminator is a MLP composed of a stack of fully connected layers followed with ReLU and dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' B DETAILED EXPERIMENTAL SETTINGS B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='1 MODEL CONFIGURATIONS We list the hyper-parameters of GeneFace in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' C ADDITIONAL EXPERIMENTS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='1 QUALITATIVE RESULTS WITH ALL BASELINES To compare the generated results of each method, we show the keyframes of one in-domain audio clip in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We have the following observations: 1) Wav2Lip achieves competitive lip-sync perfor- mance yet generates blurry mouth results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 2) MakeItTalk and PC-AVS fail to preserve the speaker’s identity, leading to unrealistic generated results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 3) LSP generates unnatural lip movement during the transition phase of different syllables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Please see our supplementary video for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Published as a conference paper at ICLR 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Table 4: Hyper-parameter list ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Hyper-parameter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='GeneFace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Variational Motion Generator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Encoder Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Decoder Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Encoder/Decoder Conv1D Kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Encoder/Decoder Conv1D Channel Size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='192 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Latent Size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Prior Flow Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Prior Flow Conv1D Kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Prior Flow Conv1D Channel Size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Sync-expert Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Sync-expert Channel Size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Post-net and Discriminator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Post-net Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Post-net Conv1D Kernel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Post-net Conv1D Channel Size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Discrimnator Layers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Discrimnator Linear Hidden Size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='Discrimnator Dropout Rate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='25 NeRF-based Renderer Head/Torso-NeRF Layers 11 Head/Torso-NeRF Hidden Size 256 Landmark/Head Color Encoder Layers 3 Landmark/Head Color Encoder Hidden Size 128 Setting L2 error on 3D landmark↓ LMD↓ GeneFace (VAE + Flow + landmark NeRF) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='0371 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='933 vanilla VAE + landmark NeRF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='0385 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='063 Regression Model + landmark NeRF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='0424 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='305 AD-NeRF N/A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='199 Table 5: Ablation study on 3D Landmark L2 error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='2 EVALUATION ON 3D LANDMARK L2 ERROR To evaluate the contribution of the variational generator to the quality of the predicted landmark, we adopt L2 error on the predicted 3D landmarks as the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We compare our vairiaitonal generator (VAE+Flow) against vanilla VAE and a simple regression model trained with MSE loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The results are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' It can be seen that removing the prior flow or using a regression-based model leads to a performance drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content='3 T-SNE VISUALIZATION FOR DOMAIN ADAPTATION To further investigate the efficacy of post-net, we utilize T-SNE to visualize the landmarks of dif- ferent domains in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The visualization results prove that there exists a significant domain gap between the LRS3 dataset and the target person video, and our post-net successfully rigs the predicted landmarks from the LRS3 domain into the target person domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' D LIMITATIONS AND FUTURE WORK There are mainly two limitations of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Firstly, we found the landmark sequence generated by variational motion generator and post-net occasionally has tiny fluctuations, which results in some artifacts such as shaking hairs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Currently, we utilize a heuristic post-processing method (Gaussian filter) to alleviate this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In future work, we will explore better modeling the temporal information in the network architecture to further improve the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Secondly, the current NeRF-based renderer is majorly based on the setting of vanilla NeRF, which results in a long training and inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' In future work, we will try to enhance the performance of the NeRF backend by combining recent progress in accelerated and light-weight NeRF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 14 Published as a conference paper at ICLR 2023 GeneFace AD-NeRF GT LSP PC-AVS MakeItTalk Wav2Lip /i/ /w/ /s/ /ɒ/ /ju:/ Audio Figure 9: The comparison of generated key frame results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' We show the phonetic symbol of the key frame and the corresponding synthesized talking heads of all baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Please zoom in for better visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' More qualitative comparisons can be found in demo video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' Figure 10: The T-SNE visualization of 3DMM landmarks in different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The green and blue points denote the ground truth landmarks in LRS3 dataset and the target person video;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' The red and yellow points represent the predicted landmarks without/with the domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} +page_content=' 15 person_train postnet pred_Irs3 vae' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NFQT4oBgHgl3EQf4Dar/content/2301.13430v1.pdf'} 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a/8dE0T4oBgHgl3EQfwgEX/content/tmp_files/2301.02632v1.pdf.txt b/8dE0T4oBgHgl3EQfwgEX/content/tmp_files/2301.02632v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e9ef877c5d345fb2af263328d438b72a2af075b --- /dev/null +++ b/8dE0T4oBgHgl3EQfwgEX/content/tmp_files/2301.02632v1.pdf.txt @@ -0,0 +1,714 @@ +arXiv:2301.02632v1 [math.GM] 30 Nov 2022 +A NOTE ON LP-KENMOTSU MANIFOLDS ADMITTING +RICCI-YAMABE SOLITONS +MOBIN AHMAD, GAZALA AND MOHD BILAL +Abstract. In the current note, we study Lorentzian para-Kenmotsu (in brief, +LP -Kenmotsu) manifolds admitting Ricci-Yamabe solitons (RYS) and gradi- +ent Ricci-Yamabe soliton (gradient RYS). At last by constructing a 5-dimensional +non-trivial example we illustrate our result. +2010 Mathematics Subject Classification. 53C20, 53C21, 53C25, 53E20. +Keywords. Lorentzian para-Kenmotsu manifolds, Ricci-Yamabe solitons, Einstein +manifolds, ν-Einstein manifolds. +1. Introduction +In 2019, a scalar combination of Ricci and Yamabe flows was proposed by the +authors G¨uler and Crasmareanu [6], this advanced class of geometric flows called +Ricci-Yamabe (RY) flow of type (σ, ρ) and is defined by +∂ +∂tg(t) + 2σS(g(t)) + ρr(t)g(t) = 0, +g(0) = g0 +for some scalars σ and ρ. +A solution to the RY flow is called RYS if it depends only on one parameter group +of diffeomorphism and scaling. A Riemannian (or semi-Riemannian) manifold M +is said to have a RYS if +£Kg + 2σS + (2Λ − ρr)g = 0, +(1.1) +where σ, ρ, Λ ∈ R (the set of real numbers). +If K is the gradient of a smooth +function v on M, then (1.1) is called the gradient Ricci-Yamabe soliton (gradient +RYS) and hence (1.1) turns to +∇2v + σS + (Λ − ρr +2 )g = 0, +(1.2) +where ∇2v is the Hessian of v. It is to be noted that a RYS of types (σ, 0) and +(0, ρ) are known as σ−Ricci soliton and ρ−Yamabe soliton, respectively. A RYS +is said to be shrinking , steady or expanding if Λ < 0, = 0 or > 0, respectively. A +RYS is said to be a +• Ricci soliton [7] if σ = 1, ρ = 0, +• Yamabe soliton [8] if σ = 0, ρ = 1, +• Einstein soliton [3] if σ = 1, ρ = −1, +As a continuation of this study, we tried to study RYS in the frame-work of +LP-Kenmotsu manifolds of dimension n. We recommend the papers [1, 2, 5, 9, 10, +13, 15, 16, 17, 18, 19] and the references therein for more details about the related +studies. +1 + +2 +MOBIN AHMAD, GAZALA AND MOHD BILAL +2. Preliminaries +An n-dimensional differentiable manifold M with structure (ϕ, ζ, ν, g) is said to +be a Lorentzian almost paracontact metric manifold, if it admits a (1, 1)-tensor field +ϕ, a contravariant vector field ζ, a 1-form ν and a Lorentzian metric g satisfying +(2.1) +ν(ζ) + 1 = 0, +(2.2) +ϕ2E = E + ν(E)ζ, +(2.3) +ϕζ = 0, +ν(ϕE) = 0, +(2.4) +g(ϕE, ϕF) = g(E, F) + ν(E)ν(F), +(2.5) +g(E, ζ) = ν(E), +(2.6) +ϕ(E, F) = ϕ(F, E) = g(E, ϕF) +for any vector fields E, F ∈ χ(M), where χ(M) is the Lie algebra of vector fields +on M. +If ζ is a killing vector field, the (para) contact structure is called a K-(para) contact. +In such a case, we have +(2.7) +∇Eζ = ϕE. +Recently, the authors Haseeb and Prasad defined and studied the following notion: +Definition 2.1. A Lorentzian almost paracontact manifold M is called Lorentzian +para-Kenmostu manifold if [11] +(2.8) +(∇Eϕ)F = −g(ϕE, F)ζ − ν(F)ϕE +for any E, F on M. +In an LP-Kenmostu manifold, we have +(2.9) +∇Eζ = −E − ν(E)ζ, +(2.10) +(∇Eν)F = −g(E, F) − ν(E)ν(F), +where ∇ denotes the Levi-Civita connection respecting to the Lorentzian metric g. +Furthermore, in an LP-Kenmotsu manifold, the following relations hold [11]: +(2.11) +g(R(E, F)G, ζ) = ν(R(E, F)G) = g(F, G)ν(E) − g(E, G)ν(F), +(2.12) +R(ζ, E)F = −R(E, ζ)F = g(E, F)ζ − ν(F)E, +(2.13) +R(E, F)ζ = ν(F)E − ν(E)F, +(2.14) +R(ζ, E)ζ = E + ν(E)ζ, +(2.15) +S(E, ζ) = (n − 1)ν(E), S(ζ, ζ) = −(n − 1), +(2.16) +Qζ = (n − 1)ζ +for any E, F, G ∈ χ(M), where R, S and Q represent the curvature tensor, the Ricci +tensor and the Q Ricci operator, respectively. + +A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS +3 +Definition 2.2. [21] An LP-Kenmotsu manifold M is said to be ν-Einstein man- +ifold if its S(̸= 0) is of the form +(2.17) +S(E, F) = ag(E, F) + bν(E)ν(F), +where a and b are smooth functions on M. In particular, if b = 0, then M is termed +as an Einstein manifold. +Remark 2.3. [12] In an LP-Kenmotsu manifold of n-dimension, S is of the form +(2.18) +S(E, F) = ( +r +n − 1 − 1)g(E, F) + ( +r +n − 1 − n)ν(E)ν(F), +where r is the scalar curvature of the manifold. +Lemma 2.4. In an n-dimensional LP-Kenmotsu manifold, we have +(2.19) +ζ(r) = 2(r − n(n − 1)), +(2.20) +(∇EQ)ζ = QE − (n − 1)E, +(2.21) +(∇ζQ)E = 2QE − 2(n − 1)E +for any E on M. +Proof. Equation (2.18) yields +(2.22) +QE = ( +r +n − 1 − 1)E + ( +r +n − 1 − n)ν(E)ζ. +Taking the covariant derivative of (2.22) with respect to F and making use of (2.9) +and (2.10), we lead to +(∇F Q)E = F(r) +n − 1(E + ν(E)ζ) − ( +r +n − 1 − n)(g(E, F)ζ + ν(E)F + 2ν(E)ν(F)ζ). +By contracting F in the foregoing equation and using trace {F → (∇F Q)E} = +1 +2E(r), we find +n − 3 +2(n − 1)E(r) = +� ζ(r) +n − 1 − (r − n(n − 1)) +� +ν(E), +which by replacing E by ζ and using (2.1) gives (2.19). We refer the readers to see +[14] for the proof of (2.20) and (2.21). +□ +Remark 2.5. From the equation (2.19), it is noticed that if an n-dimensional +LP-Kenmotsu manifold possesses the constant scalar curvature, then r = n(n − 1) +and hence (2.18) reduces to S(E, F) = (n − 1)g(E, F). Thus, the manifold under +consideration is an Einstein manifold. +3. Ricci-Yamabe solitons on LP-Kenmotsu manifolds +Let the metric of an n-dimensional LP-Kenmotsu manifold be a Ricci-Yamabe +soliton (g, K, Λ, σ, ρ), then (1.1) holds. By differentiating (1.1) covariantly with +resprct to G, we have +(∇G£Kg)(E, F) += +−2σ(∇GS)(E, F) + ρ(Gr)g(E, F). +(3.1) +Since ∇g = 0, then the following formula [20] +(£K∇Eg −∇E£Kg −∇[K,E]g)(F, G) = −g((£K∇)(E, F), G)−g((£K∇)(E, G), F) + +4 +MOBIN AHMAD, GAZALA AND MOHD BILAL +turns to +(∇E£Kg)(F, G) = g((£K∇)(E, F), G) + g((£K∇)(E, G), F). +Since the operator £K∇ is symmetric, therefore we have +2g((£K∇)(E, F), G) = (∇E£Kg)(F, G) + (∇F £Kg)(E, G) − (∇G£Kg)(E, F), +which by using (3.1) takes the form +2g((£K∇)(E, F), G) += +−2σ[(∇ES)(F, G) + (∇F S)(G, E) + (∇GS)(E, F)] ++ρ[(Er)g(F, G) + (Fr)g(G, E) + (Gr)g(E, F)]. +(3.2) +Putting F = ζ in (3.2) and using (2.5), we find +2g((£K∇)(E, ζ), G) += +−2σ[(∇ES)(ζ, G) + (∇ζS)(G, E) − (∇GS)(E, ζ)] ++ρ[(Er)ν(G) + 2(r − n(n − 1))g(E, G) − (Gr)ν(E)] +(3.3) +By virtue of (2.20) and (2.21), (3.3) leads to +2g((£K∇)(E, ζ), G) += +−4σ[S(E, G) − (n − 1)g(E, G)] ++ρ[(Er)ν(G) + 2(r − n(n − 1))g(E, G) − (Gr)ν(E)]. +By eliminating G from the foregoing equation, we have +2(£K∇)(F, ζ) += +ρg(Dr, F)ζ − ρ(Dr)ν(F) − 4σQF +(3.4) ++[4σ(n − 1) + 2ρ(r − n(n − 1))]F. +If we take r as constant, then from (2.19) we find r = n(n − 1), and hence (3.4) +reduces to +(£K∇)(F, ζ) += +−2σQF + 2σ(n − 1)F. +(3.5) +Taking covariant derivative of (3.5) with respect to E, we have +(∇E£K∇)(F, ζ) += +(£K∇)(F, E) − 2σν(E)[QF − (n − 1)F] +(3.6) +− +2σ(∇EQ)F. +Again from [20], we have +(£KR)(E, F)G = (∇E£K∇)(F, G) − (∇F £K∇)(E, G), +which by putting G = ζ and using (3.6) takes the form +(£KR)(E, F)ζ += +2σν(F)(QE − (n − 1)E) − 2σν(E)(QF − (n − 1)F) +(3.7) +−2σ((∇EQ)F − (∇F Q)E). +Putting F = ζ in (3.7) then using (2.1), (2.2), (2.20) and (2.21), we arrive at +(£KR)(E, ζ)ζ = 0. +(3.8) +The Lie derivative of R(E, ζ)ζ = −E − ν(E)ζ along K leads to +(£KR)(E, ζ)ζ − g(E, £Kζ)ζ + 2ν(£Kζ)E = −(£Kν)(E)ζ. +(3.9) +From (3.8) and (3.9), we have +(£Kν)(E)ζ = −2ν(£Kζ)E + g(E, £Kζ)ζ. +(3.10) +Taking the Lie derivative of g(E, ζ) = ν(E), we find +(£Kν)(E) = g(E, £Kζ) + (£Kg)(E, ζ). +(3.11) +By putting F = ζ in (1.1) and using (2.15), we have +(£Kg)(E, ζ) = −{2σ(n − 1) + 2Λ − ρn(n − 1)}ν(E), +(3.12) + +A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS +5 +where r = n(n − 1) being used. +The Lie derivative of g(ζ, ζ) = −1 along K we lead to +(£Kg)(ζ, ζ) = −2ν(£Kζ). +(3.13) +From (3.12) and (3.16), we find +ν(£Kζ) = −{σ(n − 1) + Λ − ρn(n − 1) +2 +}. +(3.14) +Now, combining the equations (3.10), (3.11), (3.12) and (3.17), we find +Λ = ρn(n − 1) +2 +− σ(n − 1). +(3.15) +Thus, we have +Theorem 3.1. Let (M, g) be an n-dimensional LP-Kenmotsu manifold admitting +Ricci-Yamabe soliton (g, K, Λ, σ, ρ) with constant scalar curvature tensor, then Λ = +ρn(n−1) +2 +− σ(n − 1). +For σ = 1 and ρ = 0, from (3.15) we have Λ = −(n − 1). Thus, we have the +following: +Corollary 3.2. If an n-dimensional LP-Kenmotsu manifold admits a Ricci soliton +with constant scalar curvature, then the soliton is shrinking. +For σ = 0 and ρ = 1, from (3.15) we have Λ = +n(n−1) +2 +. Thus, we have the +following: +Corollary 3.3. If an n-dimensional LP-Kenmotsu manifold admits a Yamabe +soliton with constant scalar curvature, then the soliton is shrinking. +For σ = 1 and ρ = −1, from (3.15) we have Λ = − (n2−1) +2 +. Thus, we have the +following: +Corollary 3.4. If an n-dimensional LP-Kenmotsu manifold admits an Einstein +soliton with constant scalar curvature, then the soliton is shrinking. +Now, we consider the metric of an n-dimensional LP-Kenmotsu manifold as a +Ricci-Yamabe soliton (g, ζ, Λ, σ, ρ), then from (1.1) and (2.9) we have +S(E, F) = − 1 +σ (Λ − 1 − ρr +2 )g(E, F) + 1 +σ ν(E)ν(F), +where σ ̸= 0. +(3.16) +By putting F = ζ in (3.16) and using (2.15), we find +Λ = ρr +2 − σ(n − 1). +(3.17) +Now, comparing (2.18) and (3.17), we have r = n−1 +σ ++ n(n − 1), which by using in +(3.17) it follows that Λ = −σ(n − 1) + ρ(n−1)(1+nσ) +2σ +. Thus, we have the following +theorem: +Theorem 3.5. An n-dimensional LP-Kenmotsu manifold with constant scalar +curvature admitting Ricci-Yamabe soliton (g, ζ, Λ, σ, ρ) is an ν-Einstein manifold. +Moreover, the soliton is expanding, steady or shrinking according to ρ +σ > 2σ − ρn, +ρ +σ = 2σ − ρn, or ρ +σ < 2σ − ρn. + +6 +MOBIN AHMAD, GAZALA AND MOHD BILAL +4. Gradient Ricci-Yamabe solitons on LP-Kenmotsu manifolds +Definition 4.1. A Riemannian (or semi-Riemannian) metric g on M is called a +gradient RYS, if +Hessv + σS + (Λ − ρr +2 )g = 0, +(4.1) +where Hessv denotes the Hessian of a smooth function v on M and defined by +Hessv = ∇∇v. +Let M be an n-dimensional LP-Kenmotsu manifold with g as a gradient RYS. +Then equation (4.1) can be written as +∇EDv + σQE + (Λ − ρr +2 )E = 0, +(4.2) +for all vector fields E on M, where D denotes the gradient operator of g. Taking +the covariant derivative of (4.2) with respect to F, we have +∇F ∇EDv = −σ{(∇F Q)E + Q(∇F E)} + ρF(r) +2 +E − (Λ − ρr +2 )∇F E. +(4.3) +Interchanging E and F in (4.3), we lead to +∇E∇F Dv = −σ{(∇EQ)F + Q(∇EF)} + ρE(r) +2 +F − (Λ − ρr +2 )∇EF. +(4.4) +By making use of (4.2)-(4.4), we find +R(E, F)Dv = σ{(∇F Q)E − (∇EQ)F} + ρ +2{E(r)F − F(r)E}. +(4.5) +Now, from (2.18), we find +QE = ( +r +n − 1 − 1)E + ( +r +n − 1 − n)ν(E)ζ, +which on taking covariant derivative with repect to F leads to +(∇F Q)E += +F(r) +n − 1(E + ν(E)ζ) − ( +r +n − 1 − n)(g(E, F)ζ +(4.6) ++2ν(E)ν(F)ζ + ν(E)F). +By using (4.6) in (4.5), we have +R(E, F)Dv += +(n − 1)ρ − 2σ +2(n − 1) +{E(r)F − F(r)E} + +σ +n − 1{F(r)ν(E)ζ − E(r)ν(F)ζ} +−σ( +r +n − 1 − n)(ν(E)F − ν(F)E). +(4.7) +Contracting forgoing equation along E gives +S(F, Dv) += +�(n − 1)2ρ − 2σ(n − 2) +n − 1 +� +F(r) +(4.8) ++σ(n − 3)(r − n(n − 1)) +n − 1 +ν(F). +From the equation (2.18), we can write +S(F, Dv) = ( +r +n − 1 − 1)F(v) + ( +r +n − 1 − n)ν(F)ζ(v). +(4.9) + +A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS +7 +Now, by equating (4.8) and (4.9), then putting F = ζ and using (2.1), (2.19), we +find +ζ(v) = r − n(n − 1) +n − 1 +{2(n − 1)ρ − σ(5n − 13) +n − 1 +}. +(4.10) +Taking the inner product of (4.7) with ζ, we get +F(v)ν(E) − E(v)ν(F) = ρ +2{E(r)ν(F) − F(r)ν(E)}, +which by replacing E by ζ and using (2.19), (4.10), we infer +F(v) = −(r − n(n − 1)){3ρ − σ(5n − 13) +(n − 1)2 }ν(F) − ρ +2F(r). +(4.11) +If we take r as constant, then from Remark 2.5, we get r = n(n − 1). Thus, (4.11) +leads to F(v) = 0. This implies that v is constant. Thus, the soliton under the +consideration is trivial. Hence we state: +Theorem 4.2. If the metric of an LP-Kenmotsu manifold of constant scalar curva- +ture tensor admitting a special type of vector field is gradient RYS, then the soliton +is trivial. +For v constant, (1.2) turns to +σQE = −(Λ − ρr +2 )E, +which leads to +S(E, F) = − 1 +σ (Λ − ρn(n − 1) +2 +)g(E, F), +σ ̸= 0. +(4.12) +By putting E = F = ζ in (4.12) and using (2.15), we obtain +Λ = ρn(n − 1) +2 +− σ(n − 1). +(4.13) +Corollary 4.3. If an n-dimensional LP-Kenmotsu manifold admits a gradient +Ricci soliton with the constant scalar curvature, then the manifold under the con- +sideration is an Einstein manifold and Λ = ρn(n−1) +2 +− σ(n − 1). +For σ = 1 and ρ = 0, from (4.13) we find Λ = −(n − 1). Thu, we have the +following: +Corollary 4.4. If an n-dimensional LP-Kenmotsu manifold admits a gradient +Ricci soliton with the constant scalar curvature, then the soliton is shrinking. +For σ = 1 and ρ = −1, from (4.13) we have Λ = − (n−1)(n+2) +2 +. Thus, we have +the following: +Corollary 4.5. If an n-dimensional LP-Kenmotsu manifold admits an gradient +Einstein soliton with constant scalar curvature, then the soliton is shrinking. +Example. We consider the 5-dimensional manifold M 5 = +� +(x1, x2, x3, x4, x5) ∈ R5 : x5 > 0 +� +, +where (x1, x2, x3, x4, x5) are the standard coordinates in R5. Let ̺1, ̺2, ̺3, ̺4 and +̺5 be the vector fields on M 5 given by +̺1 = ex5 ∂ +∂x1 +, ̺2 = ex5 ∂ +∂x2 +, ̺3 = ex5 ∂ +∂x3 +, ̺4 = ex5 ∂ +∂x4 +, ̺5 = +∂ +∂x5 += ζ, + +8 +MOBIN AHMAD, GAZALA AND MOHD BILAL +which are linearly independent at each point of M 5. Let g be the Lorentzian metric +defined by +g(̺i, ̺i) = 1, +for +1 ≤ i ≤ 4 +and +g(̺5, ̺5) = −1, +g(̺i, ̺j) = 0, +for +i ̸= j, +1 ≤ i, j ≤ 5. +Let ν be the 1-form defined by ν(E) = g(E, ̺5) = g(̺, ζ) for all E ∈ χ(M 5), and +let ϕ be the (1, 1)-tensor field defined by +ϕ̺1 = −̺2, ϕ̺2 = −̺1, ϕ̺3 = −̺4, ϕ̺4 = −̺3, ϕ̺5 = 0. +By applying linearity of ϕ and g, we have +ν(ζ) = g(ζ, ζ) = −1, ϕ2E = E + ν(E)ζ and g(ϕE, ϕF) = g(E, F) + ν(E)ν(F) +for all E, F ∈ χ(M 5). Thus for ̺5 = ζ, the structure (ϕ, ζ, ν, g) defines a Lorentzian +almost paracontact metric structure on M 5. Then we have +[̺i, ̺j] = −̺i, +for +1 ≤ i ≤ 4, j = 5, +[̺i, ̺j] = 0, +otherwise. +By using Koszul’s formula, we can easily find we obtain +∇̺i̺j = + + + + + +−̺5, +1 ≤ i = j ≤ 4, +−̺i, +1 ≤ i ≤ 4, j = 5, +0, +otherwise. +Also one can easily verify that +∇Eζ = −E − η(E)ζ +and +(∇Eϕ)F = −g(ϕE, F)ζ − ν(F)ϕE. +Therefore, the manifold is an LP-Kenmotsu manifold. +From the above results, we can easily obtain the non-vanishing components of R +as follows: +R(̺1, ̺2)̺1 = −̺2, R(̺1, ̺2)̺2 = ̺1, R(̺1, ̺3)̺1 = −̺3, R(̺1, ̺3)̺3 = ̺1, +R(̺1, ̺4)̺1 = −v4, R(̺1, ̺4)̺4 = ̺1, R(̺1, ̺5)̺1 = −̺5, R(̺1, ̺5)̺5 = −̺1, +R(̺2, ̺3)̺2 = −̺3, R(̺2, ̺3)̺3 = ̺2, R(̺2, ̺4)̺2 = −̺4, R(̺2, ̺4)̺4 = ̺2, +R(̺2, ̺5)̺2 = −̺5, R(̺2, ̺5)̺5 = −̺2, R(̺3, ̺4)̺3 = −̺4, R(̺3, ̺4)̺4 = ̺3, +R(̺3, ̺5)̺3 = −̺5, R(̺3, ̺5)̺5 = −̺3, R(̺4, ̺5)̺4 = −̺5, R(̺4, ̺5)̺5 = −̺4. +Also, we calculate the Ricci tensors as follows: +S(̺1, ̺1) = S(̺2, ̺2) = S(̺3, ̺3) = S(̺4, ̺4) = 4, +S(̺5, ̺5) = −4. +Therefore, we have +r = S(̺1, ̺1) + S(̺2, ̺2) + S(̺3, ̺3) + S(̺4, ̺4) − S(̺5, ̺5) = 20. +Now by taking Dv = (̺1v)̺1 + (̺2v)̺2 + (̺3v)̺3 + (̺4v)̺4 + (̺5v)̺5, we have +∇̺1Dv = (̺1(̺1v) − (̺5v))̺1 + (̺1(̺2v))̺2 + (̺1(̺3v))̺3 + (̺1(̺4v))̺4 + (̺1(̺5v) − (̺1v))̺5, +∇̺2Dv = (̺2(̺1v))̺1 + (̺2(̺2v) − (̺5v))̺2 + (̺2(̺3v))̺3 + (̺2(̺4v))̺4 + (̺2(̺5v) − (̺2v))̺5, +∇̺3Dv = (̺3(̺1v))̺1 + (̺3(̺2v))̺2 + (̺3(̺3v) − (̺5v))̺3 + (̺3(̺4v))̺4 + (̺3(̺5v) − (̺3v))̺5, +∇̺4Dv = (̺4(̺1v))̺1 + (̺4(̺2v))̺2 + (̺4(̺3v))̺3 + (̺4(̺4v) − (̺5v))̺4 + (̺4(̺5v) − (̺4v))̺5, +∇̺5Dv = (̺5(̺1v))̺1 + (̺5(̺2v))̺2 + (̺5(̺3v))̺3 + (̺5(̺4v))̺4 + (̺5(̺5v))̺5. + +A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS +9 +Thus, by virtue of (4.2), we obtain + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +̺1(̺1v) − ̺5v = −(Λ + 4σ − 10ρ), +̺2(̺2v) − ̺5v = −(Λ + 4σ − 10ρ), +̺3(̺3v) − ̺5v = −(Λ + 4σ − 10ρ), +̺4(̺4v) − ̺5v = −(Λ + 4σ − 10ρ), +̺5(̺5v) = −(Λ + 4σ − 10ρ), +̺1(̺2v) = ̺1(̺3v) = ̺1(̺4v) = 0, +̺2(̺1v) = ̺2(̺3v) = ̺2(̺4v) = 0, +̺3(̺1v) = ̺3(̺2v) = ̺3(̺4v) = 0, +̺4(̺1v) = ̺4(̺2v) = ̺4(̺3v) = 0, +̺1(̺5v) − (̺1v) = ̺2(̺5v) − (̺2v) = 0, +̺3(̺5v) − (̺3v) = ̺4(̺5v) − (̺4v) = 0. +(4.14) +Thus, the equations in (4.14) are respectively amounting to +e2x5 ∂2v +∂x2 +1 +− ∂v +∂x5 += −(Λ + 4σ − 10ρ), +e2x5 ∂2v +∂x2 +2 +− ∂v +∂x5 += −(Λ + 4σ − 10ρ), +e2x5 ∂2v +∂x2 +3 +− ∂v +∂x5 += −(Λ + 4σ − 10ρ), +e2x5 ∂2v +∂x2 +4 +− ∂v +∂x5 += −(Λ + 4σ − 10ρ), +∂2v +∂x2 +5 += −(Λ + 4σ − 10ρ), +∂2v +∂x1∂x2 += +∂2v +∂x1∂x3 += +∂2v +∂x1∂x4 += +∂2v +∂x2∂x3 += +∂2v +∂x2∂x4 += +∂2v +∂x3∂x4 += 0, +ex5 +∂2v +∂x5∂x1 ++ ∂v +∂x1 += ex5 +∂2v +∂x5∂x2 ++ ∂v +∂x2 += ex5 +∂2v +∂x5∂x3 ++ ∂v +∂x3 += ex5 +∂2v +∂x5∂x4 ++ ∂v +∂x4 += 0. +From the above equations it is observed that v is constant for Λ = −4σ + 10ρ. +Hence, equation (4.2) is satisfied. Thus, g is a gradient RYS with the soliton vector +field K = Dv, where v is constant and Λ = −4σ + 10ρ. Hence, Theorem 4.2 is +verified. +References +[1] Blaga, A. M., Solitons and geometrical structure in a perfect fluid spacetime, Rocky Mt. J. +Math. (2020). +[2] Blaga, A. M., Some geometrical aspects of Einstein, Ricci and Yamabe solitons, J. Geom. +Symmetry Phys., 52 (2019), 17-26. +[3] Catino, G. and Mazzieri, L., Gradient Einstein solitons, Nonlinear Anal., 132(2016), 66-94. +[4] Catino, G., Cremaschi, L., Djadli, Z., Mantegazza, C. and Mazzieri, L., The Ricci Bour- +guignon flow, Pacific J. Math., 28(2017), 337-370. + +10 +MOBIN AHMAD, GAZALA AND MOHD BILAL +[5] Chidananda, +S., +and +Venkatesha, +V., +Yamabe +soliton +and +Riemann +soli- +ton +on +Lorentzian +para-Sasakian +manifold, +Commun. +Korean +Math. +Soc., +https://doi.org/10.4134/CKMS.c200365. +[6] G¨uler, S. and Crasmareanu, M., Ricci-Yamabe maps for Riemannian flows and their volume +variation and volume entropy, Turk. J. Math., 43 (2019), 2631-2641. +[7] Hamilton, R. S., Lectures on Geometric Flows (Unpublished manuscript, 1989). +[8] Hamilton, R. S., The Ricci Flow on Surfaces, Mathematics and General Relativity (Santa +Cruz, CA, 1986), Contemp. Math., A.M.S., 71 (1988), 237-262. +[9] Haseeb, A. and De, U. C., η-Ricci solitons in ǫ-Kenmotsu manifolds, J. Geom. 110, 34 (2019). +[10] Haseeb, A. and Almusawa, H., Some results on Lorentzian para-Kenmotsu manifolds admit- +ting η-Ricci solitons, Palestine Journal of Mathematics, 11(2)(2022), 205-213. +[11] Haseeb, A. and Prasad, R., Certain results on Lorentzian para-Kenmotsu manifolds, Bol. +Soc. Parana. Mat., 39(3) (2021), 201-220. +[12] Haseeb, A. and Prasad, R., Some results on Lorentzian para-Kenmotsu manifolds, Bull. +Transilvania Univ. of Brasov, 13(62) (2020), no. 1, 185-198. +[13] Haseeb, A., Prasad, R. and Mofarreh, F., Sasakian manifolds admitting ∗-η-Ricci-Yamabe +solitons, Advances in Mathematical Physics, Vol. 2022, Article ID 5718736, 7 pages. doi: +https:// doi.org/10.1155/2022/5718736 +[14] Li, Y., Haseeb, A. and Ali, M., LP -Kenmotsu manifolds admitting η-Ricci solitons and +spacetime, Journal of Mathematics, 2022, Article ID 6605127, 10 pages. +[15] Lone, M. A. and Harry, I. F., Ricci Solitons on Lorentz-Sasakian space forms, Journal of +Geometry and Physics, 104547, doi: https://doi.org/10.1016/j.geomphys.2022.104547. +[16] Pankaj, Chaubey, S. K and Prasad, R., Three dimensional Lorentzian para-Kenmotsu mani- +folds and Yamabe soliton, Honam Mathematical J., 43(4) (2021), 613-626. +[17] Singh, J. P. and Khatri, M., On Ricci-Yamabe soliton and geometrical structure in a perfect +fluid spacetime, Afr. Mat., 32(2021), 1645-1656. +[18] Venkatesha, Kumara, H. A., Ricci soliton and geometrical structure in a perfect fluid space- +time with torse-forming vector field, Afr. Mat. 30 (2019), 725-736 +[19] Yoldas, H. I., On Kenmotsu manifolds admitting η-Ricci-Yamabe solitons,Int. J. Geom. Met. +Mod. Phy., 18(12) (2021), 2150189. +[20] Yano, K., Integral Formulas in Riemannian geometry, Pure and Applied Mathematics, Vol. +I, Marcel Dekker, New York, 1970. +[21] Yano, K. and Kon, M., Structures on manifolds, World Scientific, (1984). +Mobin Ahmad +Department of Mathematics, +Integral University, Kursi Road, +Lucknow-226026. +Email : mobinahmad68@gmail.com +Gazala +Department of Mathematics, +Integral University, Kursi Road, +Lucknow-226026. +Email : gazala.math@gmail.com +Mohd. Bilal +Department of Mathematical Sciences, +Umm Ul Qura University, +Makkah, Saudi Arabia. +Email: mohd7bilal@gmail.com + diff --git a/8dE0T4oBgHgl3EQfwgEX/content/tmp_files/load_file.txt b/8dE0T4oBgHgl3EQfwgEX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..95de525d33ce996df3ba6940a6e6103b9cdd58b3 --- /dev/null +++ b/8dE0T4oBgHgl3EQfwgEX/content/tmp_files/load_file.txt @@ -0,0 +1,421 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf,len=420 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='02632v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='GM] 30 Nov 2022 A NOTE ON LP-KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS MOBIN AHMAD, GAZALA AND MOHD BILAL Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' In the current note, we study Lorentzian para-Kenmotsu (in brief, LP -Kenmotsu) manifolds admitting Ricci-Yamabe solitons (RYS) and gradi- ent Ricci-Yamabe soliton (gradient RYS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' At last by constructing a 5-dimensional non-trivial example we illustrate our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' 53C20, 53C21, 53C25, 53E20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Lorentzian para-Kenmotsu manifolds, Ricci-Yamabe solitons, Einstein manifolds, ν-Einstein manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Introduction In 2019, a scalar combination of Ricci and Yamabe flows was proposed by the authors G¨uler and Crasmareanu [6], this advanced class of geometric flows called Ricci-Yamabe (RY) flow of type (σ, ρ) and is defined by ∂ ∂tg(t) + 2σS(g(t)) + ρr(t)g(t) = 0, g(0) = g0 for some scalars σ and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' A solution to the RY flow is called RYS if it depends only on one parameter group of diffeomorphism and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' A Riemannian (or semi-Riemannian) manifold M is said to have a RYS if £Kg + 2σS + (2Λ − ρr)g = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) where σ, ρ, Λ ∈ R (the set of real numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If K is the gradient of a smooth function v on M, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) is called the gradient Ricci-Yamabe soliton (gradient RYS) and hence (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) turns to ∇2v + σS + (Λ − ρr 2 )g = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2) where ∇2v is the Hessian of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' It is to be noted that a RYS of types (σ, 0) and (0, ρ) are known as σ−Ricci soliton and ρ−Yamabe soliton, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' A RYS is said to be shrinking , steady or expanding if Λ < 0, = 0 or > 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' A RYS is said to be a Ricci soliton [7] if σ = 1, ρ = 0, Yamabe soliton [8] if σ = 0, ρ = 1, Einstein soliton [3] if σ = 1, ρ = −1, As a continuation of this study, we tried to study RYS in the frame-work of LP-Kenmotsu manifolds of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' We recommend the papers [1, 2, 5, 9, 10, 13, 15, 16, 17, 18, 19] and the references therein for more details about the related studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' 1 2 MOBIN AHMAD, GAZALA AND MOHD BILAL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Preliminaries An n-dimensional differentiable manifold M with structure (ϕ, ζ, ν, g) is said to be a Lorentzian almost paracontact metric manifold, if it admits a (1, 1)-tensor field ϕ, a contravariant vector field ζ, a 1-form ν and a Lorentzian metric g satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) ν(ζ) + 1 = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2) ϕ2E = E + ν(E)ζ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='3) ϕζ = 0, ν(ϕE) = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='4) g(ϕE, ϕF) = g(E, F) + ν(E)ν(F), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5) g(E, ζ) = ν(E), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='6) ϕ(E, F) = ϕ(F, E) = g(E, ϕF) for any vector fields E, F ∈ χ(M), where χ(M) is the Lie algebra of vector fields on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If ζ is a killing vector field, the (para) contact structure is called a K-(para) contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' In such a case, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='7) ∇Eζ = ϕE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Recently, the authors Haseeb and Prasad defined and studied the following notion: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' A Lorentzian almost paracontact manifold M is called Lorentzian para-Kenmostu manifold if [11] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='8) (∇Eϕ)F = −g(ϕE, F)ζ − ν(F)ϕE for any E, F on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' In an LP-Kenmostu manifold, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='9) ∇Eζ = −E − ν(E)ζ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='10) (∇Eν)F = −g(E, F) − ν(E)ν(F), where ∇ denotes the Levi-Civita connection respecting to the Lorentzian metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Furthermore, in an LP-Kenmotsu manifold, the following relations hold [11]: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='11) g(R(E, F)G, ζ) = ν(R(E, F)G) = g(F, G)ν(E) − g(E, G)ν(F), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='12) R(ζ, E)F = −R(E, ζ)F = g(E, F)ζ − ν(F)E, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='13) R(E, F)ζ = ν(F)E − ν(E)F, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='14) R(ζ, E)ζ = E + ν(E)ζ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='15) S(E, ζ) = (n − 1)ν(E), S(ζ, ζ) = −(n − 1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='16) Qζ = (n − 1)ζ for any E, F, G ∈ χ(M), where R, S and Q represent the curvature tensor, the Ricci tensor and the Q Ricci operator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS 3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' [21] An LP-Kenmotsu manifold M is said to be ν-Einstein man- ifold if its S(̸= 0) is of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='17) S(E, F) = ag(E, F) + bν(E)ν(F), where a and b are smooth functions on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' In particular, if b = 0, then M is termed as an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' [12] In an LP-Kenmotsu manifold of n-dimension, S is of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='18) S(E, F) = ( r n − 1 − 1)g(E, F) + ( r n − 1 − n)ν(E)ν(F), where r is the scalar curvature of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' In an n-dimensional LP-Kenmotsu manifold, we have (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='19) ζ(r) = 2(r − n(n − 1)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='20) (∇EQ)ζ = QE − (n − 1)E, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='21) (∇ζQ)E = 2QE − 2(n − 1)E for any E on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='18) yields (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='22) QE = ( r n − 1 − 1)E + ( r n − 1 − n)ν(E)ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Taking the covariant derivative of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='22) with respect to F and making use of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='10), we lead to (∇F Q)E = F(r) n − 1(E + ν(E)ζ) − ( r n − 1 − n)(g(E, F)ζ + ν(E)F + 2ν(E)ν(F)ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' By contracting F in the foregoing equation and using trace {F → (∇F Q)E} = 1 2E(r), we find n − 3 2(n − 1)E(r) = � ζ(r) n − 1 − (r − n(n − 1)) � ν(E), which by replacing E by ζ and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) gives (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' We refer the readers to see [14] for the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' From the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='19), it is noticed that if an n-dimensional LP-Kenmotsu manifold possesses the constant scalar curvature, then r = n(n − 1) and hence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='18) reduces to S(E, F) = (n − 1)g(E, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus, the manifold under consideration is an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Ricci-Yamabe solitons on LP-Kenmotsu manifolds Let the metric of an n-dimensional LP-Kenmotsu manifold be a Ricci-Yamabe soliton (g, K, Λ, σ, ρ), then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' By differentiating (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) covariantly with resprct to G, we have (∇G£Kg)(E, F) = −2σ(∇GS)(E, F) + ρ(Gr)g(E, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) Since ∇g = 0, then the following formula [20] (£K∇Eg −∇E£Kg −∇[K,E]g)(F, G) = −g((£K∇)(E, F), G)−g((£K∇)(E, G), F) 4 MOBIN AHMAD, GAZALA AND MOHD BILAL turns to (∇E£Kg)(F, G) = g((£K∇)(E, F), G) + g((£K∇)(E, G), F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Since the operator £K∇ is symmetric, therefore we have 2g((£K∇)(E, F), G) = (∇E£Kg)(F, G) + (∇F £Kg)(E, G) − (∇G£Kg)(E, F), which by using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) takes the form 2g((£K∇)(E, F), G) = −2σ[(∇ES)(F, G) + (∇F S)(G, E) + (∇GS)(E, F)] +ρ[(Er)g(F, G) + (Fr)g(G, E) + (Gr)g(E, F)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2) Putting F = ζ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5), we find 2g((£K∇)(E, ζ), G) = −2σ[(∇ES)(ζ, G) + (∇ζS)(G, E) − (∇GS)(E, ζ)] +ρ[(Er)ν(G) + 2(r − n(n − 1))g(E, G) − (Gr)ν(E)] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='3) By virtue of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='21), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='3) leads to 2g((£K∇)(E, ζ), G) = −4σ[S(E, G) − (n − 1)g(E, G)] +ρ[(Er)ν(G) + 2(r − n(n − 1))g(E, G) − (Gr)ν(E)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' By eliminating G from the foregoing equation, we have 2(£K∇)(F, ζ) = ρg(Dr, F)ζ − ρ(Dr)ν(F) − 4σQF (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='4) +[4σ(n − 1) + 2ρ(r − n(n − 1))]F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If we take r as constant, then from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='19) we find r = n(n − 1), and hence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='4) reduces to (£K∇)(F, ζ) = −2σQF + 2σ(n − 1)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5) Taking covariant derivative of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5) with respect to E, we have (∇E£K∇)(F, ζ) = (£K∇)(F, E) − 2σν(E)[QF − (n − 1)F] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='6) − 2σ(∇EQ)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Again from [20], we have (£KR)(E, F)G = (∇E£K∇)(F, G) − (∇F £K∇)(E, G), which by putting G = ζ and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='6) takes the form (£KR)(E, F)ζ = 2σν(F)(QE − (n − 1)E) − 2σν(E)(QF − (n − 1)F) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='7) −2σ((∇EQ)F − (∇F Q)E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Putting F = ζ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='7) then using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='21), we arrive at (£KR)(E, ζ)ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='8) The Lie derivative of R(E, ζ)ζ = −E − ν(E)ζ along K leads to (£KR)(E, ζ)ζ − g(E, £Kζ)ζ + 2ν(£Kζ)E = −(£Kν)(E)ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='9) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='9), we have (£Kν)(E)ζ = −2ν(£Kζ)E + g(E, £Kζ)ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='10) Taking the Lie derivative of g(E, ζ) = ν(E), we find (£Kν)(E) = g(E, £Kζ) + (£Kg)(E, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='11) By putting F = ζ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='15), we have (£Kg)(E, ζ) = −{2σ(n − 1) + 2Λ − ρn(n − 1)}ν(E), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='12) A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS 5 where r = n(n − 1) being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' The Lie derivative of g(ζ, ζ) = −1 along K we lead to (£Kg)(ζ, ζ) = −2ν(£Kζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='13) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='16), we find ν(£Kζ) = −{σ(n − 1) + Λ − ρn(n − 1) 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='14) Now, combining the equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='11), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='17), we find Λ = ρn(n − 1) 2 − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='15) Thus, we have Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Let (M, g) be an n-dimensional LP-Kenmotsu manifold admitting Ricci-Yamabe soliton (g, K, Λ, σ, ρ) with constant scalar curvature tensor, then Λ = ρn(n−1) 2 − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' For σ = 1 and ρ = 0, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='15) we have Λ = −(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus, we have the following: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If an n-dimensional LP-Kenmotsu manifold admits a Ricci soliton with constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' For σ = 0 and ρ = 1, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='15) we have Λ = n(n−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus, we have the following: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If an n-dimensional LP-Kenmotsu manifold admits a Yamabe soliton with constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' For σ = 1 and ρ = −1, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='15) we have Λ = − (n2−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus, we have the following: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If an n-dimensional LP-Kenmotsu manifold admits an Einstein soliton with constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Now, we consider the metric of an n-dimensional LP-Kenmotsu manifold as a Ricci-Yamabe soliton (g, ζ, Λ, σ, ρ), then from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='9) we have S(E, F) = − 1 σ (Λ − 1 − ρr 2 )g(E, F) + 1 σ ν(E)ν(F), where σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='16) By putting F = ζ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='16) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='15), we find Λ = ρr 2 − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='17) Now, comparing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='17), we have r = n−1 σ + n(n − 1), which by using in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='17) it follows that Λ = −σ(n − 1) + ρ(n−1)(1+nσ) 2σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus, we have the following theorem: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' An n-dimensional LP-Kenmotsu manifold with constant scalar curvature admitting Ricci-Yamabe soliton (g, ζ, Λ, σ, ρ) is an ν-Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Moreover, the soliton is expanding, steady or shrinking according to ρ σ > 2σ − ρn, ρ σ = 2σ − ρn, or ρ σ < 2σ − ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' 6 MOBIN AHMAD, GAZALA AND MOHD BILAL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Gradient Ricci-Yamabe solitons on LP-Kenmotsu manifolds Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' A Riemannian (or semi-Riemannian) metric g on M is called a gradient RYS, if Hessv + σS + (Λ − ρr 2 )g = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) where Hessv denotes the Hessian of a smooth function v on M and defined by Hessv = ∇∇v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Let M be an n-dimensional LP-Kenmotsu manifold with g as a gradient RYS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Then equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1) can be written as ∇EDv + σQE + (Λ − ρr 2 )E = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2) for all vector fields E on M, where D denotes the gradient operator of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Taking the covariant derivative of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2) with respect to F, we have ∇F ∇EDv = −σ{(∇F Q)E + Q(∇F E)} + ρF(r) 2 E − (Λ − ρr 2 )∇F E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='3) Interchanging E and F in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='3), we lead to ∇E∇F Dv = −σ{(∇EQ)F + Q(∇EF)} + ρE(r) 2 F − (Λ − ρr 2 )∇EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='4) By making use of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='4), we find R(E, F)Dv = σ{(∇F Q)E − (∇EQ)F} + ρ 2{E(r)F − F(r)E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5) Now, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='18), we find QE = ( r n − 1 − 1)E + ( r n − 1 − n)ν(E)ζ, which on taking covariant derivative with repect to F leads to (∇F Q)E = F(r) n − 1(E + ν(E)ζ) − ( r n − 1 − n)(g(E, F)ζ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='6) +2ν(E)ν(F)ζ + ν(E)F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' By using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='6) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5), we have R(E, F)Dv = (n − 1)ρ − 2σ 2(n − 1) {E(r)F − F(r)E} + σ n − 1{F(r)ν(E)ζ − E(r)ν(F)ζ} −σ( r n − 1 − n)(ν(E)F − ν(F)E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='7) Contracting forgoing equation along E gives S(F, Dv) = �(n − 1)2ρ − 2σ(n − 2) n − 1 � F(r) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='8) +σ(n − 3)(r − n(n − 1)) n − 1 ν(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' From the equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='18), we can write S(F, Dv) = ( r n − 1 − 1)F(v) + ( r n − 1 − n)ν(F)ζ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='9) A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS 7 Now, by equating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='9), then putting F = ζ and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='19), we find ζ(v) = r − n(n − 1) n − 1 {2(n − 1)ρ − σ(5n − 13) n − 1 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='10) Taking the inner product of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='7) with ζ, we get F(v)ν(E) − E(v)ν(F) = ρ 2{E(r)ν(F) − F(r)ν(E)}, which by replacing E by ζ and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='10), we infer F(v) = −(r − n(n − 1)){3ρ − σ(5n − 13) (n − 1)2 }ν(F) − ρ 2F(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='11) If we take r as constant, then from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5, we get r = n(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='11) leads to F(v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' This implies that v is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus, the soliton under the consideration is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Hence we state: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If the metric of an LP-Kenmotsu manifold of constant scalar curva- ture tensor admitting a special type of vector field is gradient RYS, then the soliton is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' For v constant, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2) turns to σQE = −(Λ − ρr 2 )E, which leads to S(E, F) = − 1 σ (Λ − ρn(n − 1) 2 )g(E, F), σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='12) By putting E = F = ζ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='12) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='15), we obtain Λ = ρn(n − 1) 2 − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='13) Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If an n-dimensional LP-Kenmotsu manifold admits a gradient Ricci soliton with the constant scalar curvature, then the manifold under the con- sideration is an Einstein manifold and Λ = ρn(n−1) 2 − σ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' For σ = 1 and ρ = 0, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='13) we find Λ = −(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thu, we have the following: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If an n-dimensional LP-Kenmotsu manifold admits a gradient Ricci soliton with the constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' For σ = 1 and ρ = −1, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='13) we have Λ = − (n−1)(n+2) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus, we have the following: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' If an n-dimensional LP-Kenmotsu manifold admits an gradient Einstein soliton with constant scalar curvature, then the soliton is shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' We consider the 5-dimensional manifold M 5 = � (x1, x2, x3, x4, x5) ∈ R5 : x5 > 0 � , where (x1, x2, x3, x4, x5) are the standard coordinates in R5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Let ̺1, ̺2, ̺3, ̺4 and ̺5 be the vector fields on M 5 given by ̺1 = ex5 ∂ ∂x1 , ̺2 = ex5 ∂ ∂x2 , ̺3 = ex5 ∂ ∂x3 , ̺4 = ex5 ∂ ∂x4 , ̺5 = ∂ ∂x5 = ζ, 8 MOBIN AHMAD, GAZALA AND MOHD BILAL which are linearly independent at each point of M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Let g be the Lorentzian metric defined by g(̺i, ̺i) = 1, for 1 ≤ i ≤ 4 and g(̺5, ̺5) = −1, g(̺i, ̺j) = 0, for i ̸= j, 1 ≤ i, j ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Let ν be the 1-form defined by ν(E) = g(E, ̺5) = g(̺, ζ) for all E ∈ χ(M 5), and let ϕ be the (1, 1)-tensor field defined by ϕ̺1 = −̺2, ϕ̺2 = −̺1, ϕ̺3 = −̺4, ϕ̺4 = −̺3, ϕ̺5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' By applying linearity of ϕ and g, we have ν(ζ) = g(ζ, ζ) = −1, ϕ2E = E + ν(E)ζ and g(ϕE, ϕF) = g(E, F) + ν(E)ν(F) for all E, F ∈ χ(M 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Thus for ̺5 = ζ, the structure (ϕ, ζ, ν, g) defines a Lorentzian almost paracontact metric structure on M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Then we have [̺i, ̺j] = −̺i, for 1 ≤ i ≤ 4, j = 5, [̺i, ̺j] = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' By using Koszul’s formula, we can easily find we obtain ∇̺i̺j = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 −̺5, 1 ≤ i = j ≤ 4, −̺i, 1 ≤ i ≤ 4, j = 5, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Also one can easily verify that ∇Eζ = −E − η(E)ζ and (∇Eϕ)F = −g(ϕE, F)ζ − ν(F)ϕE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Therefore, the manifold is an LP-Kenmotsu manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' From the above results, we can easily obtain the non-vanishing components of R as follows: R(̺1, ̺2)̺1 = −̺2, R(̺1, ̺2)̺2 = ̺1, R(̺1, ̺3)̺1 = −̺3, R(̺1, ̺3)̺3 = ̺1, R(̺1, ̺4)̺1 = −v4, R(̺1, ̺4)̺4 = ̺1, R(̺1, ̺5)̺1 = −̺5, R(̺1, ̺5)̺5 = −̺1, R(̺2, ̺3)̺2 = −̺3, R(̺2, ̺3)̺3 = ̺2, R(̺2, ̺4)̺2 = −̺4, R(̺2, ̺4)̺4 = ̺2, R(̺2, ̺5)̺2 = −̺5, R(̺2, ̺5)̺5 = −̺2, R(̺3, ̺4)̺3 = −̺4, R(̺3, ̺4)̺4 = ̺3, R(̺3, ̺5)̺3 = −̺5, R(̺3, ̺5)̺5 = −̺3, R(̺4, ̺5)̺4 = −̺5, R(̺4, ̺5)̺5 = −̺4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Also, we calculate the Ricci tensors as follows: S(̺1, ̺1) = S(̺2, ̺2) = S(̺3, ̺3) = S(̺4, ̺4) = 4, S(̺5, ̺5) = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Therefore, we have r = S(̺1, ̺1) + S(̺2, ̺2) + S(̺3, ̺3) + S(̺4, ̺4) − S(̺5, ̺5) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Now by taking Dv = (̺1v)̺1 + (̺2v)̺2 + (̺3v)̺3 + (̺4v)̺4 + (̺5v)̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' we have ∇̺1Dv = (̺1(̺1v) − (̺5v))̺1 + (̺1(̺2v))̺2 + (̺1(̺3v))̺3 + (̺1(̺4v))̺4 + (̺1(̺5v) − (̺1v))̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ∇̺2Dv = (̺2(̺1v))̺1 + (̺2(̺2v) − (̺5v))̺2 + (̺2(̺3v))̺3 + (̺2(̺4v))̺4 + (̺2(̺5v) − (̺2v))̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ∇̺3Dv = (̺3(̺1v))̺1 + (̺3(̺2v))̺2 + (̺3(̺3v) − (̺5v))̺3 + (̺3(̺4v))̺4 + (̺3(̺5v) − (̺3v))̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ∇̺4Dv = (̺4(̺1v))̺1 + (̺4(̺2v))̺2 + (̺4(̺3v))̺3 + (̺4(̺4v) − (̺5v))̺4 + (̺4(̺5v) − (̺4v))̺5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ∇̺5Dv = (̺5(̺1v))̺1 + (̺5(̺2v))̺2 + (̺5(̺3v))̺3 + (̺5(̺4v))̺4 + (̺5(̺5v))̺5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' A NOTE ON LP -KENMOTSU MANIFOLDS ADMITTING RICCI-YAMABE SOLITONS 9 Thus, by virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' we obtain \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ̺1(̺1v) − ̺5v = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺2(̺2v) − ̺5v = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺3(̺3v) − ̺5v = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺4(̺4v) − ̺5v = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺5(̺5v) = −(Λ + 4σ − 10ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺1(̺2v) = ̺1(̺3v) = ̺1(̺4v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺2(̺1v) = ̺2(̺3v) = ̺2(̺4v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺3(̺1v) = ̺3(̺2v) = ̺3(̺4v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺4(̺1v) = ̺4(̺2v) = ̺4(̺3v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺1(̺5v) − (̺1v) = ̺2(̺5v) − (̺2v) = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' ̺3(̺5v) − (̺3v) = ̺4(̺5v) − (̺4v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='14) Thus, the equations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='14) are respectively amounting to e2x5 ∂2v ∂x2 1 − ∂v ∂x5 = −(Λ + 4σ − 10ρ), e2x5 ∂2v ∂x2 2 − ∂v ∂x5 = −(Λ + 4σ − 10ρ), e2x5 ∂2v ∂x2 3 − ∂v ∂x5 = −(Λ + 4σ − 10ρ), e2x5 ∂2v ∂x2 4 − ∂v ∂x5 = −(Λ + 4σ − 10ρ), ∂2v ∂x2 5 = −(Λ + 4σ − 10ρ), ∂2v ∂x1∂x2 = ∂2v ∂x1∂x3 = ∂2v ∂x1∂x4 = ∂2v ∂x2∂x3 = ∂2v ∂x2∂x4 = ∂2v ∂x3∂x4 = 0, ex5 ∂2v ∂x5∂x1 + ∂v ∂x1 = ex5 ∂2v ∂x5∂x2 + ∂v ∂x2 = ex5 ∂2v ∂x5∂x3 + ∂v ∂x3 = ex5 ∂2v ∂x5∂x4 + ∂v ∂x4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' From the above equations it is observed that v is constant for Λ = −4σ + 10ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Hence, equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='2) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=', 39(3) (2021), 201-220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' [12] Haseeb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' and Prasad, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=', Some results on Lorentzian para-Kenmotsu manifolds, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Transilvania Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' of Brasov, 13(62) (2020), no.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' doi: https:// doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='1155/2022/5718736 [14] Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=', Haseeb, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' and Ali, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=', LP -Kenmotsu manifolds admitting η-Ricci solitons and spacetime, Journal of Mathematics, 2022, Article ID 6605127, 10 pages.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' 30 (2019), 725-736 [19] Yoldas, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=', On Kenmotsu manifolds admitting η-Ricci-Yamabe solitons,Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=', 18(12) (2021), 2150189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' [20] Yano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=', Integral Formulas in Riemannian geometry, Pure and Applied Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' I, Marcel Dekker, New York, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' [21] Yano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' and Kon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=', Structures on manifolds, World Scientific, (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Mobin Ahmad Department of Mathematics, Integral University, Kursi Road, Lucknow-226026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Email : mobinahmad68@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='com Gazala Department of Mathematics, Integral University, Kursi Road, Lucknow-226026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Email : gazala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='math@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content='com Mohd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Bilal Department of Mathematical Sciences, Umm Ul Qura University, Makkah, Saudi Arabia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} +page_content=' Email: mohd7bilal@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE0T4oBgHgl3EQfwgEX/content/2301.02632v1.pdf'} 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5Center for Computational Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' University of Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Ten-nodai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 1-1-1 Tsukuba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Ibaraki 305-8577,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Japan 6Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Yale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' CT 06520 7Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The University of Texas at Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2515 Speedway Boulevard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' TX 78712,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' USA 8McDonald Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The University of Texas at Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2515 Speedway Boulevard,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' TX 78712,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' USA 9University Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Fakult¨at f¨ur Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Ludwig-Maximilians University Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Scheinerstrasse 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 81679 Munich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Germany 10Max-Planck Institut f¨ur extraterrestrische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Giessenbachstrasse 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 85748 Garching,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Germany 11South-Western Institute for Astronomy Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Yunnan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Kunming,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Yunnan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 650500,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' People’s Republic of China 12Department of Astronomy & Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' PA 16802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' USA 13Institute for Gravitation and the Cosmos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' PA 16802,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' USA ABSTRACT We present cosmological-scale 3-dimensional (3D) neutral hydrogen (Hi) tomographic maps at z = 2−3 over a total of 837 deg2 in two blank fields that are developed with Lyα forest absorptions of 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='736 background Sloan Digital Sky Survey (SDSS) quasars at z=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='08-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Using the tomographic maps, we investigate the large-scale (≳ 10 h−1cMpc) average Hi radial profiles and two-direction profiles of the line-of-sight (LoS) and transverse (TS) directions around galaxies and AGN at z = 2 − 3 identified by the Hobby-Eberly Telescope Dark Energy eXperiment (HETDEX) and SDSS surveys, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The peak of the Hi radial profile around galaxies is lower than the one around AGN, suggesting that the dark-matter halos of galaxies are less massive on average than those of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The LoS profile of AGN is narrower than the TS profile, indicating the Kaiser effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' There exist ionized outskirts at ≳ 30 h−1cMpc beyond Hi rich structures of galaxies and AGN found in the LoS profiles that can be explained by the influence of high energy photons propagating over a long distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Our findings indicate that the Hi radial profile of AGN has transitions from proximity zones (≲ a few h−1cMpc) to the Hi rich structures (∼ 1 − 30 h−1cMpc) and the ionized outskirts (≳ 30 h−1cMpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Although there is no significant dependence of AGN types (type-1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' type-2) on the Hi profiles, the peaks of the radial profiles anti-correlate with AGN luminosities, suggesting that AGN’s ionization effects are stronger than the gas mass differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Keywords: galaxies: formation — galaxies: evolution — galaxies: high-redshift — intergalactic medium 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' INTRODUCTION Galaxy formation in the Universe is closely related to the neutral hydrogen (Hi) gas in the intergalactic Corresponding author: Dongsheng Sun sunds@icrr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='jp medium (IGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Within the modern paradigm of galaxy formation, galaxies form and evolve in the filament structure of Hi gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=', Meiksin 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Mo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Cosmological hydrodynamics simulations suggest that the picture of galaxy formation and evolution is asso- ciated with large-scale baryonic gas exchange between the galaxy and the IGM (fox 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' van de Voort 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05100v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='GA] 12 Jan 2023 2 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Enormous rivers of cold gas (∼ 104 K) flow into the galaxy and trigger the star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=', Dekel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Kereˇs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2005) The cold gas is heated by star formation and then ejected by the powerful galactic- scale outflows due to feedback caused by stellar winds and supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The circulation of gas is one of the keys to under- standing galaxy formation and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The interplay of gravitational and feedback-driven processes can have surprisingly large effects on the large scale behavior of the IGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Some of the radiation produced by massive stars and black hole accretion disks can escape from the dense gaseous environments and propagate out of galaxies and photoionize the Hi gas in the circumgalac- tic medium (CGM) and even in the IGM (National Academies of Sciences, Engineering 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Mukae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Great progress has been achieved in exploring the Hi distribution around galaxies and active galactic nuclei (AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The cross-correlation of the Hi in the IGM and galaxies has been detected by Lyα absorption features in the spectra of background quasars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=', Rauch 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Faucher-Gigu`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2008a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2013) and bright star-forming galaxies (Steidel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Mawatari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Keck Baryon Structure Survey (KBSS: Rudie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Ra- kic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2014), the Very Large Telescope LBG Redshift Survey (VLRS: Crighton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Tummuangpak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2014), and other spectro- scopic programs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=', Adelberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2003, 2005) have investigated the detailed properties of the Hi distri- bution around galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' These observations target Hi gas around galaxies on the scale of the circumgalactic medium (CGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Recently, 3-dimensional (3D) Hi to- mography mapping, a powerful technique to reconstruct the large scale structure of Hi gas, has been developed by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2014, 2016, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Hi tomography map- ping is originally proposed by Pichon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2001) and Caucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2008) with the aim of reconstructing the 3D matter distribution from the Hi absorption of mul- tiple sightlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' By this technique, the COSMOS Lyα Mapping and Tomography Observations (CLAMATO) survey (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2014, 2018) has revealed Hi large scale structures with spatial resolutions of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 h−1 co- moving Megaparsec (cMpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This survey demonstrates the power of 3D Hi tomography mapping in a number of applications, including the study of a protocluster at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='44 (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2016) and the identification of cos- mic voids (Krolewski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Due to an interpola- tion algorithm (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3) used in the reconstruction of the 3D Hi tomography map, we are able to estimate the Hi distribution along lines-of-sight where there are no available background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Based on the 3D Hi to- mography map of the CLAMATO survey, Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) have reported measurements the IGM Hi–galaxy cross-correlation function (CCF) for several galaxy pop- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Due to the limited volume of the CLAMATO 3D IGM tomography data, Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) can- not construct the CCFs at scales over 24 h−1cMpc in the direction of transverse to the line-of-sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Mukae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) have investigated a larger field than the one of Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) using 3D Hi tomography mapping and report that a huge ionized structure of Hi gas associated with an extreme QSO overdensity re- gion in the EGS field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Mukae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) interpret the large ionized structure as the overlap of multiple prox- imity zones which are photoionized regions created by the enhanced ultraviolet background (UVB) of quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' However, Mukae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) found only one example of a huge ionized bubble, and no others have been reported in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Dispite the great effort made by previous studies, the limited volume of previous work prevents us from understanding how ubiquitous or rare these large ion- ized structures are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In order to answer this ques- tion, we must investigate the statistical Hi distribu- tions around galaxies and AGN at much larger spatial scales (≳ 10 h−1cMpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Although Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) derived CCFs for different populations: Lyα emitters (LAEs), Hα emitters (HAEs), [Oiii] emitters (O3Es), active galactic nuclei (AGN), and submillimeter galaxies (SMGs), on a scale of more than 20 h−1cMpc, the lim- ited sample size results in large uncertainties in the CCF at large scales and prevents definitive conclusions to be made regarding the statistical Hi distributions around galaxies and AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Another open question is the luminosity and AGN type dependence of the large scale Hi distribution around AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) have estimated the Hi distribution around AGN using the Sloan Dig- ital Sky Survey (SDSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2000) data release 9 quasar catalog (DR9Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (Pˆaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2011)) and find no dependence of the Hi distribution on AGN luminos- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this study, we investigate the luminosity depen- dence using the SDSS data release 14 quasar (DR14Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Pˆaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2018) catalog, which includes sources ∼ 2 magnitude fainter than those used by Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the AGN unification model (Antonucci & Miller 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' see also Spinoglio & Fern´andez-Ontiveros 2021), which provides a physical picture that a hot ac- cretion disk of super-massive blackhole is obscured by a dusty torus, the type-1 and type-2 classes are produced by different accretion disk viewing angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this pic- ture, the type-1 (type-2) AGN is biased to AGN with Cosmological-Scale Hi Distribution Around Galaxies and AGN 3 a wide (narrow) opening angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the case of type-1 AGN, one can directly observe the accretion disks and the broad line region, while for type-2 AGN, only the narrow line region is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Previous studies have identified the proximity effect that the IGM of type-1 AGN is statistically more ionized due to the local en- hancement of the UV background on the line-of-sight passing near the AGN (Faucher-Gigu`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Based on the unification model, the type-2 AGN ob- scured on the line of sight statistically radiates in trans- verse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The investigation of the AGN type de- pendence on the surrounding Hi can reveal the large scale Hi distribution influenced by the direction of radi- ation from the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To investigate the Hi distributions around galaxies and AGN on large scales, over tens of h−1cMpc, we need conduct a new study in a field with length of any side larger than 100 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We reconstruct a 3D Hi tomography maps of Hi distribution at z ∼ 2 − 3 in a total area of 837 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We use ≳ 15, 000 back- ground sightlines from SDSS quasars (Pˆaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Lyke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020) for the Hi tomography map recon- struction and have a large number of unbiased galaxies and AGN from the Hobby Eberly Telescope Dark En- ergy eXperiment (HETDEX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2021) and SDSS surveys for the investigations of the large scale Hi distributions around galaxies and AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Section 2 describes the details of the HETDEX survey and our spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Our foreground and background samples of galax- ies and AGN are presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The technique of creating the Hi tomography mapping and the recon- structed Hi tomography map are described in Section 4, and the observational results of Hi distributions around galaxies and AGN are given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this section, we also interpret our results in the context of previous studies, and investigate the dependence of out tomog- raphy maps on AGN type and luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We adopt a cosmological parameter set of (Ωm, ΩΛ, h) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='29, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='71, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='7) in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' DATA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' HETDEX Spectra HETDEX provides an un-targeted, wide-area, integral field spectroscopic survey, and aims to determine the evolution of dark energy in the redshift range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='88 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='52 using ∼ 1 million Lyman-α emitters (LAEs) over 540 deg2 in the northern and equatorial fields that are referred to as “Spring” and “Fall” fields, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The total survey volume is ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='9 comoving Gpc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The HETDEX spectroscopic data are gathered us- ing the 10 m Hobby-Eberly Telescope (HET;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Ramsey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2021) to collect light for the Visi- ble Integral-field Replicable Unit Spectrograph (VIRUS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Hill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2018, 2021) with 78 integral field unit (IFUs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Kelz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2014) fiber arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' VIRUS covers a wave- length, with resolving power ranging from 750 − 950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Each IFU has 448 fibers with a 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The 78 IFUs are spread over the 22 arcmin field of view, with a 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6 fill factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Here we make use of the data re- lease 2 of the HETDEX (HDR2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2023) over the Fall and Spring fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this study, we inves- tigate the fields where HETDEX survey data are taken between 2017 January and 2020 June.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The effective area is 11542 arcmin2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The estimated depth of an emission line at S/N= 5 reaches 3 − 4 × 10−17 erg cm−2 s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Subaru HSC Imaging The HETDEX-HSC imaging survey was carried out in a total time allocation of 3 nights in 2015 − 2018 (semesters S15A, S17A, and S18A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' PI: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Schulze) and 2019 − 2020 (semester S19B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' PI: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Mukae) over a ∼250 deg2 area in the Spring field, accomplishing a 5σ limit- ing magnitude of r = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The SSP-HSC program has obtained deep multi-color imaging data on the 300 deg2 sky, half of which overlaps with the HETDEX foot- prints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this study, we use the r-band imaging data from the public data release 2 (PDR2) of SSP-HSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The 5σ depth of the SSP-HSC PDR2 r-band imaging data is typically 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='7 mag for the 3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 diameter aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The data reduction of HETDEX-HSC survey and SSP- HSC program are processed with HSC pipeline software, hscPipe (Bosch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2018) version 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Because the spectral coverage width of the HETDEX survey is narrow, only 2000 ˚A, most sources appear as single-line emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Furthermore, since the Oii doublet is not resolved, we rely on the equivalent width (EW) to distinguish Lyα from Oii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The high-z Lyα emission is typically stronger than low-z [Oii] lines, due to the in- trinsic line strengths and the cosmological effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The continuum estimate from the HETDEX spectra reach about g= 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2023) and we improve on this using the deep HSC imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We estimate EW using continua measured from two sets of images taken by HSC r-band imaging survey for HET- DEX (HETDEX-HSC survey) and the Subaru Strategic Program (SSP-HSC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' and Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' find that our contamination of Oii emitters in the LAE sample to be below 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' SDSS-IV eBOSS Spectra We use quasar data from eBOSS (Dawson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2016), which is publically available in the SDSS Data Release 14 and 16 quasar catalog (DR14Q, DR16Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Pˆaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 4 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Lyke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The cosmology survey, eBOSS, is part of SDSS-IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The eBOSS quasar targets are se- lected by the XDQSOz method (Bovy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2012) and the color cut mopt − mW ISE ≥ (g − i) + 3, (1) where mopt is a weighted stacked magnitude in the g, r and i bands and mW ISE is a weighted stacked magni- tude in the W1 and W2 bands of the Wide-Field In- frared Survey (WISE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The aim of the eBOSS is to accomplish precision angular-diameter distance measurements and the Hubble parameter deter- mination at z ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 using different tracers of the underlying density fields over 7500 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Its final goal is to obtain spectra of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 million luminous red galaxies, ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='95 million emission line galaxies, ∼ 450,000 QSOs at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='9 ≤ z ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2, and the Lyman-α forest of 60,000 QSOs at z > 2 over four years of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The eBOSS program is conducted with twin SDSS spectrographs (Smee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2013), which are fed by 1,000 fibers connected from the focal plane of the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5m Sloan telescope (Gunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2006) at Apache Point Observa- tory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' SDSS spectrographs have a fixed spectral band- pass of 3600 − 10000 ˚A over the 7 deg2 field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The spectral resolution varies from 1300 at the blue end to 2600 at the red end, where one pixel corresponds to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='8 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' SAMPLES Our study aims to map the statistical distribution of Hi gas on a cosmological scale around foreground galax- ies and AGN by the 3D Hi tomography mapping tech- nique with background sources at z = 2−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We use the foreground galaxies, foreground AGN, and background sources presented in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Two of the goals of this study are to explore the de- pendence of luminosity and AGN type on the Hi distri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To examine statistical results, we need a large number of bright AGN and type-2 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Compared to moderately bright AGN and type-1 AGN, bright AGN and type-2 AGN are relatively rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To obtain a suffi- ciently large samples of bright AGN and type-2 AGN, we expand the Spring and Fall fields of the HETDEX sur- vey, from which we are able to investigate the statistical luminosity and AGN type dependence of the HI distribu- tion around AGN (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The northern extended Spring field flanking the HETDEX survey fields, referred to as the “ExSpring field”, covers over 738 deg2, while the equatorial extended Fall field flanking the HETDEX survey fields, here after “ExFall field”, covers 99 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The total area of our 3D Hi tomography mapping field is 837 deg2 in the ExSpring and ExFall fields that is re- ferred to as “our study field”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Our analysis is conducted in our study field where the foreground galaxies+AGN and the background sources overlap on the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' As an example, we present the foreground galaxies+AGN in the ExFall field at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We also present the sky distribution of the background sources within the ExFall field in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The rest of the fore- ground and background sources are shown in the Ap- pendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Foreground Galaxy Sample We make a sample of foreground galaxies from the data of the HETDEX spectra (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1) and the Sub- aru HSC images (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' With these data, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) have build a catalog of LAEs that have the rest-frame equivalent widths (EW0) of EW0 > 20 ˚A and the HETDEXs Emission Line eXplorer (ELiXer) probabilities (Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2023) larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This EW0 cut is similar to previous LAE studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=', Gronwall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Konno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This cat- alog of LAEs is composed of 15959 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Because the LAE catalog of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) consists of galaxies, type-1 AGN, and type-2 AGN, we isolate galaxies from the sources of the LAE catalog with the limited observa- tional quantities, Lyα and UV magnitude (MUV), that can be obtained from the HETDEX and Subaru/HSC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Because type-1 AGN have broad-line Lyα emis- sion, we remove sources with broad-line Lyα whose full width half maximum (FWHM) of the Lyα emission lines are greater than 1000 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To remove clear type-2 AGN from the LAE catalog, we apply a UV magnitude cut of MUV < −22 mag that is the bright end of the UV luminosity function dominated by galaxies (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We then select sources in our study field, and apply the redshift cut of z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 (as measured by the principle component analysis of multiple lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Pˆaris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2018) to match the redshift range over which we construct Hi tomography map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' These redshifts are mea- sured with Lyα emission (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2021), because Lyα is the only emission available for all of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' By these selections, we obtain 14130 galaxies from the LAE catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' These 14130 galaxies are referred to as the “Galaxy” sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Foreground AGN Samples In this subsection, we describe how we select fore- ground AGN from two sources, (a) the combination of the HETDEX spectra and the HSC imaging data and (b) the SDSS DR14Q catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The type-1 AGN are identified with the sources of (a) and (b), while the type- 2 AGN are drawn from the source of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Cosmological-Scale Hi Distribution Around Galaxies and AGN 5 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Sky distribution of the foreground AGN and galaxies at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 in the ExFall field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The squares present the positions of All-AGN sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Pink (magenta) squares represent the sources of the T1-AGN (T2-AGN) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The cyan and blue dots show the positions of the Galaxy and T1-AGN(H) sample sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The black dashed line indicates the border of the Hi tomography map in the Exfall field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Sky distribution of background AGN in the ExFall field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The gray crosses indicate background AGN that are used to reconstruct our Hi tomography map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The back dashed line has the same meaning as that in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Sample size of foreground samples at z = 2 − 3 Name of sample ExFall ExSpring Total Survey Criteria Galaxy 3431 11436 14867 HETDEX EW0 > 20 ˚A, FWHMLyα < 1000 km/s, Muv> −22 mag T1-AGN(H) 438 1349 1787 HETDEX EW0 > 20 ˚A, FWHMLyα > 1000 km/s T1-AGN 2393 12300 14693 SDSS FWHMLyα > 1000 km/s T2-AGN 436 1633 2069 SDSS FWHMLyα < 1000 km/s Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Sample size of background sample at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='08 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='67 Name of sample ExFall ExSpring Total Survey Criteria background AGN 2181 12555 14736 SDSS ⟨S/N⟩Lyαforest > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4 With the source (a) that is the same as the one stated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) have constructed the LAE catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We use the catalog of Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) to select LAEs at z ∼ 2 − 3 that fall in our study field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Applying a Lyα line width criterion of FWHM > 1000 km s−1 with the HETDEX spectra, we identify broad-line AGN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' type-1 AGN, from the LAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We thus obtain 1829 type-1 AGN that are referred to as T1-AGN(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We use the width of Lyα emission line for the selection of type-1 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This is because no other emission lines characterising AGN, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Civ, are available for all of the LAEs due to the limited wavelength coverage and the sensitivity of HETDEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Similarly, the redshifts of T1- Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg] 2 0 2 35 30 25 20 15 10 5 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg]Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg] 1 35 30 25 20 15 10 5 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg]6 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' AGN(H) objects are measured with Lyα emission whose redshifts may be shifted from the systemic redshifts by up to a few 100 km s−1 (See Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We do not select type-2 AGN from the source of (a), because we cannot identify type-2 AGN easily with the given data set of source (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' From the source (b), we obtain the other samples of foreground AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We first choose objects with a classi- fication of QSOs of the SDSS DR14Q, and remove ob- jects outside the redshift range of z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 in our study field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We obtain 23721 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For 16762 out of 23721 AGN, Lyα FWHM measurements are available from Rakshit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The other AGN without FWHM measurement are removed due to the poor qual- ity of the Lyα line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We thus use these 16762 AGN with good quality of the Lyα line to compose our AGN sam- ple, referred to as All-AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To investigate the type dependence, we classify these 16762 AGN into type-1 and type-2 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the same manner as the T1-AGN(H) sample construction, we use Lyα line width measurements of Rakshit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) for the type-1 and type-2 AGN classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For the 16762 AGN, we apply the criterion of Lyα FWHM > 1000 km s−1 (Villarroel & Korn 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Panessa & Bassani 2002) to select type-1 AGN, and obtain 14693 type-1 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Following Villarroel & Korn (2014);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Panessa & Bassani (2002), we classify type-2 AGN by the criterion of Lyα FWHM < 1000 km s−1 and obtain 2069 type- 2 AGN (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Alexandroff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Zakamska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' These type-1 and type-2 AGN are referred to as T1-AGN and T2-AGN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Table 1 presents the summary of foreground samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We obtain 14693 and 1829 type-1 AGN, which referred to as T1-AGN and T1-AGN(H), from the SDSS and HETDEX surveys, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We select 2069 type-2 AGN that are referred to as T2-AGN from the SDSS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Background Source Sample In this subsection, we describe how the background sources are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We select the background sources with the SDSS DR16Q catalog, following the three steps below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the first step, we extract QSOs in our study field from the SDSS DR16Q catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We then select QSOs falling in the range of redshifts from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='08 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The lower and upper limits of the redshift range are deter- mined by the Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Our goal is to probe Hi ab- sorbers at z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 with the Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Because the Lyα forest is observed in the rest-frame 1040 − 1185 ˚A of the background sources, we obtain the lower and up- per limits of the redshifts, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='08 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='67, by 1216 × (1 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0)/1185−1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='08 and 1216×(1+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0)/1040−1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='67, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' By this step, we have selected 26899 back- ground source candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the second step, we choose background source can- didates with good quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We calculate the average sig- nal to noise ratio, ⟨S/N⟩, in the wavelength range of the Lyα forest for the 26899 background source candidates, and select 15573 candidates with ⟨S/N⟩ greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To maximize the special resolution of the tomography map, we set the threshold, ⟨S/N⟩ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4, smaller than the value used by Mukae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This threshold is more conservative than the value, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2, used in Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the third step, we remove damped Lyα absorbers (DLAs) and broad absorption lines (BALs) from the Lyα forest of the 15573 candidates, because the DLAs and BALs cause an overestimation of the absorp- tion of the Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We identify and remove DLAs using the catalog of Chabanier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2022), which is based on the SDSS DR16Q (Lyke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We mask out the wavelength ranges contaminated by the DLAs of the Chabanier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2022) catalog (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 for the procedures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We conduct visual inspection for the 15573 candidates to remove 115 BALs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Figure 3, we show the spectrum with BALs identified by visual in- spection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this way, we obtain 15458 (= 15573 − 115) sources whose spectra are free from DLAs and BALs, which we refer to as the background source sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Ta- ble 2 lists the number of background sources in each field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Spectrum of background AGN with BALs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The black line represents the spectrum of a background source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The vertical dashed lines present the central wavelengths of the metal absorptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The yellow hatches show the wavelength ranges of the BALs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The gray hatches indi- cate the wavelength ranges not used for the reconstruction of Hi tomography maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The SDSS ID of this spectrum is 106584616, whose redshift is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='067837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' HI TOMOGRAPHY AND MAPPING 8 Flux density 6 2 4500 000S 5500 6000 6500 Wavelength [A]Cosmological-Scale Hi Distribution Around Galaxies and AGN 7 In this section we describe the process to construct Hi tomography maps with the spectra of the background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For Hi tomography, we need to obtain intrin- sic continua of the background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 ex- plains masking the biasing absorption features in the background sources, while Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3 determines the intrinsic continua of the background source spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3, we construct Hi tomography maps with the intrinsic continuum spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' DLA and Intrinsic Absorption Masking Because a DLA is an absorption system with a high neutral hydrogen column density NHI > 2 × 1020 cm−2, the intervening DLA completely absorbs a large por- tion of the Lyα forest over ∆v ∼ 103 km s−1, which gives bias in the estimates of the intrinsic continua of the background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For the spectra of the back- ground sources, we mask out the DLAs identified in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We determine the range of wavelengths for masking with the IDL code of Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The wavelength range corresponds to the equivalent width of each DLA (Draine 2011): W ∼ λα � e2 mec2 NHIfαλα �γαλα c ��1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2) In the formula, λα is the rest-frame wavelength of the hydrogen Lyα line (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 1216 ˚A), while c, e, me, fα, NHi, and γα are the speed of light, the electron charge, the electron mass, the Lyα oscillator strength, the Hi column density of the DLA, and the sum of the Einstein A coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We mask out these wavelength ranges of the background source spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Figure 5, the masked DLA is indicated by yellow hatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We also mask out the intrinsic absorption lines of the metal absorption lines, which are the other sources of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We mask SIv λ1062, Nii λ1084, Ni λ1134, and Ciii λ1176 (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2012), which are shown by the dashed lines in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Because the spectral resolu- tions of SDSS DR14Q are ∆λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='8 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 ˚A, we adopt the masking size of 10 ˚A in the observed frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Intrinsic Continuum Determination In order to obtain the intrinsic continuum of the back- ground source (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3) in the Lyα forest wavelength range (LAF-WR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 1040−1185 ˚A), we conduct mean-flux regulated principle component analysis (MF-PCA) fit- ting with the IDL code (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2012) for the back- ground sources after the masking (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' There are two steps in the MF-PCA fitting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The first step is to predict the shape of the intrinsic continuum of the background sources in the LAF-WR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We conduct least-squares principle component analysis (PCA) fitting (Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2012) to the background source spectrum in the rest frame 1216 − 1600 ˚A : fPCA(λ) = µ(λ) + 8 � j=1 cjξj(λ), (3) where λ is the rest-frame wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The values of cj are the free parameters for the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The function of µ(λ) is the average spectrum calculated from the 50 lo- cal QSO spectra in Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The function of ξj(λ) represents the jth principle component (or ‘eigen- spectrum’) out of the 8 principle components taken from the PCA template shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the second step, we predict the intrinsic continuum of the background source in the LAF-WR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Because the PCA template is obtained with the local QSO spectra, the best-fit fPCA in the LAF-WR does not include cos- mic evolution on the average transmission rate of the Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' On average, the best-fit fPCA in the LAF- WR should agree with the cosmic mean-flux evolution (Faucher-Gigu`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2008c): ⟨F(z)⟩ = exp[−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='001845(1 + z)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='924], (4) where z is the redshift of the absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We use fPCA and a correction function of a + bλ to estimate the intrinsic continuum fintrinsic(λ) for large-scale power along the line of sight with the equation: fintrinsic(λ) = fPCA(λ) × (a + bλ), (5) where a and b are the free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Because the ratio of fobs(λ)/fintrinsic(λ) should agree with the cosmic average ⟨F(z)⟩ for z = (λ/1216)−1 in the LAF-WR, we conduct least-squares-fitting to find the values of a and b providing the best fit between the mean ratio and the cosmic average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The red line shown by the bottom panel of Figure 5 presents a MF-PCA fitted continuum derived from the spectrum of one of our background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' By the MF-PCA fitting, we have obtained the esti- mates of fintrinsic(λ) for 14736 out of the 15458 back- ground sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find the other background sources show poor fitting results found by visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We do not use these background sources in the following analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 6 shows an example of poor fitting result due to the unknown absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We adopt con- tinuum fitting errors of ∼ 7%, ∼ 6%, and ∼ 4% for Lyα forests with mean S/N values of < 4, 4 − 10, and > 10, respectively (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' HI Tomography Map Reconstruction We reconstruct our Hi tomography maps by a proce- dure similar to Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We define Lyα forest 8 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Principle components and mean flux taken from Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The top panel shows the normalized mean flux of 50 local QSOs in rest-frame wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The bottom 8 panels show the 1st − 8th principle components that are used in the PCA fitting in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Each principle component is normalized to the mean flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' fluctuations δF at each pixel on the spectrum by δF = fobs/fintrinsic ⟨F(z)⟩ − 1 (6) , where fobs and fintrinsic are the observed spectrum and estimated intrinsic continuum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' ⟨F(z)⟩ is the cosmic average transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We calculate δF with our background source spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The top panel of Figure 5 shows the ‘spectrum’ of δF derived from the fobs and fintrinsic in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For the pixels in the wavelength ranges of masking (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1), we do not use δF in our further analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We thus obtain δF in 876,560 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For the the HI tomography map of the Extended Fall field, we define the cells of the Hi tomography map in the three-dimensional comoving space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We choose a volume of 30◦ × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3◦ in the longitudinal and latitudinal dimen- Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Example of a background source spectrum that was used for the reconstruction of the Hi tomography map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Bottom panel: Estimation of intrinsic continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The thin black line is the spectrum of a background source taken from the SDSS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The red and magenta lines are the re- sults of MF-PCA and PCA fitting, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The vertical dashed lines present the central wavelengths of the metal ab- sorptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The gray hatches represent the wavelength ranges that are not used for the Hi tomography map reconstruc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The yellow hatch indicates the wavelength ranges of DLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Top panel: Spectrum of δF extracted from the bottom panel in the LAF-WR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The vertical yellow and gray hatches are the same as those in the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The black and pink lines show the spectrum of δF and the error of δF at the corresponding wavelength extracted from the bottom panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The horizontal line indicates the cosmic average of Lyα forest transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as the bottom panel of Figure 5, but for the background spectrum with a poor fitting result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The red and magenta lines are the results of MF-PCA and PCA continuum fitting, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The yellow hatch indicates the wavelength range of unknown absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' sions, respectively, in the redshift range of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 < z < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The comoving size of our Hi tomography map is 2257 h−1cMpc × 233 h−1cMpc × 811 h−1cMpc in the right 5 Mean flux 0 2 1st Component 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='. 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 2nd Component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 3rd Component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 4th Component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 5th Component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 6th Component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 7th Component 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 8th Component 1000 1100 1200 1300 1400 1500 1600 Rest-frame wavelength [A]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3 AF0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6 12 Flux density 080 4000 4500 000S 5500 6000 Wavelength [A]6 Flux density 2 0 3500 4000 4500 5000 5500 Wavelength [A]Cosmological-Scale Hi Distribution Around Galaxies and AGN 9 ascension (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='), declination (Dec), and z directions, respectively in the same manner as Mukae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Our Hi tomography map has 451 × 46 × 162 cells, and one cell is a cubic with a size of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 h−1cMpc on a side, where the line-of-sight distance is estimated under the assumption of the Hubble flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We conduct a Wiener filtering scheme for reconstruct- ing the sightlines that do not have background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We use the calculation code developed by Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The solution for each cell of the reconstructed sightline is obtained by δrec F = CMD · (CDD + N)−1 · δF, (7) where CMD, CDD, and N are the map-data, data-data, and noise covariances, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We assume Gaus- sian covariances between two points r1 and r2: CMD = CDD = C(r1, r2), (8) C(r1, r2) = σ2 F exp � −(∆r∥)2 2L2 ∥ � exp � −(∆r⊥)2 2L2 ⊥ � , (9) where ∆r∥ and ∆r⊥ are the distances between r1 and r2 in the directions of parallel and transverse to the line of sight, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The values of L⊥ and L∥ are the correlation lengths for vertical and parallel to the line- of-sight (LoS) direction, respectively, and defined with L⊥ = L∥ = 15 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The value of σ2 F is the normal- ization factor that is σ2 F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2015) de- velop this Gaussian form to obtain a reasonable estimate of the true correlation function of the Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We perform the Wiener filtering reconstruction with the val- ues of δF at the 898390 pixels, using the aforementioned parameters of the Stark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2015) algorithm with a stopping tolerance of 10−3 for the pre-conditioned con- jugation gradient solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' As noted by Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2016), the boundary effect that leads to an additional error on δF occurs at the positions that are near the bound- aries of an Hi tomography map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The boundary effect is caused by the background sightlines not covering the region that contribute to the calculation of the δF values for cells near the Hi tomography map boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To avoid the boundary effect, we extend a distance of 40 h−1cMpc for each side of the Hi tomography map of the ExFall field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The resulting map is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For the HI tomography map reconstruction of the Ex- tended Spring field (hereafter ExSpring field), we per- form almost the same procedure as the one of the Ex- Fall field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The area of the ExSpring field is more than 6 times larger than that of the ExFall field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We sep- arate the ExSpring field into 8 × 3 = 24 footprints to save calculation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Each footprint covers an area of 10◦ × 5◦ in the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' and Dec directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We reconstruct the Hi tomography map one by one for the footprints of the ExSpring field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To weaken the boundary effect, we extend a distance of 40 h−1cMpc for each side of the footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The ex- tensions mean that every two adjacent footprints has an overlapping region of 80 h−1cMpc width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The width of the overlapping regions is a conservative value to weaken the boundary effect since it is much larger than the res- olution, 15 h−1cMpc, of our Hi tomography maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' By the 40 h−1cMpc extension, we reduce the uncertainty in the δF value for the edge of each footprint caused by boundary effect to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This value corresponds to the 1/10 of the typical error for each cell of the Hi tomogra- phy map (Mukae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020) The remaining additional error caused by boundary effect is negligible compared to the statistical uncertainties in the HI distributions ob- tained in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Then we follow the reconstruction procedure for the ExFall field to reconstruct HI tomog- raphy maps of the footprints and cut off all the cells within 40 h−1cMpc to the borders that are affected by the boundary effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Finally we obtain the Hi tomogra- phy map of the ExSpring field with a special volume of 3475 h−1cMpc × 1058 h−1cMpc × 811 h−1cMpc in the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=', Dec, and z directions, respectively (Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' RESULTS AND DISCUSSIONS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Average HI Profiles around AGN: Validations of our AGN Samples In this section we present the Hi profile, δF as a func- tion of distance, with the All-AGN sample sources, us- ing the reconstructed Hi tomography maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We com- pare the Hi profile of the All-AGN sample to the one of the previous study (Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We also present the comparison of the Hi profiles between T1- AGN(H) and T1-AGN samples that are made with the HETDEX and SDSS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this study, we only discuss the structures having size ≳ 15 h−1cMpc corresponding to the resolution of our 3D Hi tomography maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For the Hi profiles with the All-AGN sample, we extract δF values around the 16978 All-AGN sample sources in the Hi tomography map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We cut the Hi tomography map centered at the positions of the All- AGN sample sources, and stack the δF values to make a two dimensional (2D) map of the average δF distribution around the sources that is referred to as a 2D Hi profile of the All-AGN sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The two dimensions of the 2D Hi profile correspond to the transverse distance DTrans and the LoS Hubble distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The velocity corre- sponding to the LoS Hubble distance is referred to as the LoS velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Here we define the Lyα forest absorption fluctuation AF ≡ −δF (10) 10 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 3D Hi tomography map of the ExFall field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The color contours represent the values of δF from negative (red) to positive (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The spatial volume of the Hi tomography map is 2257×233×811 h−3cMpc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The redshift range is z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' that is an indicator of the amount of the Hi absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 9 shows the 2D Hi profile with values of AF (δF) for All-AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The solid black lines denote the contours of AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In each cell of the 2D Hi profile, we define the 1σ error with the standard deviation of AF values of the 100 mock 2D Hi profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Each mock 2D Hi profile is obtained in the same manner as the real 2D Hi profile, but with random positions of sources whose number is the same as the one of All-AGN sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Figure 9, the dotted black lines indicate the contours of the 6σ, 9σ and 12σ confidence levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find the 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5σ level detection of AF at the source position (0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The AF value at the source position indicates the averaging value over the ranges of (−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 h−1cMpc, +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 h−1cMpc) in both the LoS and transverse directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5σ level detection at the source position is suggestive that rich Hi gas exists near the All-AGN sources on average The 2D Hi profile is more extended in the transverse direction than along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We discuss this difference in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We then define a 3D distance, D, under the assump- tion of the Hubble flow in the LoS direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We derive AF as a function of D that is referred to as ”Hi radial profile”, averaging AF values of the 2D Hi profile over the 3D distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 10 shows the Hi radial profile of the All-AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find that the AF values de- crease towards a large distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This trend is consistent with the one found by Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) with the SDSS quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) have obtained the average Hi absorption distribution around the AGN taken from the SDSS data release 16 quasar (SDSS DR16Q) catalog in the field of Strip 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The criteria of the target selection for the SDSS DR16Q and SDSS DR14Q sources are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The luminosity distribution of AGN for Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) is almost the same as that of our All-AGN sample sources that are taken from the SDSS DR14Q catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We derive the average radial Hi profile of the Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) AGN sources by the same method as for our All-AGN sample, using the 3D Hi tomogra- phy map reconstructed by Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We compare the radial Hi profile of the All-AGN sample with the one derived from the 3D Hi tomography map of Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The comparison is shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Our result agrees with that of Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) within the error range at scale D > 10 h−1 cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The peak values of AF are comparable, AF ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='129 Dec[cMpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0429 2250 300 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0429 750 1500 200 1250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='129 1000 750 100 750 500 RA [cMpc 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='214 250 250 0 z[cMpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='300Cosmological-Scale Hi Distribution Around Galaxies and AGN 11 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 7, but for the ExSpring field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The spatial volume of the Hi tomography map is 3475 × 1058 × 811 h−3cMpc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' slight difference between the peak values of our and Ravoux et al.’s results can be explained by the differ- ent approaches of the estimation for the intrinsic con- tinuum adopted by Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' and us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' conduct power law fitting, which is different from the MF-PCA fitting that we used, for the intrinsic contin- uum in the wavelength range of the Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Given the low (∼ 15 h−1) spatial resolution of both our Hi to- mography map and that of Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020), neither studies are able to search for the proximity effect mak- ing a photoionization region around AGN (D’Odorico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' From the comparison shown by Figure 10, we conclude that the Hi distribution derived from our Hi tomography map is reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To check the reliability of the HETDEX survey results, we use the reliable result of the SDSS AGN to compare with the result derived by the HETDEX AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We select type-1 AGN from the HETDEX’s T1- AGN(H) and SDSS’s T1-AGN samples to make sub- samples of T1-AGN(H) and T1-AGN whose rest-frame 1350 ˚A luminosity (L1350) distributions are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For T1-AGN, the measurements directly from the SDSS spectra (Lspec 1350) are available (Rakshit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For T1-AGN(H), we do not have Lspec 1350 measurements from the HETDEX spectra, we estimate it using HSC r-band imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Since the central wavelength of the r-band imaging is rest-frame ∼ 1700˚A, we calibrate the conver- sion between r-band luminosity, Lphot UV , and Lspec 1350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We examine the 283 type-1 AGN sources that appear in both the SDSS and HETDEX surveys (and, thus, have both Lspec 1350 measurements from SDSS and r-band lumi- nosities from HSC) to calibrate the relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The results are displayed in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Lphot UV are always smaller than those of Lspec 1350 (Rakshit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Due to the blue UV slope of the spectra for the AGN both categorized in the T1-AGN(H) and T1-AGN samples, the luminosity of the rest-frame 1350 ˚A always shows a larger value than the one of rest-frame 1700 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We conduct linear fitting to the data points of Figure 11, and obtain the best-fit linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' With the best-fit linear function, we estimate Lspec 1350 values for the HET- DEX’s T1-AGN(H) sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We show the Lspec 1350 distributions of all the T1-AGN(H) and T1-AGN sample sources in the upper panel of Fig- ure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We make the sub-samples of T1-AGN and T1- AGN(H) that consist of the sources in the overlapping area of Lspec 1350 distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We present the Lspec 1350 distri- butions of the T1-AGN and T1-AGN(H) sub-samples in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='214 Dec [cMpc] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='129 11000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0429 900 800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0429 700 35Q0 600 3250 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='129 2750 500 2500 2250 400 2000 1750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='214 1500 300 RA[cMpcl 1250 200 1000 750 750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='300 10Q 500 500 250 250 0 z[cMpc]12 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2D Hi profile of the All-AGN sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The color map indicates the AF (δF) values of each cell of the 2D Hi profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The solid lines denote constant AF (δF) values in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01) starting at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The dotted lines correspond to multiples of 3σ starting at 6σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Hi radial profile of the All-AGN and Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020) AGN samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The black and gray data points and error bars show the Hi radial profiles of our All-AGN sample sources and the AGN of Ravoux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The horizontal dashed line shows the cosmic average Hi absorption, AF = 0 (δF = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' the bottom panel of Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We obtain 540 and 4338 type-1 AGN for the sub-samples of T1-AGN(H) and T1- AGN, respectively, whose Lspec 1350 distributions are shown in the bottom panel of Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We derive the Hi radial profiles for the sub-samples of T1-AGN(H) and T1-AGN sample sources, as shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Hi radial profiles of T1-AGN(H) and T1-AGN sub-sample sources are in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Relations of Lphot UV against Lspec 1350 for the sources both categorized in the T1-AGN(H) and T1-AGN samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Lphot UV and Lspec 1350 are measured from the HSC r-band imaging and SDSS spectra (Rakshit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2020), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The gray points show the distribution of Lspec 1350 − Lphot UV re- lations for the sources both categorized in the T1-AGN(H) and T1-AGN samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The black dashed line indicates the relation where Lspec 1350 = Lphot UV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The red dashed line represents the linear best fit of the blue points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' AGN Average Line-of-Sight and Transverse Hi Profiles 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 Ravoux+20 AGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 All-AGN AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0 10 20 3040506070 D [h-1cMpc]best fit 46 [erg s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 45 logL 44 45 46 logL spec [erg s-1] 1350LoS Velocity [km s-1] LoS Hubble distance 7500 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 [h-1cMpc] 5000 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 AF 2500 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0 0 204060 DTrans [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN 13 Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Top panel: Lspec 1350 distributions of the T1-AGN and T1-AGN(H) samples with blue and red histograms, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Bottom panel: Same as the top panel, but for the T1-AGN and T1-AGN(H) sub-sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Hi radial profiles of the T1-AGN and T1- AGN(H) sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The blue and red triangles show the values of AF as a function of distance, D, for the T1-AGN and T1-AGN(H) sample sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The hori- zontal dashed line shows the cosmic average Hi absorption, AF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The right y-axis shows the corresponding δF values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Based on the 2D Hi profile of the All-AGN sample (Figure 9), we find that the Hi distributions of the All- AGN sample sources are more extended in the trans- verse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this section, we present the Hi radial profiles of All-AGN sample in the LoS and transverse directions and compare these two Hi radial profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To derive the Hi radial profile of the All-AGN sample with the absolute LoS distance, which is referred to as the LoS Hi radial profile (Figure 15), we average AF val- ues of the 2D Hi profiles of All-AGN over DTrans < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 h−1cMpc (from −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 h−1cMpc to +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 h−1cMpc in the transverse direction) that corresponds to the spatial res- olution of the 2D Hi profile map, 15 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Among the 16,978 All-AGN sample sources, 10,884 sources are used as both background and foreground sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this case, the Hi absorption (AF) of these 10,884 sources at the LoS velocity ≲ −5250 km s−1 is estimated mainly from their own spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' As the discussion in Youles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2022), the redshift uncertainty of the SDSS AGN causes the overestimation of intrinsic continuum and the underestimation of AF around the metal emission lines such as Ciii λ1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This leads to a systemics toward positive AF in the Hi radial profile of LoS velocity (LoS distance) at the LoS velocity ≲ 5250 km s−1 (Figure 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Hi radial profile of LoS velocity (LoS distance) is de- rived by averaging AF values over DTrans < 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 h−1cMpc as a function of the negative and positive LoS velocity (LoS distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this study, we only use the values of AF at the LoS distance > −52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5h−1cMpc (LoS velocity > −5250 km s−1) to derive the LoS Hi radial profile of the All-AGN sample (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The scale, LoS dis- tance > −52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5h−1cMpc (LoS velocity > −5250 km s−1), is determined by the maximum wavelength of the Lyα forest we used, the smoothing scale of the Wiener filter- ing scheme, and the AGN redshift uncertainty, assumed by Youles et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' After removing the AF values af- fected the systemics in the 2D Hi profile, we present the LoS Hi radial profile of the All-AGN sample in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We estimate the Hi radial profiles of DTrans, which is referred to as the Transverse Hi radial profile,by averag- ing the AF values over the LoS velocity of (−750, +750) km s−1 whose velocity width corresponds to 15 h−1 cMpc in the Hubble-flow distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Hi radial profile of DTrans is also shown in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We compare the LoS and Transverse Hi radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The AF value decrease more rapidly in the LoS direc- tion than those in the Transverse direction (Figure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This difference may be explained by an effect similar to the Kaiser effect (Kaiser 1987), doppler shifts in AGN redshifts caused by the large-scale coherent motions of the gas towards the AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The LoS Hi radial profile is negative, AF ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='002±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0008, at the large scale, ≳ 30 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5, we discuss the negative AF values of LoS Hi radial profiles at large scale and com- pare our observational result to the models of a previous study, Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Source Dependences of the AGN Average HI Profiles In this section, we present 2D and Hi radial profiles of the AGN sub-samples to investigate how the average Hi density depends on luminosity and AGN type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' AGN Luminosity Dependence Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='15 T1-AGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='10 T1-AGN(H) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 42 43 4445 46 470.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 T1-AGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 T1-AGN(H) AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 TT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='018F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0 10 20 30 40 506070 D [h-1cMpc]14 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Hi radial profiles of LoS velocity (LoS distance) for the All-AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The black solid line shows the AF values as a function of LoS velocity (LoS distance) for the All-AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The vertical dashed line presents the posi- tion of LoS velosity = 0 km s−1 (LoS distance = 0 h−1cMpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The horizontal dashed indicates the cosmic average Hi ab- sorption, AF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The gray shaded area shows the range of the AF not used to derive LoS Hi radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' LoS and Transverse Hi radial profiles the All- AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The black and gray lines show the AF (δF) values as a function of LoS distance and DT rans, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The horizontal dashed line indicates AF (δF) = 0 (= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We study the AGN-luminosity dependence of the average Hi profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 16 presents the Lspec 1350 distribution of All-AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We make 3 sub-samples of All-AGN that are All-AGN-L3, All-AGN-L2 and All-AGN-L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The luminosity ranges of the sub- samples are 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='70 < log(Lspec 1350/[erg s−1]) < 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='41, 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='41 < log(Lspec 1350/[erg s−1]) < 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='75, and 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='75 < log(Lspec 1350/[erg s−1]) < 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='35, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The luminos- ity ranges of the 3 sub-samples are defined in a way that the numbers of the AGN are same 5695 in each subsam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We derive the 2D Hi profiles of the sub-samples in the same manner as Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1, and present the pro- files in Figures 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In these 2D Hi profiles, The bright- est sub-sample of All-AGN-L1 (the faintest sub-sample of All-AGN-L3) shows the weakest (the strongest) Hi absorptions around the source position, D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We then extract the Hi radial profiles from the 2D Hi profiles of the All-AGN sub-samples, and present the Hi radial profiles in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In this figure, we find that the peak values of AF for the All-AGN sub- samples is anti-correlates with AGN luminosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The peak AF values near the source position drops from the faintest All-AGN-L3 subsample to the brightest All- AGN-L1 subsample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The gas densities around bright AGN are higher than (or comparable to) those around faint AGN, this result would suggest that the ioniza- tion fraction of the hydrogen gas around bright AGN is higher than the one around faint AGN on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We also present the LoS and Transverse Hi radial pro- files of the All-AGN sub-samples derived by the same method as that for the All-AGN sample in Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Similar to what we found in the comparison of the Hi radial profiles for the All-AGN sub-samples, the peak values of the LoS and Transverse Hi profiles also de- crease from the faintest sub-sample, All-AGN L3, to the brightest sub-sample, All-AGN L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For the LoS (Trans- verse) Hi radial profiles at the scales beyond 25 h−1 cMpc, we do not find any significant differences in the comparison of the LoS (Transverse) Hi radial profiles for the All-AGN sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' logLspec 1350 distribution of the bright and All-AGN sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The vertical dashed lines indicate the board- ers of Lspec 1350 where log(Lspec 1350/[erg s−1]) = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='41 and 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='75, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' These three borders separate the All-AGN sam- ple into 3 sub-samples of All-AGN-L3, All-AGN-L2, and All- AGN-L1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' AGN Type Dependence LoS distance [h-1cMpc] 75 50-25 0 25 1 50 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 All-AGN LoS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 A F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 7500-5000-2500 0 2500 5000 7500 LoS velocity [km s-1]Los Velocity [km s-1] 0 2500 5000 7500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 All-AGN LoS All-AGN Trans 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 25 50 75 [ D [h-1cMpc]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='10 All-AGN-L3 Fraction All-AGN-L2 All-AGN-L1 -- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 44 45 46 47Cosmological-Scale Hi Distribution Around Galaxies and AGN 15 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 9, but for the All-AGN-L3 (top), All-AGN-L2 (middle) and All-AGN-L1 (bottom) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 13, but for the All-AGN-L3 (red), All-AGN-L2 (gray) and All-AGN-L1 (black) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We investigate the dependence of Hi profiles on type-1 and type-2 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To remove the effects of the AGN lumi- Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' LoS and Transverse Hi radial profiles of the All- AGN-L3, All-AGN-L2, and All-AGN-L1 sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The top figure (bottom figure) presents the LoS (Transverse) Hi radial profiles of the All-AGN-L3, All-AGN-L2, and All- AGN-L1 sub-samples, shown by the red, gray, and black lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The meaning of the horizontal dashed lines both in the top and bottom figures are the same as the one in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' nosity dependence (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1), we make sub-samples of T1-AGN and T2-AGN with the same Lspec 1350 distribu- tion by the same manner as the one we conduct for the selection of T1-AGN and T1-AGN(H) sub-samples in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The top panel of Figure 20 presents the Lspec 1350 distributions of T1-AGN and T2-AGN samples, while the bottom panel of Figure 20 shows those of the T1-AGN and T2-AGN sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The sub-samples of T1-AGN and T2-AGN are composed of 10329 type- 1 AGN and 1462 type-2 AGN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We derive the 2D Hi profiles from the T1-AGN and T2-AGN sub- samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The profiles are presented in Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We LoS Velocity [km s-1] LoS Hubble distance 7500 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 [h-1cMpc] 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 AF 2500 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0 0 204060 DTrans [h-1cMpc]LoS Velocity [km s-1] LoS Hubble distance 7500 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 [h-1cMpc] 5000 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 AF 2500 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0 0 204060 DTrans [h-1cMpc]Los Velocity [km s-1] LoS Hubble distance 7500 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 [h-1cMpc] 5000 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 AF 2500 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0 0 204060 DTrans [h-1cMpc]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 A1l-AGN-L3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 All-AGN-L2 AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 All-AGN-L1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0 10 20 3040506070 D [h-1cMpc]Los Velocity [km s-1] 0 2500 5000 7500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 All-AGN-L3 LoS All-AGN-L2 LoS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 All-AGN-L1 LoS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 25 50 75 D [h-1cMpc]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 All-AGN-L3 Trans All-AGN-L2 Trans 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 All-AGN-L1 Trans 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 OF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 25 50 75 D [h-1cMpc]16 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' find 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='7 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='9 σ detections at the source center po- sition (0,0) of the T1-AGN and T2-AGN sub-samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We calculate the Hi radial profiles from the 2D Hi profiles of the T1-AGN and T2-AGN sub- samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Figure 22, we compare the Hi radial profiles of the T1-AGN and T2-AGN sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' No notable difference is found within 1σ error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The peak value of AF of the T2-AGN subsample is within 1σ error of the peak value of the T1-AGN subsample near the source position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To compare the Hi distributions of type-1 and type-2 AGN in the LoS and transverse directions, we derive the LoS and Transverse Hi radial profiles of the T1-AGN and T2-AGN sub-samples and present the profiles in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Similar to the trend of the Hi radial profiles, the peak values of the LoS and Transverse Hi radial profiles for T1-AGN and T2-AGN sub-samples are not significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The comparable peak values of the LoS and Transverse Hi radial profiles suggest that the selectively different orientation and opening angles of the dusty tori of the type-1 and type-2 AGN do not significantly affect the Hi distribution at the scale ≲ 15 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' For the Hi radial profiles at the scale > 15 h−1cMpc, we find that the AF value for the LoS Hi radial pro- file of the T1-AGN sub-sample is greater than those of the T2-AGN sub-sample over the 1σ error bar at the scale around 25 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This result may hint that the type-2 AGN have a stronger power of ionization at 25 h−1cMpc than the type-1 AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The interpretation of ionization at large-scales is in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 12, but for the T1-AGN (blue) and T2-AGN (red) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 9, but for the T1-AGN (top figure) and T2-AGN (bottom figure) sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 13, but for the T1-AGN (blue) and T2-AGN (red) sub-samples and the Galaxy (gray) sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Average HI Profiles around Galaxy We derive the 2D Hi profile at the positions of the Galaxy sample sources in the same manner as the one of the All-AGN sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 24 presents the 2D Hi profile of the Galaxy sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' There is a clear 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5σ detection at the source position of (0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Similarly, we calculate the Hi radial profile from the 2D Hi profile of the Galaxy sample (Figure 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Hi radial profile of the Galaxy sample shows a trend similar to those of the All-AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Both for the Galaxy raction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='15 T1-AGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='10 T2-AGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 Fraction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 424344454647LoS Velocity [km s-1] LoS Hubble distance 7500 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 [h-1cMpc] 5000 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 AF 2500 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0 0 204060 DTrans [h-1cMpc]Los Velocity [km s-1] LoS Hubble distance 7500 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 [h-1cMpc] 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 AF 2500 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0 0 204060 DTrans [h-1cMpc]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 T1-AGN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 T2-AGN AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 Galaxy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0 10 203040506070 D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN 17 Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 19, but for the T1-AGN and T2-AGN sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' and All-AGN samples, the Hi radial profile decreases towards the large scales, reaching AF ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Figure 24, we find that the Hi distributions in the LoS and transverse directions are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' A similar difference between the values of AF in LoS and trans- verse directions of 2D Hi profiles is claimed by Mukae et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To investigate the difference between the Hi distributions in LoS and transverse directions for the Galaxy sample, we present the LoS and Transverse Hi radial profiles of the Galaxy sample in Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find that the LoS and Transverse Hi radial profiles of the Galaxy sample show different gradient of the decreasing AF at the scale D ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='75−50 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This difference can be explained by the gas version of the Kaiser effect that we discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the LoS Hi radial profile of the Galaxy sample, we find that the AF val- ues are negative on the scale of D = 25 − 70 h−1cMpc, which is similar to the negative AF values we found on the large scale of the LoS Hi radial profile for the All- AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We discuss these negative AF values on the LoS Hi radial profile of the Galaxy sample in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2D Hi profile of the Galaxy sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The color map indicates the AF (δF) values of each cell of the 2D Hi profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The dotted lines show confidence level contours of 3σ and 6σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The solid line presents the contour where AF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 (δF = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Galaxy-AGN Dependence We derive 2D Hi profiles for the T1-AGN(H) sample constructed from the HETDEX data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 24 and 25 show the 2D Hi profiles of the Galaxy and T1-AGN(H) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6σ detection around the source posi- tion for the T1-AGN(H) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 26 presents the Hi radial profiles of the Galaxy and T1-AGN(H) samples derived from the 2D Hi profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We also compare the Hi radial profiles of the Galaxy sample with those of T1- AGN and T2-AGN in Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the Hi radial profiles of the Galaxy and T1-AGN(H) samples, the AF values increase toward the source position D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Figure 26 (22), we find that the AF values of T1-AGN(H) (T1- AGN and T2-AGN) are larger than those of the galaxies at ≲ 20 h−1 cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' These AF excesses of the AGN may be explained by the hosting dark matter halos of the AGN being more massive than those of the galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) also investigate the Hi radial pro- file around AGN, and find Hi absorption decrement at the source center (≲ 5 h−1Mpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' They argure that this trend can be explained by the proximity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' On the other hand, their result is different from ours that the AF values monotonically increase with decreasing dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This difference between our and Momose et al.’s results is produced by the fact that our results for ≲ 10 h−1 cMpc are largely affected by the Hi absorption at ∼ 10 h−1 cMpc due to the coarse resolution of our Hi tomography map, 15 h−1 cMpc, in contrast with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 h−1 cMpc for the resolution of Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We then derive the LoS and Transverse radial Hi pro- file of the T1-AGN(H) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The results of the profiles 0 2500 5000 7500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 T1-AGN LoS T2-AGN LoS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0 25 50 75 D [h-1cMpc]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 T1-AGN Trans T2-AGN Trans 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 25 50 0 75 D [h-1cMpc]Los Velocity [km s-1] LoS Hubble distance 7500 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 [h-1cMpc] 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 AF 2500 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0 0 204060 DTrans [h-1cMpc]18 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' are shown in Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Similar to the LoS and Trans- verse Hi radial profiles of the All-AGN and Galaxy sam- ples, the gas version of the Kaiser effect and the nega- tive AF in the LoS direction on the scale beyond D = 25 h−1cMpc are also found in those of the T1-AGN(H) sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 17, but for the T1-AGN(H) sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 18, but for Galaxy (gray) and T1-AGN(H) (black) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Comparison with Theoretical Models There are theoretical models of Hi radial profiles around AGN that are made by Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) present their Hi radial profiles with the LoS distance in the form of cross-correlation function (CCF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We first calculate theoretical CCFs of All-AGN, fol- lowing the definition of the CCF presented in Font- Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) assume the linear cross-power spectrum of the QSOs and Lyα forest, PqF(k, z) = bq(z)[1+βq(z)µ2 k]bF(z)[1+βF(z)µ2 k]PL(k, z), (11) Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 15, but for the Galaxy and T1-AGN(H) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' where PL(k, z) is the linear matter power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Here µk is the cosine of the angle between the Fourier mode and the LoS (Kaiser 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The values of bq and bF (βq and βF) are the bias factors (redshift space distortion parameters) of the QSO and Lyα density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The redshift distortion parameter of QSO obeys the relation βq = f(Ω)/bq, where f(Ω) is the logarithmic derivative of the linear growth factor (Kaiser 1987), bq = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3 (White et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We use the condition of Lyα forest, bF(1 + βF) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='336 for bF ∝ (1 + z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='9, that is determined by observations of Lyα forest at z ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='25 (Slosar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) estimate the CCF of QSOs by the Fourier transform of PqF (Hamilton 1992): ξ(r) = ξ0(r)P0(µ) + ξ2(r)P2(µ) + ξ4(r)P4(µ), (12) where µ is the cosine of angle between the position r and the LoS in the redshift space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The values of P0, P2, and P4 are the Legendre polynomials, P0 = 1, P2 = (3µ2 − 1), and P4 = (35µ4 − 30µ2 + 3)/8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The functions of ξ0, ξ2, and ξ4 are: ξ0(r) = bqbF[1 + (βq + βF)/3 + βqβF/5]ζ(r), (13) ξ2(r) = bqbF[2/3(βq+βF)+4/7βqβF][ζ(r)− ¯ζ(r)], (14) ξ4(r) = 8/35bqbFβqβF[ζ(r) − 5/2¯ζ(r) − 7/2¯¯ζ(r)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (15) The function ζ(r) is the standard CDM linear correla- tion function in real space (Bardeen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Hamil- ton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The functions ¯ζ(r) and ¯¯ζ(r) are given by: ¯ζ(r) ≡ 3r−3 � r 0 ζ(s)s2ds, (16) LoS Velocity [km s-1] LoS Hubble distance 7500 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 [h-1cMpc] 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 AF 2500 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0 204060 DTrans [h-1cMpc]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='04 Galaxy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 T1-AGN(H) AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='018F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0 10 203040506070 D [h-1cMpc]0 2500 5000 7500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='03 Galaxy LoS Galaxy Trans 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 T1-AGN(H) LoS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='02 T1-AGN(H) Trans AF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 OF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='01 25 50 75 D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN 19 ¯¯ζ(r) ≡ 5r−5 � r 0 ζ(s)s4ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (17) Here we define ξ′(r) ≡ −ξ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (18) In Figure 28, we present Dξ′ as a function of the LoS distance for the model of Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) that is calculated under the assumption of the mean over- density of the 15 h−1cMpc corresponding to the spatial resolution of our observational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To compare our observational measurements with the model CCF of Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013), we calculate the value of ξ′ for our All-AGN sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The value of ξ′ in each cell ξ′cell is calculated by ξ′ cell = � i∈cell ωiAFi � i∈cell ωi , (19) where ωi is the weight determined by the observational errors and the intrinsic variance of the Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The value of ωi is obtained by ωi = � σ2 F(zi) + 1 ⟨S/N⟩2 × ⟨F(zi)⟩2 �−1 , (20) where σF(zi) is the intrinsic variance of the Lyα forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The value of ⟨F(zi)⟩ is the cosmic average Lyα transmis- sion (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We adopt ⟨S/N⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4 that is the criterion of the background source selection (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The intrinsic variance, σF(zi), of the Lyα forest taken from Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) is: σ2 F(zi) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='065[(1 + zi)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='25]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (21) We calculate ξ′ with our All-AGN sample via the Equations 19, 20, and 21, using the binning sizes same as those in Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We present ξ′ multiplied by D with the black squares in Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (explanation of Momose+21) For reference, we also de- rive the ξ′ for our Galaxy sample shown by the blue triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In Figure 28, we find that the Dξ′ profile of our All- AGN sample show a trend similar to the one of the model predicted by Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The ob- servational Dξ′ profile of our All-AGN sample shows a good agreement with the model Dξ′ profile of Font- Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) at the scale of D > 30 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Al- though the model Dξ′ profile of Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) is slightly higher than the Dξ′ profiles of the observa- tions at ≳ 60 h−1cMpc, the general trend of the negative Dξ′ profiles at ≳ 30 h−1cMpc are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) suggests that the negative Dξ ′ values at the large scale of ≳ 30 h−1cMpc are explained by the ion- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the model of ionization, Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) assume the spectrum of the AGN at D = 0 with Lν ∝ ν−α, where α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='5 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0) for the frequency ν over (below) the Lyman limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The luminosity of λ = 1420 ˚A is normalized as Lν = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='1 × 1030 erg/s/Hz, which is taken from the mean luminosity of the SDSS data re- lease 9 quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' No assumptions of AGN type have been made in the models of Font-Ribera+13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Based on the model of ionization, Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) calculate ξ for the homogeneous gas radiated by AGN, and obtain the function ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0065(20 h−1cMpc/D)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (22) With the ξ function, we calculate Dξ ′ that is presented with the cyan dashed curve in Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The cyan dashed curve shows the plateau at D ≥ 40 h−1cMpc with negative Dξ ′ values that is comparable with the model Dξ′ profile of Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' It indi- cates that the negative Dξ ′ values are originated from the ionization of radiation including the hard radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Similarly, the negative Dξ ′ values of our All-AGN at the large scale towards ≳ 40 h−1cMpc may be explained by the ionization of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' To distinguish the large- scale negative Dξ ′ values, which are referred to as the ‘ionized outskirts’, from the proximity zone created by the proximity effect, we plot the observational CCF of AGN obtained by Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021) in Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The AGN CCF obtained by Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' shows a de- creasing Hi absorption toward source position (D = 0 h−1cMpc) caused by the proximity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Our findings indicate that the Hi radial profile of AGN has transi- tions from proximity zones (≲ a few h−1cMpc) to the Hi structures (∼ 1 − 30 h−1cMpc) and the ionized out- skirts (≳ 30 h−1cMpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The hard radiation may pass through the Hi structure due to the small cross-section and ionizes the Hi gas in the regions of ionized out- skirts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Because of the low recombination rate, the Hi gas remains ionized in the ionized outskirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Interestingly, the Dξ′ profile of our Galaxy sample also shows negative Dξ ′ values towards ≳ 30 h−1cMpc which is similar to those of the model and our All-AGN sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This result may suggest that the Hi gas at large scale (≳ 20 h−1cMpc) around galaxies has been ion- ized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The ionizing source causing the structure of neg- ative Dξ ′ values at the large scale may not be a single galaxy, but a group of galaxies within a radius of a few cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Regions around galaxies are special as galaxies are clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Galaxies in this work are bright with MUV < −22 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The galaxies can be hosted by massive haloes, and are likely to distribute at overden- sity regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The overdensity region suggests that each galaxy can be surrounded by several satellite galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Although it is difficult for a galaxy to ionize the Hi gas 20 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' on a scale of ≳ 20 h−1cMpc, a group galaxies may have enough ionizing photons to ionize the Hi on this scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Comparison between our All-AGN and Galaxy results and the models of Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013) in the LoS CCF (ξ ′) multiplied by distance (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The black and blue points are the results derived from the All-AGN and Galaxy samples sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The orange curve is the LoS CCF of QSOs with the Lyα forest derived by Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The cyan dashed curve shows the ionization of radiation effect taken from Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The pink line presents the CCF of AGN obtained by Momose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The gray shade presents the range of the Hi structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Two white areas show the regions of proximity zone and ionized outskirt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The horizontal gray line indicates the cosmic average where Dξ ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' SUMMARY We reconstruct two 3D Hi tomography maps based on the Lyα forests in the spectra of 14763 background QSOs from the SDSS survey with no signatures of damped Lyα system or broad absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The maps cover the extended Fall and Spring fields defined by the HETDEX survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The spatial volume of the re- constructed 3D Hi tomography maps are 2257×233×811 h−3cMpc3 and 3475 × 1058 × 811 h−3cMpc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We inves- tigate Hi distribution around galaxies and AGN with samples made from HETDEX and SDSS survey results in our study field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Our results are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We derive the 2D Hi and Hi radial profiles of the All-AGN sample consisted of SDSS AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find that the 2D Hi profile is more extended in the transverse direction than along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In the Hi radial profile All-AGN sample, the values of Hi absorption, AF, decrease toward the large scale, touching to AF ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We compare the Hi radial profiles derived from the T1-AGN and T1-AGN(H) sub-samples, whose Lspec 1350 distributions are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find that the Hi radial profile of the T1-AGN sub-sample agrees with that of the T1-AGN(H) sub-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This agreement suggests that the systematic un- certainty between the SDSS and the HETDEX survey results is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We examine the dependence of the Hi profile on AGN luminosity by deriving the 2D Hi, Hi ra- dial, LoS Hi radial, and Transverse Hi radial pro- files of the All-AGN-L3 (the faintest), All-AGN- L2, and All-AGN-L1 (the brightest) sub-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find that the Hi absorption is the greatest in the lowest-luminosity AGN sub-sample, and that the Hi absorption becomes weaker with increasing AGN luminosity This result suggests that, on av- erage, if the density of Hi gas around the bright AGN is greater than (or comparable to) those of the faint AGN, the ionization fraction of Hi gas around bright AGN is higher than that around faint AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We investigate the AGN type dependence of Hi distribution around type-1 and type-2 AGN by the 2D Hi, Hi radial, LoS Hi radial, and Transverse Hi radial profiles extracted from the T1-AGN and T2-AGN sub-samples with the same Lspec 1350 distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The comparison between the Hi radial profiles of T1-AGN and T2-AGN sub-samples in- dicates that the Hi absorption around the T2- AGN sub-sample is comparable to the one of the T1-AGN sub-sample on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This trend sug- gests that, the selectively different opening angle and orientation of the dusty torus for type-1 and type-2 AGN do not have a significant impact on the Mpc-scale Hi distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We compare the Hi distributions around galax- ies and type-1 AGN with the 2D Hi, Hi radial, LoS Hi radial, and Transverse Hi radial profiles derived from the Galaxy and T1-AGN(H) sample sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The Hi absorption values, AF, around the T1-AGN(H) sample are larger than those of the Galaxy sample on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This result may be caused by the dark matter halos of type-1 AGN having a larger mass than the one of galaxies on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We find that the Hi radial profiles of the LoS dis- tance for the Galaxy and All-AGN samples show negative AF values, which means weak Hi absorp- tion, at the scale over ∼ 30 h−1cMpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' We extract LoS velocity [km s-1] 0 2000 4000 6000 8000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='50 HI structure Ionized Outskirt 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='25 DS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='00 Zone 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='25 Momose+21 CCF LoS model Ionization model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='50 Galaxy Los All-AGN LoS 0 102030 4050 60 70 80 D [h-1cMpc]Cosmological-Scale Hi Distribution Around Galaxies and AGN 21 the Dξ′ profile of our Galaxy and All-AGN sam- ples to compare with the model CCF of AGN from Font-Ribera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The general trend of the negative Dξ′ at ≳ 30 h−1cMpc is the same as the model CCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This results suggest that the Hi ra- dial profile of AGN has transitions from proximity zones (≲ a few h−1cMpc) to the Hi rich struc- tures (∼ 1−30 h−1cMpc) and the ionized outskirts (≳ 30 h−1cMpc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank Nobunari Kashikawa, Khee-Gan Lee, Akio Inoue, Rikako Ishimoto, Shengli Tang, Yongming Liang, Rieko Momose, and Koki Kakiichi for giving us helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' HETDEX is led by the University of Texas at Austin McDonald Observatory and Department of Astron- omy with participation from the Ludwig-Maximilians- Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Max-Planck-Institut f¨ur Ex- traterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Leibniz-Institut f¨ur As- trophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Institut f¨ur Astrophysik G¨ottingen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Max-Planck- Institut f¨ur Astrophysik (MPA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The University of Tokyo and Missouri University of Science and Tech- nology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In addition to Institutional support, HET- DEX is funded by the National Science Foundation (grant AST-0926815), the State of Texas, the US Air Force (AFRL FA9451-04-2- 0355), and generous sup- port from private individuals and foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The ob- servations were obtained with the Hobby-Eberly Tele- scope (HET), which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Ludwig-Maximilians-Universit¨at M¨unchen, and Georg- August-Universit¨at G¨ottingen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The HET is named in honor of its principal benefactors, William P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Hobby and Robert E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Eberly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing high performance com- puting, visualization, and storage resources that have contributed to the research results reported within this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='tacc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='edu VIRUS is a joint project of the University of Texas at Austin, Leibniz-Institut f¨ur Astrophysik Potsdam (AIP), Texas A&M University (TAMU), Max-Planck- Institut f¨ur Extraterrestrische Physik (MPE), Ludwig- Maximilians-Universit¨at Muenchen, Pennsylvania State University, Institut fur Astrophysik G¨ottingen, Univer- sity of Oxford, and the Max-Planck-Institut f¨ur As- trophysik (MPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' In addition to Institutional support, VIRUS was partially funded by the National Science Foundation, the State of Texas, and generous support from private individuals and foundations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This work is supported in part by MEXT/JSPS KAK- ENHI Grant Number 21H04489 (HY), JST FOREST Program, Grant Number JP-MJFR202Z (HY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' acknowledges financial support from the Japan Society for the Promotion of Science (JSPS) through KAKENHI grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 20K14516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' This paper is supported by World Premier Inter- national Research Center Initiative (WPI Initiative), MEXT, 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+page_content=' The different panels denote the coverages over different redshift ranges shown at the top left of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4 2 2 35 30 25 20 15 10 nz = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg] 2 2 35 30 25 20 15 10 5z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='8 2 2 35 30 25 20 15 10 5z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='8 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg] 2 品 0 品 + +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='品 中 2 35 30 25 20 15 10 5 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg]Cosmological-Scale Hi Distribution Around Galaxies and AGN 25 Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 1, but for the foreground sources in the ExSpring field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Z=2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 60 口 55 电 50 45 160 170 180 190 200 210 220 230 240Z=2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='2 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4 60 55 50 45 160 170 180 190 200 210 220 230 240Z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='4 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6 60 50 45 160 170 180 190 200 210 220 230 240 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg]26 Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Continued from Figure 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Figure 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Same as Figure 2, but for the background sources in the ExSpring field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' Z=2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='6 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='8 60 品 日 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content=' 55 50 45 160 170 180 190 200 210 220 230 240Z= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='8 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='0 60 品 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='. 55 + T 50 45 160 170 180 190 200 210 220 230 240 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg]60 Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg] 55 50 45 160 170 180 190 200 210 220 230 240 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} +page_content='[deg]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfeg2i/content/2301.05100v1.pdf'} diff --git a/DNAzT4oBgHgl3EQfiP3j/content/tmp_files/2301.01498v1.pdf.txt b/DNAzT4oBgHgl3EQfiP3j/content/tmp_files/2301.01498v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c056a5c13ab12e61cc41f78f0304c3cdee8f793b --- /dev/null +++ b/DNAzT4oBgHgl3EQfiP3j/content/tmp_files/2301.01498v1.pdf.txt @@ -0,0 +1,3983 @@ +arXiv:2301.01498v1 [math.RT] 4 Jan 2023 +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +Abstract. g-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined +by tilting theory. +We give a classification of complete g-fans of rank 2. +More explicitly, our +first main result asserts that every complete sign-coherent fan of rank 2 is a g-fan of some +finite dimensional algebra. Our proof is based on three fundamental results, Gluing Theorem, +Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level +of finite dimensional algebras. Our second main result gives a necessary and sufficient condition +for algebras of rank 2 to be g-convex. +Contents +1. +Introduction +1 +2. +Preliminaries +5 +2.1. +Preliminaries on fans +5 +2.2. +Sign-coherent fans of rank 2 +6 +3. +Basic results in silting theory +10 +3.1. +Preliminaries +10 +3.2. +Silting complexes in terms of matrices +12 +3.3. +Uniserial property of g-finite algebras +14 +4. +Gluing, Rotation and Subdivision of g-fans +15 +4.1. +Gluing fans +15 +4.2. +Rotation and Mutation +17 +4.3. +Subdivision and Extension +19 +4.4. +Proof of Theorem 1.3 +22 +4.5. +Gluing fans II +23 +5. +g-Convex algebras of rank 2 +26 +5.1. +Characterizations of g-convex algebras of rank 2 +26 +5.2. +Proof of Theorem 5.1 +27 +Acknowledgments +29 +References +29 +1. Introduction +The notion of tilting complexes is central to control equivalences of derived categories. The +class of silting complexes [KV] gives a completion of the class of tilting complexes with respect to +mutation, which is an operation to replace a direct summand of a given silting complex to construct +a new silting complex [AI]. The subclass of 2-term silting complexes enjoys remarkable properties +[AIR, DF]. They give rise to a fan in the real Grothendieck group of a finite dimensional algebra +A, see e.g. [H1, H2, Pl, B, DIJ, BST, As]. +In our previous article [AHIKM1], we introduced a g-fan Σ(A) of A and established a basic +theory of g-fans and the associated g-polytopes. A g-fan of each finite dimensional algebra A +belongs to the following special class of nonsingular fans [AHIKM1, Proposition 4.12]. +Definition 1.1. A sign-coherent fan is a pair (Σ, σ+) satisfying the following conditions. +1 + +2 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +(a) Σ is a nonsingular fan in Rd. +(b) σ+, −σ+ ∈ Σd. +(c) Take a Z-basis e1, . . . , ed of Zd such that σ+ = cone{ei | 1 ≤ i ≤ d}, and denote the orthant +corresponding to ǫ ∈ {±1}d by +Rd +ǫ := cone{ǫ(1)e1, . . . , ǫ(d)ed} = {x1e1 + · · · + xded | ǫ(i)xi ≥ 0 for each 1 ≤ i ≤ d}. +Then for each σ ∈ Σ, there exists ǫ ∈ {±1}d such that σ ⊆ Rd +ǫ. +We denote by Fansc(d) the set of complete sign-coherent fans in Rd, and by k-Fan(d) the set of +complete g-fans of finite dimensional k-algebras of rank d. Note that a g-fan Σ(A) is complete if +and only if A is g-finite (Proposition 3.9). Then we have +Fansc(d) ⊃ k-Fan(d). +It is very natural to study the following problem. +Problem 1.2. Characterize complete sign-coherent fans in Rd which can be realized as a g-fan of +some finite dimensional algebra. +This paper is devoted to give a complete answer to this problem for the case d = 2. The result +was very simple and came as a surprise to us. +Theorem 1.3 (Theorem 4.13). For each field k, we have +Fansc(2) = k-Fan(2). +Thus any complete sign-coherent fan in R2 can be realized as a g-fan of some finite dimensional +k-algebra. +We explain our method to prove Theorem 1.3. Each sign-coherent fan of rank 2 is obtained by +gluing two fans of the following form. +Σ = +•+ +− +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +Σ′ = +•+ +− +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +Recall that a finite dimensional k-algbera Λ is elementary if the k-algebra Λ/JΛ is isomorphic to +a product of k. This is automatic if Λ is basic and k is algebraically closed. We prove Gluing +Theorem 4.1, which asserts that if both Σ and Σ′ are g-fans of finite dimensional elementary k- +algebras, then so is their gluing. Therefore by symmetry, it suffices to consider sign-coherent fans +Σ of the form above. Now such Σ can be obtained from the fan +•+ +− +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +by applying subdivision in the fourth quadrant repeatedly. We prove Rotation Theorem 4.3 and +Subdivision Theorem 4.7, which imply that if Σ is a g-fan of a finite dimensional k-algebra, then +so are the subdivisions of Σ in the fourth quadrant. +Figure 1 gives fans in Fan+− +sc (2) with at most 8 facet, where each edge shows a subdivision. +Figure 2 gives examples of algebras whose g-fans are given in Figure 1. +For each finite dimensional algebra A, we define a g-polytope P(A) by gluing each simplex +associated with the cones in Σ(A). +If P(A) is convex, we call Σ(A) convex and A g-convex. +For example, Brauer tree algebras A are g-convex, and this fact plays an important role in the +classification of 2-term tilting complexes of A [AMN]. From tilting theoretic point of view, g-convex +algebras are the most fundamental. Therefore it is important to study the following problem. +Problem 1.4. Classify convex g-fans in Rd. + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +3 +Σ00 +Σ111 +Σ2121 +Σ1212 +Σ31221 +Σ22131 +Σ12213 +Σ21312 +Σ13122 +Σ412221 +Σ321321 +Σ313131 +Σ312312 +Σ231231 +Σ222141 +Σ221412 +Σ122214 +Σ123123 +Σ214122 +Σ131313 +Σ213213 +Σ132132 +Σ141222 +Figure 1. Fans in Fan+− +sc (2) with at most 8 facets +An answer to the case d = 2 was given in [AHIKM1, Theorem 6.3]. There are precisely 7 convex +g-fans up to isomorphism of g-fans. +•+ +− +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +•+ +− +❄❄❄❄❄ +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +•+ +− +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +•+ +− +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❖❖❖❖❖❖❖ +❄❄❄❄❄ +•+ +− +❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄ +❖❖❖❖❖❖❖ +❄❄❄❄❄ +•+ +− +❄❄❄❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖❖❖❖❖❖❖ +❄❄❄❄❄ +•+ +− +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄ +✴✴✴✴✴✴✴ +❄❄❄❄❄ +❖❖❖❖❖❖❖ +❄❄❄❄❄ +More precisely, in the last Section 5, we show that there are 16 convex g-fans in Fansc(2) +Σa;b with a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}. +We also give a characterization of algebras whose g-fans are one of them. Let t(ΛM) (respectively, +t(MΛ)) be the minimal number of generators of a left (respectively, right) Λ-module M. + +4 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +Σ00 +k [ • +• ] +Σ111 +k +� +• +• +a � +� +Σ2121 +k +� +• +• +a � +c +� +� +⟨c2⟩ +Σ1212 +k +� +• +• +a � +c +� +� +⟨c2⟩ +Σ31221 +k + + +• +• +a � +c1 +� +c2 +� + + +⟨c2 +1, c2 +2, c2c1, ac2⟩ +Σ22131 +k + + +• +• +a � +b +� +c0 +� +c1 +� + + +�b2, c2 +0, c2 +1, c1c0, +ba − ac0 +� +Σ12213 +k + + +• +• +a � +c1 � +c2 +� + + +� +c2 +1, c2 +2, c1c2, c2a +� +Σ21312 +k +� +• +• +a � +b +� +c +� +� +⟨b2, c2, bac⟩ +Σ13122 +k + + +• +• +a � +c0 � +c1 +� +b +� + + +�b2, c2 +0, c2 +1, c0c1, +ab − c0a +� +Σ412221 +k + + +• +• +a � +c1 +� +c2 +� +c3 +� + + +�c2 +1, c2 +2, c2 +3, c3c2, c3c1, +c2c1, c1c3, ac2, ac3 +� +Σ321321 +k + + +• +• +a � +b +� +c0 +� +c1 +� +c2 +� + + +� b2, c2 +0, c2 +1, c2 +2, ac0, +ac2, c2c1, c2c0, c0c2, +c0c1c0, ba − ac1c0 +� +Σ313131 +k + + +• +• +a � +b +� +c0 +� +c1 +� +c2 +� + + +� +b2, c2 +0, c2 +1, c2 +2, ac2, +c2c1, c2c0, c1c0, c0c2, +c0c1c2, ba − ac0 +� +Σ312312 +k + + +• +• +a � +b +� +c1 +� +c2 +� + + +�b2, c2 +1, c2 +2, ac2, +bac1, c2c1 +� +Σ231231 +k + + +• +• +a � +b0 � +b1 +� +c0 +� +c1 +� +c2 +� + + +�b2 +0, b2 +1, c2 +0, c2 +1, c2 +2, ac2, b1b0, +c2c1c2, c2c0, c1c0, c0c2, +b0a − ac0, b1a − ac1c2 +� +Σ222141 +k + + +• +• +a � +b0 � +b1 +� +c0 +� +c1 +� +c2 +� + + +�b2 +0, b2 +1, c2 +0, c2 +1, c2 +2, b1b0, +c2c1, c2c0, c1c0, c0c2 +b0a − ac0, b1a − ac1 +� +Σ221412 +k + + +• +• +a � +b0 � +b1 +� +c0 +� +c1 +� + + +�b2 +0, b2 +1, c2 +0, c2 +1, b1b0, +b0b1, c1c0, b1ac0, +b1ac1, b0a − ac0 +� +Σ122214 +k + + +• +• +a � +c1 � +c2 +� +c3 +� + + +�c2 +1, c2 +2, c2 +3, c2c3, c1c3 +c1c2, c3c1, c2a, c3a +� +Σ123123 +k + + +• +• +a � +c0 � +c1 +� +c2 +� +b +� + + +� b2, c2 +0, c2 +1, c2 +2, c0a, +c2a, c1c2, c0c2, c2c0, +c0c1c0, ab − c0c1a +� +Σ214122 +k + + +• +• +a � +c0 � +c1 +� +b0 +� +b1 +� + + +�b2 +0, b2 +1, c2 +0, c2 +1, b0b1, +b1b0, c0c1, c0ab1, +c1ab1, ab0 − c0a +� +Σ131313 +k + + +• +• +a � +c0 � +c1 +� +c2 +� +b +� + + +� +b2, c2 +0, c2 +1, c2 +2, c2a, +c1c2, c0c2, c0c1, c2c0, +c2c1c0, ab − c0a +� +Σ213213 +k + + +• +• +a � +c1 � +c2 +� +b +� + + +�b2, c2 +1, c2 +2, c2a, +c1ab, c1c2 +� +Σ132132 +k + + +• +• +a � +c0 � +c1 +� +c2 +� +b0 +� +b1 +� + + +�b2 +0, b2 +1, c2 +0, c2 +1, c2 +2, c2a, b0b1, +c2c1c2, c0c2, c0c1, c2c0, +ab0 − c0a, ab1 − c2c1a +� +Σ141222 +k + + +• +• +a � +c0 � +c1 +� +c2 +� +b0 +� +b1 +� + + +�b2 +0, b2 +1, c2 +0, c2 +1, c2 +2, b0b1, +c2c0, c1c2, c0c2, c0c1, +ab0 − c0a, ab1 − c1a +� +Figure 2. Algebras whose g-fans are given in Figure 1 + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +5 +Theorem 1.5 (Theorem 5.1). Let A be a basic finite dimensional algebra, {e1, e2} a complete set +of primitive orthogonal idempotents in A, and Pi = eiA (i = 1, 2). +(a) A is g-convex if and only if Σ(A) = Σa;b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}. +(b) Let (l, r) := (t(e1Ae1e1Ae2), t(e1Ae2e2Ae2)). Then we have the following statements. +• Σ(A) = Σ00;b for some b if and only if (l, r) = (0, 0). +• Σ(A) = Σ111;b for some b if and only if (l, r) = (1, 1). +• Σ(A) = Σ1212;b for some b if and only if (l, r) = (1, 2) and t(Rxe1Ae1) = 2 hold for some +left generator x of e1Ae2 and Rx := {a ∈ e1Ae1 | ax ∈ xAe2}. +• Σ(A) = Σ2121;b for some b if and only if (l, r) = (2, 1) and t(e2Ae2Lx) = 2 hold for some +right generator x of e1Ae2 and Lx := {b ∈ e2Ae2 | xb ∈ e1Ax}. +Σ00;b +• +P2 +P1 +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +⑧⑧⑧⑧⑧⑧ +Σ111;b +• +P2 +P1 +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +Σ1212;b +• +P2 +P1 +❄ +❄ +❄ +❄ +❄ +❄ +✴✴✴✴✴✴✴✴✴ +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +Σ2121;b +• +P2 +P1 +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +Further, in a forthcoming paper [AHIKM2], we will give a complete answer to Problem 1.4 for +d = 3. +2. Preliminaries +2.1. Preliminaries on fans. We recall some fundamental materials on fans. We refer the reader +to e.g. [F, BR, BP] for these materials. +A convex polyhedral cone σ is a set of the form σ = {�s +i=1 rivi | ri ≥ 0}, where v1, . . . , vs ∈ Rd. +We denote it by σ = cone{v1, . . . , vs}. Note that {0} is regarded as a convex polyhedral cone. We +collect some notions concerning convex polyhedral cones. Let σ be a convex polyhedral cone. +• The dimension of σ is the dimension of the linear space generated by σ. +• We say that σ is strongly convex if σ ∩ (−σ) = {0} holds, i.e., σ does not contain a linear +subspace of positive dimension. +• We call σ rational if each vi can be taken from Qd. +• We denote by ⟨·, ·⟩ the usual inner product. +A supporting hyperplane of σ is a hyperplane +{v ∈ σ | ⟨u, v⟩ = 0} in Rd given by some u ∈ Rd satisfying σ ⊂ {v ∈ Rd | ⟨u, v⟩ ≥ 0}. +• A face τ of σ is the intersection of σ with a supporting hyperplane of σ. +In what follows, a cone means a strongly convex rational polyhedral cone for short. +Definition 2.1. A fan Σ in Rd is a collection of cones in Rd such that +(a) each face of a cone in Σ is also contained in Σ, and +(b) the intersection of two cones in Σ is a face of each of those two cones. +For each i ≥ 0, we denote by Σi the subset of cones of dimension i. For example, Σ0 consists of +the trivial cone {0}. We call each element in Σ1 a ray of Σ. +We collect some notions concerning fans used in this paper. Let Σ be a fan in Rd. +• We call Σ finite if it consists of a finite number of cones. +• We call Σ complete if � +σ∈Σ σ = Rd. +• We call Σ nonsingular (or smooth) if each maximal cone in Σ is generated by a Z-basis for Zd. +We prepare some notions which will be used in this paper. +Definition 2.2. Let Σ be a nonsingular fan in Rd. We call Σ pairwise positive if the following +condition is satisfied. + +6 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +• For each two adjacent maximal cones σ, τ ∈ Σd, take Z-basis {v1, . . . , vd−1, vd} and {v1, . . . , vd−1, v′ +d} +of Zd such that σ = cone{v1, . . . , vd−1, vd} and τ = cone{v1, . . . , vd−1, v′ +d}. Then vd + v′ +d belongs +to cone{v1, . . . , vd−1}. +Definition 2.3. Let Σ and Σ′ be fans in Rd and Rd′ respectively. +(1) An isomorphism Σ ≃ Σ′ of fans is an isomorphism Zd ≃ Zd′ of abelian groups such that +the induced linear isomorphism Rd → Rd′ gives a bijection Σ ≃ Σ′ between cones. +(2) Let (Σ, σ+) and (Σ′, σ′ ++) be sign-coherent fans in Rd and Rd′ respectively. An isomorphism +of sign-coherent fans is an isomorphism f : Σ ≃ Σ′ of fans such that {f(σ+), f(−σ+)} = +{σ′ ++, −σ′ ++}. +2.2. Sign-coherent fans of rank 2. In this subsection, we introduce some terminologies of sign- +coherent fans of rank 2, and discuss some fundamental properties. +Let Σ be a complete nonsingular fan of rank 2. We denote the rays of Σ by +v1, v2, . . . , vn−1, vn = v0 +(2.1) +which are indexed in a clockwise orientation. For each 1 ≤ i ≤ n, since Σ is nonsingular, there +exists an integer ai satisfying +aivi = vi−1 + vi+1 for each 1 ≤ i ≤ n. +We call the sequence of integers +s(Σ) = (a1, . . . , an) +(2.2) +the defining sequence of Σ. In fact, Σ is uniquely determined by its defining sequence. A fan with +defining sequence (a1, . . . , an) is denoted by +Σ(a1, . . . , an). +Remark 2.4. [F, Section 2.5] An integer sequence (a1, . . . , an) is a defining sequence of nonsingular +complete fan of rank 2 if and only if it satisfies +n +� +i=1 +ai = 3n − 12 and +�0 +−1 +1 +a1 +� �0 +−1 +1 +a2 +� +· · · +�0 +−1 +1 +an +� += +�1 +0 +0 +1 +� +. +Definition 2.5. We denote by Fansc(2) the set of all (possibly infinite) fans Σ satisfying that +• Σ is a sign-coherent fans (Definition 1.1) of rank 2 with positive and negative cones +σ+ := cone{(1, 0), (0, 1)} and σ− := cone{(−1, 0), (0, −1)} respectively, +• each ray is a face of precisely two facets. +We denote the subset of complete fans by +Fansc(2) ⊂ Fansc(2). +For Σ ∈ Fansc(2), we denote the rays of Σ in a clockwise orientation by +Σ1 = {v1 := (1, 0), v2, . . . , vn−1, vn = v0 := (0, 1)}. +Then there exists 2 ≤ i ≤ n − 2 such that vi = (0, −1) and vi+1 = (−1, 0). +• +vn=v0=(0,1) +v1=(1,0) +vi=(0,−1) +vi+1=(−1,0) ++ +− +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +In this case, it is more convenient to rewrite (2.2) as +s(Σ) = (a1, a2, . . . , ai; an, an−1, . . . , ai+1). +Thus we mainly use the notation +Σ(a1, . . . , ai; an, . . . , ai+1) = Σa1,...,ai;an,...,ai+1 + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +7 +instead of Σ(a1, . . . , an). +We consider subsets +Fan ++− +sc (2) +⊂ +Fansc(2) +∪ +∪ +Fan+− +sc (2) +⊂ +Fansc(2) +which consist of fans Σ containing σ−+ := cone{(−1, 0), (0, 1)}, i.e. Σ has the following form. +• +σ−+ ++ +− +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +Thus the rays and the facets of Σ ∈ Fan+− +sc (2) are written as +Σ1 += +{v1 = (1, 0), v2, . . . , vn−2 = (0, −1), vn−1 = (−1, 0), vn = v0 = (0, 1)}, +(2.3) +Σ2 += +{σ1, . . . , σn−3, σn−2 = σ−, σn−1 = σ−+, σn = σ+}. +(2.4) +Similarly, we define Fan +−+ +sc (2) and Fan−+ +sc (2) as the subsets of Fansc(2) and Fansc(2) respectively +which consist of fans containing σ+− := cone{(1, 0), (0, −1)}. +The following observations are clear. +Lemma 2.6. The following assertions hold. +(1) The correspondence Σ �→ {−σ | σ ∈ Σ} gives bijections Fan ++− +sc (2) → Fan +−+ +sc (2) and Fan+− +sc (2) → +Fan−+ +sc (2). +(2) Let Σ ∈ Fansc(2). Then Σ ∈ Fan+− +sc (2) (respectively, Σ ∈ Fan−+ +sc (2)) holds if and only if s(Σ) +has the form +(b1, . . . , bm; 0, 0) (respectively, (0, 0; b1, . . . , bm)). +In this case, bi ≥ 0 holds for any 1 ≤ i ≤ m. +Definition 2.7. For Σ ∈ Fan ++− +sc (2) and Σ′ ∈ Fan +−+ +sc (2), we define Σ ∗ Σ′ ∈ Fansc(2) by +(Σ ∪ Σ′)1 +:= +Σ1 ∪ Σ′ +1 +(Σ ∪ Σ′)2 +:= +(Σ2 \ {σ−+}) ∪ (Σ′ +2 \ {σ+−}) . +Σ = +• ++ +− +? +σ−+ +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +Σ′ = +• ++ +− +! +σ+− +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +Σ ∗ Σ′ = +• ++ +− +? +! +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +Then, we clearly have +Fansc(2) += +Fan ++− +sc (2) ∗ Fan +−+ +sc (2) := {Σ ∗ Σ′ | Σ ∈ Fan ++− +sc (2), Σ′ ∈ Fan +−+ +sc (2)}, +Fansc(2) += +Fan+− +sc (2) ∗ Fan−+ +sc (2) := {Σ ∗ Σ′ | Σ ∈ Fan+− +sc (2), Σ′ ∈ Fan−+ +sc (2)}. +(2.5) +Definition 2.8. Let Σ be a (possibly infinite) nonsingular fan of rank 2. For a cone σ := cone{u, v} +of Σ, we define a new nonsingular fan Dσ(Σ) by +Dσ(Σ)1 += +Σ1 ∪ {cone{u + v}}, +Dσ(Σ)2 += +(Σ2 \ {σ}) ⊔ {cone{u, u + v}, cone{v, u + v}}. +We call Dσ(Σ) the subdivision of Σ at σ. +Σ = +• +.......................................... +σ +❨ +❨ +❨ +❨ +❨ +❨ +❨ +❡ +❡ +❡ +❡ +❡ +❡ +❡ +Dσ(Σ) = +• +.......................................... ❨ +❨ +❨ +❨ +❨ +❨ +❨ +❡ +❡ +❡ +❡ +❡ +❡ +❡ +❡❡❡❡❡❡❡ +❨❨❨❨❨❨❨ + +8 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +For a sequence a = (a1 . . . , an) and 1 ≤ i ≤ n, we define a new sequence by +Di(a) = (a1, . . . , ai−1, ai + 1, 1, ai+1 + 1, ai+2, . . . , an). +(2.6) +For a complete nonsingular fan Σ with rays (2.1) and σi := cone{vi, vi+1} for 1 ≤ i ≤ n, we have +s ◦ Dσi(Σ) = Di ◦ s(Σ). +(2.7) +Example 2.9. Figure 1 gives fans in Fan+− +sc (2) with at most 8 facets, where +Σa1,...,an := Σ(a1, . . . , an; 0, 0) +and each edge shows a subdivision. Figure 2 gives examples of algebras whose g-fans are given +in Figure 1. For example, Σ111 is the g-vector fan of a cluster algebra of type A2 [FZ1, FZ2]. +Similarly, Σ1212 and Σ2121 are the g-vector fans of cluster algebras of type B2, and Σ131313 and +Σ313131 are the g-vector fans of cluster algebras of type G2. +Later we need the following observation (cf. [F, Section 4.3]). +Proposition 2.10. Each fan in Fan+− +sc (2) can be obtained from Σ(0, 0; 0, 0) by a sequence of +subdivisions. +Σ(0, 0; 0, 0) = +•+ +− +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +To prove this, we need the following preparation. +Lemma 2.11 (cf. [F, p.43]). Let Σ ∈ Fan+− +sc (2) and s(Σ) = (a1, . . . , an−2; 0, 0). If n ≥ 5, then +there exists 2 ≤ i ≤ n − 3 satisfying ai = 1. +Proof. Let vi = (xi, yi) ∈ Z2 for 1 ≤ i ≤ n. Assume that n ≥ 5 and ai ≥ 2 for any 2 ≤ i ≤ n − 3. +We claim that xi+1 ≥ xi holds for each 1 ≤ i ≤ n − 3. In fact, n ≥ 5 implies x2 ≥ 1 = x1. Then +we have +xi+1 = aixi − xi−1 ≥ 2xi − xi−1 ≥ xi +for each 2 ≤ i ≤ n − 3, and the claim follows inductively. Consequently 1 = x1 ≤ x2 ≤ · · · ≤ +xn−2 = 0 holds, a contradiction. +□ +We are ready to prove Proposition 2.10. +Proof of Proposition 2.10. Let F ⊂ Fan+− +sc (2) be the set of fans obtained from Σ(0, 0; 0, 0) by a +sequence of subdivisions. It suffices to show Fan+− +sc (2) = F. +We will show that each Σ ∈ Fan+− +sc (2) belongs to F by using induction on n = #Σ2. +Clearly n ≥ 4 holds. If n = 4, then Σ = Σ(0, 0; 0, 0) ∈ F. +Suppose that Σ with #Σ2 = n ≥ 5 belongs to Fan+− +sc (2). In terms of (2.3) and (2.4), there +exists 2 ≤ i ≤ n − 3 satisfying vi = vi−1 + vi+1 by Lemma 2.11. Since vi−1, vi+1 forms a Z-basis +of Z2, we obtain a new fan Σ′ ∈ Fan+− +sc (2) by +Σ′ +1 +:= +Σ1 \ {vi}, +Σ′ +2 +:= +(Σ2 \ {σi−1, σi}) ∪ {σ} for σ := cone{vi−1, vi+1}. +Since #Σ′ +2 = n − 1, the induction hypothesis implies Σ′ ∈ F. Thus Σ = Dσ(Σ′) ∈ F holds. +□ +Remark 2.12. For each n ≥ 1, we have a bijection +{Σ ∈ Fan+− +sc (2) | #Σ2 = n + 3} ≃ {the ways to parenthesize n factors completely}, +where parentheses show how cones in the fourth quadrant are obtained by iterated subdivisions. +For example, Σ141222 in Figure 1 has 5 cones σ1, . . . , σ5 in the fourth quadrant in terms of (2.4), +and they are parenthesized as σ1(((σ2σ3)σ4)σ5). In particular, we have +#{Σ ∈ Fan+− +sc (2) | #Σ2 = n + 3} = 1 +n +�2n − 2 +n − 1 +� +. + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +9 +We also have a bijection +{Σ ∈ Fan+− +sc (2) | #Σ2 = n + 3} ≃ {Triangulations of a regular (n + 1)-gon}, +where Σa1,...,an+1 corresponds to a triangulation satisfying the following condition: Let 1, 2, . . ., n+ +1 be the vertices of the regular (n + 1)-gon in a clockwise direction, and ai (1 ≤ i ≤ n + 1) the +number of triangles containing the vertex i in the triangulation. For example, Σ141222 corresponds +to the following triangulation, where 1 is the top vertex. +We introduce piecewise linear transformation of sign coherent fan of rank 2. This is a general- +ization of mutation of g-vectors of cluster algebras of rank 2 [FZ2, NZ], and also a special case of +so called combinatorial mutation [ACGK, FH]. +Definition 2.13. For Σ ∈ Fan ++− +sc (2) with σ+ = cone{(0, 1), (1, 0)}, take σ = cone{(1, 0), (ℓ, −1)} ∈ +Σ2. Define a new sign-coherent fan Σ′ by +Σ′ +1 +:= +(Σ1 \ {(0, 1)}) ∪ {(−ℓ, 1)} +Σ′ +2 +:= +(Σ2 \ {σ+, σ−+}) ∪ {−σ, cone{(−ℓ, 1), (1, 0)}}, +where the positive and negative cones of Σ′ are σ and −σ respectively. +Σ = +• +(0,1) +(1,0) +(ℓ,−1) +(0,−1) +(−1,0) ++ +− +σ +σ−+ +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄❄❄❄❄❄ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +ρ(Σ) ≃ Σ′ = +• +(−ℓ,1) +(1,0) +(ℓ,−1) +(0,−1) +(−1,0) ++ +− +σ +−σ +❄❄❄❄❄❄ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +We define the rotation ρ(Σ) ∈ Fan ++− +sc (2) of Σ as the image of Σ′ by a linear transformation of R2 +mapping (1, 0) �→ (0, 1) and (ℓ, −1) �→ (1, 0). +We give basic properties of rotation, where the name “rotation” is explained by (a) below. +Proposition 2.14. Let Σ ∈ Fan+− +sc (2) with facets (2.4) and s(Σ) = (a1, . . . , an−2; 0, 0). +(a) We have +s(ρ(Σ)) = (a2, . . . , an−2, a1; 0, 0). +In particular, ρn−2(Σ) = Σ holds, and therefore ρ is an invertible operation. +(b) For each 1 ≤ i ≤ n − 3, we have +Dσi(Σ) = ρn−3−i ◦ Dσn−3 ◦ ρi+1(Σ). +Proof. (a) Recall Σ1 = {v1, . . . , vn} and aivi = vi−1 + vi+1 for 1 ≤ i ≤ n. Moreover +ρ(Σ)1 = {w1, . . . , wn} where wi := vi+1 (i ̸= n − 1), wn−1 := −v2. +Hence we have +wi−1 + wi+1 += +vi + vi+2 = ai+1vi+1 = ai+1wi for i ̸= n − 2, n, +wn−1 + w1 += +−v2 + v2 = 0 · wn, +wn−3 + wn−1 += +vn−2 − v2 = −(vn + v2) = −a1v1 = a1vn−1 = a1wn−2. +Thus s(ρ(Σ)) = (a2, . . . , an−2, a1; 0, 0) as desired. +(b) By (a), we have s ◦ ρi+1(Σ) = (ai+2, . . . , an−2, a1, . . . , ai+1; 0, 0). Thus +s ◦ Dσn−3 ◦ ρi+1(Σ) +(2.7) += Dn−3 ◦ s ◦ ρi+1(Σ) = (ai+2, . . . , an−2, a1, . . . , ai−1, ai + 1, 1, ai+1 + 1; 0, 0). + +10 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +By (a) again, we have +s ◦ ρn−3−i ◦ Dσn−3 ◦ ρi+1(Σ) += +(a1, . . . , ai−1, ai + 1, 1, ai+1 + 1, ai+2, . . . , an−2; 0, 0) += +Di ◦ s(Σ) +(2.7) += s ◦ Dσi(Σ). +Since a fan is uniquely determined by its defining sequence, the assertion follows. +□ +3. Basic results in silting theory +3.1. Preliminaries. Let A be a finite dimensional algebra over a field k. Let K0(proj A) be the +Grothendieck group of the additive category proj A, which is identified with the Grothendieck +group of the triangulated category Kb(proj A). +We recall basic results on silting theory from +[AI, AIR, AHIKM1]. First we recall the definition of 2-term silting complexes. +Definition 3.1. Let T = (T i, di) ∈ Kb(proj A). +(a) T is called presilting if HomKb(proj A)(T, T [ℓ]) = 0 for all positive integers ℓ. +(b) T is called silting if it is presilting and Kb(proj A) = thick T . +(c) T is called 2-term if T i = 0 for all i ̸= 0, −1. In this case, the class [T ] = [T 0] − [T −1] ∈ +K0(proj A) of T is called the g-vector of T . +(d) An element of K0(proj A) is rigid if it is a g-vector of some 2-term presilting complex. +We denote by siltA (respectively, psiltA, 2-siltA, 2-psiltA) the set of isomorphism classes of basic +silting (respectively, presilting, 2-term silting, 2-term presilting) complexes of Kb(proj A). Note +that a 2-term presilting complex T is silting if and only if |T | = |A| holds. +For T, U ∈ siltA, we write T ≥ U if HomKb(proj A)(T, U[ℓ]) = 0 holds for all positive integers ℓ. +Then (siltA, ≥) is a partially ordered set [AI]. +In this paper, the subposet (2-siltA, ≥) of (siltA, ≥) plays a central role. +It is known that +Hasse(2-siltA) is n-regular for n := |A|. More precisely, let T = T1 ⊕ · · · ⊕ Tn ∈ 2-siltA with +indecomposable Ti. For each 1 ≤ i ≤ n, there exists precisely one T ′ ∈ 2-siltA such that T ′ = +T ′ +i ⊕ (� +j̸=i Tj) for some T ′ +i ̸= Ti. In this case, we call T ′ mutation of T at Ti and write +T ′ = µTi(T ) = µi(T ). +In this case, either T > T ′ or T ′ < T holds. We denote T ′ by µ− +i (T ) (respectively, µ+ +i (T )) if +T > T ′ and call it left mutation (respectively, right mutation). The following result is fundamental +in silting theory. +Proposition 3.2. Let T, T ′ ∈ 2-siltA. Take a decomposition T = T1⊕· · ·⊕Tn with indecomposable +Ti. Then the following conditions are equivalent. +(a) T > T ′, and T and T ′ are mutation of each other. +(b) There is an arrow T → T ′ in Hasse(2-siltA). +(c) T ′ = T ′ +i ⊕ (� +j̸=i Tj) and there is a triangle +Ti +f−→ Ui → T ′ +i → Ti[1] +such that f is a minimal left (add � +j̸=i Tj)-approximation. +(d) T ′ = T ′ +i ⊕ (� +j̸=i Tj) and there is a triangle +Ti → Ui +g−→ T ′ +i → Ti[1] +such that g is a minimal right (add � +j̸=i Tj)-approximation. +The triangles in (c) and (d) are isomorphic, and called an exchange triangle. +To introduce the g-fan of a finite dimensional k-algebra A, we consider the real Grothendieck +group of A: +K0(proj A)R := K0(proj A) ⊗Z R ≃ R|A|. + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +11 +Definition 3.3. For T = T1 ⊕ · · · ⊕ Tℓ ∈ 2-psiltA with indecomposable Ti, let +C(T ) +:= +{ +ℓ +� +i=1 +ai[Ti] | a1, . . . , aℓ ≥ 0} ⊂ K0(proj A)R, +C≤1(T ) +:= +{ +ℓ +� +i=1 +ai[Ti] | a1, . . . , aℓ ≥ 0, +ℓ +� +i=1 +ai ≤ 1} ⊂ K0(proj A)R. +We call the set +Σ(A) := {C(T ) | T ∈ 2-psiltA} +of cones the g-fan of A. We also define the g-polytope P(A) of A by +P(A) := +� +T ∈2-siltA +C≤1(T ). +We say that A is g-convex if the g-polytope P(A) is convex. +Notice that Σ(A) can be an infinite set. We give the following basic properties of g-fans. +Proposition 3.4. Let A be a finite dimensional algebra over a field k and n := |A|. +(a) Σ is a pairwise positive sign-coherent fan whose positive (respectively, negative) cone is given +by σ+ := C(A) (respectively, σ− := C(A[1])). +(b) Any cone in Σ(A) is a face of a cone of dimension n. +(c) Any cone in Σ(A) of dimension n − 1 is a face of precisely two cones of dimension n. +The following basic observation will be used frequently. +Proposition 3.5. Let Λ be a finite dimensional algebra with orthogonal primitive idempotents +1 = e1 + e2. Under the identification P1 = (1, 0) and P2 = (0, 1), the following assertions hold. +(a) cone{(−1, 0), (0, 1)} ∈ Σ(Λ) if and only if e2Λe1 = 0. +(b) cone{(1, 0), (0, −1)} ∈ Σ(Λ) if and only if e1Λe2 = 0. +Proposition 3.5 is explained by the following picture. +e2Λe1 = 0 ⇔ +• ++ +− +P1 +P2 +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +e2Λe1 = 0 ⇔ +• ++ +− +P1 +P2 +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +Proof. We only prove (a): Σ(A) ∈ Fan+− +sc (2) if and only if P1[1] ⊕ P2 ∈ 2-siltA if and only if +HomKb(proj A)(P1, P2) = 0 if and only if e2Λe1 = 0. +□ +We end this subsection with recalling the sign decomposition technique studied in [Ao, AHIKM1]. +We have to introduce the following notations. +Definition 3.6. Let A be a basic finite dimensional algebra over a field k with |A| = n, and +1 = e1 + · · · + en the orthogonal primitive idempotents. For ǫ ∈ {±1}n, we define +K0(proj A)ǫ,R := cone(ǫi[eiA] | i ∈ {1, . . ., n}) +and a subfan of Σ(A) by +Σǫ(A) := {σ ∈ Σ(A) | σ ⊂ K0(proj A)ǫ,R}. +Define idempotents of A by +e+ +ǫ := +� +ǫi=1 +ei and e− +ǫ := +� +ǫi=−1 +ei. +We denote by Aǫ the subalgebra of A given by +Aǫ := +� e+ +ǫ Ae+ +ǫ +e+ +ǫ Ae− +ǫ +0 +e− +ǫ Ae− +ǫ +� +. + +12 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +Define an ideal Iǫ of Aǫ by +Iǫ := +� rad(e+ +ǫ Ae+ +ǫ ) ∩ Anne+ +ǫ Ae+ +ǫ (e+ +ǫ Ae− +ǫ ) +0 +0 +rad(e− +ǫ Ae− +ǫ ) ∩ Ann(e+ +ǫ Ae− +ǫ )e− +ǫ Ae− +ǫ +� +. +The following result is often very useful to calculate Σǫ(A). +Proposition 3.7. [AHIKM1, Example 4.26] For each ideal I of Aǫ contained in Iǫ, the isomor- +phisms − ⊗Aǫ A : K0(proj Aǫ)R ≃ K0(proj A)R and − ⊗Aǫ (Aǫ/I) : K0(proj Aǫ)R ≃ K0(proj Aǫ/I)R +gives an isomorphism of fans +Σǫ(A) ≃ Σǫ(Aǫ/I). +The following finiteness condition plays a central role in this paper. +Definition 3.8. Let A be a finite dimensional algebra over a field k. We say that A is g-finite if +#2-siltA < ∞. (This is called τ-tilting finite in [DIJ].) +Proposition 3.9. A is g-finite (or equivalently, Σ(A) is finite) if and only if Σ(A) is complete. +3.2. Silting complexes in terms of matrices. In this subsection, we give basic properties of +2-term presilting complexes. Throughout this subsection, we assume the following. +Assumption 3.10. For rings A and B and an Aop ⊗k B-module X which is finitely generated on +both sides, let +Λ := +� A +X +0 +B +� +. +Equivalently, Λ is a ring with orthogonal idempotents 1 = e1 + e2 satisfying e2Λe1 = 0. In fact, +we can recover Λ from A := e1Λe1, B := e2Λe2 and X := e1Λe2 by the equality above. +Consider projective Λ-modules +P1 := [A X], P2 := [0 B] ∈ proj Λ. +For s, t ≥ 0, we denote by Ms,t(X) the set of s × t matrices with entries in X. Then we have an +isomorphism +Ms,t(X) ≃ HomΛ(P ⊕t +2 , P ⊕s +1 ) +sending x ∈ Ms,t(X) to the left multiplication x(·) : P ⊕t +2 +→ P ⊕s +1 . Thus we have a 2-term complex +Px := (P ⊕t +2 +x(·) +−−→ P ⊕s +1 ) ∈ per Λ. +The following observation is basic. +Proposition 3.11. Let s, t, u, v ≥ 0, x ∈ Ms,t(X) and y ∈ Mu,v(X). +(a) Then we have an exact sequence +Mu,s(A) ⊕ Mv,t(B) +[(·)x y(·)] +−−−−−−→ Mu,t(X) → Homper Λ(Px, Py[1]) → 0. +(b) In particular, Px is presilting if and only if Ms,t(X) = Ms(A)x + xMt(B) holds. +Proof. The assertion (a) follows from an exact sequence +HomΛ(P ⊕s +1 +, P ⊕u +1 +) ⊕ HomΛ(P ⊕t +2 , P ⊕v +2 +) +[(·)x y(·)] +−−−−−−→ HomΛ(P ⊕t +2 , P ⊕u +1 +) → Homper Λ(Px, Py[1]) → 0. +The assertion (b) is immediate from (a). +□ +The following construction of silting complexes of Λ will be used frequently, where t(XB) (re- +spectively, t(AX)) is the minimal number of generators of X as a right B-module (respectively, +left A-module). +Proposition 3.12. In Assumption 3.10, assume that A and B are local k-algebras. +(a) Σ(Λ) contains cone{(0, −1), (1, −r)} for r := t(XB) = dim(X/XJB)B/JB. +(b) Σ(Λ) contains cone{(1, 0), (ℓ, −1)} for ℓ := t(AX) = dimA/JA(X/JAX). + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +13 +(c) Let g1, . . . , gr be a minimal set of generators of the B-module X. Then µ+ +1 (Λ[1]) = Pg ⊕P2[1] ∈ +2-siltΛ holds for g := [g1 · · · gr] ∈ M1,r(X). +(d) Let h1, . . . , hℓ a minimal set of generators of the Aop-module X. Then µ− +2 (Λ) = P1 ⊕ Ph ∈ +2-siltΛ holds for h := +� h1 +... +hℓ +� +∈ Mℓ,1(X). +By Propositions 3.5 and 3.12, a part of Σ(Λ) has the following form. +Σ(Λ) = +• +P2 +P1 +Ph=ℓP1−P2 +Pg=P1−rP2 +P2[1] +P1[1] ++ +− +µ− +2 (Λ) +µ+ +1 (Λ[1]) +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄❄❄❄❄❄ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +✯✯✯✯✯✯✯✯✯✯✯✯✯ +✴✴✴✴✴✴✴✴✴ +Proof. We only prove (a)(c) since (b)(d) are the duals. A minimal right (add P2[1])-approximation +of P1[1] is given by +g(·) : P2[1]⊕r → P1[1]. +Thus the mutation of Λ[1] at P1[1] is Pg ⊕ P2. +□ +Now we assume that B is a local algebra. We fix a minimal set of generators g1, . . . , gr of the +right B-module X and set +g := [g1 · · · gr] ∈ M1,r(X) and g := [g1 · · · gr] ∈ M1,r(X/XJB), +where (·) is a canonical surjection X ։ X/XJB. Then we have an isomorphism +g(·) : Mr,1(B/JB) ≃ X/XJB, +and we define a map π : X → Mr,1(B/JB) by +π := (X +(·) +−→ X/XJB +(g(·))−1 +−−−−−→ Mr,1(B/JB)). +For each s, t ≥ 0, an entry-wise application of π gives a map +π : Ms,t(X) → Ms,t(Mr,1(B/JB)) = Mrs,t(B/JB). +In other words, for the identity matrix Is ∈ Ms(k) and gIs := +� g +O +... +O +g +� +∈ Ms(M1,r(k)) = +Ms,rs(k), we have +x = (gIs)π(x) for each x ∈ Ms,t(X). +(3.1) +Define a morphism of k-algebras +φ : Ms(A) → Mrs(B/JB) by a(gIs) = (gIs)φ(a). +Later we will use the following observation. +Proposition 3.13. In Assumption 3.10, assume that B is a local algebra. Let s, t ≥ 0. +(a) π : Ms,t(X) → Mrs,t(B/JB) is a morphism of Ms(A)op ⊗k Mt(B)-modules, where we regard +Mrs,t(B/JB) as an Ms(A)op-module via φ. +(b) Let x ∈ Ms,t(X). If Px is presilting, then π(x) ∈ Mrs,t(B/JB) has full rank. +Proof. (a) For any a ∈ Ms(A), x ∈ Ms,t(X) and b ∈ Mt(B), we need to show π(axb) = φ(a)π(x)b. +In fact, +(gIs)φ(a)π(x)b = a(gIs)π(x)b +(3.1) += axb = axb +(3.1) += (gIs)π(axb) +gives the desired equality since gIs(·) is injective. + +14 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +(b) By Proposition 3.11(b), we have Ms,t(X) = Ms(A)x + xMt(B). Applying π, we have +Mrs,t(B/JB) = π(Ms(A)x + xMt(B)) +(a) += +φ(Ms(A))π(x) + π(x)Mt(B) +⊂ +Mrs(B/JB)π(x) + π(x)Mt(B/JB). +Thus the right-hand side is Mrs,t(B/JB). This clearly implies that π(x) has full rank. +□ +For completeness, we also give the dual statement of Proposition 3.13. Now we assume that A +is a local algebra. We fix a minimal set of generators h1, . . . , hℓ of the left A-module X and set +h := +� h1 +... +hℓ +� +∈ Mℓ,1(X) and h := +� h1 +... +hℓ +� +∈ Mℓ,1(X/JAX), +where by abuse of notations, (·) is a canonical surjection X ։ X/JAX. Then we have an isomor- +phism (·)h : M1,ℓ(A/JA) ≃ X/JAX. By abuse of notations, let +π := (X +(·) +−→ X/JAX +((·)h)−1 +−−−−−→ M1,ℓ(A/JA)). +For each s, t ≥ 0, an entry-wise application of π gives a map +π : Ms,t(X) → Ms,t(M1,ℓ(A/JA)) = Ms,ℓt(A/JA). +Define a morphism of k-algebras +φ : Mt(B) → Mℓt(A/JA) by (hIt)b = φ(b)(hIs). +We have the following dual of Proposition 3.13. +Proposition 3.14. In Assumption 3.10, assume that A is a local algebra. Let s, t ≥ 0. +(a) π : Ms,t(X) → Ms,ℓt(A/JA) is a morphism of Ms(A)op ⊗k Mt(B)-modules, where we regard +Ms,ℓt(A/JA) as an Mt(B)-module via φ. +(b) Let x ∈ Ms,t(X). If Px is presilting, then π(x) ∈ Ms,ℓt(A/JA) has full rank. +3.3. Uniserial property of g-finite algebras. As an application of results in the previous sub- +section, we prove the following result, which is not used in the rest of this paper. +Theorem 3.15. Let Λ be a finite dimensional elementary k-algebra, and 1 = e1 + · · · + en the +orthogonal primitive idempotents. If Λ is g-finite, then for each 1 ≤ i ̸= j ≤ n, eiΛej/eiΛejJΛej +is a uniserial (eiΛei)op-module and eiΛej/eiJΛeiJΛej is a uniserial ejΛej-module. +Thanks to sign decomposition, we can deduce Theorem 3.15 from the following result. +Theorem 3.16. Let A and B be local k-algebras with k ≃ A/JA ≃ B/JB. If X is a Aop ⊗k B- +module such that +� +A +X +0 +B +� +is g-finite, then X/XJB is a uniserial Aop-module and X/JAX is a +uniserial B-module. +Proof of Theorem 3.16⇒Theorem 3.15. Since Λ is g-finite, so is Γ := (ei + ej)Λ(ei + ej). +By +Proposition 3.7, Γ+− = +� eiΛei +eiΛej +0 +ejΛej +� +is also g-finite. Thus the assertion follows from Theorem +3.16. +□ +In the rest of this subsection, we prove Theorem 3.16. +The following observation plays a key role in the proof, where we identify K0(proj Λ) with Z2 +via [A X] �→ (1, 0), [0 B] �→ (0, 1). +Lemma 3.17. Let Λ := +� A +X +0 +k +� +. Assume that (1, −1) ∈ K0(proj Λ) is rigid. +(a) There exists h ∈ X such that X = Ah. + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +15 +(b) Let Λ′ := +� A +JAX +0 +k +� +and t ≥ 1. If (1, −t) ∈ K0(proj Λ) is rigid, then (1, 1 − t) ∈ K0(proj Λ′) +is rigid. +Proof. (a) By Proposition 3.11(b), there exists h ∈ X satisfying X = Ah + hk = Ah. +(b) By Proposition 3.11(b), there exists [x1 x2 · · · xt] ∈ M1,t(X) such that +M1,t(X) = A[x1 · · · xt] + [x1 · · · xt]Mt(k). +(3.2) +As in Section 3.2, the element h gives surjections +π := (X +(·) +−→ X/JAX +((·)h)−1 +−−−−−→ A/JA = k) and π : M1,t(X) → M1,t(k). +By Proposition 3.14, π(x) ∈ M1,t(k) has full rank. By changing indices if necessary, we can assume +x1 ∈ A×h. Multiplying an element in A× from left, we can assume x1 = h. Multiplying an element +in GLt(k) from right, we can assume xi ∈ JAh for each 2 ≤ i ≤ t. We claim +M1,t−1(JAX) = A[x2 · · · xt] + [x2 · · · xt]Mt−1(k). +In fact, fix any [y2 · · · yt] ∈ M1,t−1(JAX). By (3.2) there exist a ∈ A and b = [bij]1≤i,j≤t ∈ Mt(k) +such that +[0 y2 · · · yt] = a[h x2 · · · xt] + [h x2 · · · xt]b. +(3.3) +Applying π, we obtain +[0 0 · · · 0] = a[1 0 · · · 0] + [1 0 · · · 0]b in M1,t(k). +Thus we obtain b12 = · · · = b1t = 0. Looking at the i-th entries for 2 ≤ i ≤ t of (3.3), we have +[y2 · · · yt] = a[x2 · · · xt] + [x2 · · · xt][bij]2≤i,j≤n. +Thus the claim follows. +□ +We are ready to prove Theorem 3.16. +Proof of Theorem 3.16. We prove that X/XJB is a uniserial Aop-module under a weaker assump- +tion that (1, −t) ∈ K0(proj Λ) is rigid for each t ≥ 1. Since Λ := +� A +X/XJB +0 +k +� +is a factor +algebra of Λ, the element (1, −t) ∈ K0(proj Λ) is rigid for each t ≥ 1. Replacing Λ by Λ, we can +assume that +B = k and Λ = +� A +X +0 +k +� +. +We use induction on dimk X. +By Lemma 3.17(a), the Aop-module X has a unique maximal +submodule JAX. Let Λ′ = +� A +JAX +0 +k +� +. By Lemma 3.17(b), (1, −t) ∈ K0(proj Λ′) is rigid for +each t ≥ 1. +By induction hypothesis, JAX is a uniserial Aop-module. +Therefore X is also a +uniserial Aop-module. +□ +4. Gluing, Rotation and Subdivision of g-fans +4.1. Gluing fans. Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive +idempotents 1 = e1 + e2 ∈ Λ and 1 = e′ +1 + e′ +2 ∈ Λ′. In this subsection, we prove the following +Gluing Theorem, where we identify K0(proj Λ) and K0(proj Λ′) with Z2 by e1Λ = (1, 0) = e′ +1Λ′ and +e2Λ = (0, 1) = e′ +2Λ′. +Theorem 4.1 (Gluing Theorem). Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal +primitive idempotents 1 = e1 + e2 ∈ Λ and 1 = e′ +1 + e′ +2 ∈ Λ′. Assume e1Λe2 = 0 and e′ +2Λ′e′ +1 = 0, +or equivalently, Σ(Λ) ∈ Fan ++− +sc (2) and Σ(Λ′) ∈ Fan +−+ +sc (2) (Proposition 3.5). Then, there exists an +elementary k-algebra Γ such that +Σ(Γ) = Σ(Λ) ∗ Σ(Λ′). +(4.1) + +16 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +Theorem 4.1 is explained by the following picture. +Σ(Λ) = +• ++ +− +? +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +Σ(Λ′) = +• ++ +− +! +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +Σ(Γ) = +• ++ +− +? +! +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +The construction of Γ is as follows: We can write +Λ = +� +A +X +0 +B +� +and Λ′ = +� +C +0 +Y +D +� +, +where A, B, C, D are local k-algebras, X is an Aop ⊗k B-module, and Y is an Dop ⊗k C-module. +Since Λ and Λ′ are elementary, we have k ≃ A/JA ≃ B/JB ≃ C/JC ≃ D/JD. Let A ×k C be a +fiber product of canonical surjections (·) : A → k and (·) : C → k, that is, +A ×k C := {(a, c) ∈ A × C | a = c}. +Let B ×k D be a fibre product of (·) : B → k and (·) : D → k. Using the projections A ×k C → A +and B ×k D → B, we regard X as an (A ×k C)op ⊗k (B ×k D)-module, and using the projections +A ×k C → C and B ×k D → D, we regard Y as an (B ×k D)op ⊗k (A ×k C)-module. +We prove that the algebra +Γ := +� A ×k C +X +Y +B ×k D +� +satisfies Σ(Γ) = Σ(Λ) ∗ Σ(Λ′), where the multiplication of the elements of X and those of Y are +defined to be zero. +Proof of Theorem 4.1. It suffices to prove +Σ+−(Γ) = Σ+−(Λ) and Σ−+(Γ) = Σ−+(Λ′). +For ǫ = (+, −), we have Γǫ = +� +A ×k C +X +0 +B ×k D +� +. The ideal I := +� +rad C +0 +0 +rad D +� +of Γǫ is +contained in Iǫ, and we have an isomorphism Γǫ/I ≃ Λ of k-algebras. Applying Proposition 3.7 to +Γ, we get Σ+−(Γ) = Σ+−(Λ). By the same argument, Σ−+(Γ) = Σ−+(Λ′) holds, as desired. +□ +Example 4.2. Let Λ and Λ′ be the following algebras. +Λ := +k + + +1 +2 +a3 � +a4 +� +a2 +� +a1 +� + + +⟨a2 +1, a2 +2, a2 +4, a2a1, a2a3 − a3a4⟩, +Λ′ := +k +� +1 +2 +b1 +� +b2 +� +� +⟨b2 +2⟩ +By Examples 4.6 and 4.11 below, we have +Σ(Λ) = Σ13122;00 = +Σ(Λ′) = Σ00;1212 = +Let A = e1Λe1, X = e1Λe2, B = e2Λe2, C = e1Λ′e1, Y = e2Λ′e1, D = e2Λ′e2 and Γ = +� A ×k C +X +Y +B ×k D +� +. Then +Γ = +k + + +1 +2 +a3 � +a4 +� +a2 +� +a1 +� b1 +� +b2 +� + + +⟨a2 +1, a2 +2, a2 +4, a2a1, a2a3 − a3a4, b2 +2⟩ + ⟨aibj, bjai | i ∈ {1, 2, 3, 4}, j ∈ {1, 2}⟩ + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +17 +By Gluing Theorem 4.1, we have +Σ(Γ) = Σ(Λ) ∗ Σ(Λ′) = Σ13122;1212 = +4.2. Rotation and Mutation. In this subsection, we explain a connection between the rotation +of a fan given in Definition 2.13 and mutation of a 2-term silting complex. +The following main result in this section shows that mutation of an algebra induce the rotation +of the g-fan, where we identify K0(proj Λ) with Z2 by e1Λ = (1, 0) and e2Λ = (0, 1). +Theorem 4.3 (Rotation Theorem). Let Λ be a finite dimensional k-algebra of rank 2 with or- +thogonal primitive idempotents 1 = e1 + e2. Assume e1Λe2 = 0, or equivalently, Σ(Λ) ∈ Fan ++− +sc (2) +(Proposition 3.5). Then, there exists a finite dimensional k-algebra Γ such that +Σ(Γ) = ρ(Σ(Λ)). +Furthermore, if Λ is elementary, then Γ can be taken to be elementary. +Theorem 4.3 is explained by the following picture. +Σ(Λ) = +• ++ +− +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄❄❄❄❄❄ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +Σ(Γ) ≃ +• ++ +−❄❄❄❄❄❄ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +To prove Theorem 4.3, we need the following preparation. +Let A be a basic finite dimensional algebra over a field k with |A| = n, and 1 = e1 + · · · + en +the orthogonal primitive idempotents. For 1 ≤ i ≤ n and δ ∈ {±1}, consider a half space +Rn +i,δ := {x1e1 + · · · + xden ∈ Rn | δxi ≥ 0} +and define a subfan of Σ by +Σi,δ := {σ ∈ Σ | σ ⊂ Rn +i,δ}. +On the other hand, for elements T ≥ T ′ in siltA, we consider the interval +[T ′, T ] := {U ∈ siltA | T ≥ U ≥ T ′}. +The following result provides a correspondence of a part of two g-fans. +Proposition 4.4. For 1 ≤ i ≤ n, let B := EndA(µ− +i (A)), where µ− +i (A) = Ti ⊕ (� +j̸=i P A +j ). +(a) [AHIKM1, Threom 4.26] There exists a triangle functor F : Kb(proj A) → Kb(proj B) which +satisfies F(Ti) ≃ P B +i +and F(P A +j ) ≃ P B +j +for each j ̸= i and gives an isomorphism K0(proj A) ≃ +K0(proj B) and an isomorphism of fans +Σi,−(A) ≃ Σi,+(B). +(b) There are isomorphisms (1 − ei)A(1 − ei) ≃ (1 − ei)B(1 − ei) and A/(1 − ei) ≃ B/(1 − ei) of +k-algebras. +Proof. (b) Although this is known to experts, we give a proof for convenience of the reader. The first +isomorphism is clear. To prove the second one, notice that A/(1 − ei) = EndKb(proj A)(P A +i )/[A/P A +i ] +and B/(1−ei) = EndKb(proj A)(Ti)/[T/Ti] hold, where [X] denotes the ideal consisting of morphisms +factoring through add X. Let Pi +f−→ Q +g−→ Ti +h−→ Pi[1] be an exchange triangle. Let a ∈ eiAei = +EndKb(proj A)(Pi). Since f is a minimal left (add A/Pi)-approximation of Pi, we obtain the following +commutative diagram. +Pi +f +� +a� +Q +g +� +� +Ti +h � +b� +Pi[1] +a[1] +� +Pi +f +� Q +g +� Ti +h � Pi[1] + +18 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +It is routine to check that the desired isomorphism A/(1 − ei)A = EndKb(proj A)(P A +i )/[A/P A +i ] ≃ +B/(1 − ei) = EndKb(proj A)(Ti)/[T/Ti] is given by a �→ b. +□ +We are ready to prove Theorem 4.3. +Proof of Theorem 4.3. Let T = P Λ +1 ⊕ T2 := µ− +2 (Λ) and E := EndKb(proj Λ)(T ). By Proposition +4.4(a), we have a triangle functor F : Kb(proj Λ) → Kb(proj E) which satisfies +F(P Λ +1 ) = P E +1 +and F(T2) = P E +2 +and induces an isomorphism F : K0(proj Λ) ≃ K0(proj E) and an isomorphism of fans +F : Σ2,−(Λ) ≃ Σ2,+(E). +• +Σ(Λ) +P Λ +2 +P Λ +1 +T2 +T ++ +− +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄❄❄❄❄❄ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +• +Σ(E) +P E +2 [1] +P E +1 +P E +2 +P E +1 [1] ++ +− +E +E[1] +❄❄❄❄❄❄ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +• +Σ(Γ) +P Γ +2 [1] +P Γ +1 +P Γ +2 +P Γ +1 [1] ++ +− +Γ +Γ[1] +❄❄❄❄❄❄ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +Applying Theorem 3.7 to E, we obtain a k-algebra Γ := E−+ such that +e1Γe2 = 0 and Σ−+(Γ) = Σ−+(E). +Therefore under the isomorphism K0(proj Γ) ≃ Z2 given by P Γ +1 �→ (0, 1) and P Γ +2 �→ (1, 0), we have +Σ(Γ) = ρ(Σ(Λ)), as desired. +It remains to prove the last assertion. By Proposition 4.4(a), we have isomorphisms e1Ee1 ≃ +e1Λe1 and Λ/(e1) ≃ E/(e1) of k-algebras. Thus, if Λ is elementary, then so are E and Γ. +□ +We give two examples of Theorem 4.3. The first one satisfies E = Γ. +Example 4.5. Let Λ be the following algebra. Then Σ(Λ) is the following fan by Example 4.11 +below. +Λ = +k +� +1 +2 +a � +b +� +� +⟨b2⟩ +Σ(Λ) = Σ1212 = +We set µ2(Λ) = T = T1 ⊕ T2 := [e2Λ +a· +−→ e1Λ] ⊕ e1Λ and E := EndKb(proj Λ)(T ). Then we have +Γ = E = +k +� +1 +2 +a � +b +� +� +⟨b2⟩ +and Σ(Γ) = ρ(Σ(Λ)) = Σ2121 = +The second example satisfies E ̸= Γ. +Example 4.6. Let Λ be the following algebra. Then Σ(Λ) is the following fan by Example 4.12 +below. +Λ = +k +� +1 +2 +a � +b +� +c +� +� +⟨b2, c2, bac⟩ +Σ(Λ) = Σ21312 = +We set µ2(Λ) = T = T1 ⊕ T2 := [e2Λ ( a· +ac·) +−−−−→ e1Λ⊕2] ⊕ e1Λ and E := EndKb(proj Λ)(T ), where we +switch the indices 1 and 2 unlike the proof of Theorem 4.3. Then +E = +k +� +1 +2 +a � +a′ +� +b +� +� +⟨b2, a′b, a′aa′⟩ +and Σ(E) = Σ13122;111 = + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +19 +where new arrows a, a′ and b are morphisms in Kb(proj Λ) given by commutative diagrams +0 +e1Λ +e2Λ +e1Λ⊕2 +� +� +( a· +ac·) � +( 0 +1) +� +e2Λ +e1Λ⊕2 +0 +e1Λ +( a· +ac·) � +� +� +( 0 b· ) +� +e2Λ +e1Λ⊕2 +e2Λ +e1Λ⊕2 +( a· +ac·) � +c· � +( a· +ac·) � +( 0 1 +0 0) +� +respectively. Let Γ := E+− = +� e1Ee1 +e1Ee2 +0 +e2Ee2 +� += +� ⟨e1, b, aa′, baa′⟩k +⟨a, ba, aa′a, baa′a⟩k +0 +⟨e2, a′a⟩k +� +. +Then +Γ = +k + + +1 +2 +a � +c +� +b +� +b′ +� + + +⟨b2, b′2, c2, b′b, b′a − ac⟩ and Σ(Γ) = ρ(Σ(Λ)) = Σ13122 = +where b′ := aa′ and c := a′a. +4.3. Subdivision and Extension. In this section, we realize subdivisions of g-fans of rank 2 by +extensions of algebras. The following theorem is a main result of this section, where we identify +K0(proj Λ) with Z2 by e1Λ = (1, 0) and e2Λ = (0, 1). +Theorem 4.7 (Subdivision Theorem). Let Λ be a finite dimensional elementary k-algebra with +orthogonal primitive idempotents 1 = e1+e2. Assume e1Λe2 = 0, or equivalently, Σ(Λ) ∈ Fan ++− +sc (2) +(Proposition 3.5). Then, for cones σ = C(µ+ +1 (Λ[1])) and σ′ := C(µ− +2 (Λ)) of Σ(Λ), there exist finite +dimensional elementary k-algebras Γ and Γ′ such that +Σ(Γ) = Dσ(Σ(Λ)) and Σ(Γ′) = Dσ′(Σ(Λ)). +Theorem 4.7 is explained by the following picture. +Σ(Λ) = +• +P2 +P1 +P2[1] +P1[1] ++ +− +µ− +2 (Λ) +µ+ +1 (Λ[1]) +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄❄❄❄❄❄ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +✯✯✯✯✯✯✯✯✯✯✯✯✯ +✴✴✴✴✴✴✴✴✴ +Σ(Γ) = +• ++ +− +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄❄❄❄❄❄ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +✯✯✯✯✯✯✯✯✯✯✯✯✯ +✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ +✯✯✯✯✯✯✯✯✯✯✯✯✯ +Σ(Γ′) = +• ++ +− +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄❄❄❄❄❄ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +✯✯✯✯✯✯✯✯✯✯✯✯✯ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +✴✴✴✴✴✴✴✴✴ +In the rest, we only prove the existence of Γ since the existence of Γ′ is the dual. +The construction of Γ is as follows: +Construction 4.8. By Proposition 3.5, we can write +Λ = +� A +X +0 +B +� +. +where A, B are local k-algebras and X is an Aop ⊗k B-module. Since Λ is elementary, we have +k ≃ A/JA ≃ B/JB. Let +X := X/XJB. +Then the k-dual DX is an A-module, and we regard it as an Aop-module by using the action of k +through the natural surjection A → k. Let +C := A ⊕ DX +be a trivial extension algebra of A by DX. Let +(·) : A → k, (·) : B → k and (·) : X → X + +20 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +be canonical surjections. We regard +Y := +� k +X +� +as a Cop ⊗k B-module by +(a, f) · [ α +x ] · b := +� +aαb+f(x)b +axb +� +for (a, f) ∈ C = A ⊕ DX, [ α +x ] ∈ Y = +� k +X +� +and b ∈ B. +Then we set +Γ := +� +C +Y +0 +B +� +. +In the rest of this subsection, we prove Theorem 4.7. We set +Q1 := [C Y ], Q2 := [0 B] ∈ proj Γ. +For y ∈ Ms,t(Y ) ≃ HomΓ(Q⊕t +2 , Q⊕s +1 ), we define +Qy := [Q⊕t +2 +y(·) +−−→ Q⊕s +1 ] ∈ Kb(proj Γ). +We fix a minimal set of generators g1, . . . , gr of the B-module X. Then (g1, . . . , gr) forms a +k-basis of X = X/XJB. Set +g := [g1 · · · gr] ∈ M1,r(X) and g := [g1 · · · gr] ∈ M1,r(X/XJB). +We need the following easy observation. +Lemma 4.9. Σ(Γ) contains cone{(0, 1), (1, −r−1)} and cone{(1, −r−1), (1, −r)}. More explicitly, +let +� 0 +g +� +∈ M1,r(Y ) and +� 0 1 +g 0 +� +∈ M1,r+1(Y ). +Then Q� 0 1 +g 0 +� ⊕ Q2[1] and Q� 0 +g +� ⊕ Q� 0 1 +g 0 +� belong to 2-siltΓ. +Proof. A minimal set of generators of the B-module Y is given by the r+1 columns of +� 0 1 +g 0 +� +. Thus +Q� 0 1 +g 0 +� ⊕ Q2[1] ∈ 2-siltΓ holds by Proposition 3.12. +In the rest, we prove that T := Q� 0 +g +� ⊕ Q� 0 1 +g 0 +� is basic silting. By the first statement, Q� 0 1 +g 0 +� +is indecomposable. If Q� 0 +g +� is not indecomposable, then |T | is bigger than two, a contradiction. +Thus T is basic. +We will show that T is presilting by using Proposition 3.11(b). By our choice of g, we have +gMr,1(B) = X and (DX)g = M1,r(k). +Thus we have +� 0 +g +� +Mr(B) = M1,r([ 0 +X ]) and (DX) +� 0 +g +� += M1,r([ k +0 ]), and hence +C +� 0 +g +� ++ +� 0 +g +� +Mr(B) ⊃ (DX) +� 0 +g +� ++ +� 0 +g +� +Mr(B) = M1,r([ 0 +X ]) + M1,r([ k +0 ]) = M1,r(Y ). +This clearly implies +C +� 0 +g +� ++ +� 0 1 +g 0 +� +Mr+1,r(B) = M1,r(Y ), +and a similar argument implies +C +� 0 1 +g 0 +� ++ +� 0 +g +� +Mr,r+1(B) = M1,r+1(Y ). +Thus Proposition 3.11(b) implies that T is presilting, as desired. +□ +As in Section 3.2, the element g gives a surjection +π := (X +(·) +−→ X +(g(·))−1 +−−−−−→ Mr,1(B) = Mr,1(k)), +which extends to the map π : Ms,t(X) → Mrs,t(k) for each s, t ≥ 0. +The following observation is crucial. +Proposition 4.10. Let s, t ≥ 0. For x ∈ Ms,t(X), consider [ 0 +x ] ∈ Ms,t(Y ). + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +21 +(a) Px is indecomposable in Kb(proj Λ) if and only if Q[ 0 +x] is indecomposable in Kb(proj Γ). +(b) If Q[ 0 +x] is a presilting complex of Γ, then Px is a presilting complex of Λ. +(c) The converse of (b) holds if t ≤ rs. +(d) The restriction of Σ(Γ) to {(x, y) ∈ R2 | 0 ≤ −y ≤ rx} coincides with that of Σ(Λ). +Proof. Notice that Γ is the trivial extension Λ ⊕ I of Λ by the following ideal I of Γ: +I := +� +DX +k +0 +0 +� +. +(a) Since Px ≃ Q[ 0 +x] ⊗Γ Λ and Q[ 0 +x] ≃ Px ⊗Λ Γ, the assertion follows immediately. +(b) Since Λ = Γ/I and Q[ 0 +x] ⊗Γ Λ ≃ Px, the assertion follows. +(c) Assume that Px is a presilting complex of Λ. Then by Proposition 3.11(b), we have +Ms,t(X) = Ms(A)x + xMt(B). +(4.2) +Again by Proposition 3.11(b), it suffices to show the equality +V := Ms(C) [ 0 +x ] + [ 0 +x ] Mt(B) = Ms,t( +� k +X +� +). +Since V ⊃ Ms(A) [ 0 +x ] + [ 0 +x ] Mt(B) +(4.2) += Ms,t([ 0 +X ]) holds, it suffices to show +V ⊃ Ms,t([ k +0 ]). +(4.3) +By our assumption t ≤ rs and Proposition 3.13(b), π(x) has rank t and the map +(·)π(x) : Ms,rs(k) → Ms,t(k) +(4.4) +is surjective. We denote by g∗ +1, . . . , g∗ +r the basis of DX which is dual to g1, . . . , gr. Then the map +(·) +� g∗ +1 +... +g∗ +r +� +: M1,r(k) ≃ DX is a bijection, and we denote its inverse by +π′ : DX ≃ M1,r(k). +It gives a bijection π′ : Ms(DX) ≃ Ms,rs(k). We have a commutative diagram +Ms(DX) × Ms,t(X) +π′×π +� +eval. +�❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +❱ +Ms,rs(k) × Mrs,t(k) +mult. +� +Ms,t(k) +where eval. is given by the evaluation map DX × X → DX × X → k. Thus the commutativity of +the diagram above and the surjectivity of (4.4) shows that the map +(·)x : Ms(DX) → Ms,t(k) +is also surjective. Therefore the desired claim (4.3) follows from +V ⊃ Ms(C) [ 0 +x ] ⊃ Ms(DX) [ 0 +x ] = Ms,t([ k +0 ]). +□ +(d) Immediate from (c). +We are ready to prove Theorem 4.7. +Proof of Theorem 4.7. The assertion follows from Lemma 4.9 and Proposition 4.10(d). +□ +We give two examples of Subdivision Theorem 4.7. + +22 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +Example 4.11. Let Λ be the following algebra. Then Σ(Λ) is the following fan. +Λ = k[1 → 2] +Σ(Λ) = Σ111 = +Applying Theorem 4.7 to Λ, we get +Γ := +� +k ⊕ Dk +� k +k +� +0 +k +� += +k +� +1 +2 +� +b +� +� +⟨b2⟩ +and Σ(Γ) = D3(Σ(Λ)) = Σ1212 = +Example 4.12. Let Λ be the following algebra. Then Σ(Λ) is the following fan by Example 4.5. +Λ = +k +� +1 +2 +a � +b +� +� +⟨b2⟩ +Σ(Λ) = Σ2121 = +Applying Theorem 4.7 to Λ, we get +Γ := +� +k ⊕ D(ka) +� +k +⟨a,ab⟩k +� +0 +⟨e2, b⟩k +� += +k +� +1 +2 +a � +c +� +b +� +� +⟨b2, c2, cab⟩ +and Σ(Γ) = D4(Σ(Λ)) = Σ21312 = +4.4. Proof of Theorem 1.3. Let k be a field. For a finite dimensional k-algebras Λ of rank 2, +we regard the g-fan Σ(Λ) as a fan in R2 by isomorphism K0(proj Λ) ≃ R2 given by P1 �→ (1, 0) and +P2 �→ (0, 1). We denote by +k-Fan(2) +the subset of Fansc(2) consisting of g-fans of finite dimensional k-algebras of rank 2. Let k-Fanel(2) +be the subset of k-Fan(2) consisting of g-fans of finite dimensional elementary k-algebras of rank +2. +The following is a main result of this paper. +Theorem 4.13. For any field k, we have +k-Fanel(2) = k-Fan(2) = Fansc(2). +(4.5) +That is, any sign-coherent fan in R2 can be realized as a g-fan Σ(Λ) of some finite dimensional +elementary k-algebra Λ. +Proof. It suffices to show Fansc(2) = k-Fanel(2). Let +k-Fan+− +el (2) := k-Fanel(2) ∩ Fan+− +sc (2) and k-Fan−+ +el (2) := k-Fanel(2) ∩ Fan−+ +sc (2). +By Gluing Theorem 4.1, we have +k-Fanel(2) = k-Fan+− +el (2) ∗ k-Fan−+ +el (2). +By Rotation Theorem 4.3, k-Fan+− +el (2) is closed under rotations. By Theorem 4.7 and Proposition +2.14(b), k-Fan+− +el (2) is closed under subdivisions. Since Σ(0, 0; 0, 0) = Σ(k × k) ∈ k-Fan+− +el (2), +Proposition 2.10 implies +k-Fan+− +el (2) = Fan+− +sc (2). +Similarly, we have k-Fan−+ +el (2) = Fan−+ +sc (2). Consequently, we have +Fansc(2) +(2.5) += Fan+− +sc (2) ∗ Fan−+ +sc (2) = k-Fan+− +el (2) ∗ k-Fan−+ +el (2) = k-Fanel(2). +□ +For given Σ ∈ Fansc(2), our proof of Theorem 4.13 gives a concrete algorithm to construct a +finite dimensional k-algebra Λ satisfying Σ(Λ) = Σ. We demonstrate it in the following example. +Example 4.14. We construct a finite dimensional k-algebra Γ satisfying Σ(Γ) = Σ13122;1212 by +the following three steps. + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +23 +(I) We obtain a finite dimensional k-algebra +Λ = +k + + +1 +2 +a3 � +a4 +� +a2 +� +a1 +� + + +⟨a2 +1, a2 +2, a2 +4, a2a1, a2a3 − a3a4⟩ +satisfying Σ(Λ) = Σ13122;00 by using Rotation Theorem 4.3 and Subdivision Theorem 4.7 as +follows. +Σ00 +D2 +� +Σ111 +D3 +Ex.4.11 +� +Σ1212 +ρ +Ex.4.5 +� +Σ2121 +D4 +Ex.4.12 +� +Σ21312 +ρ +Ex.4.6 +� +Σ13122 +(II) Similarly, we obtain a finite dimensional k-algebra +Λ′ := +k +� +1 +2 +b1 +� +b2 +� +� +⟨b2 +2⟩ +satisfying Σ(Λ′) = Σ(0, 0; 1, 2, 1, 2). +(III) We obtain a finite dimensional k-algebra +Γ = +k + + +1 +2 +a3 � +a4 +� +a2 +� +a1 +� b1 +� +b2 +� + + +⟨a2 +1, a2 +2, a2 +4, a2a1, a2a3 − a3a4, b2 +2⟩ + ⟨aibj, bjai | i ∈ {1, 2, 3, 4}, j ∈ {1, 2}⟩ +satisfying Σ(Γ) = Σ(1, 3, 1, 2, 2; 1, 2, 1, 2) by applying Gluing Theorem 4.1 to Λ and Λ′, see +Example 4.2. +Σ(Λ) = +Σ(Λ′) = +Σ(Γ) = Σ(Λ) ∗ Σ(Λ′) = +4.5. Gluing fans II. In this subsection, we study another type of gluing g-fans. Results in this +subsection will not be used in the rest of this paper. +Theorem 4.15. Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive idem- +potents 1 = e1 + e2 ∈ Λ and 1 = e′ +1 + e′ +2 ∈ Λ′ satisfying e1Λe2 = 0, e′ +1Λ′e′ +2 = 0, +σ = cone{(0, −1), (1, −1)} ∈ Σ(Λ) +and σ′ = cone{(1, −1), (1, 0)} ∈ Σ(Λ′). +(4.6) +Then, there exists an elementary k-algebra Γ such that +Σ2(Γ) = (Σ2(Λ) \ {σ}) ∪ (Σ2(Λ′) \ {σ′}). +Theorem 4.15 is explained by the following picture. +Σ(Λ) = +• ++ +− +? +P1 +P2 +σ +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +Σ(Λ′) = +• ++ +− +! +P ′ +1 +P ′ +2 +σ′ +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +Σ(Γ) = +• ++ +− +! +? +Q1 +Q2 +❄❄❄❄❄❄ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ + +24 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +The assumption (4.6) is equivalent to that the defining sequences can be written as +Σ(Λ) = Σ(a1, . . . , an−1, 1; 0, 0) and Σ(Λ′) = Σ(1, b2, . . . , bm; 0, 0). +In this case, the defining sequence of Σ(Γ) is given by +Σ(Γ) = Σ(a1, . . . , an−2, an−1 + b2 − 1, b3, . . . , bm; 0, 0). +The rest of this section is devoted to proving Theorem 4.15. By our assumption, we can write +Λ = +� +A +X +0 +B +� +and P1 := [A X], P2 := [0 B] ∈ proj Λ, +Λ′ = +� C +Y +0 +D +� +and P ′ +1 := [C Y ], P ′ +2 := [0 D] ∈ proj Λ′, +where +• A, B, C, D are local k-algebras such that k ≃ A/JA ≃ B/JB ≃ C/JC ≃ D/JD. +• X is an Aop ⊗k B-module and Y is an Cop ⊗k D-module. +• There exist g ∈ X and h ∈ Y such that X = gB ̸= 0 and Y = Ch ̸= 0 by Proposition 3.12. +The construction of Γ is as follows: Let A ×k C (respectively, B ×k D) be a fibre product of +canonical surjections A → k and C → k (respectively, B → k and D → k). As in Section 3.2, we +consider maps +π : X → X/XJB +(g(·))−1 +−−−−−→ B/JB = k and π′ : Y → Y/JCY +((·)h)−1 +−−−−−→ C/JC = k. +(4.7) +Let X×kY be a fibre product of π : X → k and π′ : Y → k. Then X×kY is a (A×kC)op⊗k(B×kD)- +module, and let +Γ := +� A ×k C +X ×k Y +0 +B ×k D +� +and Q1 := [A ×k C X ×k Y ], Q2 := [0 B ×k D] ∈ proj Γ. +Consider ideals of Γ by +I = +� JC +JCY +0 +JD +� +and I′ = +� JA +XJB +0 +JB +� +. +Then there exist isomorphisms of k-algebras +Γ/I ≃ Λ and Γ/I′ ≃ Λ′. +(4.8) +As in Section 3.2, for s, t ≥ 0, x ∈ Ms,t(X), y ∈ Ms,t(Y ) and (x′, y′) ∈ Ms,t(X ×k Y ), we define +Px +:= +(P ⊕t +2 +x(·) +−−→ P ⊕s +1 ) ∈ per Λ, +P ′ +y +:= +(P ′ +2 +⊕t +y(·) +−−→ P ′ +1 +⊕s) ∈ per Λ′ +Q(x,y) +:= +(Q⊕t +2 +(x′,y′)(·) +−−−−−−→ Q⊕s +1 ) ∈ per Γ. +Proposition 4.16. Let s, t ≥ 0 and (x, y) ∈ Ms,t(X ×k Y ). If Q(x,y) is a presilting complex of Γ, +then Px is a presilting complex of Λ and P ′ +y is a presilting complex of Λ′. +Proof. By (4.8) and Q(x,y) ⊗Γ Λ = Px, the complex Px is presilting. The complex P ′ +y is presilting +similarly. +□ +Define maps (·) : A → C and (·) : B → D as the compositions of canonical maps +(·) : A +(·) +−→ k ⊂ C and (·) : B +(·) +−→ k ⊂ D. +Using π and π′ in (4.7), define maps (·) : X → Y and (·) : Y → X by +(·) : X +π−→ k +(·)h +−−→ kh ⊂ Y +and (·) : Y +π′ +−→ k +(·)g +−−→ kg ⊂ X. +(4.9) +Then the first projection X ×k Y → X, (x, y) �→ x has a section given by +X → X ×k Y, x �→ (x, x), + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +25 +and the second projection X ×k Y → Y , (x, y) �→ y has a section given by +Y → X ×k Y, y �→ (y, y). +The following is a crucial result. +Proposition 4.17. The following assertions hold. +(a) Let s ≥ t. For x ∈ Ms,t(X), consider (x, x) ∈ Ms,t(X ⊗k Y ). Then Px is a presilting complex +of Λ if and only if Q(x,x) is a presilting complex of Γ. +(b) Let s ≤ t. For y ∈ Ms,t(Y ), consider (y, y) ∈ Mc,d(X ⊗k Y ). Then P ′ +y is a presilting complex +of Λ′ if and only if Q(y,y) is a presilting complex of Γ. +Proof. It suffices to prove (a) since (b) is dual to (a). +The “if” part is clear from Proposition 4.16. +We prove the “only if” part. By Proposition 3.11, it suffices to show +Ms,t(X ×k Y ) = Ms(A ×k C)(x, x) + (x, x)Mt(B ×k D). +Since +X ×k Y = {(0, y) | y ∈ JCY } + {(z, z) | z ∈ X}, +it suffices to show the following assertions. +(i) For each y ∈ Ms,t(JCY ), we have (0, y) ∈ Ms(A ×k C)(x, x). +(ii) For each z ∈ Ms,t(X), we have (z, z) ∈ Ms(A ×k C)(x, x) + (x, x)Mt(B ×k D). +We prove (i). Since Px is presilting, π(x) ∈ Ms,t(k) has full rank by Proposition 3.13. Since s ≥ t, +the map (·)π(x) : Ms(k) → Ms,t(k) is surjective. Applying JC ⊗k −, the map (·)π(x) : Ms(JC) → +Ms,t(JC) is also surjective, and so is the composition +(·)x +(4.9) += (·)π(x)h : Ms(JC) +(·)π(x) +−−−−→ Ms,t(JC) +(·)h +−−→ Ms,t(JCY ). +Therefore there exists c ∈ Ms(JC) such that y = cx. Then (0, c) ∈ Ms(A ×k C) satisfies +(0, c)(x, x) = (0, y). +We prove (ii). Since Px is presilting, we have Ms,t(X) = Ms(A)x + xMt(B) by Proposition 3.11. +Thus there exist a ∈ Ms(A) and b ∈ Mt(B) such that z = ax + xb. Then +(a, a)(x, x) + (x, x)(b, b) = (ax + xb, ax + xb) = (z, z). +Thus the assertion follows. +□ +Now we are ready to prove Theorem 4.15. +Proof of Theorem 4.15. By Proposition 3.5, each of Σ(Λ), Σ(Λ′) and Σ(Γ) contains cone{(−1, 0), (0, 1)}. +By Propositions 4.16 and 4.17, the following assertions hold. +(i) Let s ≥ t. Then there exists x ∈ Ms,t(X) such that Px is presilting if and only if there exists +(x, y) ∈ Ms,t(X ×k Y ) such that Q(s,t) is presilting. +(ii) Let s ≤ t. Then there exists y ∈ Ms,t(Y ) such that P ′ +y is presilting if and only if there exists +(x, y) ∈ Ms,t(X ×k Y ) such that Q(s,t) is presilting. +Therefore the claim follows. +□ +Example 4.18. Let Λ and Λ′ be the following algebras. +Λ := +k +� +1 +2 +a � +b +� +� +⟨b2⟩ +Λ′ := +k + + +1 +2 +a � +d +� +c1 +� +c2 +� + + +⟨c2 +1, c2 +2, d2, c1c2, c1a − ad⟩ + +26 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +By Examples 4.5 and 4.6, we have +Σ(Λ) = Σ2121 = +Σ(Λ′) = Σ13122 = +Let Γ := +� e1Λe1 ×k e1Λ′e1 +e1Λe2 ×k e1Λ′e2 +0 +e2Λe2 ×k e2Λ′e2 +� +. Then we have +Γ = +k + + +1 +2 +a � +b +� +d +� +c1 +� +c2 +� + + +⟨b2, c2 +1, c2 +2, d2, c1c2, c1a − ad, c2ab, bd, db⟩ and Σ(Γ) = Σ214122 = +5. g-Convex algebras of rank 2 +In this section, we will characterize algebras of rank 2 which have convex g-polygons. +5.1. Characterizations of g-convex algebras of rank 2. Let e, e′ be pairwise orthogonal prim- +itive idempotents in A and x ∈ eAe′. Then we use the following notations. +• x ∈ eAe′ is a left generator (respectively, right generator) of eAe′ if eAx = eAe′ (respectively, +xAe′ = eAe′). +• Define subalgebras Lx ⊂ e′Ae′ and Rx ⊂ eAe as follows (see Lemma 5.5). +Rx := {a ∈ eAe | ax ∈ xAe′} and Lx := {a ∈ e′Ae′ | xa ∈ eAx}. +Recall that, for an algebra Λ and a right (respectively, left) Λ-module M, we denote by t(MΛ) +(respectively, t(ΛM)) the minimal number of generators of M. +Theorem 5.1. Let A be a basic finite dimensional algebra, {e1, e2} a complete set of primitive +orthogonal idempotents in A and Pi = eiA (i = 1, 2). +(a) A is g-convex if and only if Σ(A) = Σa;b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}. +(b) Let (l, r) := (t(e1Ae1e1Ae2), t(e1Ae2e2Ae2)). Then we have the following statements. +• Σ(A) = Σ00;b for some b if and only if (l, r) = (0, 0). +• Σ(A) = Σ111;b for some b if and only if (l, r) = (1, 1). +• Σ(A) = Σ1212;b for some b if and only if (l, r) = (1, 2) and t(Rxe1Ae1) = 2 hold for some +left generator x of e1Ae2. +• Σ(A) = Σ2121;b for some b if and only if (l, r) = (2, 1) and t(e2Ae2Lx) = 2 hold for some +right generator x of e1Ae2. +Σ00;b +• +P2 +P1 +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +⑧⑧⑧⑧⑧⑧ +Σ111;b +• +P2 +P1 +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +Σ1212;b +• +P2 +P1 +❄ +❄ +❄ +❄ +❄ +❄ +✴✴✴✴✴✴✴✴✴ +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +Σ2121;b +• +P2 +P1 +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +Remark 5.2. For a left (respectively, right) generator x of e1Ae2, Rx (respectively, Lx) is unique +up to conjugacy. In particular, t(Rxe1Ae1) (respectively, t(e2Ae2Lx)) does not depend on the choice +of a left (respectively, right) generator x. + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +27 +Example 5.3. (a) Here, we give algebras which realize 7 convex g-fans up to isomorphism of +g-fans. We define Ai = kQ/I (i ∈ {1, 2, 3, 4, 5, 6, 7}) as follows. +A1 = +k +� +1 +2 +a � +b� +c +� +d +� +� +⟨ab, ad, ba, bc, c2, d2⟩ +A2 = +k +� +1 +2 +a � +b� +d +� +� +⟨ab, ba, d2⟩ +A3 = +� +1 +2 +a � +b� +d +� +� +⟨ab, ba, ad, d2⟩ +A4 = +k +� +1 +2 +b� +d +� +� +⟨d2⟩ +A5 = +k +� +1 +2 +a � +b� +� +⟨ab, ba⟩ +A6 = k +� +1 +2 +b� +� +A7 = k +� +1 2 +� +Then the g-fans Σ(Ai) (i ∈ {1, 2, 3, 4, 5, 6, 7}) are given by the following table. +i +1 +2 +3 +4 +5 +6 +7 +Σ(Ai) +•+ +− +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +✴✴✴✴✴✴✴ +❄❄❄❄❄ +❖❖❖❖❖❖❖ +•+ +− +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄❄❄❄❄ +❄❄❄❄❄ +❖❖❖❖❖❖❖ +•+ +− +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❖❖❖❖❖❖❖ +•+ +− +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❖❖❖❖❖❖❖ +•+ +− +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +❄❄❄❄❄ +•+ +− +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +❄❄❄❄❄ +•+ +− +❄❄❄❄❄ +⑧⑧⑧⑧⑧ +⑧⑧⑧⑧⑧ +❄❄❄❄❄ +(b) Let K/k be a field extension with degree two, and A be a k-algebra +�k +K +0 +K +� +with e1 = +�1 +0 +0 +0 +� +, +e2 = +�0 +0 +0 +1 +� +. We write K = k(t) and set x := +�0 +1 +0 +0 +� +∈ e1Ae2, u = +�0 +0 +0 +t +� +∈ e2Ae2. Then we +have Lx = +�0 +0 +0 +k +� +, u ̸∈ Lx, and the following equations hold. +• e1Ae2 = +� +0 +K +0 +0 +� += xAe2 = e1Ax + e1Axu +• e2Ae2 = +�0 +0 +0 +K +� += Lx + uLx +Further, we have e2Ae1 = 0. Therefore, Theorem 5.1 implies that Σ(A) has the following form. +• +P2 +P1 +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄❄❄❄❄❄ +⑧⑧⑧⑧⑧⑧ +5.2. Proof of Theorem 5.1. In this subsection, we prove Theorem 5.1. The following observation +shows Theorem 5.1(a) and gives another proof of [AHIKM1, Theorem 6.3]. +Proposition 5.4. Let A be as in Theorem 5.1. Then A is g-convex if and only if Σ(A) = Σa;b for +some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}. +Proof. The “if” part is clear. +Conversely, assume that A is g-convex and Σ(A) = Σa;b with +a = (a1, . . . , an) and b = (b1, . . . , bm). Then ai ≤ 2 and bj ≤ 2 hold for each i, j. Using Proposition +2.10, it is easy to check that a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)} holds (see Figure 1). +□ +Next we show the following. +Lemma 5.5. Let x ∈ e1Ae2. Then Lx is a subalgebra of e2Ae2, and Rx is a subalgebra of e1Ae1. +Proof. This is a special case of the following easy fact: Let A, B be rings, M an (A, B)-module, +and x ∈ M. Then {b ∈ B | xb ∈ Ax} is a subring of B. +□ +Now we give a key observation. As in Section 3.2, for s, t ≥ 0, x ∈ Ms,t(e1Ae2), we define +Px := (e2A⊕t +x(·) +−−→ e1A⊕s) ∈ Kb(proj A). +Proposition 5.6. Assume t(e1Ae1e1Ae2) = 1. +For a left generator x ∈ e1Ae2, the following +conditions are equivalent. + +28 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +(1) Σ(A) contains cone{(1, −1), (1, −2)}. +(2) t(Rxe1Ae2) = 2. +(3) e1Ay + xM1,2(e2Ae2) = M1,2(e1Ae2) holds for some u ∈ e1Ae1 \ Rx and y := [x ux]. +Proof. Notice that Px is indecomposable presilting by Proposition 3.12. +(1)⇒(2) If t(Rxe1Ae1) = 1, then e1Ae1 = Rx holds. Thus e1Ae2 = e1Ax ⊂ xAe2 holds, and +thus x is a right generator. By Proposition 3.12, Px ⊕ P2[1] ∈ 2-siltA holds, a contradiction to +cone{(1, −1), (1, −2)} ∈ Σ(A). Thus it suffices to prove t(Rxe1Ae1) ≤ 2. +Since cone{(1, −1), (1, −2)} ∈ Σ(A), there exists y = [x1 x2] ∈ M1,2(e1Ae2) such that Px ⊕ Py +is silting. By Proposition 3.11, we have +M1,2(e1Ae2) = e1Ay + yM2,2(e2Ae2), +(5.1) +M1,2(e1Ae2) = e1Ay + xM1,2(e2Ae2). +(5.2) +Looking at the first entry of (5.1), at least one of x1 and x2 does not belong to rade1Ae1 e1Ae2. +Without loss of generality, assume x1 /∈ rade1Ae1 e1Ae2. Then there exists a ∈ (e1Ae1)× such that +x = ax1. Since Py ≃ Pay, we can assume x1 = x by replacing y by ay. Since x is a left generator, +there exists u ∈ e1Ae1 such that x2 = ux. Consequently, we can assume +y = [x ux]. +For each a ∈ e1Ae1, (5.2) implies that there exist a′ ∈ e1Ae1 and b, b′ ∈ e2Ae2 such that +[0 ax] = a′[x ux] + x[b b′]. +Then a′ and a−a′u are in Rx, and hence a = a′u+(a−a′u) ∈ Rxu+Rx. Thus e1Ae1 = Rx +Rxu +and t(Rxe1Ae1) ≤ 2 hold, as desired. +(2)⇒(3) Since t(Rxe1Ae2) = 2 and Rx ̸⊂ radRx e1Ae1, there exists u ∈ e1Ae1 \ Rx such that +Rxu + Rx = e1Ae1. +Multiplying x from the right, we have Rxux + Rxx = e1Ax = e1Ae2. Since Rxx ⊂ xAe2, we +have +Rxux + xAe2 = e1Ae2. +(5.3) +To prove (3), take any [z w] ∈ M1,2(e1Ae2). Since x is a left generator, there exists a ∈ e1Ae1 +such that z = ax. By (5.3), there exist r ∈ Rx and b ∈ e2Ae2 such that w − aux = rux + xb. By +definition of Rx, there exists c ∈ e2Ae2 such that rx = xc. Then we have +[z w] = (a + r)[x ux] + x[−c b] ∈ e1Ay + xM1,2(e2Ae2). +(3)⇒(1) By Proposition 3.11, the following assertions hold. +• Px is presilting if and only if (i) e1Ax + xAe2 = e1Ae2. +• Py is presilting if and only if (ii) e1Ay + yM2,2(e2Ae2) = M1,2(e1Ae2). +• HomKb(proj A)(Px, Py[1]) = 0 if and only if (iii) e1Ax + yM2,1(e2Ae2) = e1Ae2. +• HomKb(proj A)(Py, Px[1]) = 0 if and only if (iv) e1Ay + xM1,2(e2Ae2) = M1,2(e1Ae2). +It is clear that (iv) implies (ii), and (i) implies (iii). By looking at the first entry of the row vector, +(iv) implies (i). +Our assumption (3) implies that (iv) holds, and hence (i)-(iii) also hold. +Thus Px ⊕ Py is +presilting. It remains to show that Py is indecomposable. Suppose that Py is decomposable. By +considering g-vector, we have that Py ≃ e2A[1] ⊕ Pz for some z ∈ e1Ae2. Since [Pz] = [Px], we +have Pz ≃ Px by [DIJ, Theorem 6.5(a)]. This shows that e2A[1] ⊕ Px is silting. By Proposition +3.12, we have xAe2 = e1Ae2 and Rx = eAe. This contradicts u ̸∈ Rx. +□ +We are ready to prove Theorem 5.1(b). +Proof of Theorem 5.1(b). The first and second statements follow from Proposition 3.5 and Propo- +sition 3.12. + +FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 +29 +We prove the third statement. By Proposition 3.12, cone{(1, 0), (1, −1)} ∈ Σ(A) if and only if +t(e1Ae1e1Ae2) = 1, and cone{(0, −1), (1, −2)} ∈ Σ(A) if and only if t(e1Ae2e2Ae2) = 2. Thus the +assertion follows from Proposition 5.6. +The fourth statement is the dual of the third statement. +□ +Acknowledgments +T.A is supported by JSPS Grants-in-Aid for Scientific Research JP19J11408. A.H is supported +by JSPS Grant-in-Aid for Scientists Research (C) 20K03513. O.I is supported by JSPS Grant- +in-Aid for Scientific Research (B) 16H03923, (C) 18K3209 and (S) 15H05738. R.K is supported +by JSPS Grant-in-Aid for Young Scientists (B) 17K14169. Y.M is supported by Grant-in-Aid for +Scientific Research (C) 20K03539. +References +[ACGK] M. Akhtar, T. 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IMRN 2013, +no. 10, 2368–2420. + +30 +TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO +Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, +Osaka 565-0871, Japan +Email address: aoki-t@ist.osaka-u.ac.jp +Department of Pure and Applied Mathematics, Graduate School of Information Science and Tech- +nology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan +Email address: higashitani@ist.osaka-u.ac.jp +Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo +153-8914, Japan +Email address: iyama@ms.u-tokyo.ac.jp +Department of Information Science and Engineering, Okayama University of Science, 1-1 Ridaicho, +Kita-ku, Okayama 700-0005, Japan +Email address: r-kase@ous.ac.jp +Faculty of Liberal Arts, Sciences and Global Education / Graduate School of Science, Osaka Met- +ropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan +Email address: yuya.mizuno@omu.ac.jp + diff --git a/DNAzT4oBgHgl3EQfiP3j/content/tmp_files/load_file.txt b/DNAzT4oBgHgl3EQfiP3j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b5951570aaa26cd5d2a4eb783ff5572686cf6ea9 --- /dev/null +++ b/DNAzT4oBgHgl3EQfiP3j/content/tmp_files/load_file.txt @@ -0,0 +1,1986 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf,len=1985 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='01498v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='RT] 4 Jan 2023 FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' g-fan of a finite dimensional algebra is a fan in its real Grothendieck group defined by tilting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We give a classification of complete g-fans of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' More explicitly, our first main result asserts that every complete sign-coherent fan of rank 2 is a g-fan of some finite dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Our proof is based on three fundamental results, Gluing Theorem, Rotation Theorem and Subdivision Theorem, which realize basic operations on fans in the level of finite dimensional algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Our second main result gives a necessary and sufficient condition for algebras of rank 2 to be g-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Preliminaries 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Preliminaries on fans 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Sign-coherent fans of rank 2 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Basic results in silting theory 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Preliminaries 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Silting complexes in terms of matrices 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Uniserial property of g-finite algebras 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Gluing, Rotation and Subdivision of g-fans 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Gluing fans 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Rotation and Mutation 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Subdivision and Extension 19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Gluing fans II 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' g-Convex algebras of rank 2 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Characterizations of g-convex algebras of rank 2 26 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 27 Acknowledgments 29 References 29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Introduction The notion of tilting complexes is central to control equivalences of derived categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The class of silting complexes [KV] gives a completion of the class of tilting complexes with respect to mutation, which is an operation to replace a direct summand of a given silting complex to construct a new silting complex [AI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The subclass of 2-term silting complexes enjoys remarkable properties [AIR, DF].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' They give rise to a fan in the real Grothendieck group of a finite dimensional algebra A, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [H1, H2, Pl, B, DIJ, BST, As].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In our previous article [AHIKM1], we introduced a g-fan Σ(A) of A and established a basic theory of g-fans and the associated g-polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A g-fan of each finite dimensional algebra A belongs to the following special class of nonsingular fans [AHIKM1, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A sign-coherent fan is a pair (Σ, σ+) satisfying the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 1 2 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO (a) Σ is a nonsingular fan in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) σ+, −σ+ ∈ Σd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (c) Take a Z-basis e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ed of Zd such that σ+ = cone{ei | 1 ≤ i ≤ d}, and denote the orthant corresponding to ǫ ∈ {±1}d by Rd ǫ := cone{ǫ(1)e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ǫ(d)ed} = {x1e1 + · · · + xded | ǫ(i)xi ≥ 0 for each 1 ≤ i ≤ d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then for each σ ∈ Σ, there exists ǫ ∈ {±1}d such that σ ⊆ Rd ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote by Fansc(d) the set of complete sign-coherent fans in Rd, and by k-Fan(d) the set of complete g-fans of finite dimensional k-algebras of rank d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Note that a g-fan Σ(A) is complete if and only if A is g-finite (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have Fansc(d) ⊃ k-Fan(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It is very natural to study the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Characterize complete sign-coherent fans in Rd which can be realized as a g-fan of some finite dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' This paper is devoted to give a complete answer to this problem for the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The result was very simple and came as a surprise to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each field k, we have Fansc(2) = k-Fan(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus any complete sign-coherent fan in R2 can be realized as a g-fan of some finite dimensional k-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We explain our method to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Each sign-coherent fan of rank 2 is obtained by gluing two fans of the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ = + − ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ Σ′ = + − ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ Recall that a finite dimensional k-algbera Λ is elementary if the k-algebra Λ/JΛ is isomorphic to a product of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' This is automatic if Λ is basic and k is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We prove Gluing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1, which asserts that if both Σ and Σ′ are g-fans of finite dimensional elementary k- algebras, then so is their gluing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Therefore by symmetry, it suffices to consider sign-coherent fans Σ of the form above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Now such Σ can be obtained from the fan + − ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ by applying subdivision in the fourth quadrant repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We prove Rotation Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3 and Subdivision Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7, which imply that if Σ is a g-fan of a finite dimensional k-algebra, then so are the subdivisions of Σ in the fourth quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Figure 1 gives fans in Fan+− sc (2) with at most 8 facet, where each edge shows a subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Figure 2 gives examples of algebras whose g-fans are given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each finite dimensional algebra A, we define a g-polytope P(A) by gluing each simplex associated with the cones in Σ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If P(A) is convex, we call Σ(A) convex and A g-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For example, Brauer tree algebras A are g-convex, and this fact plays an important role in the classification of 2-term tilting complexes of A [AMN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' From tilting theoretic point of view, g-convex algebras are the most fundamental.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Therefore it is important to study the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Classify convex g-fans in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 3 Σ00 Σ111 Σ2121 Σ1212 Σ31221 Σ22131 Σ12213 Σ21312 Σ13122 Σ412221 Σ321321 Σ313131 Σ312312 Σ231231 Σ222141 Σ221412 Σ122214 Σ123123 Σ214122 Σ131313 Σ213213 Σ132132 Σ141222 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Fans in Fan+− sc (2) with at most 8 facets An answer to the case d = 2 was given in [AHIKM1, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' There are precisely 7 convex g-fans up to isomorphism of g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' + − ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ + − ❄❄❄❄❄ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ + − ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ + − ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❖❖❖❖❖❖❖ ❄❄❄❄❄ + − ❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄ ❖❖❖❖❖❖❖ ❄❄❄❄❄ + − ❄❄❄❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖❖❖❖❖❖❖ ❄❄❄❄❄ + − ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄ ✴✴✴✴✴✴✴ ❄❄❄❄❄ ❖❖❖❖❖❖❖ ❄❄❄❄❄ More precisely, in the last Section 5, we show that there are 16 convex g-fans in Fansc(2) Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b with a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We also give a characterization of algebras whose g-fans are one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let t(ΛM) (respectively, t(MΛ)) be the minimal number of generators of a left (respectively, right) Λ-module M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 4 TOSHITAKA AOKI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' AKIHIRO HIGASHITANI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' OSAMU IYAMA,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b1ac1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b0a − ac0 � Σ122214 k \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a � c1 � c2 � c3 � \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb �c2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2c3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c1c3 c1c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c3c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c3a � Σ123123 k \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a � c0 � c1 � c2 � b � \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb � b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c1c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c1c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ab − c0c1a � Σ214122 k \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a � c0 � c1 � b0 � b1 � \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb �b2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b0b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b1b0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0ab1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c1ab1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ab0 − c0a � Σ131313 k \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a � c0 � c1 � c2 � b � \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb � b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c1c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2c1c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ab − c0a � Σ213213 k \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a � c1 � c2 � b � \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb �b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c1ab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c1c2 � Σ132132 k \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a � c0 � c1 � c2 � b0 � b1 � \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb �b2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b0b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2c1c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ab0 − c0a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ab1 − c2c1a � Σ141222 k \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a � c0 � c1 � c2 � b0 � b1 � \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb �b2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b0b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c2c0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c1c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' c0c1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ab0 − c0a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ab1 − c1a � Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Algebras whose g-fans are given in Figure 1 FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 5 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A be a basic finite dimensional algebra, {e1, e2} a complete set of primitive orthogonal idempotents in A, and Pi = eiA (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) A is g-convex if and only if Σ(A) = Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Let (l, r) := (t(e1Ae1e1Ae2), t(e1Ae2e2Ae2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(A) = Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some b if and only if (l, r) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(A) = Σ111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some b if and only if (l, r) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(A) = Σ1212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some b if and only if (l, r) = (1, 2) and t(Rxe1Ae1) = 2 hold for some left generator x of e1Ae2 and Rx := {a ∈ e1Ae1 | ax ∈ xAe2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(A) = Σ2121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some b if and only if (l, r) = (2, 1) and t(e2Ae2Lx) = 2 hold for some right generator x of e1Ae2 and Lx := {b ∈ e2Ae2 | xb ∈ e1Ax}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b P2 P1 ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ⑧⑧⑧⑧⑧⑧ Σ111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b P2 P1 ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ Σ1212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b P2 P1 ❄ ❄ ❄ ❄ ❄ ❄ ✴✴✴✴✴✴✴✴✴ ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ Σ2121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b P2 P1 ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ Further, in a forthcoming paper [AHIKM2], we will give a complete answer to Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4 for d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Preliminaries on fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We recall some fundamental materials on fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We refer the reader to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [F, BR, BP] for these materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A convex polyhedral cone σ is a set of the form σ = {�s i=1 rivi | ri ≥ 0}, where v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vs ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote it by σ = cone{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Note that {0} is regarded as a convex polyhedral cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We collect some notions concerning convex polyhedral cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let σ be a convex polyhedral cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The dimension of σ is the dimension of the linear space generated by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We say that σ is strongly convex if σ ∩ (−σ) = {0} holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=', σ does not contain a linear subspace of positive dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call σ rational if each vi can be taken from Qd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote by ⟨·, ·⟩ the usual inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A supporting hyperplane of σ is a hyperplane {v ∈ σ | ⟨u, v⟩ = 0} in Rd given by some u ∈ Rd satisfying σ ⊂ {v ∈ Rd | ⟨u, v⟩ ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A face τ of σ is the intersection of σ with a supporting hyperplane of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In what follows, a cone means a strongly convex rational polyhedral cone for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A fan Σ in Rd is a collection of cones in Rd such that (a) each face of a cone in Σ is also contained in Σ, and (b) the intersection of two cones in Σ is a face of each of those two cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each i ≥ 0, we denote by Σi the subset of cones of dimension i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For example, Σ0 consists of the trivial cone {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call each element in Σ1 a ray of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We collect some notions concerning fans used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Σ be a fan in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call Σ finite if it consists of a finite number of cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call Σ complete if � σ∈Σ σ = Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call Σ nonsingular (or smooth) if each maximal cone in Σ is generated by a Z-basis for Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We prepare some notions which will be used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Σ be a nonsingular fan in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call Σ pairwise positive if the following condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 6 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO For each two adjacent maximal cones σ, τ ∈ Σd, take Z-basis {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vd−1, vd} and {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vd−1, v′ d} of Zd such that σ = cone{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vd−1, vd} and τ = cone{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vd−1, v′ d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then vd + v′ d belongs to cone{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vd−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Σ and Σ′ be fans in Rd and Rd′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (1) An isomorphism Σ ≃ Σ′ of fans is an isomorphism Zd ≃ Zd′ of abelian groups such that the induced linear isomorphism Rd → Rd′ gives a bijection Σ ≃ Σ′ between cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (2) Let (Σ, σ+) and (Σ′, σ′ +) be sign-coherent fans in Rd and Rd′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' An isomorphism of sign-coherent fans is an isomorphism f : Σ ≃ Σ′ of fans such that {f(σ+), f(−σ+)} = {σ′ +, −σ′ +}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Sign-coherent fans of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this subsection, we introduce some terminologies of sign- coherent fans of rank 2, and discuss some fundamental properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Σ be a complete nonsingular fan of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote the rays of Σ by v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vn−1, vn = v0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) which are indexed in a clockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each 1 ≤ i ≤ n, since Σ is nonsingular, there exists an integer ai satisfying aivi = vi−1 + vi+1 for each 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call the sequence of integers s(Σ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2) the defining sequence of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In fact, Σ is uniquely determined by its defining sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A fan with defining sequence (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an) is denoted by Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [F, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5] An integer sequence (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an) is a defining sequence of nonsingular complete fan of rank 2 if and only if it satisfies n � i=1 ai = 3n − 12 and �0 −1 1 a1 � �0 −1 1 a2 � · · �0 −1 1 an � = �1 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote by Fansc(2) the set of all (possibly infinite) fans Σ satisfying that Σ is a sign-coherent fans (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) of rank 2 with positive and negative cones σ+ := cone{(1, 0), (0, 1)} and σ− := cone{(−1, 0), (0, −1)} respectively, each ray is a face of precisely two facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote the subset of complete fans by Fansc(2) ⊂ Fansc(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For Σ ∈ Fansc(2), we denote the rays of Σ in a clockwise orientation by Σ1 = {v1 := (1, 0), v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vn−1, vn = v0 := (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then there exists 2 ≤ i ≤ n − 2 such that vi = (0, −1) and vi+1 = (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' vn=v0=(0,1) v1=(1,0) vi=(0,−1) vi+1=(−1,0) + − ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ In this case, it is more convenient to rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2) as s(Σ) = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' an, an−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ai+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus we mainly use the notation Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ai+1) = Σa1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=',ai;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='an,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=',ai+1 FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 7 instead of Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We consider subsets Fan +− sc (2) ⊂ Fansc(2) ∪ ∪ Fan+− sc (2) ⊂ Fansc(2) which consist of fans Σ containing σ−+ := cone{(−1, 0), (0, 1)}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ has the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' σ−+ + − ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ Thus the rays and the facets of Σ ∈ Fan+− sc (2) are written as Σ1 = {v1 = (1, 0), v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vn−2 = (0, −1), vn−1 = (−1, 0), vn = v0 = (0, 1)}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3) Σ2 = {σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , σn−3, σn−2 = σ−, σn−1 = σ−+, σn = σ+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4) Similarly, we define Fan −+ sc (2) and Fan−+ sc (2) as the subsets of Fansc(2) and Fansc(2) respectively which consist of fans containing σ+− := cone{(1, 0), (0, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following observations are clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (1) The correspondence Σ �→ {−σ | σ ∈ Σ} gives bijections Fan +− sc (2) → Fan −+ sc (2) and Fan+− sc (2) → Fan−+ sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (2) Let Σ ∈ Fansc(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Σ ∈ Fan+− sc (2) (respectively, Σ ∈ Fan−+ sc (2)) holds if and only if s(Σ) has the form (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , bm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) (respectively, (0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , bm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this case, bi ≥ 0 holds for any 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For Σ ∈ Fan +− sc (2) and Σ′ ∈ Fan −+ sc (2), we define Σ ∗ Σ′ ∈ Fansc(2) by (Σ ∪ Σ′)1 := Σ1 ∪ Σ′ 1 (Σ ∪ Σ′)2 := (Σ2 \\ {σ−+}) ∪ (Σ′ 2 \\ {σ+−}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ = + − ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' σ−+ ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ Σ′ = + − !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' σ+− ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ Σ ∗ Σ′ = + − ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ Then, we clearly have Fansc(2) = Fan +− sc (2) ∗ Fan −+ sc (2) := {Σ ∗ Σ′ | Σ ∈ Fan +− sc (2), Σ′ ∈ Fan −+ sc (2)}, Fansc(2) = Fan+− sc (2) ∗ Fan−+ sc (2) := {Σ ∗ Σ′ | Σ ∈ Fan+− sc (2), Σ′ ∈ Fan−+ sc (2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Σ be a (possibly infinite) nonsingular fan of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For a cone σ := cone{u, v} of Σ, we define a new nonsingular fan Dσ(Σ) by Dσ(Σ)1 = Σ1 ∪ {cone{u + v}}, Dσ(Σ)2 = (Σ2 \\ {σ}) ⊔ {cone{u, u + v}, cone{v, u + v}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call Dσ(Σ) the subdivision of Σ at σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='. σ ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❡ ❡ ❡ ❡ ❡ ❡ ❡ Dσ(Σ) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='. ❨ ❨ ❨ ❨ ❨ ❨ ❨ ❡ ❡ ❡ ❡ ❡ ❡ ❡ ❡❡❡❡❡❡❡ ❨❨❨❨❨❨❨ 8 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO For a sequence a = (a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an) and 1 ≤ i ≤ n, we define a new sequence by Di(a) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ai−1, ai + 1, 1, ai+1 + 1, ai+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6) For a complete nonsingular fan Σ with rays (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) and σi := cone{vi, vi+1} for 1 ≤ i ≤ n, we have s ◦ Dσi(Σ) = Di ◦ s(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7) Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Figure 1 gives fans in Fan+− sc (2) with at most 8 facets, where Σa1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=',an := Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) and each edge shows a subdivision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Figure 2 gives examples of algebras whose g-fans are given in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For example, Σ111 is the g-vector fan of a cluster algebra of type A2 [FZ1, FZ2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Similarly, Σ1212 and Σ2121 are the g-vector fans of cluster algebras of type B2, and Σ131313 and Σ313131 are the g-vector fans of cluster algebras of type G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Later we need the following observation (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [F, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Each fan in Fan+− sc (2) can be obtained from Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) by a sequence of subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) = + − ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ To prove this, we need the following preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [F, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Σ ∈ Fan+− sc (2) and s(Σ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If n ≥ 5, then there exists 2 ≤ i ≤ n − 3 satisfying ai = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let vi = (xi, yi) ∈ Z2 for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Assume that n ≥ 5 and ai ≥ 2 for any 2 ≤ i ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We claim that xi+1 ≥ xi holds for each 1 ≤ i ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In fact, n ≥ 5 implies x2 ≥ 1 = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have xi+1 = aixi − xi−1 ≥ 2xi − xi−1 ≥ xi for each 2 ≤ i ≤ n − 3, and the claim follows inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Consequently 1 = x1 ≤ x2 ≤ · · · ≤ xn−2 = 0 holds, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ We are ready to prove Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let F ⊂ Fan+− sc (2) be the set of fans obtained from Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) by a sequence of subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It suffices to show Fan+− sc (2) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We will show that each Σ ∈ Fan+− sc (2) belongs to F by using induction on n = #Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Clearly n ≥ 4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If n = 4, then Σ = Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Suppose that Σ with #Σ2 = n ≥ 5 belongs to Fan+− sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In terms of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4), there exists 2 ≤ i ≤ n − 3 satisfying vi = vi−1 + vi+1 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since vi−1, vi+1 forms a Z-basis of Z2, we obtain a new fan Σ′ ∈ Fan+− sc (2) by Σ′ 1 := Σ1 \\ {vi}, Σ′ 2 := (Σ2 \\ {σi−1, σi}) ∪ {σ} for σ := cone{vi−1, vi+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since #Σ′ 2 = n − 1, the induction hypothesis implies Σ′ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus Σ = Dσ(Σ′) ∈ F holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each n ≥ 1, we have a bijection {Σ ∈ Fan+− sc (2) | #Σ2 = n + 3} ≃ {the ways to parenthesize n factors completely}, where parentheses show how cones in the fourth quadrant are obtained by iterated subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For example, Σ141222 in Figure 1 has 5 cones σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , σ5 in the fourth quadrant in terms of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4), and they are parenthesized as σ1(((σ2σ3)σ4)σ5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In particular, we have #{Σ ∈ Fan+− sc (2) | #Σ2 = n + 3} = 1 n �2n − 2 n − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 9 We also have a bijection {Σ ∈ Fan+− sc (2) | #Σ2 = n + 3} ≃ {Triangulations of a regular (n + 1)-gon}, where Σa1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=',an+1 corresponds to a triangulation satisfying the following condition: Let 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=', n+ 1 be the vertices of the regular (n + 1)-gon in a clockwise direction, and ai (1 ≤ i ≤ n + 1) the number of triangles containing the vertex i in the triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For example, Σ141222 corresponds to the following triangulation, where 1 is the top vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We introduce piecewise linear transformation of sign coherent fan of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' This is a general- ization of mutation of g-vectors of cluster algebras of rank 2 [FZ2, NZ], and also a special case of so called combinatorial mutation [ACGK, FH].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For Σ ∈ Fan +− sc (2) with σ+ = cone{(0, 1), (1, 0)}, take σ = cone{(1, 0), (ℓ, −1)} ∈ Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Define a new sign-coherent fan Σ′ by Σ′ 1 := (Σ1 \\ {(0, 1)}) ∪ {(−ℓ, 1)} Σ′ 2 := (Σ2 \\ {σ+, σ−+}) ∪ {−σ, cone{(−ℓ, 1), (1, 0)}}, where the positive and negative cones of Σ′ are σ and −σ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='0) (ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='−1) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='−1) (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='0) + − σ σ−+ ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❄❄❄❄❄❄ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ρ(Σ) ≃ Σ′ = (−ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='0) (ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='−1) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='−1) (−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='0) + − σ −σ ❄❄❄❄❄❄ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ We define the rotation ρ(Σ) ∈ Fan +− sc (2) of Σ as the image of Σ′ by a linear transformation of R2 mapping (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0) �→ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 1) and (ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' −1) �→ (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We give basic properties of rotation, where the name “rotation” is explained by (a) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Σ ∈ Fan+− sc (2) with facets (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4) and s(Σ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) We have s(ρ(Σ)) = (a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−2, a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In particular, ρn−2(Σ) = Σ holds, and therefore ρ is an invertible operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) For each 1 ≤ i ≤ n − 3, we have Dσi(Σ) = ρn−3−i ◦ Dσn−3 ◦ ρi+1(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) Recall Σ1 = {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , vn} and aivi = vi−1 + vi+1 for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Moreover ρ(Σ)1 = {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , wn} where wi := vi+1 (i ̸= n − 1), wn−1 := −v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Hence we have wi−1 + wi+1 = vi + vi+2 = ai+1vi+1 = ai+1wi for i ̸= n − 2, n, wn−1 + w1 = −v2 + v2 = 0 · wn, wn−3 + wn−1 = vn−2 − v2 = −(vn + v2) = −a1v1 = a1vn−1 = a1wn−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus s(ρ(Σ)) = (a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−2, a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) By (a), we have s ◦ ρi+1(Σ) = (ai+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−2, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ai+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus s ◦ Dσn−3 ◦ ρi+1(Σ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7) = Dn−3 ◦ s ◦ ρi+1(Σ) = (ai+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−2, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ai−1, ai + 1, 1, ai+1 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 10 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO By (a) again, we have s ◦ ρn−3−i ◦ Dσn−3 ◦ ρi+1(Σ) = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , ai−1, ai + 1, 1, ai+1 + 1, ai+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) = Di ◦ s(Σ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7) = s ◦ Dσi(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since a fan is uniquely determined by its defining sequence, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Basic results in silting theory 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A be a finite dimensional algebra over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let K0(proj A) be the Grothendieck group of the additive category proj A, which is identified with the Grothendieck group of the triangulated category Kb(proj A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We recall basic results on silting theory from [AI, AIR, AHIKM1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' First we recall the definition of 2-term silting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let T = (T i, di) ∈ Kb(proj A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) T is called presilting if HomKb(proj A)(T, T [ℓ]) = 0 for all positive integers ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) T is called silting if it is presilting and Kb(proj A) = thick T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (c) T is called 2-term if T i = 0 for all i ̸= 0, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this case, the class [T ] = [T 0] − [T −1] ∈ K0(proj A) of T is called the g-vector of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (d) An element of K0(proj A) is rigid if it is a g-vector of some 2-term presilting complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote by siltA (respectively, psiltA, 2-siltA, 2-psiltA) the set of isomorphism classes of basic silting (respectively, presilting, 2-term silting, 2-term presilting) complexes of Kb(proj A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Note that a 2-term presilting complex T is silting if and only if |T | = |A| holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For T, U ∈ siltA, we write T ≥ U if HomKb(proj A)(T, U[ℓ]) = 0 holds for all positive integers ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then (siltA, ≥) is a partially ordered set [AI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this paper, the subposet (2-siltA, ≥) of (siltA, ≥) plays a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It is known that Hasse(2-siltA) is n-regular for n := |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' More precisely, let T = T1 ⊕ · · · ⊕ Tn ∈ 2-siltA with indecomposable Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each 1 ≤ i ≤ n, there exists precisely one T ′ ∈ 2-siltA such that T ′ = T ′ i ⊕ (� j̸=i Tj) for some T ′ i ̸= Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this case, we call T ′ mutation of T at Ti and write T ′ = µTi(T ) = µi(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this case, either T > T ′ or T ′ < T holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote T ′ by µ− i (T ) (respectively, µ+ i (T )) if T > T ′ and call it left mutation (respectively, right mutation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following result is fundamental in silting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let T, T ′ ∈ 2-siltA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Take a decomposition T = T1⊕· · ·⊕Tn with indecomposable Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then the following conditions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) T > T ′, and T and T ′ are mutation of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) There is an arrow T → T ′ in Hasse(2-siltA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (c) T ′ = T ′ i ⊕ (� j̸=i Tj) and there is a triangle Ti f−→ Ui → T ′ i → Ti[1] such that f is a minimal left (add � j̸=i Tj)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (d) T ′ = T ′ i ⊕ (� j̸=i Tj) and there is a triangle Ti → Ui g−→ T ′ i → Ti[1] such that g is a minimal right (add � j̸=i Tj)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The triangles in (c) and (d) are isomorphic, and called an exchange triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' To introduce the g-fan of a finite dimensional k-algebra A, we consider the real Grothendieck group of A: K0(proj A)R := K0(proj A) ⊗Z R ≃ R|A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 11 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For T = T1 ⊕ · · · ⊕ Tℓ ∈ 2-psiltA with indecomposable Ti, let C(T ) := { ℓ � i=1 ai[Ti] | a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , aℓ ≥ 0} ⊂ K0(proj A)R, C≤1(T ) := { ℓ � i=1 ai[Ti] | a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , aℓ ≥ 0, ℓ � i=1 ai ≤ 1} ⊂ K0(proj A)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We call the set Σ(A) := {C(T ) | T ∈ 2-psiltA} of cones the g-fan of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We also define the g-polytope P(A) of A by P(A) := � T ∈2-siltA C≤1(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We say that A is g-convex if the g-polytope P(A) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Notice that Σ(A) can be an infinite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We give the following basic properties of g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A be a finite dimensional algebra over a field k and n := |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) Σ is a pairwise positive sign-coherent fan whose positive (respectively, negative) cone is given by σ+ := C(A) (respectively, σ− := C(A[1])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Any cone in Σ(A) is a face of a cone of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (c) Any cone in Σ(A) of dimension n − 1 is a face of precisely two cones of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following basic observation will be used frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ be a finite dimensional algebra with orthogonal primitive idempotents 1 = e1 + e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Under the identification P1 = (1, 0) and P2 = (0, 1), the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) cone{(−1, 0), (0, 1)} ∈ Σ(Λ) if and only if e2Λe1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) cone{(1, 0), (0, −1)} ∈ Σ(Λ) if and only if e1Λe2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' e2Λe1 = 0 ⇔ + − P1 P2 ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ e2Λe1 = 0 ⇔ + − P1 P2 ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We only prove (a): Σ(A) ∈ Fan+− sc (2) if and only if P1[1] ⊕ P2 ∈ 2-siltA if and only if HomKb(proj A)(P1, P2) = 0 if and only if e2Λe1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ We end this subsection with recalling the sign decomposition technique studied in [Ao, AHIKM1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We have to introduce the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A be a basic finite dimensional algebra over a field k with |A| = n, and 1 = e1 + · · · + en the orthogonal primitive idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For ǫ ∈ {±1}n, we define K0(proj A)ǫ,R := cone(ǫi[eiA] | i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=', n}) and a subfan of Σ(A) by Σǫ(A) := {σ ∈ Σ(A) | σ ⊂ K0(proj A)ǫ,R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Define idempotents of A by e+ ǫ := � ǫi=1 ei and e− ǫ := � ǫi=−1 ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote by Aǫ the subalgebra of A given by Aǫ := � e+ ǫ Ae+ ǫ e+ ǫ Ae− ǫ 0 e− ǫ Ae− ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 12 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO Define an ideal Iǫ of Aǫ by Iǫ := � rad(e+ ǫ Ae+ ǫ ) ∩ Anne+ ǫ Ae+ ǫ (e+ ǫ Ae− ǫ ) 0 0 rad(e− ǫ Ae− ǫ ) ∩ Ann(e+ ǫ Ae− ǫ )e− ǫ Ae− ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following result is often very useful to calculate Σǫ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [AHIKM1, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='26] For each ideal I of Aǫ contained in Iǫ, the isomor- phisms − ⊗Aǫ A : K0(proj Aǫ)R ≃ K0(proj A)R and − ⊗Aǫ (Aǫ/I) : K0(proj Aǫ)R ≃ K0(proj Aǫ/I)R gives an isomorphism of fans Σǫ(A) ≃ Σǫ(Aǫ/I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following finiteness condition plays a central role in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A be a finite dimensional algebra over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We say that A is g-finite if #2-siltA < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (This is called τ-tilting finite in [DIJ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=') Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A is g-finite (or equivalently, Σ(A) is finite) if and only if Σ(A) is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Silting complexes in terms of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this subsection, we give basic properties of 2-term presilting complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Throughout this subsection, we assume the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For rings A and B and an Aop ⊗k B-module X which is finitely generated on both sides, let Λ := � A X 0 B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Equivalently, Λ is a ring with orthogonal idempotents 1 = e1 + e2 satisfying e2Λe1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In fact, we can recover Λ from A := e1Λe1, B := e2Λe2 and X := e1Λe2 by the equality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Consider projective Λ-modules P1 := [A X], P2 := [0 B] ∈ proj Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For s, t ≥ 0, we denote by Ms,t(X) the set of s × t matrices with entries in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have an isomorphism Ms,t(X) ≃ HomΛ(P ⊕t 2 , P ⊕s 1 ) sending x ∈ Ms,t(X) to the left multiplication x(·) : P ⊕t 2 → P ⊕s 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus we have a 2-term complex Px := (P ⊕t 2 x(·) −−→ P ⊕s 1 ) ∈ per Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following observation is basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let s, t, u, v ≥ 0, x ∈ Ms,t(X) and y ∈ Mu,v(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) Then we have an exact sequence Mu,s(A) ⊕ Mv,t(B) [(·)x y(·)] −−−−−−→ Mu,t(X) → Homper Λ(Px, Py[1]) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) In particular, Px is presilting if and only if Ms,t(X) = Ms(A)x + xMt(B) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The assertion (a) follows from an exact sequence HomΛ(P ⊕s 1 , P ⊕u 1 ) ⊕ HomΛ(P ⊕t 2 , P ⊕v 2 ) [(·)x y(·)] −−−−−−→ HomΛ(P ⊕t 2 , P ⊕u 1 ) → Homper Λ(Px, Py[1]) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The assertion (b) is immediate from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ The following construction of silting complexes of Λ will be used frequently, where t(XB) (re- spectively, t(AX)) is the minimal number of generators of X as a right B-module (respectively, left A-module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10, assume that A and B are local k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) Σ(Λ) contains cone{(0, −1), (1, −r)} for r := t(XB) = dim(X/XJB)B/JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Σ(Λ) contains cone{(1, 0), (ℓ, −1)} for ℓ := t(AX) = dimA/JA(X/JAX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 13 (c) Let g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , gr be a minimal set of generators of the B-module X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then µ+ 1 (Λ[1]) = Pg ⊕P2[1] ∈ 2-siltΛ holds for g := [g1 · · · gr] ∈ M1,r(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (d) Let h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , hℓ a minimal set of generators of the Aop-module X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then µ− 2 (Λ) = P1 ⊕ Ph ∈ 2-siltΛ holds for h := � h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' hℓ � ∈ Mℓ,1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12, a part of Σ(Λ) has the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(Λ) = P2 P1 Ph=ℓP1−P2 Pg=P1−rP2 P2[1] P1[1] + − µ− 2 (Λ) µ+ 1 (Λ[1]) ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❄❄❄❄❄❄ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ✯✯✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴✴ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We only prove (a)(c) since (b)(d) are the duals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A minimal right (add P2[1])-approximation of P1[1] is given by g(·) : P2[1]⊕r → P1[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus the mutation of Λ[1] at P1[1] is Pg ⊕ P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Now we assume that B is a local algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We fix a minimal set of generators g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , gr of the right B-module X and set g := [g1 · · · gr] ∈ M1,r(X) and g := [g1 · · · gr] ∈ M1,r(X/XJB), where (·) is a canonical surjection X ։ X/XJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have an isomorphism g(·) : Mr,1(B/JB) ≃ X/XJB, and we define a map π : X → Mr,1(B/JB) by π := (X (·) −→ X/XJB (g(·))−1 −−−−−→ Mr,1(B/JB)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each s, t ≥ 0, an entry-wise application of π gives a map π : Ms,t(X) → Ms,t(Mr,1(B/JB)) = Mrs,t(B/JB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In other words, for the identity matrix Is ∈ Ms(k) and gIs := � g O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' O g � ∈ Ms(M1,r(k)) = Ms,rs(k), we have x = (gIs)π(x) for each x ∈ Ms,t(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) Define a morphism of k-algebras φ : Ms(A) → Mrs(B/JB) by a(gIs) = (gIs)φ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Later we will use the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10, assume that B is a local algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let s, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) π : Ms,t(X) → Mrs,t(B/JB) is a morphism of Ms(A)op ⊗k Mt(B)-modules, where we regard Mrs,t(B/JB) as an Ms(A)op-module via φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Let x ∈ Ms,t(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If Px is presilting, then π(x) ∈ Mrs,t(B/JB) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) For any a ∈ Ms(A), x ∈ Ms,t(X) and b ∈ Mt(B), we need to show π(axb) = φ(a)π(x)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In fact, (gIs)φ(a)π(x)b = a(gIs)π(x)b (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) = axb = axb (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) = (gIs)π(axb) gives the desired equality since gIs(·) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 14 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO (b) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11(b), we have Ms,t(X) = Ms(A)x + xMt(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Applying π, we have Mrs,t(B/JB) = π(Ms(A)x + xMt(B)) (a) = φ(Ms(A))π(x) + π(x)Mt(B) ⊂ Mrs(B/JB)π(x) + π(x)Mt(B/JB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus the right-hand side is Mrs,t(B/JB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' This clearly implies that π(x) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ For completeness, we also give the dual statement of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Now we assume that A is a local algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We fix a minimal set of generators h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , hℓ of the left A-module X and set h := � h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' hℓ � ∈ Mℓ,1(X) and h := � h1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' hℓ � ∈ Mℓ,1(X/JAX), where by abuse of notations, (·) is a canonical surjection X ։ X/JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have an isomor- phism (·)h : M1,ℓ(A/JA) ≃ X/JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By abuse of notations, let π := (X (·) −→ X/JAX ((·)h)−1 −−−−−→ M1,ℓ(A/JA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each s, t ≥ 0, an entry-wise application of π gives a map π : Ms,t(X) → Ms,t(M1,ℓ(A/JA)) = Ms,ℓt(A/JA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Define a morphism of k-algebras φ : Mt(B) → Mℓt(A/JA) by (hIt)b = φ(b)(hIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We have the following dual of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10, assume that A is a local algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let s, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) π : Ms,t(X) → Ms,ℓt(A/JA) is a morphism of Ms(A)op ⊗k Mt(B)-modules, where we regard Ms,ℓt(A/JA) as an Mt(B)-module via φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Let x ∈ Ms,t(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If Px is presilting, then π(x) ∈ Ms,ℓt(A/JA) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Uniserial property of g-finite algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' As an application of results in the previous sub- section, we prove the following result, which is not used in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ be a finite dimensional elementary k-algebra, and 1 = e1 + · · · + en the orthogonal primitive idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If Λ is g-finite, then for each 1 ≤ i ̸= j ≤ n, eiΛej/eiΛejJΛej is a uniserial (eiΛei)op-module and eiΛej/eiJΛeiJΛej is a uniserial ejΛej-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thanks to sign decomposition, we can deduce Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='15 from the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A and B be local k-algebras with k ≃ A/JA ≃ B/JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If X is a Aop ⊗k B- module such that � A X 0 B � is g-finite, then X/XJB is a uniserial Aop-module and X/JAX is a uniserial B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16⇒Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Λ is g-finite, so is Γ := (ei + ej)Λ(ei + ej).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7, Γ+− = � eiΛei eiΛej 0 ejΛej � is also g-finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus the assertion follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ In the rest of this subsection, we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following observation plays a key role in the proof, where we identify K0(proj Λ) with Z2 via [A X] �→ (1, 0), [0 B] �→ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ := � A X 0 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Assume that (1, −1) ∈ K0(proj Λ) is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) There exists h ∈ X such that X = Ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 15 (b) Let Λ′ := � A JAX 0 k � and t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If (1, −t) ∈ K0(proj Λ) is rigid, then (1, 1 − t) ∈ K0(proj Λ′) is rigid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11(b), there exists h ∈ X satisfying X = Ah + hk = Ah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11(b), there exists [x1 x2 · · · xt] ∈ M1,t(X) such that M1,t(X) = A[x1 · · · xt] + [x1 · · · xt]Mt(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2) As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2, the element h gives surjections π := (X (·) −→ X/JAX ((·)h)−1 −−−−−→ A/JA = k) and π : M1,t(X) → M1,t(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='14, π(x) ∈ M1,t(k) has full rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By changing indices if necessary, we can assume x1 ∈ A×h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Multiplying an element in A× from left, we can assume x1 = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Multiplying an element in GLt(k) from right, we can assume xi ∈ JAh for each 2 ≤ i ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We claim M1,t−1(JAX) = A[x2 · · · xt] + [x2 · · · xt]Mt−1(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In fact, fix any [y2 · · · yt] ∈ M1,t−1(JAX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2) there exist a ∈ A and b = [bij]1≤i,j≤t ∈ Mt(k) such that [0 y2 · · · yt] = a[h x2 · · · xt] + [h x2 · · · xt]b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3) Applying π, we obtain [0 0 · · · 0] = a[1 0 · · · 0] + [1 0 · · · 0]b in M1,t(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus we obtain b12 = · · · = b1t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Looking at the i-th entries for 2 ≤ i ≤ t of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3), we have [y2 · · · yt] = a[x2 · · · xt] + [x2 · · · xt][bij]2≤i,j≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ We are ready to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We prove that X/XJB is a uniserial Aop-module under a weaker assump- tion that (1, −t) ∈ K0(proj Λ) is rigid for each t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Λ := � A X/XJB 0 k � is a factor algebra of Λ, the element (1, −t) ∈ K0(proj Λ) is rigid for each t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Replacing Λ by Λ, we can assume that B = k and Λ = � A X 0 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We use induction on dimk X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='17(a), the Aop-module X has a unique maximal submodule JAX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ′ = � A JAX 0 k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='17(b), (1, −t) ∈ K0(proj Λ′) is rigid for each t ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By induction hypothesis, JAX is a uniserial Aop-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Therefore X is also a uniserial Aop-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Gluing, Rotation and Subdivision of g-fans 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Gluing fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive idempotents 1 = e1 + e2 ∈ Λ and 1 = e′ 1 + e′ 2 ∈ Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this subsection, we prove the following Gluing Theorem, where we identify K0(proj Λ) and K0(proj Λ′) with Z2 by e1Λ = (1, 0) = e′ 1Λ′ and e2Λ = (0, 1) = e′ 2Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 (Gluing Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive idempotents 1 = e1 + e2 ∈ Λ and 1 = e′ 1 + e′ 2 ∈ Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Assume e1Λe2 = 0 and e′ 2Λ′e′ 1 = 0, or equivalently, Σ(Λ) ∈ Fan +− sc (2) and Σ(Λ′) ∈ Fan −+ sc (2) (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then, there exists an elementary k-algebra Γ such that Σ(Γ) = Σ(Λ) ∗ Σ(Λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) 16 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(Λ) = + − ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ Σ(Λ′) = + − !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ Σ(Γ) = + − ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ The construction of Γ is as follows: We can write Λ = � A X 0 B � and Λ′ = � C 0 Y D � , where A, B, C, D are local k-algebras, X is an Aop ⊗k B-module, and Y is an Dop ⊗k C-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Λ and Λ′ are elementary, we have k ≃ A/JA ≃ B/JB ≃ C/JC ≃ D/JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A ×k C be a fiber product of canonical surjections (·) : A → k and (·) : C → k, that is, A ×k C := {(a, c) ∈ A × C | a = c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let B ×k D be a fibre product of (·) : B → k and (·) : D → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Using the projections A ×k C → A and B ×k D → B, we regard X as an (A ×k C)op ⊗k (B ×k D)-module, and using the projections A ×k C → C and B ×k D → D, we regard Y as an (B ×k D)op ⊗k (A ×k C)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We prove that the algebra Γ := � A ×k C X Y B ×k D � satisfies Σ(Γ) = Σ(Λ) ∗ Σ(Λ′), where the multiplication of the elements of X and those of Y are defined to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It suffices to prove Σ+−(Γ) = Σ+−(Λ) and Σ−+(Γ) = Σ−+(Λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For ǫ = (+, −), we have Γǫ = � A ×k C X 0 B ×k D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The ideal I := � rad C 0 0 rad D � of Γǫ is contained in Iǫ, and we have an isomorphism Γǫ/I ≃ Λ of k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7 to Γ, we get Σ+−(Γ) = Σ+−(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By the same argument, Σ−+(Γ) = Σ−+(Λ′) holds, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ and Λ′ be the following algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Λ := k \uf8ee \uf8ef\uf8ef\uf8f0 1 2 a3 � a4 � a2 � a1 � \uf8f9 \uf8fa\uf8fa\uf8fb ⟨a2 1, a2 2, a2 4, a2a1, a2a3 − a3a4⟩, Λ′ := k � 1 2 b1 � b2 � � ⟨b2 2⟩ By Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11 below, we have Σ(Λ) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='00 = Σ(Λ′) = Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1212 = Let A = e1Λe1, X = e1Λe2, B = e2Λe2, C = e1Λ′e1, Y = e2Λ′e1, D = e2Λ′e2 and Γ = � A ×k C X Y B ×k D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Γ = k \uf8ee \uf8ef\uf8ef\uf8f0 1 2 a3 � a4 � a2 � a1 � b1 � b2 � \uf8f9 \uf8fa\uf8fa\uf8fb ⟨a2 1, a2 2, a2 4, a2a1, a2a3 − a3a4, b2 2⟩ + ⟨aibj, bjai | i ∈ {1, 2, 3, 4}, j ∈ {1, 2}⟩ FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 17 By Gluing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1, we have Σ(Γ) = Σ(Λ) ∗ Σ(Λ′) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1212 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Rotation and Mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this subsection, we explain a connection between the rotation of a fan given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13 and mutation of a 2-term silting complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following main result in this section shows that mutation of an algebra induce the rotation of the g-fan, where we identify K0(proj Λ) with Z2 by e1Λ = (1, 0) and e2Λ = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3 (Rotation Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ be a finite dimensional k-algebra of rank 2 with or- thogonal primitive idempotents 1 = e1 + e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Assume e1Λe2 = 0, or equivalently, Σ(Λ) ∈ Fan +− sc (2) (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then, there exists a finite dimensional k-algebra Γ such that Σ(Γ) = ρ(Σ(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Furthermore, if Λ is elementary, then Γ can be taken to be elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(Λ) = + − ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❄❄❄❄❄❄ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ Σ(Γ) ≃ + −❄❄❄❄❄❄ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3, we need the following preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A be a basic finite dimensional algebra over a field k with |A| = n, and 1 = e1 + · · · + en the orthogonal primitive idempotents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For 1 ≤ i ≤ n and δ ∈ {±1}, consider a half space Rn i,δ := {x1e1 + · · · + xden ∈ Rn | δxi ≥ 0} and define a subfan of Σ by Σi,δ := {σ ∈ Σ | σ ⊂ Rn i,δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' On the other hand, for elements T ≥ T ′ in siltA, we consider the interval [T ′, T ] := {U ∈ siltA | T ≥ U ≥ T ′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following result provides a correspondence of a part of two g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For 1 ≤ i ≤ n, let B := EndA(µ− i (A)), where µ− i (A) = Ti ⊕ (� j̸=i P A j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) [AHIKM1, Threom 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='26] There exists a triangle functor F : Kb(proj A) → Kb(proj B) which satisfies F(Ti) ≃ P B i and F(P A j ) ≃ P B j for each j ̸= i and gives an isomorphism K0(proj A) ≃ K0(proj B) and an isomorphism of fans Σi,−(A) ≃ Σi,+(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) There are isomorphisms (1 − ei)A(1 − ei) ≃ (1 − ei)B(1 − ei) and A/(1 − ei) ≃ B/(1 − ei) of k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Although this is known to experts, we give a proof for convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The first isomorphism is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' To prove the second one, notice that A/(1 − ei) = EndKb(proj A)(P A i )/[A/P A i ] and B/(1−ei) = EndKb(proj A)(Ti)/[T/Ti] hold, where [X] denotes the ideal consisting of morphisms factoring through add X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Pi f−→ Q g−→ Ti h−→ Pi[1] be an exchange triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let a ∈ eiAei = EndKb(proj A)(Pi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since f is a minimal left (add A/Pi)-approximation of Pi, we obtain the following commutative diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Pi f � a� Q g � � Ti h � b� Pi[1] a[1] � Pi f � Q g � Ti h � Pi[1] 18 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO It is routine to check that the desired isomorphism A/(1 − ei)A = EndKb(proj A)(P A i )/[A/P A i ] ≃ B/(1 − ei) = EndKb(proj A)(Ti)/[T/Ti] is given by a �→ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ We are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let T = P Λ 1 ⊕ T2 := µ− 2 (Λ) and E := EndKb(proj Λ)(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4(a), we have a triangle functor F : Kb(proj Λ) → Kb(proj E) which satisfies F(P Λ 1 ) = P E 1 and F(T2) = P E 2 and induces an isomorphism F : K0(proj Λ) ≃ K0(proj E) and an isomorphism of fans F : Σ2,−(Λ) ≃ Σ2,+(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(Λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P Λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='T2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❄❄❄❄❄❄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='⑧ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❄❄❄❄❄❄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ Σ(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2 [1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 [1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='E[1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❄❄❄❄❄❄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ Σ(Γ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2 [1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 [1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='Γ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='Γ[1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❄❄❄❄❄❄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❖ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❲ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='Applying Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7 to E, we obtain a k-algebra Γ := E−+ such that e1Γe2 = 0 and Σ−+(Γ) = Σ−+(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Therefore under the isomorphism K0(proj Γ) ≃ Z2 given by P Γ 1 �→ (0, 1) and P Γ 2 �→ (1, 0), we have Σ(Γ) = ρ(Σ(Λ)), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It remains to prove the last assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4(a), we have isomorphisms e1Ee1 ≃ e1Λe1 and Λ/(e1) ≃ E/(e1) of k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus, if Λ is elementary, then so are E and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ We give two examples of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The first one satisfies E = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ be the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Σ(Λ) is the following fan by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Λ = k � 1 2 a � b � � ⟨b2⟩ Σ(Λ) = Σ1212 = We set µ2(Λ) = T = T1 ⊕ T2 := [e2Λ a· −→ e1Λ] ⊕ e1Λ and E := EndKb(proj Λ)(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have Γ = E = k � 1 2 a � b � � ⟨b2⟩ and Σ(Γ) = ρ(Σ(Λ)) = Σ2121 = The second example satisfies E ̸= Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ be the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Σ(Λ) is the following fan by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Λ = k � 1 2 a � b � c � � ⟨b2, c2, bac⟩ Σ(Λ) = Σ21312 = We set µ2(Λ) = T = T1 ⊕ T2 := [e2Λ ( a· ac·) −−−−→ e1Λ⊕2] ⊕ e1Λ and E := EndKb(proj Λ)(T ), where we switch the indices 1 and 2 unlike the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then E = k � 1 2 a � a′ � b � � ⟨b2, a′b, a′aa′⟩ and Σ(E) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='111 = FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 19 where new arrows a, a′ and b are morphisms in Kb(proj Λ) given by commutative diagrams 0 e1Λ e2Λ e1Λ⊕2 � � ( a· ac·) � ( 0 1) � e2Λ e1Λ⊕2 0 e1Λ ( a· ac·) � � � ( 0 b· ) � e2Λ e1Λ⊕2 e2Λ e1Λ⊕2 ( a· ac·) � c· � ( a· ac·) � ( 0 1 0 0) � respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Γ := E+− = � e1Ee1 e1Ee2 0 e2Ee2 � = � ⟨e1, b, aa′, baa′⟩k ⟨a, ba, aa′a, baa′a⟩k 0 ⟨e2, a′a⟩k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Γ = k \uf8ee \uf8ef\uf8ef\uf8f0 1 2 a � c � b � b′ � \uf8f9 \uf8fa\uf8fa\uf8fb ⟨b2, b′2, c2, b′b, b′a − ac⟩ and Σ(Γ) = ρ(Σ(Λ)) = Σ13122 = where b′ := aa′ and c := a′a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Subdivision and Extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this section, we realize subdivisions of g-fans of rank 2 by extensions of algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following theorem is a main result of this section, where we identify K0(proj Λ) with Z2 by e1Λ = (1, 0) and e2Λ = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7 (Subdivision Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ be a finite dimensional elementary k-algebra with orthogonal primitive idempotents 1 = e1+e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Assume e1Λe2 = 0, or equivalently, Σ(Λ) ∈ Fan +− sc (2) (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then, for cones σ = C(µ+ 1 (Λ[1])) and σ′ := C(µ− 2 (Λ)) of Σ(Λ), there exist finite dimensional elementary k-algebras Γ and Γ′ such that Σ(Γ) = Dσ(Σ(Λ)) and Σ(Γ′) = Dσ′(Σ(Λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='Σ(Λ) = P2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P2[1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='P1[1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='µ− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2 (Λ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='µ+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 (Λ[1]) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='❚ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='✴✴✴✴✴✴✴✴✴ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='In the rest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' we only prove the existence of Γ since the existence of Γ′ is the dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The construction of Γ is as follows: Construction 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5, we can write Λ = � A X 0 B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' where A, B are local k-algebras and X is an Aop ⊗k B-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Λ is elementary, we have k ≃ A/JA ≃ B/JB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let X := X/XJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then the k-dual DX is an A-module, and we regard it as an Aop-module by using the action of k through the natural surjection A → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let C := A ⊕ DX be a trivial extension algebra of A by DX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let (·) : A → k, (·) : B → k and (·) : X → X 20 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO be canonical surjections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We regard Y := � k X � as a Cop ⊗k B-module by (a, f) · [ α x ] · b := � aαb+f(x)b axb � for (a, f) ∈ C = A ⊕ DX, [ α x ] ∈ Y = � k X � and b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we set Γ := � C Y 0 B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In the rest of this subsection, we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We set Q1 := [C Y ], Q2 := [0 B] ∈ proj Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For y ∈ Ms,t(Y ) ≃ HomΓ(Q⊕t 2 , Q⊕s 1 ), we define Qy := [Q⊕t 2 y(·) −−→ Q⊕s 1 ] ∈ Kb(proj Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We fix a minimal set of generators g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , gr of the B-module X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then (g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , gr) forms a k-basis of X = X/XJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Set g := [g1 · · · gr] ∈ M1,r(X) and g := [g1 · · · gr] ∈ M1,r(X/XJB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We need the following easy observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(Γ) contains cone{(0, 1), (1, −r−1)} and cone{(1, −r−1), (1, −r)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' More explicitly, let � 0 g � ∈ M1,r(Y ) and � 0 1 g 0 � ∈ M1,r+1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Q� 0 1 g 0 � ⊕ Q2[1] and Q� 0 g � ⊕ Q� 0 1 g 0 � belong to 2-siltΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A minimal set of generators of the B-module Y is given by the r+1 columns of � 0 1 g 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus Q� 0 1 g 0 � ⊕ Q2[1] ∈ 2-siltΓ holds by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In the rest, we prove that T := Q� 0 g � ⊕ Q� 0 1 g 0 � is basic silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By the first statement, Q� 0 1 g 0 � is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If Q� 0 g � is not indecomposable, then |T | is bigger than two, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus T is basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We will show that T is presilting by using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By our choice of g, we have gMr,1(B) = X and (DX)g = M1,r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus we have � 0 g � Mr(B) = M1,r([ 0 X ]) and (DX) � 0 g � = M1,r([ k 0 ]), and hence C � 0 g � + � 0 g � Mr(B) ⊃ (DX) � 0 g � + � 0 g � Mr(B) = M1,r([ 0 X ]) + M1,r([ k 0 ]) = M1,r(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' This clearly implies C � 0 g � + � 0 1 g 0 � Mr+1,r(B) = M1,r(Y ), and a similar argument implies C � 0 1 g 0 � + � 0 g � Mr,r+1(B) = M1,r+1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11(b) implies that T is presilting, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2, the element g gives a surjection π := (X (·) −→ X (g(·))−1 −−−−−→ Mr,1(B) = Mr,1(k)), which extends to the map π : Ms,t(X) → Mrs,t(k) for each s, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following observation is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let s, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For x ∈ Ms,t(X), consider [ 0 x ] ∈ Ms,t(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 21 (a) Px is indecomposable in Kb(proj Λ) if and only if Q[ 0 x] is indecomposable in Kb(proj Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) If Q[ 0 x] is a presilting complex of Γ, then Px is a presilting complex of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (c) The converse of (b) holds if t ≤ rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (d) The restriction of Σ(Γ) to {(x, y) ∈ R2 | 0 ≤ −y ≤ rx} coincides with that of Σ(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Notice that Γ is the trivial extension Λ ⊕ I of Λ by the following ideal I of Γ: I := � DX k 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) Since Px ≃ Q[ 0 x] ⊗Γ Λ and Q[ 0 x] ≃ Px ⊗Λ Γ, the assertion follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Since Λ = Γ/I and Q[ 0 x] ⊗Γ Λ ≃ Px, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (c) Assume that Px is a presilting complex of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11(b), we have Ms,t(X) = Ms(A)x + xMt(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2) Again by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11(b), it suffices to show the equality V := Ms(C) [ 0 x ] + [ 0 x ] Mt(B) = Ms,t( � k X � ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since V ⊃ Ms(A) [ 0 x ] + [ 0 x ] Mt(B) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2) = Ms,t([ 0 X ]) holds, it suffices to show V ⊃ Ms,t([ k 0 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3) By our assumption t ≤ rs and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13(b), π(x) has rank t and the map (·)π(x) : Ms,rs(k) → Ms,t(k) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote by g∗ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , g∗ r the basis of DX which is dual to g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then the map (·) � g∗ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' g∗ r � : M1,r(k) ≃ DX is a bijection, and we denote its inverse by π′ : DX ≃ M1,r(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It gives a bijection π′ : Ms(DX) ≃ Ms,rs(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We have a commutative diagram Ms(DX) × Ms,t(X) π′×π � eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' �❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ ❱ Ms,rs(k) × Mrs,t(k) mult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' � Ms,t(k) where eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' is given by the evaluation map DX × X → DX × X → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus the commutativity of the diagram above and the surjectivity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4) shows that the map (·)x : Ms(DX) → Ms,t(k) is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Therefore the desired claim (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3) follows from V ⊃ Ms(C) [ 0 x ] ⊃ Ms(DX) [ 0 x ] = Ms,t([ k 0 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ (d) Immediate from (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The assertion follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='9 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ We give two examples of Subdivision Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 22 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ be the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Σ(Λ) is the following fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Λ = k[1 → 2] Σ(Λ) = Σ111 = Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7 to Λ, we get Γ := � k ⊕ Dk � k k � 0 k � = k � 1 2 � b � � ⟨b2⟩ and Σ(Γ) = D3(Σ(Λ)) = Σ1212 = Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ be the following algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Σ(Λ) is the following fan by Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Λ = k � 1 2 a � b � � ⟨b2⟩ Σ(Λ) = Σ2121 = Applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7 to Λ, we get Γ := � k ⊕ D(ka) � k ⟨a,ab⟩k � 0 ⟨e2, b⟩k � = k � 1 2 a � c � b � � ⟨b2, c2, cab⟩ and Σ(Γ) = D4(Σ(Λ)) = Σ21312 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let k be a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For a finite dimensional k-algebras Λ of rank 2, we regard the g-fan Σ(Λ) as a fan in R2 by isomorphism K0(proj Λ) ≃ R2 given by P1 �→ (1, 0) and P2 �→ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We denote by k-Fan(2) the subset of Fansc(2) consisting of g-fans of finite dimensional k-algebras of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let k-Fanel(2) be the subset of k-Fan(2) consisting of g-fans of finite dimensional elementary k-algebras of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following is a main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For any field k, we have k-Fanel(2) = k-Fan(2) = Fansc(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5) That is, any sign-coherent fan in R2 can be realized as a g-fan Σ(Λ) of some finite dimensional elementary k-algebra Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It suffices to show Fansc(2) = k-Fanel(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let k-Fan+− el (2) := k-Fanel(2) ∩ Fan+− sc (2) and k-Fan−+ el (2) := k-Fanel(2) ∩ Fan−+ sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Gluing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1, we have k-Fanel(2) = k-Fan+− el (2) ∗ k-Fan−+ el (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Rotation Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3, k-Fan+− el (2) is closed under rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='14(b), k-Fan+− el (2) is closed under subdivisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) = Σ(k × k) ∈ k-Fan+− el (2), Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10 implies k-Fan+− el (2) = Fan+− sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Similarly, we have k-Fan−+ el (2) = Fan−+ sc (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Consequently, we have Fansc(2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5) = Fan+− sc (2) ∗ Fan−+ sc (2) = k-Fan+− el (2) ∗ k-Fan−+ el (2) = k-Fanel(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ For given Σ ∈ Fansc(2), our proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13 gives a concrete algorithm to construct a finite dimensional k-algebra Λ satisfying Σ(Λ) = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We demonstrate it in the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We construct a finite dimensional k-algebra Γ satisfying Σ(Γ) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1212 by the following three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 23 (I) We obtain a finite dimensional k-algebra Λ = k \uf8ee \uf8ef\uf8ef\uf8f0 1 2 a3 � a4 � a2 � a1 � \uf8f9 \uf8fa\uf8fa\uf8fb ⟨a2 1, a2 2, a2 4, a2a1, a2a3 − a3a4⟩ satisfying Σ(Λ) = Σ13122;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='00 by using Rotation Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3 and Subdivision Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ00 D2 � Σ111 D3 Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11 � Σ1212 ρ Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5 � Σ2121 D4 Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12 � Σ21312 ρ Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6 � Σ13122 (II) Similarly, we obtain a finite dimensional k-algebra Λ′ := k � 1 2 b1 � b2 � � ⟨b2 2⟩ satisfying Σ(Λ′) = Σ(0, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 1, 2, 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (III) We obtain a finite dimensional k-algebra Γ = k \uf8ee \uf8ef\uf8ef\uf8f0 1 2 a3 � a4 � a2 � a1 � b1 � b2 � \uf8f9 \uf8fa\uf8fa\uf8fb ⟨a2 1, a2 2, a2 4, a2a1, a2a3 − a3a4, b2 2⟩ + ⟨aibj, bjai | i ∈ {1, 2, 3, 4}, j ∈ {1, 2}⟩ satisfying Σ(Γ) = Σ(1, 3, 1, 2, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 1, 2, 1, 2) by applying Gluing Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 to Λ and Λ′, see Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(Λ) = Σ(Λ′) = Σ(Γ) = Σ(Λ) ∗ Σ(Λ′) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Gluing fans II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this subsection, we study another type of gluing g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Results in this subsection will not be used in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ and Λ′ be elementary k-algebras of rank 2 with orthogonal primitive idem- potents 1 = e1 + e2 ∈ Λ and 1 = e′ 1 + e′ 2 ∈ Λ′ satisfying e1Λe2 = 0, e′ 1Λ′e′ 2 = 0, σ = cone{(0, −1), (1, −1)} ∈ Σ(Λ) and σ′ = cone{(1, −1), (1, 0)} ∈ Σ(Λ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6) Then, there exists an elementary k-algebra Γ such that Σ2(Γ) = (Σ2(Λ) \\ {σ}) ∪ (Σ2(Λ′) \\ {σ′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='15 is explained by the following picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(Λ) = + − ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' P1 P2 σ ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ Σ(Λ′) = + − !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' P ′ 1 P ′ 2 σ′ ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ Σ(Γ) = + − !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Q1 Q2 ❄❄❄❄❄❄ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ 24 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO The assumption (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6) is equivalent to that the defining sequences can be written as Σ(Λ) = Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−1, 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0) and Σ(Λ′) = Σ(1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , bm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this case, the defining sequence of Σ(Γ) is given by Σ(Γ) = Σ(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an−2, an−1 + b2 − 1, b3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , bm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The rest of this section is devoted to proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By our assumption, we can write Λ = � A X 0 B � and P1 := [A X], P2 := [0 B] ∈ proj Λ, Λ′ = � C Y 0 D � and P ′ 1 := [C Y ], P ′ 2 := [0 D] ∈ proj Λ′, where A, B, C, D are local k-algebras such that k ≃ A/JA ≃ B/JB ≃ C/JC ≃ D/JD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' X is an Aop ⊗k B-module and Y is an Cop ⊗k D-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' There exist g ∈ X and h ∈ Y such that X = gB ̸= 0 and Y = Ch ̸= 0 by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The construction of Γ is as follows: Let A ×k C (respectively, B ×k D) be a fibre product of canonical surjections A → k and C → k (respectively, B → k and D → k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2, we consider maps π : X → X/XJB (g(·))−1 −−−−−→ B/JB = k and π′ : Y → Y/JCY ((·)h)−1 −−−−−→ C/JC = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7) Let X×kY be a fibre product of π : X → k and π′ : Y → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then X×kY is a (A×kC)op⊗k(B×kD)- module, and let Γ := � A ×k C X ×k Y 0 B ×k D � and Q1 := [A ×k C X ×k Y ], Q2 := [0 B ×k D] ∈ proj Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Consider ideals of Γ by I = � JC JCY 0 JD � and I′ = � JA XJB 0 JB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then there exist isomorphisms of k-algebras Γ/I ≃ Λ and Γ/I′ ≃ Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='8) As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2, for s, t ≥ 0, x ∈ Ms,t(X), y ∈ Ms,t(Y ) and (x′, y′) ∈ Ms,t(X ×k Y ), we define Px := (P ⊕t 2 x(·) −−→ P ⊕s 1 ) ∈ per Λ, P ′ y := (P ′ 2 ⊕t y(·) −−→ P ′ 1 ⊕s) ∈ per Λ′ Q(x,y) := (Q⊕t 2 (x′,y′)(·) −−−−−−→ Q⊕s 1 ) ∈ per Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let s, t ≥ 0 and (x, y) ∈ Ms,t(X ×k Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' If Q(x,y) is a presilting complex of Γ, then Px is a presilting complex of Λ and P ′ y is a presilting complex of Λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='8) and Q(x,y) ⊗Γ Λ = Px, the complex Px is presilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The complex P ′ y is presilting similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Define maps (·) : A → C and (·) : B → D as the compositions of canonical maps (·) : A (·) −→ k ⊂ C and (·) : B (·) −→ k ⊂ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Using π and π′ in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='7), define maps (·) : X → Y and (·) : Y → X by (·) : X π−→ k (·)h −−→ kh ⊂ Y and (·) : Y π′ −→ k (·)g −−→ kg ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='9) Then the first projection X ×k Y → X, (x, y) �→ x has a section given by X → X ×k Y, x �→ (x, x), FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 25 and the second projection X ×k Y → Y , (x, y) �→ y has a section given by Y → X ×k Y, y �→ (y, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following is a crucial result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) Let s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For x ∈ Ms,t(X), consider (x, x) ∈ Ms,t(X ⊗k Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Px is a presilting complex of Λ if and only if Q(x,x) is a presilting complex of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Let s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For y ∈ Ms,t(Y ), consider (y, y) ∈ Mc,d(X ⊗k Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then P ′ y is a presilting complex of Λ′ if and only if Q(y,y) is a presilting complex of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It suffices to prove (a) since (b) is dual to (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The “if” part is clear from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We prove the “only if” part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11, it suffices to show Ms,t(X ×k Y ) = Ms(A ×k C)(x, x) + (x, x)Mt(B ×k D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since X ×k Y = {(0, y) | y ∈ JCY } + {(z, z) | z ∈ X}, it suffices to show the following assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (i) For each y ∈ Ms,t(JCY ), we have (0, y) ∈ Ms(A ×k C)(x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (ii) For each z ∈ Ms,t(X), we have (z, z) ∈ Ms(A ×k C)(x, x) + (x, x)Mt(B ×k D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We prove (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Px is presilting, π(x) ∈ Ms,t(k) has full rank by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since s ≥ t, the map (·)π(x) : Ms(k) → Ms,t(k) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Applying JC ⊗k −, the map (·)π(x) : Ms(JC) → Ms,t(JC) is also surjective, and so is the composition (·)x (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='9) = (·)π(x)h : Ms(JC) (·)π(x) −−−−→ Ms,t(JC) (·)h −−→ Ms,t(JCY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Therefore there exists c ∈ Ms(JC) such that y = cx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then (0, c) ∈ Ms(A ×k C) satisfies (0, c)(x, x) = (0, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Px is presilting, we have Ms,t(X) = Ms(A)x + xMt(B) by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus there exist a ∈ Ms(A) and b ∈ Mt(B) such that z = ax + xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then (a, a)(x, x) + (x, x)(b, b) = (ax + xb, ax + xb) = (z, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Now we are ready to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5, each of Σ(Λ), Σ(Λ′) and Σ(Γ) contains cone{(−1, 0), (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='16 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='17, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (i) Let s ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then there exists x ∈ Ms,t(X) such that Px is presilting if and only if there exists (x, y) ∈ Ms,t(X ×k Y ) such that Q(s,t) is presilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (ii) Let s ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then there exists y ∈ Ms,t(Y ) such that P ′ y is presilting if and only if there exists (x, y) ∈ Ms,t(X ×k Y ) such that Q(s,t) is presilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Therefore the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let Λ and Λ′ be the following algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Λ := k � 1 2 a � b � � ⟨b2⟩ Λ′ := k \uf8ee \uf8ef\uf8ef\uf8f0 1 2 a � d � c1 � c2 � \uf8f9 \uf8fa\uf8fa\uf8fb ⟨c2 1, c2 2, d2, c1c2, c1a − ad⟩ 26 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO By Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6, we have Σ(Λ) = Σ2121 = Σ(Λ′) = Σ13122 = Let Γ := � e1Λe1 ×k e1Λ′e1 e1Λe2 ×k e1Λ′e2 0 e2Λe2 ×k e2Λ′e2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have Γ = k \uf8ee \uf8ef\uf8ef\uf8f0 1 2 a � b � d � c1 � c2 � \uf8f9 \uf8fa\uf8fa\uf8fb ⟨b2, c2 1, c2 2, d2, c1c2, c1a − ad, c2ab, bd, db⟩ and Σ(Γ) = Σ214122 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' g-Convex algebras of rank 2 In this section, we will characterize algebras of rank 2 which have convex g-polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Characterizations of g-convex algebras of rank 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let e, e′ be pairwise orthogonal prim- itive idempotents in A and x ∈ eAe′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' x ∈ eAe′ is a left generator (respectively, right generator) of eAe′ if eAx = eAe′ (respectively, xAe′ = eAe′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Define subalgebras Lx ⊂ e′Ae′ and Rx ⊂ eAe as follows (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Rx := {a ∈ eAe | ax ∈ xAe′} and Lx := {a ∈ e′Ae′ | xa ∈ eAx}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Recall that, for an algebra Λ and a right (respectively, left) Λ-module M, we denote by t(MΛ) (respectively, t(ΛM)) the minimal number of generators of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A be a basic finite dimensional algebra, {e1, e2} a complete set of primitive orthogonal idempotents in A and Pi = eiA (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) A is g-convex if and only if Σ(A) = Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (b) Let (l, r) := (t(e1Ae1e1Ae2), t(e1Ae2e2Ae2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(A) = Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some b if and only if (l, r) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(A) = Σ111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some b if and only if (l, r) = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(A) = Σ1212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some b if and only if (l, r) = (1, 2) and t(Rxe1Ae1) = 2 hold for some left generator x of e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ(A) = Σ2121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some b if and only if (l, r) = (2, 1) and t(e2Ae2Lx) = 2 hold for some right generator x of e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Σ00;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b P2 P1 ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ⑧⑧⑧⑧⑧⑧ Σ111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b P2 P1 ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ Σ1212;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b P2 P1 ❄ ❄ ❄ ❄ ❄ ❄ ✴✴✴✴✴✴✴✴✴ ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ Σ2121;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b P2 P1 ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For a left (respectively, right) generator x of e1Ae2, Rx (respectively, Lx) is unique up to conjugacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In particular, t(Rxe1Ae1) (respectively, t(e2Ae2Lx)) does not depend on the choice of a left (respectively, right) generator x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 27 Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (a) Here, we give algebras which realize 7 convex g-fans up to isomorphism of g-fans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We define Ai = kQ/I (i ∈ {1, 2, 3, 4, 5, 6, 7}) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A1 = k � 1 2 a � b� c � d � � ⟨ab, ad, ba, bc, c2, d2⟩ A2 = k � 1 2 a � b� d � � ⟨ab, ba, d2⟩ A3 = � 1 2 a � b� d � � ⟨ab, ba, ad, d2⟩ A4 = k � 1 2 b� d � � ⟨d2⟩ A5 = k � 1 2 a � b� � ⟨ab, ba⟩ A6 = k � 1 2 b� � A7 = k � 1 2 � Then the g-fans Σ(Ai) (i ∈ {1, 2, 3, 4, 5, 6, 7}) are given by the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' i 1 2 3 4 5 6 7 Σ(Ai) + − ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ✴✴✴✴✴✴✴ ❄❄❄❄❄ ❖❖❖❖❖❖❖ + − ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❄❄❄❄❄ ❄❄❄❄❄ ❖❖❖❖❖❖❖ + − ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❖❖❖❖❖❖❖ + − ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❖❖❖❖❖❖❖ + − ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ + − ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ❄❄❄❄❄ + − ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ (b) Let K/k be a field extension with degree two,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' and A be a k-algebra �k K 0 K � with e1 = �1 0 0 0 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' e2 = �0 0 0 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' We write K = k(t) and set x := �0 1 0 0 � ∈ e1Ae2, u = �0 0 0 t � ∈ e2Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have Lx = �0 0 0 k � , u ̸∈ Lx, and the following equations hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' e1Ae2 = � 0 K 0 0 � = xAe2 = e1Ax + e1Axu e2Ae2 = �0 0 0 K � = Lx + uLx Further, we have e2Ae1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Therefore, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1 implies that Σ(A) has the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' P2 P1 ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' In this subsection, we prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The following observation shows Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1(a) and gives another proof of [AHIKM1, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let A be as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then A is g-convex if and only if Σ(A) = Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b for some a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The “if” part is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Conversely, assume that A is g-convex and Σ(A) = Σa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='b with a = (a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , an) and b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' , bm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then ai ≤ 2 and bj ≤ 2 hold for each i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='10, it is easy to check that a, b ∈ {(0, 0), (1, 1, 1), (1, 2, 1, 2), (2, 1, 2, 1)} holds (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Next we show the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Let x ∈ e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then Lx is a subalgebra of e2Ae2, and Rx is a subalgebra of e1Ae1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' This is a special case of the following easy fact: Let A, B be rings, M an (A, B)-module, and x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then {b ∈ B | xb ∈ Ax} is a subring of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Now we give a key observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2, for s, t ≥ 0, x ∈ Ms,t(e1Ae2), we define Px := (e2A⊕t x(·) −−→ e1A⊕s) ∈ Kb(proj A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Assume t(e1Ae1e1Ae2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For a left generator x ∈ e1Ae2, the following conditions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 28 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO (1) Σ(A) contains cone{(1, −1), (1, −2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (2) t(Rxe1Ae2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (3) e1Ay + xM1,2(e2Ae2) = M1,2(e1Ae2) holds for some u ∈ e1Ae1 \\ Rx and y := [x ux].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Notice that Px is indecomposable presilting by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (1)⇒(2) If t(Rxe1Ae1) = 1, then e1Ae1 = Rx holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus e1Ae2 = e1Ax ⊂ xAe2 holds, and thus x is a right generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12, Px ⊕ P2[1] ∈ 2-siltA holds, a contradiction to cone{(1, −1), (1, −2)} ∈ Σ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus it suffices to prove t(Rxe1Ae1) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since cone{(1, −1), (1, −2)} ∈ Σ(A), there exists y = [x1 x2] ∈ M1,2(e1Ae2) such that Px ⊕ Py is silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11, we have M1,2(e1Ae2) = e1Ay + yM2,2(e2Ae2), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1) M1,2(e1Ae2) = e1Ay + xM1,2(e2Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2) Looking at the first entry of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1), at least one of x1 and x2 does not belong to rade1Ae1 e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Without loss of generality, assume x1 /∈ rade1Ae1 e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then there exists a ∈ (e1Ae1)× such that x = ax1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Py ≃ Pay, we can assume x1 = x by replacing y by ay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since x is a left generator, there exists u ∈ e1Ae1 such that x2 = ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Consequently, we can assume y = [x ux].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' For each a ∈ e1Ae1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='2) implies that there exist a′ ∈ e1Ae1 and b, b′ ∈ e2Ae2 such that [0 ax] = a′[x ux] + x[b b′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then a′ and a−a′u are in Rx, and hence a = a′u+(a−a′u) ∈ Rxu+Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus e1Ae1 = Rx +Rxu and t(Rxe1Ae1) ≤ 2 hold, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (2)⇒(3) Since t(Rxe1Ae2) = 2 and Rx ̸⊂ radRx e1Ae1, there exists u ∈ e1Ae1 \\ Rx such that Rxu + Rx = e1Ae1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Multiplying x from the right, we have Rxux + Rxx = e1Ax = e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since Rxx ⊂ xAe2, we have Rxux + xAe2 = e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3) To prove (3), take any [z w] ∈ M1,2(e1Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since x is a left generator, there exists a ∈ e1Ae1 such that z = ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='3), there exist r ∈ Rx and b ∈ e2Ae2 such that w − aux = rux + xb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By definition of Rx, there exists c ∈ e2Ae2 such that rx = xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Then we have [z w] = (a + r)[x ux] + x[−c b] ∈ e1Ay + xM1,2(e2Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' (3)⇒(1) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='11, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Px is presilting if and only if (i) e1Ax + xAe2 = e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Py is presilting if and only if (ii) e1Ay + yM2,2(e2Ae2) = M1,2(e1Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' HomKb(proj A)(Px, Py[1]) = 0 if and only if (iii) e1Ax + yM2,1(e2Ae2) = e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' HomKb(proj A)(Py, Px[1]) = 0 if and only if (iv) e1Ay + xM1,2(e2Ae2) = M1,2(e1Ae2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It is clear that (iv) implies (ii), and (i) implies (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By looking at the first entry of the row vector, (iv) implies (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Our assumption (3) implies that (iv) holds, and hence (i)-(iii) also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus Px ⊕ Py is presilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' It remains to show that Py is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Suppose that Py is decomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By considering g-vector, we have that Py ≃ e2A[1] ⊕ Pz for some z ∈ e1Ae2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Since [Pz] = [Px], we have Pz ≃ Px by [DIJ, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' This shows that e2A[1] ⊕ Px is silting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12, we have xAe2 = e1Ae2 and Rx = eAe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' This contradicts u ̸∈ Rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ We are ready to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The first and second statements follow from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='5 and Propo- sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' FANS AND POLYTOPES IN TILTING THEORY II: g-FANS OF RANK 2 29 We prove the third statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='12, cone{(1, 0), (1, −1)} ∈ Σ(A) if and only if t(e1Ae1e1Ae2) = 1, and cone{(0, −1), (1, −2)} ∈ Σ(A) if and only if t(e1Ae2e2Ae2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Thus the assertion follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' The fourth statement is the dual of the third statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' □ Acknowledgments T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='A is supported by JSPS Grants-in-Aid for Scientific Research JP19J11408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='H is supported by JSPS Grant-in-Aid for Scientists Research (C) 20K03513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='I is supported by JSPS Grant- in-Aid for Scientific Research (B) 16H03923, (C) 18K3209 and (S) 15H05738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='K is supported by JSPS Grant-in-Aid for Young Scientists (B) 17K14169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='M is 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Treffinger, Wall and chamber structure for finite-dimensional algebras, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 354 (2019), 106746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [DIJ] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Demonet, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Iyama, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Jasso, τ-tilting finite algebras, bricks, and g-vectors, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' IMRN 2019, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 3, 852–892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [DF] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Derksen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Fei, General presentations of algebras, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 278 (2015), 210–237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [FZ1] S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 1, 63–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [FZ2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Fomin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Zelevinsky, Cluster algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Coefficients, Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' IMRN 2021, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 12, 9567–9607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [F] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Fulton, Introduction to Toric Varieties, Ann of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Studies 131, Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Press, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [H1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Hille, On the volume of a tilting module, Abh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Hamburg 76 (2006), 261–277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [H2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Hille, Tilting Modules over the Path Algebra of Type A, Polytopes, and Catalan Numbers, Lie algebras and related topics, 91–101, Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=', 652, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=', Providence, RI, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [KV] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Keller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Vossieck, Aisles in derived categories, Deuxi`eme Contact Franco-Belge en Alg`ebre (Faulx-les- Tombes, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Belg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' S´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' A 40 (1988), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 2, 239–253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [NZ] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Nakanishi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Zelevinsky, On tropical dualities in cluster algebras, Algebraic groups and quantum groups, 217–226, Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=', 565, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=', Providence, RI, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' [Pl] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Plamondon, Generic bases for cluster algebras from the cluster category, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' IMRN 2013, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 10, 2368–2420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content=' 30 TOSHITAKA AOKI, AKIHIRO HIGASHITANI, OSAMU IYAMA, RYOICHI KASE, AND YUYA MIZUNO Graduate School of Information Science and Technology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan Email address: aoki-t@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='jp Department of Pure and Applied Mathematics, Graduate School of Information Science and Tech- nology, Osaka University, 1-5 Yamadaoka, Suita, Osaka 565-0871, Japan Email address: higashitani@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='jp Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-8914, Japan Email address: iyama@ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='jp Department of Information Science and Engineering, Okayama University of Science, 1-1 Ridaicho, Kita-ku, Okayama 700-0005, Japan Email address: r-kase@ous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='jp Faculty of Liberal Arts, Sciences and Global Education / Graduate School of Science, Osaka Met- ropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan Email address: yuya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='mizuno@omu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} +page_content='jp' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAzT4oBgHgl3EQfiP3j/content/2301.01498v1.pdf'} diff --git a/EdFJT4oBgHgl3EQfCCz-/content/tmp_files/2301.11428v1.pdf.txt b/EdFJT4oBgHgl3EQfCCz-/content/tmp_files/2301.11428v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c6b127dc073fd1641603bcb3d298bd126de246a5 --- /dev/null +++ b/EdFJT4oBgHgl3EQfCCz-/content/tmp_files/2301.11428v1.pdf.txt @@ -0,0 +1,21942 @@ +Recursive characterisations of random matrix +ensembles and associated combinatorial objects +Anas A. Rahman +ORCID iD: 0000-0003-2317-6685 +Doctor of Philosophy – Science +January, 2022 +The School of Mathematics and Statistics +The University of Melbourne +Submitted in total fulfilment for the degree of +Doctor of Philosophy – Science +arXiv:2301.11428v1 [math-ph] 26 Jan 2023 + + +Abstract +Mixed moments and cumulants of random matrices have been studied extensively over the +last half-century with applications in a large variety of fields ranging from enumerative geometry +to quantum mechanics. It is particularly interesting to expand the cumulants in 1/N, where N +denotes matrix size, and study the coefficients of these expansions along with their generating +functions, which we call correlator expansion coefficients. We give an overview of the recursive +characterisations of random matrix ensembles that are currently at the forefront of random matrix +theory by way of studying two classes of ensembles using two different types of recursive schemes: +Established theory on Selberg correlation integrals is used to derive linear differential equations +on the eigenvalue densities and resolvents of the classical matrix ensembles, which lead to 1-point +recursions, understood to be analogues of the Harer–Zagier recursion, for the expansion coefficients +of the associated 1-point cumulants, while loop equation analysis is used to recursively compute some +leading order correlator expansion coefficients pertaining to certain products of random matrices +that have recently come into interest due to their connections to Muttalib–Borodin ensembles and +integrals of Harish-Chandra–Itzykson–Zuber type. We also show how the aforementioned differential +equations can be used to characterise the large N limiting statistics of the classical matrix ensembles’ +eigenvalue densities in the global and edge scaling regimes. +A common theme between the two classes of ensembles studied in this thesis is that their +representative random matrices can, for the most part, be constructed from Ginibre matrices (matrices +whose entries are independently and identically distributed normal variables), which allows for +their mixed moments and cumulants to be interpreted as enumerations of particular ribbon graphs, +topological and combinatorial maps, and/or topological and combinatorial hypermaps. A major +part of this thesis is devoted to a comprehensive review of how the Isserlis–Wick theorem implies +these interpretations for the mixed moments and cumulants of the Gaussian and Laguerre ensembles, +which leads on to original discussion on how the relevant theory extends to the matrix products +mentioned above. Thus, the loop equations derived in this thesis have the added value of solving +certain problems in enumerative combinatorics. +In order to make this thesis self-contained and to properly motivate our original contributions, +a decent portion of our development constitutes a pedagogically detailed survey of the contiguous +literature. It is therefore expected that this thesis will serve as a valuable resource for readers +wanting a well-rounded introduction to classical matrix ensembles, matrix product ensembles, 1-point +recursions, loop equations, Selberg correlation integrals, and ribbon graphs. +i + +Declaration +This is to certify that: +1. the thesis comprises only my original work towards the degree of Doctor of Philosophy +except where indicated in the preface; +2. due acknowledgement has been made in the text to all other material used; and +3. the thesis is fewer than 100 000 words in length, exclusive of bibliographies, tables, +maps, and appendices. +Anas A. Rahman +January 22, 2022 +ii + +Preface +The introductory content of Chapter 1 is wholly original to this thesis. So too are the contents +of Chapter 2 preceding §2.1.2, whereas the content of §2.1.2 is drawn from the work “Linear +differential equations for the resolvents of the classical matrix ensembles” [279] done by +the present author in collaboration with Peter Forrester and published by Random Matrices: +Theory and Applications in 2021. Section 2.2 contains a mild enhancement of content from the +work “Relations between moments for the Jacobi and Cauchy random matrix ensembles” +[130] done by the present author in collaboration with Peter Forrester and published by the +Journal of Mathematical Physics in 2021. The contribution of the present author towards the +works [279], [130] is equal to that of Forrester. +Thereafter, the contents of Sections 2.3, 2.4, and 3.1 are amalgamations of content from +[279], [130] reworked slightly for coherence and supplemented with additional, original +results for completeness; Appendix B is taken from [279]. In §3.3.3, Section 4.2, Section 4.3, +Appendix C, and Appendix D, we give a heavily detailed presentation of the ongoing work +“Combinatorics and loop equations for antisymmetrised and Hermitised matrix product +ensembles” [75] of the present author and Stephane Dartois. The results pertaining to [75] are +born of vigorous discussions between the present author and Dartois, but the corresponding +outcomes displayed in this thesis are due solely to the present author. +The 2017 article [142] by the present author, Peter Forrester, and Nicholas Witte is briefly +reviewed in §4.1.1, but is work mostly completed prior to commencement of this thesis. Thus, +it is cited appropriately where necessary. All other contents of this thesis not addressed in +the above are original to this thesis and have not been published elsewhere. +The candidature of the author was financially supported by the Australian Government +Research Training Program Scholarship, the ARC Centre of Excellence for Mathematical and +Statistical Frontiers, and the ARC Grant DP210102887. +iii + +Acknowledgements +First and foremost, I would like to thank my supervisors Peter J. Forrester, Mario Kieburg, +and Paul T. Norbury for the many years of patient guidance through the various topics +encountered in this thesis. In particular, Peter’s uncanny ability to recall obscure tidbits from +his vast knowledgebase and provide insightful feedback on written text combined with his +generous willingness to do so upon request at almost any time of day, regardless of frequency, +has most certainly been a privilege to experience. It would have been impossible for me +to learn the amount of random matrix theory that I did in the last few years without the +countless hours of effort put in by Peter. Likewise, I would like to thank my co-supervisors +Mario and Paul for introducing me to topics in analysis and geometry, respectively, which +have contributed to my growth as a mathematician. +Next, I would like to thank those who would explain things to me for hours on end or +just lend me a friendly ear without any obligation to do so: My various interactions with +Stephane Dartois, Jesper Ipsen, Shi-Hao Li, Anthony Mays, Norman Do, David Ridout, Nora +Ganter, Thomas Quella, Arun Ram, Paul Fijn, and Gufang Zhao have enlightened me on +the value and sheer joy of belonging to an academic community. Furthermore, Stephane’s +skill in explaining intricate topics in a sympathetic and illuminating manner has made it an +absolute pleasure to collaborate with him. +As for peers with whom I shared the experience of being a graduate student, I am +grateful to Allan Trinh, Jiyuan Zhang, Srivatsa Badariprasad, Wee Chaimanowong, Eric +Shen, Campbell Wheeler, and Behrooz Niknami for years of friendship and interesting +conversations. It is not unusual to spend a majority of your graduate studies in solitude, +especially when working in such a specialised field. Therefore, I consider myself very lucky +to have shared an office with two friends, Allan and Jiyuan, with whom I could discuss +random matrix theory to my heart’s content. On the other hand, conversations with friends +from outside random matrix theory have given me opportunities to take a break from +thinking about my work while affording me a chance to learn about interesting topics far +removed from my field. +Finally on the academic side, I would like to note that my mathematical maturity has +benefitted greatly from the seminars, conferences, and summer schools that I have been able +iv + +to participate in — I am indebted to the people who make these events possible. In that vein, +I am similarly indebted to the academic support staff at our ACEMS node and the School +of Mathematics and Statistics for their timely help with all administrative issues, such as +sorting out IT access, arranging travel funds, or organising seminars. +In terms of non-academic support, I would like to thank my friends and family for their +love and support, without which I could not have completed this journey. I am grateful to +my mother for instilling in me the confidence to pursue my passions from a young age. To +my family and the friends that I have not been able to see very often, I apologise and thank +you for your understanding of my absence. It is in the nature of time for it to grow scarce +and it is truly a shame whenever one must spend it alone. +Most of all, my utmost gratitude is reserved for my beautiful fianc´ee Johanna, who has +quelled my worries in my times of woe; who has shown seemingly blind faith in my abilities +when I had little faith of my own; who has been understanding at times when I have been +unavailable; and who has been the source of my first smile of each day. Thank you. +v + + +Contents +Abstract +i +Declaration +ii +Preface +iii +Acknowledgements +iv +1 +Introduction +1 +1.1 +Random Matrix Ensembles +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +1 +1.1.1 +Quantities of interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +6 +1.2 +Classical Matrix Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +1.2.1 +The Cauchy and circular Jacobi ensembles . . . . . . . . . . . . . . . . +19 +1.2.2 +Invariant matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . +22 +1.2.3 +Skew-orthogonal polynomial ensembles +. . . . . . . . . . . . . . . . . +30 +1.2.4 +Classical β ensembles +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +1.3 +Matrix Product Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +41 +1.3.1 +Complex Wishart product ensembles and their correlation kernels . . +44 +1.3.2 +Muttalib–Borodin and biorthogonal ensembles +. . . . . . . . . . . . . +51 +1.3.3 +Integrals of Harish-Chandra–Itzykson–Zuber type +. . . . . . . . . . . +58 +1.4 +Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +60 +2 +Differential Equations for the Classical Matrix Ensembles +65 +2.1 +Selberg Correlation Integrals +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +68 +2.1.1 +The Selberg and Dixon–Anderson integrals +. . . . . . . . . . . . . . . +68 +vii + +2.1.2 +Further preliminaries concerning Selberg correlation integrals +. . . . +72 +2.2 +Relating the Cauchy and Jacobi Ensembles Through Analytic Continuation . +74 +2.2.1 +Relating the symmetric Cauchy and shifted Jacobi ensembles . . . . . +75 +2.2.2 +Relating the non-symmetric Cauchy and shifted Jacobi ensembles . . +77 +2.3 +Linear Differential Equations for the Eigenvalue Densities and Resolvents . . +80 +2.3.1 +Differential equations for the Jacobi ensembles . . . . . . . . . . . . . . +82 +2.3.2 +Differential equations for the shifted Jacobi and Cauchy ensembles . . +87 +2.3.3 +Differential equations for the Laguerre ensembles . . . . . . . . . . . . +92 +2.3.4 +Differential equations for the Gaussian ensembles . . . . . . . . . . . . +94 +2.4 +Scalings of the Differential Equations +. . . . . . . . . . . . . . . . . . . . . . . +98 +2.4.1 +Global scaled differential equations . . . . . . . . . . . . . . . . . . . . +99 +2.4.2 +Soft and hard edge scaled differential equations . . . . . . . . . . . . . +109 +3 +Characterisations of the Moments and Cumulants +117 +3.1 +Recurrence Relations for the Moments of the Classical Matrix Ensembles +. . +122 +3.1.1 +Recurrences for the spectral moments . . . . . . . . . . . . . . . . . . . +123 +3.1.2 +1-point recursions for the moment expansion coefficients +. . . . . . . +131 +3.2 +Established Results on the Moments of the Classical Matrix Ensembles +. . . +144 +3.2.1 +Results from skew-orthogonal polynomial theory . . . . . . . . . . . . +144 +3.2.2 +Results from symmetric function theory +. . . . . . . . . . . . . . . . . +145 +3.2.3 +Results in terms of hypergeometric orthogonal polynomials . . . . . . +147 +3.3 +The Isserlis–Wick Theorem and Ribbon Graphs +. . . . . . . . . . . . . . . . . +151 +3.3.1 +Moments of the Gaussian unitary and orthogonal ensembles . . . . . +154 +3.3.2 +Moments of the Laguerre unitary and orthogonal ensembles +. . . . . +174 +3.3.3 +Moments of the Hermitised and antisymmetrised matrix products . . +185 +4 +Loop Equations for the Matrix Product Ensembles +204 +4.1 +A Brief Introduction to Loop Equations . . . . . . . . . . . . . . . . . . . . . . +207 +4.1.1 +Loop equations for the classical β ensembles . . . . . . . . . . . . . . . +210 +4.1.2 +From loop equations to the topological recursion . . . . . . . . . . . . +215 +4.2 +Loop Equations for the Antisymmetrised Laguerre Ensemble . . . . . . . . . +223 +viii + +4.2.1 +Loop equations on the ˜U(J1) +n +. . . . . . . . . . . . . . . . . . . . . . . . +226 +4.2.2 +Loop equations on the ˜W(J1) +n +and W(J1),l +n +. . . . . . . . . . . . . . . . . +233 +4.2.3 +Discussion on W(J1),0 +1 +and W(J1),0 +2 +. . . . . . . . . . . . . . . . . . . . . +239 +4.3 +Loop Equations for the Hermitised Laguerre Ensemble . . . . . . . . . . . . . +244 +4.3.1 +Loop equations on the ˜U(H1) +n +. . . . . . . . . . . . . . . . . . . . . . . . +247 +4.3.2 +Loop equations on the ˜W(H1) +n +and W(H1),l +n +. . . . . . . . . . . . . . . . . +251 +4.3.3 +Discussion on W(H1),0 +1 +and W(H1),0 +2 +. . . . . . . . . . . . . . . . . . . . . +254 +4.4 +Concluding Remarks and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . +257 +4.4.1 +Larger matrix products and topological recursion . . . . . . . . . . . . +257 +4.4.2 +On 1-point recursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . +259 +4.4.3 +Combinatorics associated with the Jacobi unitary ensemble . . . . . . +260 +Bibliography +263 +Appendix A Quaternionic Matrices and Pfaffians +291 +Appendix B +Particular Stieltjes Transforms +294 +Appendix C Loop Equations on Moments +296 +C.1 +Loop Equations on the ˜m(J1) +k1,...,kn +. . . . . . . . . . . . . . . . . . . . . . . . . . . +296 +C.2 +Loop Equations on the ˜m(H1) +k1,...,kn +. . . . . . . . . . . . . . . . . . . . . . . . . . . +299 +Appendix D Evaluation of Some Cumulants at Leading Order +304 +D.1 Evaluation of c(J1),0 +k1 +, c(J1),1 +k1 +, and c(J1),0 +k1,k2 +. . . . . . . . . . . . . . . . . . . . . . . +304 +D.2 Evaluation of c(H1),0 +k1 +, c(H1),2 +k1 +, and c(H1),0 +k1,k2 +. . . . . . . . . . . . . . . . . . . . . . +306 +ix + +List of Figures +1.1 +Limiting eigenvalue density of the Gaussian ensembles . . . . . . . . . . . . . +18 +1.2 +Limiting eigenvalue density of the Laguerre ensembles . . . . . . . . . . . . . +18 +1.3 +Limiting eigenvalue density of the Jacobi ensembles . . . . . . . . . . . . . . . +18 +3.1 +A 3-ribbon graph with a M¨obius half-twisted ribbon +. . . . . . . . . . . . . . +120 +3.2 +A labelled planar ribbon graph with viable edge and vertex markings . . . . +135 +3.3 +A pair of marked, planar ribbon graphs that correspond to each other . . . . +137 +3.4 +The ribbon graphs that contribute to the value of m(GUE) +4 +. . . . . . . . . . . . +156 +3.5 +The surfaces corresponding to the ribbon graphs of Figure 3.4 . . . . . . . . . +158 +3.6 +A ribbon graph and the compact orientable surface it represents +. . . . . . . +159 +3.7 +The surface representation of W(GUE),0 +1 +(x) . . . . . . . . . . . . . . . . . . . . . +160 +3.8 +A ribbon graph contributing to the computation of m(GUE) +4,3,3 +. . . . . . . . . . . +162 +3.9 +Some ribbon graphs that contribute to the fourth GOE spectral moment . . . +167 +3.10 A topological map representation of a ribbon graph . . . . . . . . . . . . . . . +169 +3.11 Rooted topological maps for the fourth GUE spectral moment . . . . . . . . . +170 +3.12 A ribbon graph and topological map with labelled quarter-edges . . . . . . . +172 +3.13 An oriented planar graph with labelled half-edges . . . . . . . . . . . . . . . . +173 +3.14 The ribbon graphs that contribute to the second LUE spectral moment . . . . +175 +3.15 A locally orientable bicoloured ribbon graph . . . . . . . . . . . . . . . . . . . +179 +3.16 A locally orientable bicoloured topological map . . . . . . . . . . . . . . . . . +181 +3.17 A bicoloured topological map and its hypermap analogue . . . . . . . . . . . +182 +3.18 A topological hypermap with a twisted edge . . . . . . . . . . . . . . . . . . . +183 +3.19 Equivalent topological hypermaps with twisted edges +. . . . . . . . . . . . . +184 +x + +3.20 A 3-coloured ribbon graph contributing to the computation of m(H2) +2,1,1 . . . . . +188 +3.21 A 3-coloured ribbon graph contributing to the computation of m(J2) +2 +. . . . . +193 +3.22 A topological hypermap contributing to the computation of c(H1) +2,1,1 . . . . . . . +199 +4.1 +A construction of a two-sheeted covering of the complex plane . . . . . . . . +217 +4.2 +A diagrammatic interpretation of the topological recursion . . . . . . . . . . . +221 +4.3 +The topological hypermaps corresponding to limN→∞ ˜c(J1) +4 +/N . . . . . . . . . +243 +4.4 +The topological hypermaps corresponding to limN→∞ ˜c(J1) +2,2 +. . . . . . . . . . . +244 +4.5 +The topological hypermaps corresponding to c(H1),0 +4 +, c(H1),2 +2 +, and c(H1),0 +1,1 +. . . . +256 +xi + + +Chapter 1 +Introduction +The overarching theme of this thesis is that of recursive structures within random matrix +theory. The focus is on two types of recursions: 1-point recursions for moment expansion +coefficients and loop equations for n-point correlators. A method of deriving 1-point recursions +from differential equations satisfied by the eigenvalue densities of the classical matrix ensembles +is demonstrated in Chapters 2 and 3, while the method of loop equations is demonstrated +through an application to certain matrix product ensembles in Chapter 4. These two classes +of random matrix ensembles are respectively introduced in Sections 1.2 and 1.3, along with +auxiliary details drawn from the literature so as to give our development sufficient context. +Before introducing these ensembles, we review some pertinent fundamentals of random +matrix theory. +1.1 +Random Matrix Ensembles +Borrowing language from statistical mechanics, an ensemble can be thought of as a virtual +collection of all possible states that a given system can be in, with more probable states +appearing more often in the collection. Then, if one draws uniformly from the ensemble, +one is more likely to retrieve more probable states. One way to obtain such an ensemble is to +generate copies of the system of interest ad infinitum, with the final infinite collection being +named the ensemble. With this intuition in mind, we henceforth work with the following +rigorous definition, in the case that the underlying system is a matrix. +1 + +CHAPTER 1. Introduction +Definition 1.1. A random matrix ensemble E = (S, P) is a set S of matrices of fixed size +equipped with a probability density function (p.d.f.) P : S → [0, ∞). To say that a matrix +X is drawn from the random matrix ensemble E is to say that it is a random variable with +values in S and p.d.f. P. +Remark 1.1. It is commonplace to define a random matrix ensemble by prescribing p.d.f.s +Pij on the independent entries Xij of a representative random matrix X ∈ E along with a +specification of how the other entries of X depend on these independent entries. In such +cases, the p.d.f. on S is the joint probability density function (j.p.d.f.) formed by taking the +product +P(X) = +∏ +i,j s.t. {Xij} ind. +Pij(Xij). +(1.1.1) +Definition 1.1 allows for a broad range of examples: For many random matrix ensembles, +the set S of say M × N matrices is described by applying constraints to the set of real +matrices MM×N(R), the set of complex matrices MM×N(C), or the set of quaternionic +matrices MM×N(H) (see Appendix A for an introduction to quaternionic matrices), though +the definition above does not forbid more exotic matrix entries like octonions [264], [124] or p- +adic numbers [261], to name a few examples — the Frobenius theorem [146], [97] directs our +attention to matrices with entries in R, C, or H due to their being the only finite-dimensional +associative division algebras over the real numbers, up to isomorphism. The constraints +used to construct S are usually those that require the elements of S to exhibit features such +as orthogonality, unitarity, (skew-)symmetry, Hermiticity, positive definiteness, invariance +under complex conjugation, or combinations thereof. Thus, S is oftentimes simply the set of +symmetric, Hermitian, or skew-symmetric matrices, but in physical settings is more likely to +be some (possibly trivial) quotient of products of the classical matrix groups O(N), U(N), +and Sp(N), of which ten such amalgamations stand out from the viewpoint of modelling +Hamiltonians subject to symmetry constraints [24], [283]; these are in correspondence with +the ten infinite families of matrix Lie algebras. Another type of constraint that has been +explored in addition to these is that of requiring entries Xij of the representative random +matrix X ∈ E to vanish when the N-periodic distance between i and j is greater than some +threshold called the band width. These so-called random band matrices relate to the study of +Anderson localisation-delocalisation transitions [48]. +2 + +1.1. Random Matrix Ensembles +When it comes to determining the p.d.f. P on S, there are again many options that +have garnered interest throughout the literature. There are far too many to cover here in +totality, but let us highlight the diversity of topics in random matrix theory through a few +examples. A simple yet non-trivial example is that of Bernoulli matrices B whose entries +Bij are all independently and identically distributed (i.i.d.) Bernoulli random variables ±1: +S = MN×N(R) and Pij(Bij) = 1 +2[δ(Bij − 1) + δ(Bij + 1)], where δ is the Dirac delta [208], [297]. +If we instead require that B be symmetric, its diagonal entries be zero, and its independent +hence upper triangular entries have p.d.f. +Pij(Bij) = c +N δ(Bij − 1) + +� +1 − c +N +� +δ(Bij), +0 < c ≡ c(N) ≪ N, +(1.1.2) +B belongs to the ensemble of Erd˝os–R´enyi matrices [103], [102], which are the adjacency +matrices of Erd˝os–R´enyi random graphs [104], [157]; the parameter c is the mean connectivity +of the nodes. This ensemble is a type of sparse random matrix ensemble [282], [214], [323], which +is an umbrella term for a variety of random matrix ensembles whose shared characteristic +is that for each such ensemble, the independent entries Bij of their representative random +matrix B (now allowed to be complex and/or have aforementioned constraints relaxed) have +p.d.f. Pij of the form given in equation (1.1.2) but with δ(Bij − 1) replaced by any p.d.f. of +interest. This brings us to the question of which p.d.f.s are ‘interesting’. +Moving on from Bernoulli and sparse random matrix ensembles, let us consider a more +general applications-based perspective where one would like to model real world systems +such as ecologies [236], heavy atom spectra [313], [98], [316], wireless communication +channels [301], random quantum states [68], log-gases [119], or quantum transport [33], +among many others. From this viewpoint, where one chooses to eschew knowledge of the +finer details of the real world system at hand in favour of treating them statistically, there +are essentially three ideologies that dominate the field. The first is to assume as little as +possible about the system and work in full generality, focusing on universal results. In this +line of thinking, one studies Wigner matrix ensembles [313], [315], [294] or, more recently, +general Wigner-type matrix ensembles [6]: Let X be drawn from either the real symmetric or +complex Hermitian N × N Wigner matrix ensemble. Then, its entries Xij are independently +distributed for 1 ⩽ i ⩽ j ⩽ N, with mean zero and all moments having an upper bound +independent of i, j. In addition, the diagonal entries have identical variance while the upper +3 + +CHAPTER 1. Introduction +triangular entries have variance one. If we relax the condition on the variances, X is instead +of general Wigner-type. Many random matrix ensembles fall into the class of Wigner matrix +ensembles due to the freedom enjoyed by the variables Xij. For example, results proved for +Wigner matrix ensembles hold when the matrix entries are i.i.d. centred normal variables. +The second of the three dominant ideologies is that of studying uniform distributions, +which corresponds to a principle often seen in statistical mechanics: In the absence of any +relevant information, one should assign equal probability to all states of an ensemble (cf. the +microcanonical ensemble, dice, a deck of cards, etc.). Our discussion up to this point has +focused on random matrix ensembles distributed according to p.d.f.s P defined with respect +to the flat Lebesgue measure induced by the canonical embedding of S into Euclidean space; +e.g., for the Bernoulli matrix ensemble, we have implicitly taken +dB = +N +∏ +i,j=1 +dBij, +while for the complex Hermitian Wigner matrix ensemble, we have taken +dX = +N +∏ +i=1 +dXii +∏ +1⩽j 0, +the corrections ρl(λ) integrate to zero on their supports and are thus signed densities. In +short, the following diagram does not commute. +ρ(λ) +˜ρ(λ) +� +ρl(λ) +�∞ +l=0 +W1(x) +1 +N ˜W1(x) +� +Wl +1(x) +�∞ +l=0 +scaling cN +Stieltjes transform +Stieltjes transform +moments +Stieltjes transform +scaling cN +Sokhotski–Plemelj +Sokhotski–Plemelj +1/N expansion +Sokhotski–Plemelj +Let us now turn our attention to Chapter 4, wherein we derive three types of loop +equations. The first set of loop equations are satisfied by the unconnected n-point correlators +Un(x1, . . . , xn) := +� +n +∏ +i=1 +Tr +1 +xi − X +� +P(X) +, +(1.1.26) +which act as generating functions for the mixed moments +mk1,...,kn := +� +n +∏ +i=1 +Tr Xki +� +P(X) +, +k1, . . . , kn ∈ N. +(1.1.27) +Related to the mixed moments are the mixed cumulants cκi, which are defined implicitly by +the moment-cumulants relation [237, Ch. 2] +mk1,...,kn = +∑ +K⊢{k1,...,kn} ∏ +κi∈K +cκi, +(1.1.28) +where K ⊢ {k1, . . . , kn} means that K is a partition of {k1, . . . , kn}, i.e., K = {κi}m +i=1 for some +1 ⩽ m ⩽ n such that the disjoint union κ1 ⊔ · · · ⊔ κm is equal to {k1, . . . , kn}. These mixed +10 + +1.1. Random Matrix Ensembles +cumulants are generated by the so-called connected n-point correlators +Wn(x1, . . . , xn) := +∞ +∑ +k1,...,kn=0 +ck1,...,kn +xk1+1 +1 +· · · xkn+1 +n +(1.1.29) += +∑ +G⊢{x1,...,xn} +(−1)#G−1(#G − 1)! ∏ +Gi∈G +U#Gi(Gi) +(1.1.30) += Un(x1, Jn) − ∑ +∅̸=J⊆Jn +Wn−#J(x1, Jn \ J)U#J(J), +Jn = (x2, . . . , xn), +(1.1.31) +where #S denotes the size of the set S [289], [119, pg. 187], [320, pp. 8–9]2. Setting n = 1, 2 +in equation (1.1.30) shows that +W1(x) = U1(x), +W2(x1, x2) = CovP(X) +� +Tr +1 +x1 − X, Tr +1 +x2 − X +� +, +where +CovP(X)( f (X), g(X)) := +� +( f (X) − ⟨ f (X)⟩P(X))(g(X) − ⟨g(X)⟩P(X)) +� +P(X) +denotes the covariance of the linear statistics f (X) and g(X) with respect to P(X); a similar +formula is found for n = 3, but the analogous structures for n ⩾ 4 are more complicated. +Equation (1.1.31) is used in Chapter 4 to transform the set of loop equations on the un- +connected correlators Un into a second set of loop equations satisfied by the connected +correlators Wn. +Our interest in the connected correlators Wn over their unconnected counterparts Un is +that the former is of lower order in N due to the cancelling of leading order terms brought on +by the inclusion-exclusion structure of equation (1.1.30). Hence, the connected correlators Wn +have the advantage of admitting large N expansions of the form (1.1.21). This is significant +because the loop equations on both the connected and unconnected correlators fail to close +and are thus unsolvable, but combining them with the large N expansion (1.1.21) yields a +third set of loop equations for the correlator expansion coefficients Wl +n that form a triangular +recursive system. Considering n = 1 for simplicity, this means that Wl +1 can be computed +through (at most) +�(1 + l/2)2� +applications of the loop equations (see, e.g., [142]). Although +complexity grows strongly with l, it is quite feasible to write down W0 +1, . . . , W3 +1 and thus +calculate the scaled spectral moments up to a corresponding order in 1/N. To be precise, +2The otherwise formal series (1.1.29) converges when |x1|, . . . , |xn| > +max +λ∈supp ρ|λ|; cf. equation (1.1.18). The +characterisations (1.1.30), (1.1.31) in terms of the Un follow from (the inverse of) the relation (1.1.28). +11 + +CHAPTER 1. Introduction +let ˜mk denote the spectral moments of ˜ρ(λ) (i.e., the scaled spectral moments of ρ(λ)) and +implicitly define the moment expansion coefficients ˜Mk,l according to +˜mk = +∞ +∑ +l=0 +˜Mk,l N−l. +(1.1.32) +Then, if Coeff(x, k) denotes the action of expanding in 1/x and extracting the coefficient of +1/xk, the following diagram commutes. +˜W1(x) +Wl +1(x) +˜mk +˜Mk,l +Coeff(N, l) +Coeff(x, k + 1) +Coeff(x, k + 1) +Coeff(N, l) +Remark 1.2. We recognise that the use of the adjectives ‘unconnected’ and ‘connected’ has yet +to be motivated. This terminology alludes to the fact that the mixed moments and cumulants +often relate to problems of counting topological surfaces that are respectively unconnected +or connected; we address the details of this relationship in Chapter 3. +Having defined the observables of interest in full generality, it is now time to introduce +the specific forms that are relevant to the contents of Chapters 2–4. +1.2 +Classical Matrix Ensembles +Let us first define the classical matrix ensembles that will be at the focus of Chapter 2. +Definition 1.4. Recalling Definition 1.3, let G be drawn from the N × N real, complex, or +quaternionic Ginibre ensemble. Then, the random matrix H = 1 +2(G† + G) represents the +corresponding Gaussian (also known as Hermite) ensemble [314], [98]. If we instead take G to +be M × N Ginibre, then W = G†G is said to be drawn from the (M, N) Laguerre (also known +as Wishart) ensemble [317]. We say that Y is an element of the (M1, M2, N) real, complex, or +quaternionic Jacobi ensemble if it is of the form Y = W1(W1 + W2)−1 where the Wi belong to +the corresponding (Mi, N) Laguerre ensembles with Mi ⩾ N [249, Ch. 3]. +12 + +1.2. Classical Matrix Ensembles +The real Gaussian ensemble was first introduced as a statistical model for heavy atom +spectra by Wigner [314] in 1957, while its complex and quaternionic counterparts were +introduced by Dyson [98] in 1962 with the same application in mind; in this latter work, +Dyson also highlighted, and made application of, analogies with the statistical mechanics +of log-gases. These and related papers from the surrounding literature are conveniently +collated and reviewed in the 1965 book [275] of Porter. The Laguerre (Wishart) ensemble, +on the other hand, has been a mainstay of multivariate statistical analysis since the 1928 +work [317] of Wishart showing how to change variables from the entries of an M × N real +Ginibre matrix G to the entries of the Wishart–Laguerre matrix W = GTG defined above. +The textbook [249] details applications of the real and complex Laguerre ensembles in +multivariate statistics, including its role in estimating covariance matrices and supplying null +hypotheses for principal component analysis. Around the turn of the century, the Laguerre +ensembles found further applications in the fields of wireless communications [301] and +quantum transport [33] (one such application is briefly discussed early in Section 3.1). The +review [33] also outlines how problems of quantum transport served as early motivation +for studying the Jacobi ensembles. Another appreciable motivation was given a decade +later in the 2008 work [197] of Johnstone showing how the eigenvalue statistics of the Jacobi +ensembles can be used to determine null hypotheses for multivariate analysis of variance. +The matrices given in Definition 1.4 are all self-adjoint and thus have N real eigenvalues. +Taking either the real, complex, or quaternionic case for definiteness, the entries of the +Hermite–Gaussian random matrix H are independent (up to the constraint of H being +self-adjoint) centred normal variables with the diagonal entries having variance 1 +2 and the +real components of the upper triangular entries having variance 1 +4. Thus, it can be observed +that the Gaussian ensembles are examples of Wigner matrix ensembles. The diagonal entries +of the Wishart–Laguerre random matrix W are i.i.d. chi-squared variables with p.d.f. +P(L) +ii (Wii) = +1 +Γ(βM/2)WβM/2−1 +ii +e−Wii. +The p.d.f.s of the off-diagonal entries of W and indeed all entries of the Jacobi random matrix +Y are too complicated to present here. On the contrary, the p.d.f.s of the random matrices +H, W, and Y are themselves quite elegant. +13 + +CHAPTER 1. Introduction +Proposition 1.1. Let κ := β/2, a = κ(M1 − N + 1) − 1, and b = κ(M2 − N + 1) − 1. From the +p.d.f. (1.1.6) of the Ginibre ensembles, we can change variables to obtain the p.d.f.s of the N × N +Gaussian, (M1, N) Laguerre, and (M1, M2, N) Jacobi ensembles, respectively [119, Ch. 1,3]: +P(G)(H) = +1 +Z(G) +N,β +exp +�− Tr H2� +, +(1.2.1) +P(L)(W) = +1 +Z(L) +N,a +(Det W)a exp (− Tr W) , +(1.2.2) +P(J)(Y) = +1 +Z(J) +N,a,b +(Det Y)a (Det(IN − Y))b , +(1.2.3) +where the ZN are normalisation constants, often referred to as partition functions, whose explicit +forms are inconsequential to us but can be obtained from the method described at the end of §2.1.1. +Remark 1.3. With β = 1, 2, 4 corresponding to F = R, C, H, the random matrix ensembles +introduced in Definition 1.4 are specified by assigning the p.d.f.s (1.2.1)–(1.2.3) to the matrix +sets +S(G) = {H ∈ MN×N(F) | H = H†}, +(1.2.4) +S(L) = {W ∈ S(G) | W positive definite}, +(1.2.5) +S(J) = {Y ∈ S(L) | IN − Y positive definite}, +(1.2.6) +respectively. +Remark 1.4. Apart from the matrix Y given in Definition 1.4, there are some other important +constructions that are known to yield random matrices Y′, Y′′, Y′′′, which all have p.d.f. (1.2.3) +with a, b as prescribed in Proposition 1.1 [119, Sec. 3.6], [330], [117]. +1. Let X be a Haar-distributed element of either O(M1 + M2) (β = 1), U(M1 + M2) +(β = 2), or Sp(2(M1 + M2)) (β = 4) for positive integers M1, M2. Take T to be the +M1 × N truncation of X with Tij = Xij for 1 ⩽ i ⩽ M1, 1 ⩽ j ⩽ N, and M1, M2 ⩾ N. +Define Y′ = T†T. +2. For i = 1, 2, let Gi be Mi × N Ginibre matrices with Mi ⩾ N and define +X = +� +�G1 +G2 +� +� , +˜I = +� +�IM1 +0M2 +� +� . +Set Y′′ = X† ˜IX(X†X)−1. +14 + +1.2. Classical Matrix Ensembles +3. In the notation of Definition 1.4, let Y′′′ := (W1 + W2)−1/2W1(W1 + W2)−1/2. +The term ‘classical matrix ensemble’ originally referred to the Gaussian, Laguerre and +Jacobi ensembles due to their relation to the classical orthogonal polynomials, which we +briefly review in §1.2.3. However, the family of classical matrix ensembles has one more +constituent according to the contemporary definition. +Definition 1.5. A random matrix ensemble is said to be a classical matrix ensemble [119, §5.4.3], +[138] if its eigenvalue j.p.d.f. is of the form +p(w)(λ1, . . . , λN; β) = +1 +N (w) +N,β +N +∏ +i=1 +w(λi) |∆N(λ)|β, +β = 1, 2, 4, +(1.2.7) +where N (w) +N,β is a normalisation constant with values given in [93], [129] (alternatively, see +§2.1.1), ∆N(λ) is the Vandermonde determinant, as specified by equation (1.1.9), and the +weight function w(λ) is such that +d +dλ log w(λ) = w′(λ) +w(λ) = f (λ) +g(λ), +(1.2.8) +with f and g polynomials of degree at most one and two, respectively. +Up to fractional linear transformations λ �→ (a11λ + a12)/(a21λ + a22) with [aij] ∈ GL2(R), +there are precisely four distinct weights satisfying the criterion (1.2.8): +w(λ) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +e−λ2, +Gaussian, +λae−λχλ>0, +Laguerre, +λa(1 − λ)bχ0<λ<1, +Jacobi, +(1 − iλ)η(1 + iλ)η, +generalised Cauchy, +(1.2.9) +where the indicator function χA equals one when A is true and zero otherwise; its presence +is due to the fact that the Laguerre random matrix W = G†G is positive definite. Taking +a, b as prescribed in Proposition 1.1 relates the eigenvalue j.p.d.f. (1.2.7) with the Gaussian, +Laguerre, and Jacobi weights (1.2.9) to the p.d.f.s (1.2.1)–(1.2.3). More generally, one usually +considers a, b > −1 real and η = −κ(N − 1) − 1 − α with α ∈ C and Re(α) > −1/2 for +convergence issues [39], though other values can be considered through analytic continuation +— in this general setting, the function (1.2.7) continues to be referred to as an eigenvalue +15 + +CHAPTER 1. Introduction +j.p.d.f. even if the variables {λi}N +i=1 it governs can no longer be interpreted as eigenvalues +of an actual random matrix. We will see in §1.2.4 that there is a further generalisation +of equation (1.2.7) where β is taken to be positive real parameter. Like with the circular +ensembles’ eigenvalue j.p.d.f. (1.1.8), we will suppress the β argument of the eigenvalue +j.p.d.f. (1.2.7) when its value is clear from context. Moreover, we will opt to use superscripts +(G), (L), (J), (Cy) to distinguish the four cases given in equation (1.2.9). +The eigenvalue densities corresponding to the eigenvalue j.p.d.f.s of Definition 1.5 are +given by equation (1.1.10). Existing derivations of their explicit forms for finite N in terms +of (skew-)orthogonal polynomials are presented in §1.2.3. For now, we review the large N +limiting forms of the corresponding global scaled eigenvalue densities ˜ρ(λ). +Definition 1.6. Recalling that κ = β/2, take cN = +√ +κN, κN, 1 in equation (1.1.22) for the +Gaussian, Laguerre, and Jacobi cases, respectively [319], [142]. Hence, define the global scaled +eigenvalue densities of the classical matrix ensembles to be +˜ρ(G)(λ) := +� +κ +N ρ(G)( +√ +κNλ), +(1.2.10) +˜ρ(L)(λ) := κ ρ(L)(κNλ), +(1.2.11) +˜ρ(J)(λ) := 1 +N ρ(J)(λ), +(1.2.12) +˜ρ(Cy)(λ) := 1 +N ρ(Cy)(λ) +��� +η=−κ(N+ˆαN−1)−1, +(1.2.13) +with ˆα constant in N. +There is some freedom in choosing the scaling cN, but the required characteristic of +the densities (1.2.10)–(1.2.13) is that their large N limiting forms have compact connected +support on which they integrate to one. The Jacobi ensembles’ eigenvalue densities already +have compact connected support [0, 1], so to obtain an appropriate expression for ˜ρ(J)(λ), +one need only renormalise ρ(J)(λ) so that it integrates to one on said support. On the other +hand, scaling the eigenvalue densities of the (generalised) Cauchy ensembles according to +equation (1.1.22) cannot produce a density with compact support as there is no choice of +parameter cN that leads to fast enough decay as N, |λ| → ∞. Instead, one needs to take +η = −κ(N − 1) − 1 − α and change variables to relate to the circular Jacobi ensemble, at +which point it can be seen that when α ∝ N, one obtains a compact support in the large +16 + +1.2. Classical Matrix Ensembles +N limit. The details of this phenomenon and some other properties that set the Cauchy +ensembles apart from the other classical ensembles are discussed in §1.2.1. +Proposition 1.2. Recall that the N → ∞ limit of ˜ρ(λ) is equal to ρ0(λ), the leading order term of +the smoothed eigenvalue density. In the Gaussian case, it is given by the celebrated Wigner semi-circle +law [315], +ρ(G),0(λ) = +√ +2 − λ2 +π +χ|λ|< +√ +2. +(1.2.14) +If we suppose that γi := limN→∞ Mi/N exists for i = 1, 2, then we have ρ(L),0(λ) given by the +Marˇcenko-Pastur law [234], +ρ(L),0(λ) = (1 − γ1)δ(λ)χγ1<1 + +� +(λ(MP) ++ +− λ)(λ − λ(MP) +− +) +2πλ +χλ(MP) +− +<λ<λ(MP) ++ +, +(1.2.15) +and ρ(J),0(λ) given by the so-called Wachter law [307], +ρ(J),0(λ) = (γ1 + γ2) +� +(λ(Wac) ++ +− λ)(λ − λ(Wac) +− +) +2πλ(1 − λ) +χλ(Wac) +− +<λ<λ(Wac) ++ +, +(1.2.16) +with +λ(MP) +± +:= (1 ± √γ1)2 , +(1.2.17) +λ(Wac) +± +:= +�� +γ1 +γ1 + γ2 +� +1 − +1 +γ1 + γ2 +� +± +� +1 +γ1 + γ2 +� +1 − +γ1 +γ1 + γ2 +��2 +. +(1.2.18) +Writing ˆα = ˆα1 + iˆα2 with ˆα1, ˆα2 real and constant in N, one surmises from the equivalent result for +the circular Jacobi ensemble [49, Eq. (5.6)] that in the Cauchy case, +ρ(Cy),0(λ) = ˆα1 +� +(λ(Cy) ++ +− λ)(λ − λ(Cy) +− +) +π(1 + λ2) +χλ(Cy) +− +<λ<λ(Cy) ++ , +(1.2.19) +where +λ(Cy) +± +:= +−ˆα2(ˆα1 + 1) ± +� +(ˆα2 +1 + ˆα2 +2)(2ˆα1 + 1) +ˆα2 +1 +. +(1.2.20) +Note 1.1. Due to how κ is involved in the scalings of Definition 1.6, all of the large N limiting +forms given in Proposition 1.2 are independent of β. It is known that the same does not hold +for the 1/N correction terms ρ1(λ), as will be seen in §2.4.1. +17 + +CHAPTER 1. Introduction +The limiting densities ρ(G),0(λ), ρ(L),0(λ), and ρ(J),0(λ) take on parameter-independent +forms when a, b are constant in N since then, γ1 = γ2 = 1. They display, respectively, two +soft edges, a soft and hard edge, and two hard edges. Thus, from a topological viewpoint, +all possible combinations of soft and hard edges on a connected interval are represented by +the eigenvalue densities of the Gaussian, Laguerre, and Jacobi ensembles in the relatively +simple regime a, b = O(1). Here, a soft edge refers to an endpoint λ of supp ρ0 with the +property that, for all large N, cNλ lies within the interior of supp ρ, while a hard edge refers to +such an endpoint when cNλ is also an endpoint of supp ρ [139], [118]. In the present setting +concerning classical matrix ensembles, ρ0(λ) exhibits square root profiles at its soft edges +owing to tails extending past said edges being scaled to negligibility, whereas its hard edges +are located at inverse equare root singularities corresponding to strict constraints enforced +by the indicator functions seen in equation (1.2.9); see Figures 1.1–1.3 below. +Figure 1.1: ρ(G),0(λ) +Figure 1.2: ρ(L),0(λ)|γ1=1 +Figure 1.3: ρ(J),0(λ)|γ1=γ2=1 +The other regime of interest is when the parameters a and/or b are proportional to N [33], +[56], [305], [266], [229], [240], [241], [70], [71]. This situation is a little more complicated, as +the limiting forms ρ(L),0(λ) and ρ(J),0(λ) are no longer parameter-independent. In fact, when +a (b) is linear in N, we have γ1 > 1 (γ2 > 1); consequently, the endpoint λ− (λ+) turns from +a hard edge to a soft edge, a phenomenon which is discussed further in §2.4.2. In the Cauchy +case, one sees that ρ(Cy),0(λ) also has soft edges when we take α = ˆακN with ˆα = O(1), but +unlike the ρ0(λ) of the other classical matrix ensembles, this is not a degeneration of a hard +edge: When α is constant in N, the Cauchy ensembles technically do not have a global scaled +eigenvalue density, but the large N limiting form of its equivalent is +lim +N→∞ ˜ρ(Cy)(λ) +��� +ˆα=0 = ρ(Cy),0(λ) +��� +ˆα1=ˆα2=0 = +1 +π(1 + λ2). +(1.2.21) +As this is supported on the real line, we do not see a hard nor soft edge. Instead, one sees a +spectrum singularity at infinity, which is a term we formally introduce in §1.2.1. +18 + +0.4 +0.2 +2 +-1 +70.6 +0.4 +0.2 +2 +3 +4 +51.5 +0.51.2. Classical Matrix Ensembles +Another reason for focusing on the Gaussian, Laguerre, and Jacobi ensembles while +relegating the Cauchy ensembles to secondary interest is that when considering fractional +linear transformations taken from GL2(C) rather than GL2(R), the Cauchy weight can be +seen to be equivalent to the Jacobi weight. Indeed, a simple change of variables shows that +w(Cy)(λ) = 4Re(η) w(J) � 1 +2(1 − iλ) +� ��� +a=b=η. +(1.2.22) +Thus, our results for the Jacobi ensembles translate to the corresponding Cauchy ensembles +through analytic continuation in the parameters a, b. In this vein, the proofs of Chapter 2 +concerning the Jacobi ensembles also transfer to the Gaussian and Laguerre ensembles via +Lebesgue’s dominated convergence theorem combined with the following limiting procedure. +Lemma 1.1. Let w(G)(λ), w(L)(λ), and w(J)(λ) be given by equation (1.2.9) in the Gaussian, +Laguerre, and Jacobi cases, respectively. Then, +lim +b→∞ ba w(J)(λ/b) = w(L)(λ) +(1.2.23) +and if we set a = b = L, +lim +L→∞ 42L w(J) � +1 +2(1 + λ/ +√ +L) +� += w(G)(λ). +(1.2.24) +Moreover, it can be observed that w(L)(λ2)|a=0 = w(G)(λ), so there is a hierarchy placing +the Jacobi ensembles above the Laguerre ensembles, which, in turn, sit above the Gaussian +ensembles. This corresponds to the loss of parameter b and then a. +Some of the tools used in the proofs of Chapter 2 apply to the more general classical β +ensembles, so we defer their introductions to §1.2.4. Other properties of the classical matrix +ensembles that we wish to review follow from their status as (skew-)orthogonal polynomial +ensembles (SOPEs), which are themselves examples of invariant matrix ensembles. These classes +of random matrix ensembles are discussed in §1.2.3 and §1.2.2, respectively. Before doing so, +let us first look at the Cauchy ensembles in more depth. +1.2.1 +The Cauchy and circular Jacobi ensembles +For specific values of a, b and with β ∈ {1, 2, 4}, we have seen from Definition 1.4 that p(G), +p(L), and p(J) are j.p.d.f.s of the eigenvalues of random matrices that are constructed from +19 + +CHAPTER 1. Introduction +Ginibre matrices. In contrast, for fixed β ∈ {1, 2, 4}, we can interpret p(Cy) as the eigenvalue +j.p.d.f. of a random matrix related to an appropriate circular ensemble. To see this, let U be +drawn from the N × N circular orthogonal, unitary, or symplectic ensemble. Then, define +the Hermitian random matrix A via the Cayley transformation +A := iU + IN +U − IN += i(U + IN)(U − IN)−1 ∈ MN×N(C). +(1.2.25) +The eigenvalues {λi}N +i=1 of A are related to the eigenvalues {eiθi}N +i=1 of U by the stereographic +projection +λi = cot(θi/2). +(1.2.26) +Applying the associated inverse mapping eiθi = (λi + i)/(λi − i) to the eigenvalue j.p.d.f. +(1.1.8) of the circular ensembles shows that the eigenvalues of A are distributed according to +the eigenvalue j.p.d.f. (1.2.7) with the Cauchy weight (1.2.9), when η = −κ(N − 1) − 1 (i.e., +when α = 0). +When studying the eigenvalue j.p.d.f. (1.2.7), there is no reason to restrict our parameters +to only those that relate the j.p.d.f. to a realisable random matrix. In fact, for most of this +thesis, we consider the Laguerre and Jacobi ensembles with a, b > −1 real. The natural +way to generalise the Cauchy ensembles’ eigenvalue j.p.d.f.s is to introduce the complex +parameter α with Re(α) > −1/2, and write η = −κ(N − 1) − 1 − α. This ensures that +w(Cy)(λ) continues to be real-valued on the real line and that +N (Cy) +N,β = +� +RN p(Cy)(λ1, . . . , λN; β) dλ1 · · · dλN +is finite (we require Re(α) > −1/2 for finiteness due to the non-compact domain of integra- +tion). Applying the stereographic projection (1.2.26) to p(Cy) with α ̸= 0 yields the eigenvalue +j.p.d.f. of the circular Jacobi ensemble, which is an extension of the circular ensembles. +Definition 1.7. Taking β > 0 real, let α ∈ C, α1 := Re(α) > −1/2, and α2 := Im(α). The size- +N circular Jacobi ensemble is the system of ‘eigenvalues’3 on the unit circle {eiθ | θ ∈ [0, 2π)} +distributed according to the eigenvalue j.p.d.f. +p(cJ)(eiθ1, . . . , eiθN; β) := +1 +N (cJ) +N,β +N +∏ +i=1 +w(cJ)(eiθi) |∆N(eiθ)|β, +(1.2.27) +3Although calling p(cJ) an eigenvalue j.p.d.f. is a priori artifical in some sense, we will see in §1.2.4 that +random matrices have been constructed whose eigenvalues are indeed distributed according to p(cJ) [49]. Similar +matrix constructions for the eigenvalue j.p.d.f. (1.2.7) with β > 0 general real are also outlined in §1.2.4. +20 + +1.2. Classical Matrix Ensembles +where N (cJ) +N,β is a normalisation constant, ∆N(eiθ) is the Vandermonde determinant (1.1.9), +and the circular Jacobi weight [49], [226] is given by +w(cJ)(eiθ) := (1 − e−iθ)α(1 − eiθ)α = eα2(θ−π) |1 − eiθ|2α1. +(1.2.28) +The weight w(cJ) is said to have a spectrum singularity of Fisher-Hartwig type [113], [30], +[325] at θ = 0; when α1 ̸= 0, it is of root-type (it is actually log w(cJ)(eiθ) that is singular) and +when α2 ̸= 0, it is of jump-type. Accordingly, we say that the Cauchy ensembles display a +spectrum singularity in the regime |λ| → ∞. Thus, studying the Cauchy ensembles allows us +to shine light on aspects of the spectrum singularity scaling regime, supplementing similar +studies of the soft and hard edges exhibited by the other classical ensembles. Earlier, we +mentioned that the spectrum singularity of the Cauchy ensembles transforms into a pair of +soft edges when we take α ∝ N — the same behaviour has been recorded for the circular +Jacobi ensemble [226]. This mechanism is interesting to us because when we have a spectrum +singularity, our methods involving the resolvent W1(x) cannot be used since not enough +spectral moments m(Cy) +k +converge. However, this ceases to be an issue when we set α = ˆακN +and work with ˜ρ(Cy)(λ) as defined in (1.2.13). To be precise, note that when λ1, . . . , λN → ∞, +|∆N(λ)|β ∼ +N +∏ +i=1 +λβ(N−1) +i +, +w(Cy)(λi)|η=−κ(N−1)−1−α ∼ λ−β(N−1)−2−2α1 +i +, +so by equation (1.1.15), m(Cy) +k +is convergent only when −1 < k < 2α1 + 1. This upper bound +disappears upon setting α1 = ˆα1κN, as introduced in Proposition 1.2, and taking N → ∞. +In this thesis, the connection between the Cauchy and circular Jacobi ensembles will not be +used to study the Cauchy ensembles, but will rather serve as motivation for doing so. Instead, +as mentioned earlier, our results concerning the Cauchy ensembles flow from equivalent +results for the Jacobi ensembles. Along the way, we will also formulate certain intermediate +results in terms of the shifted Jacobi ensembles, which are defined to have eigenvalue j.p.d.f. +(1.2.7) with weight +w(sJ)(λ) := (1 − λ)a(1 + λ)bχ−1<λ<1 = w(J) � 1 +2(1 − λ) +� +. +(1.2.29) +Since results for the Cauchy and shifted Jacobi ensembles descend from those for the (un- +shifted) Jacobi ensembles, we present only those results where the relevant transformation +gives rise to significant simplifications. Hence, we will report primarily on the symmetric +Cauchy and shifted Jacobi ensembles, which corresponds to taking α ∈ R and a = b. +21 + +CHAPTER 1. Introduction +1.2.2 +Invariant matrix ensembles +Given a group G and a group action A : G × S → S, a random matrix ensemble E = (S, P) +is said to be an A-invariant matrix ensemble if the probability measure dP(X) := P(X) dX +(recall that in the case of the circular ensembles, we do not prescribe a p.d.f. P(X), but work +with a probability measure dP(X) directly) satisfies +dP(A(g, X)) = dP(X) +(1.2.30) +for every g ∈ G. For example, the CUE of Definition 1.2 is invariant under left and right +multiplication by elements of U(N), while the COE and CSE have invariant structures +induced by those of the CUE [98, I, Thrms. 1,5]. +The adjoint action is by far the most prolific group action studied in random matrix +theory. Indeed, the term invariant matrix ensemble, with A unspecified, refers to invariance +under the mapping X �→ gXg−1 for all g drawn from some appropriate compact group G; +when such an invariant matrix ensemble is assumed to be irreducible and the matrix set +underlying it is such that S ⊆ MN×N(R), MN×N(C), or MN×N(H), the group G must be +one of O(N), U(N), or Sp(2N) [97]. Then, the invariant matrix ensemble at hand is more +precisely referred to as an orthogonal (O(N)), unitary (U(N)), or symplectic (Sp(2N)) ensemble, +and if the generic entries of the elements of both S, G belong to the same number system, the +statistics of the ensemble are basis-independent (conjugation by an element of G is equivalent +to a change of basis). The adjectives orthogonal, unitary, and symplectic have already been +introduced in the circular and Ginibre cases in Definitions 1.2 and 1.3, where we saw that +they correspond to β = 1, 2, and 4, respectively. It can moreover be checked that these +invariances extend to the Cauchy matrix A defined in equation (1.2.25) and we show below +that the real, complex, and quaternionic matrices of Definition 1.4 have orthogonal, unitary, +and symplectic invariance, respectively. Thus, with G, L, J, Cy labelling the classical weights +(1.2.9) and O, U, S the invariance classes, we will for the remainder of this thesis refer, e.g., +to the real Gaussian ensemble as the GOE and the β = 4 Cauchy ensemble as the CySE. +Invariance of the circular, Ginibre, Gaussian, Laguerre, and Jacobi ensembles +In the CUE case, dP(X) is the Haar measure on U(N), so it is trivially invariant under +the mapping X �→ UXU† for U ∈ U(N) — it is a slightly more astute observation that +22 + +1.2. Classical Matrix Ensembles +this invariance translates to invariance of the COE (CSE) under the adjoint action of O(N) +(Sp(2N)) [98, I] (recall the forms of S given in Definition 1.2). Otherwise, the probability +measure dP(X) can be seen to be invariant under the adjoint action of a group G if one is able +to establish the invariance of the p.d.f. P(X) and the Lebesgue measure dX separately. It is +immediate that P(Gin)|M=N(X), P(G)(X), P(L)(X), and P(J)(X), as specified in Definition 1.3 +and Proposition 1.1, are invariant under the appropriate adjoint actions (conjugation of X +by elements of O(N), U(N), Sp(2N) corresponding to β = 1, 2, 4) since the functions Det X, +Tr X†X, and Tr Xk (k ∈ N) all have this property. More involved is the invariance of the +Lebesgue measure dX, where the strategy used depends on whether or not the associated +matrix set S consists purely of self-adjoint matrices. In regards to Remark 1.3, one may use +the diagonalisation formula X = UΛU† and some calculus of differential forms to see that +the Lebesgue measure on the sets of self-adjoint matrices S(G), S(L), S(J) (1.2.4)–(1.2.6) can +be written in a form that is manifestly invariant under the relevant adjoint action. +Proposition 1.3. Given an N × N real (β = 1), complex (β = 2), or quaternionic (β = 4) +self-adjoint matrix variable X, diagonalise it as X = UΛU†, where Λ = diag(λ1, . . . , λN) is a +diagonal matrix of eigenvalues, and U is a corresponding matrix of eigenvectors; the spectral theorem +dictates that U is an element of O(N) in the real case, U(N) in the complex case, or Sp(2N) in the +quaternionic case. To make the mapping X �→ UΛU† bijective, we insist that the eigenvalues be listed +in increasing order and that the first row of U be non-negative real. The canonical Lebesgue measure +dX = +N +∏ +i=1 +dXii +∏ +1⩽j −1, seeing as then the eigenvalue +j.p.d.f. induced by equation (1.2.42) is integrable and supported on an interval in the real +line. Moreover, we are able to give the Cauchy analogue of these p.d.f.s. +Proposition 1.4. Let β = 1, 2, 4, equivalently κ = 1/2, 1, 2, correspond to fixing the field F as +R, C, H. Then, matrices X drawn from S(G) (1.2.4), the set of self-adjoint N × N matrices, with +p.d.f. +P(Cy)(X) := +1 +Z(Cy) +N,β,α +Det(IN + X2)−κ(N−1)−1−α +(1.2.48) +have eigenvalues distributed according the eigenvalue j.p.d.f. p(Cy) as defined by equation (1.2.7) with +Cauchy weight (1.2.9). The partition function Z(Cy) +N,β,α is given by N (Cy) +N,β (2.1.14) multiplied by the +appropriate volume (1.2.45) as prescribed by equation (1.2.44). +Thus, there exist invariant matrix ensembles that correspond to the eigenvalue j.p.d.f. (1.2.7) +for every classical weight (1.2.9) and all valid parameters a, b, α. All that is lost due to this +formulation is that the relevant random matrices can no longer be related to the Ginibre +ensembles in a natural way. Rather, we will see in §1.2.4 that there are constructions involving +tridiagonal matrices which hold for continuous parameter values. +Remark 1.8. Using equation (1.2.42), one is able to interpret a given multivariable symmetric +function p(λ1, . . . , λN) as the eigenvalue j.p.d.f. of a self-adjoint real, complex, or quaternionic +random matrix X = UΛU†, with Λ = diag(λ1, . . . , λN) and U ∈ O(N), U(N), Sp(2N) Haar- +distributed according to dµHaar(U) (1.2.33). Indeed, it is known from symmetric function +theory [232] that the factor P(Λ) in the right-hand side of equation (1.2.42) is a function of +{Tr Λk}N +k=0, hence {Tr Xk}N +k=0. Thus, the matrix X has a well-defined p.d.f. P(X) on a subset +of the space of self-adjoint matrices, assuming that p(λ1, . . . , λN) is sufficiently integrable. +29 + +CHAPTER 1. Introduction +Since the upcoming (skew-)orthogonal polynomial ensembles and classical β ensembles +are specified by eigenvalue j.p.d.f.s of the type considered in the above remark, they can be +interpreted as invariant matrix ensembles. As we end this subsection, let us remark that the +matrix product ensembles discussed in §1.3.1 are also invariant matrix ensembles due to the +invariance of the Ginibre and Laguerre ensembles (see, e.g., [185] and references therein). +1.2.3 +Skew-orthogonal polynomial ensembles +The subclass of invariant matrix ensembles considered in this subsection are those concerning +self-adjoint matrices X with p.d.f.s P(X) such that one has the decomposition +P(X) = +N +∏ +i=1 +w(λi), +(1.2.49) +where {λi}N +i=1 are the indistinguishable eigenvalues of X and w(λ) is some continuous +weight function that is real-valued on R and decays fast enough as |λ| → ∞. Equivalently, +we are interested in j.p.d.f.s of the form (1.2.7), but with the weights no longer constrained +to be classical. +Definition 1.8. Let β = 1, 2, or 4, fix N ∈ N, and let w(λ) be a continuous, non-negative, +real-valued function such that w(λ) = o(λ−β(N−1)−1). Then, the j.p.d.f. +p(w)(λ1, . . . , λN; β) = +1 +N (w) +N,β +N +∏ +i=1 +w(λi) |∆N(λ)|β +(1.2.50) +describes a size-N orthogonal polynomial ensemble (OPE) when β = 2, skew-orthogonal polynomial +ensemble of real type (R-SOPE) when β = 1, or skew-orthogonal polynomial ensemble of quaternion +type (H-SOPE) when β = 4 [238, Ch. 5]. As usual, N (w) +N,β is a normalisation constant and +∆N(λ) is the Vandermonde determinant (1.1.9). +Note 1.3. The condition w(λ) = o(λ−β(N−1)−1) ensures that the j.p.d.f. specified by the above +definition has a well-defined (convergent) normalisation constant, but it does not necessarily +have convergent spectral moments. As an example, the Cauchy ensembles of §1.2.1 exhibit +these features. +To understand the nomenclature introduced in Definition 1.8, a few key observations +regarding the Vandermonde determinant need to be made. +30 + +1.2. Classical Matrix Ensembles +Lemma 1.2. Following [238, Ch. 5], [119, Ch. 6], let {qj(λ)}∞ +j=0 be a set of monic polynomials such +that qj(λ) is degree j for each j. Then, +∆N(λ) := Det +� +λj−1 +i +�N +i,j=1 = Det +� +qj−1(λi) +�N +i,j=1 , +(1.2.51) +∆N(λ)4 = Det +� +� +λj−1 +i +(j − 1)λj−2 +i +� +� +i=1,...,N +j=1,...,2N += Det +� +�qj−1(λi) +q′ +j−1(λi) +� +� +i=1,...,N +j=1,...,2N +, +(1.2.52) +and when N is even (N odd can be treated by setting sgn(λN+1 − λi) = 1 in the Pfaffian factor), +|∆N(λ)| = ∆N(λ) Pf +� +sgn(λj − λi) +�N +i,j=1 , +(1.2.53) +where sgn(0) = 0 and otherwise, sgn(λ) = λ/|λ|. +Proof sketch. To obtain the right-hand side of equation (1.2.51) from the definition of the +Vandermonde determinant presented on its immediate left, one simply notes that the +jth column of the right-hand side is equal to the jth column of the Vandermonde matrix +[λj−1 +i +]N +i,j=1 plus columns to its left. Since adding columns does not change the determinant, +one can replace the the columns [λj−1 +i +]N +i=1 with [qj−1(λi)]N +i=1 by successively iterating through +j = 2, 3, . . . , N. The middle expression of equation (1.2.52) can be obtained through induction +on N, and the right-hand side follows from a column-operation argument similar to that for +equation (1.2.51). Equation (1.2.53) follows from the right-hand side of equation (1.1.9) and +the identity [76] (valid for even N, see Appendix A) +∏ +1⩽i 0, rather than just β = 1, 2, 4. Thus, the classical β ensembles are specified by the +35 + +CHAPTER 1. Introduction +eigenvalue j.p.d.f. +p(w)(λ1, . . . , λN; β) = +1 +N (w) +N,β +N +∏ +i=1 +w(λi) |∆N(λ)|β, +β = 2κ > 0, +(1.2.81) +with w(λ) a classical weight (1.2.9). In addition to taking β to be positive real, we also +consider the classical weights with parameters a, b > −1 real and η = −κ(N − 1) − 1 − α, +α ∈ C with Re(α) > −1/2. At the end of §1.2.2, we argued that even at this level of generality, +these eigenvalue j.p.d.f.s describe the eigenvalues of certain ensembles of self-adjoint random +matrices. However, generating matrix representatives of these ensembles is not practical +since the p.d.f.s P(X) of the matrices X do not translate to j.p.d.f.s on the entries of X. Hence, +there was a natural motivation to construct alternative matrix models for the classical β +ensembles, with a focus on specifying p.d.f.s on matrix entries. +Theorem 1.4 (Dumitriu–Edelman ’02, Killip–Nenciu ’04). Recall that random variables x ∼ χ(k) +and y ∼ B(s, t) are respectively chi and beta distributed if they have p.d.f.s +2k/2−1 +Γ(k/2) xk−1e−x2/2χx>0, +21−s−tΓ(s + t) +Γ(s)Γ(t) +(1 − y)s−1(1 + y)t−1χ−1 0. These matrices are themselves products of two block-diagonal matrices whose +entries have fully specified p.d.f.s, so they can be generated numerically in a straightforward +manner. This work was extended in [49] through a certain α-deformation to produce matrix +models for the circular Jacobi ensemble defined in §1.2.1. Through the Cayley transformation +(1.2.25), the matrix models for the circular Jacobi ensemble essentially serve as such for the +Cauchy β ensemble. +Remark 1.9. +1. Applying Householder transformations to Ginibre matrices gives rise +to Hessenberg matrices, with the upper triangular entries still normally distributed +elements of the respective number system. Thus, without the self-adjoint symmetry of +the GOE, etc., the simplicity of (1.2.83) is lost and there is no general-β matrix model +for the Ginibre ensembles. +38 + +1.2. Classical Matrix Ensembles +2. The eigenvalue densities corresponding to the classical β ensembles have the large N +limiting forms given in Proposition 1.2 with the identifications (see Proposition 1.1) +γ1 = 1 + lim +N→∞ +a − κ + 1 +κN +, +γ2 = 1 + lim +N→∞ +b − κ + 1 +κN +. +(1.2.88) +This can be understood from [49, Eq. (5.6)] in the Cauchy case, and the loop equation +analysis done in [319], [142], otherwise (see also references therein). +The classical β ensembles have been studied extensively throughout the last couple of +decades, leading to a vast collection of results: see [222], [80], [280], [43], [319], [218] and +references therein for a diverse but non-exhaustive sample. A result that is of great use to +us is that the spectral moments of the classical β ensembles can be expanded in either N or +1/N, in line with equation (1.1.32). +Lemma 1.3 (Dumitriu–Edelman ’06, Dumitriu–Paquette ’12). Let mk be the kth spectral moment +of the Gaussian, Laguerre, or Jacobi β ensemble, as specified by equations (1.1.13) and (1.1.15). In the +first two cases, mk is a degree-(k + 1) polynomial in N (identically zero in the Gaussian case with k +odd due to the weight w(G)(λ) being an even function) [94], while in the Jacobi case, mk/N has a +large N expansion in 1/N [95]. For later use, we implicitly define the (N-independent) polynomial +and series coefficients Mk,l as follows: +m(G) +2k = +k +∑ +l=0 +M(G) +k,l N1+k−l, +(1.2.89) +m(L) +k += +k +∑ +l=0 +M(L) +k,l N1+k−l, +(1.2.90) +m(J) +k += +∞ +∑ +l=0 +M(J) +k,l N1−l. +(1.2.91) +The existence of these expansions was established in [94], [95] using Jack polynomial +theory (see also the later work [242] and the discussion in Section 3.2). The Cauchy and +shifted Jacobi spectral moments m(Cy) +k +, m(sJ) +k +can be written in terms of the Jacobi spectral +moments m(J) +k +(see Section 2.2 and the proof of Lemma 1.4 below) so that equation (1.2.91) +translates to a similar expansion for the m(Cy) +k +, m(sJ) +k +. +Classical even-β ensembles +The methods used in Chapter 2 are not suitable for treating the classical general-β ensembles, +but do extend past the β = 1, 2, 4 paradigm corresponding to the orthogonal, unitary, and +39 + +CHAPTER 1. Introduction +symplectic ensembles. Indeed, we are technically able to study a scheme in between these two +which we will call the classical even-β ensembles. These are simply the classical β ensembles +with β constrained to be a positive even integer. The significance of β being even is that the +Vandermonde factor |∆N(λ)|β in equation (1.2.81) is then a polynomial in the eigenvalues +{λi}N +i=1. This was seen to be a critical requirement for some results related to Selberg +correlation integrals, the 1/r2 quantum many-body system, and the Calogero–Sutherland +model (see [119, Sec. 13.2], [123, Sec. 2] for a review). In particular, the works [116], [81], +[120], [133] on various subsets of the classical even-β ensembles contain (edge scalings of) +expressions for the relevant eigenvalue densities ρ(λ) in terms of hypergeometric functions +and β-dimensional integrals, which we use as points of comparison in §2.4.2. +One issue with studying the classical even-β ensembles is that the case β = 1 is not +included. As outlined earlier, this case is very important due to it corresponding to real +symmetric matrices with orthogonal invariance. It turns out that we are able to extend our +results in Chapter 2 to the classical β ensembles with β such that 4/β is an even integer; this +allows for treatment of the β = 1 case, along with classical β ensembles corresponding to +some non-integer rational values of β such as β = 2/3. This extension is possible because the +works [94], [95] cited in Lemma 1.3 tell us that the coefficients Mk,l therein are palindromic +polynomials in −1/κ = −2/β, upon appropriate scaling of the parameters a, b. To be +precise, since the even-β analysis in Chapter 2 focuses on the spectral moments mk and their +generating functions W1(x), it is amenable to the following β ↔ 4/β duality relations. +Lemma 1.4. Let us highlight the dependence of the spectral moments mk (1.1.13) and resolvents +W1(x) (1.1.17) on the parameters N, κ = β/2, a, b, α by listing them as arguments (recall that we +write the parameter η in the Cauchy weight (1.2.9) as η = −κ(N − 1) − 1 − α with α ∈ C and +Re(α) > −1/2). Then, the kth spectral moments of the classical β ensembles satisfy the duality +relations [94], [95] (see also [149, App. A] by A. Borodin and V. Gorin) +m(G) +2k (N, κ) = (−κ)k−1m(G) +2k (−κN, 1/κ), +(1.2.92) +m(L) +k (N, κ, a) = (−κ)k−1m(L) +k (−κN, 1/κ, −a/κ), +(1.2.93) +m(J) +k (N, κ, a, b) = −κ−1m(J) +k (−κN, 1/κ, −a/κ, −b/κ), +(1.2.94) +m(Cy) +k +(N, κ, α) = −κ−1m(Cy) +k +(−κN, 1/κ, −α/κ). +(1.2.95) +40 + +1.3. Matrix Product Ensembles +These extend to the duality relations [319], [142] +W(G) +1 +(x; N, κ) = −iκ−3/2W(G) +1 +(ix/√ +κ; −κN, 1/κ), +(1.2.96) +W(L) +1 +(x; N, κ, a) = κ−2W(L) +1 +(−x/κ; −κN, 1/κ, −a/κ), +(1.2.97) +W(J) +1 (x; N, κ, a, b) = −κ−1W(J) +1 (x; −κN, 1/κ, −a/κ, −b/κ), +(1.2.98) +W(Cy) +1 +(x; N, κ, α) = −κ−1W(Cy) +1 +(x; −κN, 1/κ, −α/κ). +(1.2.99) +Proof. The references cited above do not study the Cauchy case. However, the duality +relations for the Jacobi β ensemble translate to the Cauchy β ensemble through the identities +presented in Section 2.2: Identity (2.2.3) expresses the spectral moments m(sJ) +k +of the shifted +Jacobi β ensemble (the classical β ensemble with weight w(sJ)(λ) = (1 − λ)a(1 + λ)bχ−1<λ<1, +as introduced at the end of §1.2.1) in terms of the m(J) +k +above, which shows that m(sJ) +k +satisfies +the duality (1.2.94). Then, Proposition 2.4 combined with Carlson’s theorem (cf. the proof of +Corollary 2.2 and see Remark 2.5) shows that +m(Cy) +k +(N, κ, α) = (−1)k/2m(sJ) +k +(N, κ, a, b) +��� +a=b=−κ(N−1)−1−α, +(1.2.100) +so that equation (1.2.94) implies (1.2.95). To obtain the duality relation (1.2.99), one can +repeat these arguments with either equation (1.1.17) or (1.1.20) specifying the resolvent. +1.3 +Matrix Product Ensembles +From the viewpoint of using random matrices to model transformations such as quantum +operators, scattering channels, or evolution operators for certain systems, it is quite natural +to use products of random matrices to model repeated applications of such transformations. +Hence, the theory of matrix product ensembles has applications in the study of wireless +communications [253], [301], quantum transport [33], the stability and chaoticity of large +dynamical systems [69], [184], finance [47], and the stability of neural networks [272], [172]. +There are also interesting connections to Fuss–Catalan and Raney numbers [273], [127]. +Investigations into products of random matrices can be traced back to the 1950s, with +many fundamental results and techniques being developed in the works [153], [175], [147], +[270], [278], [263] (among others) carried out over the following four decades; see [177] +(group theory) and [69] (statistical physics) for textbook treatments. There has since been +41 + +CHAPTER 1. Introduction +a consistent level of research on products of random matrices (some notable works being +[189], [191] and the applications cited above, along with the inception and success of free +probability theory [306], [244]), but the last decade has seen a remarkable surge in interest on +this topic. The beginning of this latter era is marked by the derivation of correlation kernels +(recall Theorem 1.3) encapsulating detailed knowledge of the eigenvalue and singular value +statistics of certain matrix product ensembles, where the products contain a finite number of +factors and each factor is a finite-sized matrix — most results from before this era pertain to +regimes where either the matrix sizes or number of factors are taken to infinity. The reader +is referred to [8], [182] for reviews on this period of time. +At the beginning of this chapter, it was said that two types of random matrix ensembles +will be studied in this thesis. The first is the family of classical matrix ensembles reviewed in +the previous section, while the second consists of the particular matrix product ensembles +that we now introduce. +Definition 1.10. Let m, N0 ∈ N, fix ν0 := 0, ν1, . . . , νm ∈ N, and define Ni := N0 + νi with +N := Nm. Recalling Definitions 1.3 and 1.4 of the Ginibre and Gaussian ensembles, let H +be an N0 × N0 GUE matrix and for 1 ⩽ i ⩽ m, let Gi be drawn independently from the +Ni−1 × Ni complex Ginibre ensemble. Then, the N × N product +Hm := G† +m · · · G† +1HG1 · · · Gm +(1.3.1) +represents the (N0, . . . , Nm−1, N) Hermitised matrix product ensemble [143]. When m = 1, we +alternatively say that H1 represents the (N0, N) Hermitised Laguerre ensemble. +Definition 1.11. Let m, N0/2 ∈ N, fix ν0 := 0, ν1, . . . , νm ∈ N, and define Ni := N0 + 2νi +with N := Nm. Recalling Definition 1.3 of the Ginibre ensembles, let JN0 be the elementary +antisymmetric matrix defined by equation (1.1.5) and for 1 ⩽ i ⩽ m, let Gi be drawn +independently from the Ni−1 × Ni real Ginibre ensemble. Then, the N × N product +Jm := GT +m · · · GT +1 JN0G1 · · · Gm +(1.3.2) +represents the (N0, . . . , Nm−1, N) antisymmetrised matrix product ensemble [144]. When m = 1, +we alternatively say that J1 represents the (N0, N) antisymmetrised Laguerre ensemble. +We think of these ensembles as perturbations of the more widely studied Wishart product +ensembles [8], [182], which are defined as follows. +42 + +1.3. Matrix Product Ensembles +Definition 1.12. As in Definition 1.10, let m, N0 ∈ N, fix ν0 := 0, ν1, . . . , νm ∈ N, and define +Ni := N0 + νi with N := Nm. For 1 ⩽ i ⩽ m, let Gi be drawn independently from the +Ni−1 × Ni real, complex, or quaternionic Ginibre ensemble, as specified by Definition 1.3 +(fixing the number system across all Gi). Then, the N × N product +Wm := G† +m · · · G† +1G1 · · · Gm +(1.3.3) +represents the (N0, . . . , Nm−1, N) real, complex, or quaternionic Wishart product ensemble, respec- +tively. Setting m = 1 recovers the (N0, N) Laguerre ensemble, in keeping with Definition 1.4. +Our present goal in studying the Hermitised and antisymmetrised matrix product ensem- +bles is to better understand the ramifications of perturbing the Wishart product ensembles +in the corresponding ways (inserting a GUE matrix H or elementary antisymmetric matrix +JN0 in the middle of the relevant products), in terms of both changes to the eigenvalue +statistics and changes to the effectiveness of existing techniques for analysing these statistics. +In particular, we would like to see how the recent loop equation analysis of Dartois and +Forrester [74] for the (N, N, N) complex Wishart product ensemble extends to the Hermi- +tised and antisymmetrised matrix product ensembles. Thus, in Chapter 4, we derive loop +equations characterising the correlator expansion coefficients Wl +n of the Hermitised and +antisymmetrised Laguerre ensembles (i.e., the product ensembles of Definitions 1.10 and 1.11 +with m = 1) after they have been scaled such that their connected n-point correlators Wn +(1.1.29) admit large N expansions of the form (1.1.21) — we argue the validity of these large +N expansions in §3.3.3 by giving combinatorial interpretations (which are interesting in +their own right) to the mixed cumulants generated by the Wn. Our decision to restrict the +analysis of Chapter 4 to the m = 1 ensembles stems from the fact that taking m to be any +larger results in loop equations that are too unwieldy to fit within the scope of this thesis. +Nonetheless, the loop equations derived in Chapter 4 have rich enough structure for us to +make meaningful comparison with the loop equations presented in [74]. We are also able to +glean insight on the expected structure of the loop equations pertaining to arbitrary m ∈ N. +Exploring the consequences of perturbing the Wishart product ensembles is not our only +motivation for studying the Hermitised and antisymmetrised matrix product ensembles, +as this could be done by considering any number of alternative, comparable perturbations +of said ensembles. Rather, our secondary motivation for studying the Hermitised and +43 + +CHAPTER 1. Introduction +antisymmetrised matrix product ensembles is that they have recently been shown to be +related to certain Muttalib–Borodin ensembles and a generalisation, respectively analogue, +of the Harish-Chandra–Itzykson–Zuber (HCIZ) integral [143], [144]. Hence, we are able +to compare the results of Chapter 4 against explicit functional forms for the eigenvalue +j.p.d.f.s of the Hermitised and antisymmetrised matrix product ensembles that were obtained +in [143], [144]. Moreover, it is expected that the results of Chapter 4 will help advance +our understanding of the broader connection between products of random matrices and +Muttalib–Borodin ensembles that is seen in the literature [216], [141], [182], [321], [135], [143]. +Let us now review, in order, the complex Wishart product ensembles, Muttalib–Borodin +ensembles, and HCIZ-type integrals. We keep our review brief by focusing mostly on aspects +of these topics that connect them to the ensembles that are studied in this thesis. +1.3.1 +Complex Wishart product ensembles and their correlation kernels +If one wishes to capture the essence of multiplying generic random matrices, it is reasonable +to study products of Ginibre matrices: On one hand, prescribing p.d.f.s on the matrix factors +makes the situation relatively tractable and allows the use of some powerful analytical tools. +On the other hand, there is no benefit in imposing symmetry constraints on the involved +matrices, since products of self-adjoint matrices are not necessarily self-adjoint. Thus, a +large portion of recent work in the field has been on products of Ginibre matrices; see +[58], [7], [217], [183] and references therein for a sample. Note that products of truncated +Haar-distributed unitary matrices (cf. the first point of Remark 1.4) and products involving +inverse Ginibre matrices have also drawn considerable interest [185], [13], [122], [3]. +In studying the spectral properties of products of Ginibre matrices, one is interested +in either the eigenvalues, which are generally complex, or the singular values, which are +non-negative real (cf. Remark 1.6). Both have been studied in, e.g., [7], [185], [3] and [12], +[11], respectively. Due to their relative simplicity, our interest lies in the squared singular +values, which are equivalent to the eigenvalues of the Wishart product matrices Wm (1.3.3). +Remark 1.10. Like the eigenvalues of Wm, the eigenvalues of the matrix product Hm (1.3.1) +are real, since it is Hermitian. In a similar vein, the eigenvalues of the antisymmetric matrix +Jm (1.3.2) lie on the imaginary axis in complex conjugate pairs. +44 + +1.3. Matrix Product Ensembles +Let us now present the analogue of Theorem 1.3 for the complex Wishart product +ensembles, which we obtain by combining results of [11] and [217]. +Proposition 1.6. Let p(cWm)(λ1, . . . , λN0) be the j.p.d.f. of the non-zero eigenvalues of the complex +Wishart product matrix Wm (1.3.3) and let us recall [31] that the Meijer G-function is defined as +Gm,n +p,q +�a1, . . . , ap +b1, . . . , bq +���� z +� +:= +1 +2πi +� +γ +∏m +j=1 Γ(bj + u) ∏n +j=1 Γ(1 − aj − u) +∏ +q +j=m+1 Γ(1 − bj − u) ∏ +p +j=n+1 Γ(aj + u)z−u du, +(1.3.4) +where the choice of integration contour γ is too involved to specify here. Then, for 1 ⩽ k ⩽ N0, the +k-point correlation function (cf. equation (1.2.72)) associated with p(cWm)(λ1, . . . , λN0) has the form +ρ(cWm) +k +(λ1, . . . , λk; N0) := +N0! +(N0 − k)! +� +RN0−k p(cWm)(λ1, . . . , λN0) dλk+1 · · · dλN0 +(1.3.5) += Det +� +K(cWm) +N0,ν1,...,νm(λi, λj) +�k +i,j=1 , +(1.3.6) +with correlation kernel given by +K(cWm) +N0,ν1,...,νm(x, y) = +� 1 +0 G1,0 +1,m+1 +� +N0 +0, −ν1, . . . , −νm +���� ux +� +Gm,1 +1,m+1 +� +−N0 +ν1, . . . , νm, 0 +���� uy +� +du. +(1.3.7) +Equation (1.3.6) tells us that the complex Wishart product ensembles are determinantal +point processes, much like what was seen in §1.2.3 for the β = 2 classical matrix ensembles. +However, while this property of the β = 2 classical matrix ensembles is a consequence of +their being orthogonal polynomial ensembles, the same is not true for the complex Wishart +product ensembles. Instead, the complex Wishart product ensembles, together with the +Hermitised and antisymmetrised matrix product ensembles, are biorthogonal ensembles [38] +(see the next subsection for a formal definition of this term). In fact, all three of these matrix +product ensembles belong to a further subclass of biorthogonal ensembles that are simply +called polynomial ensembles4 [216]; these so-called polynomial ensembles are characterised by +their eigenvalue j.p.d.f.s having the form +p(λ1, . . . , λN0) = +1 +NN0 +∆N0(λ) Det +� +wj−1(λi) +�N0 +i,j=1 , +λ1, . . . , λN0 ∈ R, +(1.3.8) +4In the case of the antisymmetrised matrix product ensembles, it is common practice to study the positive +eigenvalues of iJm, since these are real and independent, in keeping with Remark 1.10. Technically, it is the +positive eigenvalues of iJm that constitute a biorthogonal ensemble and it is the squares of these eigenvalues +that exhibit the structure (1.3.8) of polynomial ensembles. +45 + +CHAPTER 1. Introduction +where NN0 is a normalisation constant, ∆N0(λ) is the Vandermonde determinant (1.1.9), and +the family of weight functions {wj−1(λ)}N0 +j=1 is such that the above j.p.d.f. is well-defined. +Proposition 1.7. The j.p.d.f.s p(cWm)(λ1, . . . , λN0) and p(Hm)(λ1, . . . , λN0) governing the non-zero +eigenvalues of the complex Wishart product matrix Wm (1.3.3) and Hermitised matrix product Hm +(1.3.1), respectively, are given by equation (1.3.8) with weights [11], [143] +w(cWm) +j +(λ) = Gm,0 +0,m +� +− +ν1 + j, ν2, . . . , νm +���� λ +� +, +(1.3.9) +w(Hm) +j +(λ) = (sgn λ)j +m +∏ +i=1 +2νi−1 +√π G2m+1,0 +0,2m+1 +� +− +ν1 +2 , ν1+1 +2 , . . . , νm +2 , νm+1 +2 +, j +2 +���� +λ2 +4m +� +, +(1.3.10) +where we recall the definition of the Meijer G-function from Proposition 1.6. Similarly, the positive +eigenvalues of iJm (1.3.2) are distributed according to the eigenvalue j.p.d.f. [144] +p(iJm)(λ1, . . . , λN0/2) = +1 +N (iJm) +N0/2 +∏ +1⩽k −1 +(cf. the discussion surrounding Proposition 1.4). Moreover, the results of these propositions +are invariant under permutations of the parameters ν1, . . . , νm due to the weak commutation +relation established in [185]. +A point of intrigue regarding the antisymmetrised matrix product ensembles is that they +have recently been shown in [144] to be closely related to the complex Wishart product +ensembles generalised through the induced Ginibre matrices described above. +Proposition 1.8. Let Jm be as specified in Definition 1.11 and let λ1, . . . , λN0/2 denote the positive +eigenvalues of iJm. Then, by [144, Cor. 1.2], the variables λ′ +j = λ2 +j /4m (1 ⩽ j ⩽ N0/2) are +statistically equivalent to the eigenvalues of the product ˆG† +2m · · · ˆG† +1 ˆG1 · · · ˆG2m of (N0/2) × (N0/2) +induced Ginibre matrices with p.d.f.s (1.3.13) +P(indG)( ˆG2i−1; N0/2, 2, νi), +P(indG)( ˆG2i; N0/2, 2, νi − 1/2), +1 ⩽ i ⩽ m. +In short, the j.p.d.f. of the positive eigenvalues of iJm is given by +p(iJm)(λ1, . . . , λN0/2) = 2N0(1/2−m)λ1 · · · λN0/2 +× p(cW2m)(λ2 +1/4m, . . . , λ2 +N0/2/4m) +���� ν2i−1�→νi, +ν2i�→νi−1/2 +�m +i=1 +, +(1.3.14) +where the right-hand side can be read off by reparametrising equations (1.3.8) and (1.3.9). +In addition to highlighting a relationship between the weights w(cWm) +j +(λ) (1.3.9) and +w(iJm) +j +(λ) (1.3.12), the above proposition tells us that the positive eigenvalues of iJm constitute +a determinantal point process whose k-point correlation functions have the form (1.3.6) with +correlation kernel given by +K(iJm) +N0/2,ν1,...,νm(x, y) = 21−2m√xy K(cW2m) +N0/2,ν1,ν1−1/2,...,νm,νm−1/2(x2/4m, y2/4m), +(1.3.15) +where the right-hand side is specified by equation (1.3.7). Let us mention here that it was +shown in [143] that the non-zero eigenvalues of the Hermitised matrix product Hm also +constitute a determinantal point process; we do not present the relevant correlation kernel +here, but note that it is made explicit in [143]. +47 + +CHAPTER 1. Introduction +Scalings of the eigenvalue densities and correlation kernels +Recall from Proposition 1.2 that in the case of the classical matrix ensembles, the N → ∞ +limits ρ0(λ) of the global scaled eigenvalue densities ˜ρ(λ) (1.1.22) have distinct forms for +each classical weight with, e.g., the Gaussian and Laguerre weights corresponding to the +Wigner semi-circle and Marˇcenko–Pastur laws, respectively. It turns out that the analogous +limiting densities (now taking N0 → ∞) of the matrix product ensembles considered in this +section relate to the Fuss–Catalan distribution [273], [182] (see also the independent works +of Biane [35, App. A] and Neuschel [262] for a Plancherel–Rotach-like parametrisation in +terms of elementary functions) +ρ(FCm)(λ) = +1 +√ +2π +mm−3/2 +(m + 1)m+1/2 Gm,0 +m,m +� +1 +m, 0, − 1 +m, . . . , − m−2 +m +− +1 +m+1, − +2 +m+1, . . . , − +m +m+1 +���� +mm +(m + 1)m+1 λ +� +, +(1.3.16) +which is so named due to its spectral moments being the Fuss–Catalan numbers [167]: +m(FCm) +k +:= +� +R λkρ(FCm)(λ) dλ = +1 +mk + 1 +�(m + 1)k +k +� +, +k ∈ N. +(1.3.17) +Note that due to properties of the Meijer G-function (1.3.4), the density ρ(FCm)(λ) has compact +support [0, (m + 1)m+1/mm], on which it integrates to unity. +Definition 1.13. Let ρ(cWm)(λ), ρ(Hm)(λ), and ρ(iJm)(λ) denote the eigenvalue densities +(1.1.10), equivalently the 1-point correlation functions (1.3.5), associated with the eigenvalue +j.p.d.f.s p(cWm)(λ1, . . . , λN0), p(Hm)(λ1, . . . , λN0), and p(iJm)(λ1, . . . , λN0/2) specified in Propo- +sition 1.7, respectively. Then, in analogy with Definition 1.6, and partially following [182], +[143], define the corresponding global scaled eigenvalue densities (1.1.22) to be +˜ρ(cWm)(λ) := Nm−1 +0 +ρ(cWm)(Nm +0 λ), +(1.3.18) +˜ρ(Hm)(λ) := +1 +√ +2 Nm−1/2 +0 +ρ(Hm)( 1 +√ +2 Nm+1/2 +0 +λ), +(1.3.19) +˜ρ(iJm)(λ) := 2Nm−1 +0 +ρ(iJm)(Nm +0 λ). +(1.3.20) +Furthermore, recall the notation ρ0(λ) = limN0→∞ ˜ρ(λ) introduced in §1.1.1. +It has been shown in [29], [273], [143] that for fixed ν1, . . . , νm > −1, the large N0 limits +of ˜ρ(cWm)(λ) and ˜ρ(Hm)(λ) are given in terms of the Fuss–Catalan distribution (1.3.16) by +ρ(cWm),0(λ) = ρ(FCm)(λ), +(1.3.21) +ρ(Hm),0(λ) = |λ|ρ(FC2m+1)(λ2). +(1.3.22) +48 + +1.3. Matrix Product Ensembles +Setting x = y in equation (1.3.15) so that it may be read as a relation between the correspond- +ing eigenvalue densities, and then combining this relation with equation (1.3.21) shows that +(see also [127] for the m = 1 case) +ρ(iJm),0(λ) = 2λρ(FC2m)(λ2)χλ>0. +(1.3.23) +It can immediately be seen that, aside from subtleties concerning the parameter m, ρ(Hm),0(λ) +is essentially the symmetric extension of ρ(iJm),0(λ) to the negative real axis. Moreover, +both of these densities relate to ρ(cWm),0(λ) via the mapping λ �→ +√ +λ, much like how the +Wigner semi-circle (1.2.14) and Marˇcenko–Pastur (1.2.15) laws are related by the equality +ρ(G),0(λ) = 2|λ| ρ(L),0(2λ2) when the Laguerre parameter a is fixed. In particular, recall that +ρ(cW1),0(λ) = ρ(L),0(λ) and ρ(H0),0(λ) = ρ(G),0(λ/ +√ +2), by definition. +The global scaling regimes just discussed are such that the limiting densities ρ0(λ) have +compact connected supports. In contrast, one is also interested in local scaling regimes, where +the eigenvalue density ρ(λ) is centred on a particular point λ0 ∈ supp ρ and then scaled so +that the spacing between this eigenvalue and its neighbours is of order unity. The resulting +statistics depends on whether λ0 is positioned at a soft edge, a hard edge, or within the bulk. +In the case of the complex Wishart product ensembles with ν1, . . . , νm > −1 fixed, λ0 is said +to be at the soft edge if limN0→∞ λ0/Nm +0 = (m + 1)m+1/mm, the hard edge if limN0→∞ λ0/Nm +0 +vanishes, and in the bulk if limN0→∞ λ0/Nm +0 ∈ (0, (m + 1)m+1/mm). (The analogous scaling +regimes of ρ(Hm)(λ) and ρ(iJm)(λ) are likewise defined by comparing similar scaled limits, +with scalings prescribed by equations (1.3.19), (1.3.20), to ±(m + 1)(m+1)/2/mm/2.) +Proposition 1.9. Let m ∈ N and ν1, . . . , νm > −1 be fixed. Furthermore, let +x0 ∈ (0, (m + 1)m+1/mm), +cb = Nm−1 +0 +/ρ(FCm)(x0), +x+ = +� +N0 +m +�m +(m + 1)m+1, +c+ = Nm−2/3 +0 +21/3mm−1 +(m+1)m+2/3 . +(1.3.24) +Then, the hard edge, bulk, and soft edge limiting forms of the correlation kernel (1.3.7) are respectively +given by [228], [217] +lim +N0→∞ +1 +N0 +K(cWm) +N0,ν1,...,νm (x/N0, y/N0) = K(Meijer) +ν1,...,νm (x, y), +(1.3.25) +lim +N0→∞ cbK(cWm) +N0,ν1,...,νm (cbx + Nm +0 x0, cby + Nm +0 x0) = K(sine)(x, y), +(1.3.26) +lim +N0→∞ c+K(cWm) +N0,ν1,...,νm (c+x + x+, c+y + x+) = K(Airy)(x, y), +(1.3.27) +49 + +CHAPTER 1. Introduction +where +K(Meijer) +ν1,...,νm (x, y) = +� 1 +0 G1,0 +0,m+1 +� +− +0, −ν1, . . . , −νm +���� ux +� +Gm,0 +0,m+1 +� +− +ν1, . . . , νm, 0 +���� uy +� +du, +(1.3.28) +K(sine)(x, y) = sin π(x − y) +π(x − y) +, +(1.3.29) +K(Airy)(x, y) = Ai(x)Ai′(y) − Ai′(x)Ai(y) +x − y +(1.3.30) +are the Meijer G-, sine, and Airy kernels, respectively (recall that the Meijer G-function is partially +defined in equation (1.3.4) and that Ai(x) denotes the Airy function). +Remark 1.11. Recall from the discussion following Figures 1.1–1.3 that the hard edge of +the Laguerre ensembles degenerates to a soft edge upon letting the Laguerre parameter a +grow linearly with N. A similar phenomenon is exhibited by the complex Wishart product +ensembles [11], whose hard edge ceases to be if all of the parameters ν1, . . . , νm scale with +N0. If one writes νi = ˆνiN0 + δi with ˆνi, δi = O(1) for 1 ⩽ i ⩽ m and further enforces +ˆν1 = · · · = ˆνl = 0, but ˆνl+1, . . . , ˆνm > 0 for some 1 ⩽ l ⩽ m, then it is known from [182] that +equation (1.3.25) should be replaced with +lim +N0→∞ c−K(cWm) +N0,ν1,...,νm(c−x, c−y) = K(Meijer) +ν1,...,νl (x, y), +c− = ˆνl+1 · · · ˆνmNm−l−1 +0 +. +(1.3.31) +It follows from Propositions 1.8 and 1.9 that the local correlations of the positive eigen- +values of iJm are characterised by the Meijer G-, sine, and Airy kernels, in the appropriate +scaling regimes. Likewise, it is mentioned (without proof) in [143] that the bulk and soft edge +scaling limits of the Hermitised matrix product ensembles are also described by the sine and +Airy kernels. More interesting, however, is the microscopic behaviour of these ensembles at +the origin. There, one observes the so-called double-sided hard edge phenomenon. +Proposition 1.10. Let m ∈ N and ν1, . . . , νm > −1 be fixed. Moreover, let K(Hm) +N0,ν1,...,νm(x, y) denote +the correlation kernel corresponding to p(Hm)(λ1, . . . , λN0) in analogy with equation (1.3.6). Then, +for x, y ∈ R \ {0}, one has the limit [143] +lim +N0→∞ +1 +√ +N0/2K(Hm) +N0,ν1,...,νm +� +x +√ +N0/2, +y +√ +N0/2 +� += sgn(xy)|x| +4m +K(Meijer) +1 +2, ν1 +2 , ν1+1 +2 +,..., νm +2 , νm+1 +2 +� x2 +4m , y2 +4m +� ++ |y| +4m K(Meijer) +− 1 +2 , ν1 +2 , ν1−1 +2 +,..., νm +2 , νm−1 +2 +� x2 +4m , y2 +4m +� +. +(1.3.32) +50 + +1.3. Matrix Product Ensembles +Remark 1.12. It is important to note that the term ‘double-sided hard edge’ mentioned +above merely refers to the fact that the local correlations at the origin of the spectra of +the Hermitised matrix product ensembles relate to the hard edge of the complex Wishart +product ensembles via the mapping λ �→ λ2 and that said spectra do not actually exhibit +an edge at the origin. In fact, it is shown in [143] that setting m = 0 in the right-hand side +of equation (1.3.32) recovers the sine kernel, which corresponds to bulk statistics. Let us +also take this opportunity to point out that the limiting densities ρ(cWm),0(λ) of the complex +Wishart product ensembles do not always have inverse square root singularities at the hard +edge, in contrast to what is known for the classical matrix ensembles. Rather, in the setting +of Remark 1.11, ρ(cWm),0(λ) diverges like λ−l/(l+1) as λ → 0 [58]. +It is well known [115], [39], [119, Ch. 7] that the local bulk, soft edge, and hard edge +statistics of the β = 2 classical matrix ensembles are respectively governed by the sine (1.3.29), +Airy (1.3.30), and Bessel kernels (the required edge scalings are given in §2.4.2), with the +latter defined as +K(Bessel) +a +(x, y) = Ja(√x)√yJ′ +a(√y) − √xJ′ +a(√x)Ja(√y) +2(x − y) +, +(1.3.33) +where Ja(x) is the Bessel function and a is the exponent seen in the Laguerre and Jacobi +weights (1.2.9). Thus, the microscopic bulk and soft edge statistics of the matrix product en- +sembles discussed in this section are within the same universality classes as those pertaining +to the β = 2 classical matrix ensembles, but this is not the case at the hard edge (although, +the Meijer G-kernel (1.3.28) reduces to the Bessel kernel upon setting m = 1, as one would +expect [8]). In this way, as well as their being biorthogonal ensembles rather than orthogonal +polynomial ensembles, our matrix product ensembles display universal structures that are +fundamentally different to those seen in the classical setting. +1.3.2 +Muttalib–Borodin and biorthogonal ensembles +In their 1988 work [239] on multichannel disordered wires, Mello, Pereyra, and Kumar +(building on the 1982 work [87] of Dorokhov) derived a Fokker–Planck equation that came +to be known as the DMPK equation. This equation characterises the eigenvalues of the +associated transfer matrix, which encode physical statistics of the wire at hand, such as its +51 + +CHAPTER 1. Introduction +quantum conductance and electron localisation length. The DMPK equation was shown in +the 1993 work [34] of Beenakker and Rejaei to be solved by the eigenvalue j.p.d.f. +p(DMPK)(λ1, . . . , λN) = +1 +N (DMPK) +N,s +N +∏ +i=1 +exp (−VDMPK(λi; s)) ∆N(λ) +× +∏ +1⩽j 0, +(1.3.34) +where ∆N(λ) is the Vandermonde determinant (1.1.9), s is a length scale, N (DMPK) +N,s +is a +normalisation constant, and +VDMPK(λ; s) = N +s arcsinh2( +√ +λ) +� +1 + O(N−1) +� +(1.3.35) +is a background potential. Two years later, Muttalib proposed [254] the study of simplifica- +tions of the above model that correspond to eigenvalue j.p.d.f.s of the form +p(λ1, . . . , λN) = +1 +NN,θ +N +∏ +i=1 +exp (−V(λi)) ∆N(λ) +∏ +1⩽j 0, (1.3.36) +where NN,θ is a normalisation constant, θ ∈ N is a deformation parameter (the ensembles +can be thought of as deformations of the θ = 1 cases, which correspond to the OPEs +discussed in §1.2.3), and V(λ) is a general background potential such that the j.p.d.f. (1.3.36) +is well-defined — this structure relates to the j.p.d.f. (1.3.34) in the limit θ → 0+ [135]. +It turns out that the relation between Muttalib’s model and the DMPK equation was not +pursued much further by the physics community, but varying forms of equation (1.3.36) +appeared nonetheless in a range of works [230], [66], [65], [216], [127] over the following two +decades. A significant motivation for many of these works was that the associated models +were shown to be exactly solvable by Muttalib [254] and Borodin [38]: Muttalib used the +Konhauser theory of biorthogonal polynomials [209] (this theory was later realised [15] to +have been studied much earlier in [84], [79]) to prove that the ensembles corresponding +to equation (1.3.36) are determinantal point processes; Borodin extended this work by +introducing the so-called biorthogonal Hermite, Laguerre, and Jacobi ensembles (see below) +and giving explicit formulae for their correlation kernels (cf. equation (1.3.6)). Thus, in +the recent work [135], the name Muttalib–Borodin ensemble was given to the systems of +positive real eigenvalues governed by j.p.d.f.s of the form (1.3.36), with θ now a positive real +deformation parameter. +52 + +1.3. Matrix Product Ensembles +Definition 1.14. Having fixed N ∈ N, a biorthogonal ensemble [38] is a system of eigenvalues +{λi}N +i=1 supported on a possibly infinite interval I ⊆ R with eigenvalue j.p.d.f. of the form +p(b,w)(λ1, . . . , λN) = +1 +N (b,w) +N +N +∏ +i=1 +w(λi) Det +� +fj(λi) +�N +i,j=1 Det +� +gj(λi) +�N +i,j=1 , +(1.3.37) +where N (b,w) +N +is a normalisation constant, w(λ) is an admissible weight supported on I (that +is, w(λ) is continuous, non-negative, and real-valued over I [209]; cf. Definition 1.8), and +the sequences of real-valued functions { fj(λ)}N +j=1 and {gj(λ)}N +j=1, along with w(λ), are such +that the above j.p.d.f. is well-defined. +Remark 1.13. Polynomial ensembles are examples of biorthogonal ensembles since setting +N = N0, +w(λi) = 1, +fj(λi) = λj−1 +i +, +gj(λi) = wj−1(λi) +in equation (1.3.37) recovers the structure (1.3.8). Likewise, Muttalib–Borodin ensembles +(1.3.36) can be seen to be biorthogonal ensembles by setting +w(λi) = exp(−V(λi)), +fj(λi) = λj−1 +i +, +gj(λi) = λθ(j−1) +i +in equation (1.3.37). +Rewriting equation (1.3.36) by absorbing the factors exp(−V(λi)) +into the determinant ∏1⩽j0, +Laguerre, +λa(1 − λ)bχ0<λ<1, +Jacobi. +(1.3.45) +54 + +1.3. Matrix Product Ensembles +Here, N (w) +N,θ is again a normalisation constant and θ > 0, a, b > −1 are real parameters. +As one might expect, the j.p.d.f. (1.3.44) is a generalisation of the form (1.3.36); they are +equivalent if all eigenvalues are positive and/or if θ is an odd integer. +The correlation kernels (1.3.41) governing the statistics of the Muttalib–Borodin ensembles +with weights (1.3.45) (with b = 0) were made explicit in [38], along with their (double-sided) +hard edge limiting forms. The associated biorthogonal polynomials were studied earlier in +[210], [233], [296]. More recently, Forrester and Ipsen [125] have shown how Selberg integral +theory can be used to obtain explicit formulae for the biorthogonal polynomials related to +the weights (1.3.45), in addition to the so-called Jacobi prime, generalised symmetric Jacobi, +and generalised Cauchy weights. Incidentally, the biorthogonal functions relating to the +correlation kernels K(cWm) +N0,ν1,...,νm(x, y), K(iJm) +N0/2,ν1,...,νm(x, y), and K(Hm) +N0,ν1,...,νm(x, y) discussed in the +previous subsection have also been made explicit in the works [11], [143], [144]. +Relations to matrix product ensembles +Our interest lies in the Laguerre and Hermite Muttalib–Borodin ensembles, as they have +been shown [141], [182], [143] to approximate the complex Wishart and Hermitised matrix +product ensembles upon applying the asymptotic formula [231, Sec. 5.7] +Gm,0 +0,m +� +− +ν1, . . . , νm +���� λ +� +∼ +λ→∞ +1 +√m +� 2π +λ1/m +�(m−1)/2 +λ(ν1+···+νm)/pe−mλ1/m � +1 + O(λ1/m) +� +(1.3.46) +to the Meijer G-functions in Proposition 1.7 and appropriately changing variables. +Proposition 1.11. Let m, N0 ∈ N and ν1, . . . , νm > −1 be fixed (with N0 even when working +with iJm). Furthermore, let p(H,θ)(λ1, . . . , λN; a) and p(L,θ)(λ1, . . . , λN; a) denote the eigenvalue +j.p.d.f. (1.3.44) with w(λ) being the generalised Hermite and Laguerre weights (1.3.45), respectively. +Then, substituting the large λ approximation (1.3.46) into the formulae of Proposition 1.7 shows that +in the λ1, . . . , λN0 → ∞ limit with ¯ν := ν1 + · · · + νm and γi = λi/√ +2m + 1 (1 ⩽ i ⩽ N0), +p(cWm) �� +λ1 +m +�m +, . . . , +� λN0 +m +�m� +≈ +N0 +∏ +i=1 +� +λi +m +�1−m +p(L,m) � +λ1, . . . , λN0; ¯ν + m−1 +2 +� +, +(1.3.47) +p(iJm) �� +λ1 +m +�m +, . . . , +� λN0/2 +m +�m� +≈ +N0/2 +∏ +i=1 +� +λi +m +�1−m +p(L,2m) � +λ1, . . . , λN0/2; 2¯ν + m−1 +2 +� +, +(1.3.48) +p(Hm) � +2mγ2m+1 +1 +, . . . , 2mγ2m+1 +N0 +� +≈ +N0 +∏ +i=1 +γ−2m +i +2m√ +2m + 1 +p(H,2m+1) (λ1, . . . , λN0; 2¯ν + m) . +(1.3.49) +55 + +CHAPTER 1. Introduction +Remark 1.14. It is shown in [144, Cor. 4.4] that the approximation (1.3.48) is in fact exact +for m = 1. This complements the fact that the approximations (1.3.47) and (1.3.49) are also +exact for m = 1 and m = 0, respectively. (Comparing equations (1.2.7), (1.2.9) to equations +(1.3.44), (1.3.45) reveals that p(L,1)(λ1, . . . , λN; a) and p(H,1)(λ1, . . . , λN; 0) are respectively the +eigenvalue j.p.d.f.s of the Laguerre and Gaussian unitary ensembles.) +The works [141], [182] contain formulae similar to the approximation (1.3.47), while the +approximation (1.3.49) is given in [143]. Indeed, the approximation (1.3.49) represents the +success of the latter work in relating the Hermite Muttalib–Borodin ensembles to products +of random matrices with explicit p.d.f.s on their entries. Thus, [143] completed the task of +finding matrix model realisations of the Muttalib–Borodin ensembles with (semi-)classical +weights (1.3.45), supplementing analogous realisations of the Laguerre and Jacobi Muttalib– +Borodin ensembles given in the earlier works [5], [141], [135], [65]. We note, in particular, that +the Jacobi Muttalib–Borodin ensembles relate to products of Ginibre matrices and truncations +of Haar-distributed unitary matrices (see [205], [135, §2.2, §3.2] and references therein). +In light of the scalings discussed in §1.3.1, a question of obvious interest regarding +Proposition 1.11 is, “How accurate are the approximations (1.3.47)–(1.3.49) in the large N0 +limit and where do they break down?” In other words, “When taking N0 → ∞, how do the +eigenvalue statistics of the matrix product ensembles discussed in §1.3.1 differ from those +of the Muttalib–Borodin ensembles that they relate to via Proposition 1.11?” It turns out +that the global, bulk, and soft edge scaling statistics of the Muttalib–Borodin ensembles +pertaining to the right-hand sides of the approximations (1.3.47)–(1.3.49) are equivalent to +those of the corresponding matrix product ensembles [38], [327], [127], [135], [143] — that is, +the respective statistics are yet again described by the Fuss–Catalan distribution (1.3.16), the +sine kernel (1.3.29), and the Airy kernel (1.3.30). This makes sense at a heuristic level since, +by the discussion preceding Proposition 1.9, the eigenvalues of the matrix product ensembles +considered in Proposition 1.11 that are not at the (double-sided) hard edge grow like Nm +0 +and are thus large enough, in the N0 → ∞ limit, to fall within the scope of said proposition. +As one might expect, Proposition 1.11 breaks down at the origin, but not too drastically: +Let K(L,θ) +N +(x, y; a) denote the correlation kernel (1.3.41) corresponding via equation (1.3.40) to +the eigenvalue j.p.d.f. p(L,θ)(λ1, . . . , λN; a) specified in Proposition 1.11. Borodin [38] showed +56 + +1.3. Matrix Product Ensembles +that the hard edge scaling limit of this kernel is given by +K(a,θ)(x, y) := lim +N→∞ N−1/θK(L,θ) +N +(N−1/θx, N−1/θy; a) += θxa +� 1 +0 J(a+1)/θ,1/θ(xt) Ja+1,θ((yt)θ)ta dt, +where +Ja,b(x) = +∞ +∑ +j=0 +(−x)j +j!Γ(a + jb) +is Wright’s generalised Bessel function [324]. It has been shown [216, Thrm. 5.1] for θ ∈ N, +as relevant to Proposition 1.11 (they also show an analogous result for 1/θ ∈ N that we do +not display here), that +x1/θ−1K(a,θ)(θx1/θ, θy1/θ) = K(Meijer) +a+1 +θ +−1, a+2 +θ +−1,..., a+θ +θ +−1(y, x), +(1.3.50) +where K(Meijer) +ν1,...,νm (x, y) is the Meijer G-kernel (1.3.28) describing the hard edge statistics of the +complex Wishart product ensembles via equation (1.3.25). In the Hermite case, Borodin +[38] showed that scaling the correlation kernel corresponding to the eigenvalue j.p.d.f. +p(H,θ)(λ1, . . . , λN; a) to the origin results in a limiting kernel that is related to K(a,θ)(x, y) in +a similar fashion to equation (1.3.32). Thus, in analogy with Proposition 1.10, the Hermite +Muttalib–Borodin ensembles exhibit a double-sided hard edge at the origin that is described +by the Meijer G-kernel (some discussion on this point is also given in [143]). +Let us highlight the remarkable fact that applying the large eigenvalue approximation +(1.3.46) to our matrix product ensembles of interest does not disrupt the small eigenvalue +statistics enough to change the fact that their (double-sided) hard edge statistics are described +by the Meijer G-kernel — one need only reparametrise said kernel. Thus, it is understood in +the present day that the matrix product ensembles reviewed in this section, together with +their approximations in terms of Muttalib–Borodin ensembles, belong to a universality class +of eigenvalue ensembles that are characterised by having their bulk, soft edge, and (double- +sided) hard edge statistics described respectively by the sine kernel, the Airy kernel, and +the Meijer G-kernel with various parametrisations. This universality class has consequently +garnered great interest, with recent works showing that it contains other matrix product +ensembles [122], [216], [205], along with the class of Muttalib–Borodin ensembles with θ > 0 +real and general weight of the form w(λ) = λae−NV(λ)χλ>0 [215], [247], [311] (see also [67], +[60] for related results on this latter class of Muttalib–Borodin ensembles). +57 + +CHAPTER 1. Introduction +1.3.3 +Integrals of Harish-Chandra–Itzykson–Zuber type +The Harish-Chandra–Itzykson–Zuber (HCIZ) integral formula is the evaluation +� +U(N) eTr(AUBU†) d ˜µ′ +Haar(U) = +N−1 +∏ +i=1 +i! +Det[eajbk]N +j,k=1 +∆N(a)∆N(b) , +(1.3.51) +where A and B are N × N complex Hermitian matrices with eigenvalues {aj}N +j=1 and {bj}N +j=1, +∆N(λ) is the Vandermonde determinant (1.1.9), and d ˜µ′ +Haar(U) = dµ′ +Haar(U)/vol(U(N)) is +the Haar probability measure on U(N). (The measure dµ′ +Haar(U) is specified by equation +(1.2.35) with β = 2, while vol(U(N)) = � +U(N) dµ′ +Haar(U) can be obtained by setting β = 2 in +equation (1.2.45) and multiplying the result by [vol(U(1))]N = [2π]N.) The HCIZ integral +formula (1.3.51) is so named due to it being a special case of the general group integral +studied by Harish-Chandra in the 1957 work [175], and for its independent derivation and +introduction to random matrix theory by Itzykson and Zuber in the 1980 work [187]. +The HCIZ integral and its numerous analogues (some of which can be inferred from +Harish-Chandra’s original group integral [175] as shown in, e.g., [150], [276]) have seen +direct applications in the study of quantum chromodynamics [203], [188], [9], [204] and have +otherwise been studied more broadly for their connections to Lie theory, combinatorics, +harmonic analysis, and probability theory [177], [295], [328]. In random matrix theory, +HCIZ-type integrals were shown in the early to mid-2000s to be powerful tools for studying +complex Gaussian sample covariance matrices (i.e., products Σ−1W with W a complex +Wishart–Laguerre matrix, as specified in Definition 1.4, and Σ a fixed covariance matrix) [37], +[26], [288] and sums involving random matrices [194], [202], [131]. +A breakthrough was made in the 2013 work [12] of Akemann et al., who showed how +the formula (1.3.51) can be used to obtain the eigenvalue j.p.d.f. p(cWm)(λ1, . . . , λN) of +the complex Wishart product ensembles (recall Definition 1.12) with arbitrary m ∈ N and +ν1 = · · · = νm = 0 (so that all of the Ginibre matrices G1, . . . , Gm in equation (1.3.3) are N × N) +— their method was subsequently extended in [11] to allow for arbitrary ν1, . . . , νm > −1 +through the use of induced Ginibre matrices (cf. the discussion below Proposition 1.7). The +success of these works in applying the HCIZ integral formula to arbitrarily large products +of random matrices prompted a series of studies on other random matrix products that +could likewise be treated using HCIZ-type integrals. Thus, it was soon found that products +58 + +1.3. Matrix Product Ensembles +of two coupled random matrices [10], [227] and products of arbitrarily many truncated +Haar-distributed unitary matrices [205] can be studied using appropriate generalisations of +the HCIZ integral formula (in fact, the authors of this latter work introduced a previously +unknown HCIZ-type integral formula as part of their development). +Let us now draw attention to an analogue and a generalisation of the HCIZ integral +formula (1.3.51) that are of particular interest to us: For N an even integer, the orthogonal +Harish-Chandra integral formula is the result +� +O(N) e +1 +2 Tr(AUBUT) d ˜µ′ +Haar(U) = 2−N/2 +N/2−1 +∏ +i=1 +(2i)! +Det[2 cosh(ajbk)]N/2 +j,k=1 +∏1⩽j0 completely determined by the choice +of a0. This latter feature is true of all the differential equation characterisations we will +obtain for ρ(x; N, β) in this chapter. It holds true because the underlying differential- +difference equation (2.1.17) is effectively a (multi-dimensional) first order recurrence. +67 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +The third order differential equation for the GUE eigenvalue density (equation (2.0.3) +with β = 2) can be traced back to the work of Lawes and March [221]. There, the setting is +that of interpreting the GUE eigenvalue density as the squared ground state wave function of +spinless non-interacting fermions in one dimension, in the presence of a harmonic confining +potential. In the random matrix theory literature, this result appeared in the work of G¨otze +and Tikhomirov [164], making use of an earlier result of Haagerup and Thorbjørnsen [169] +characterising the two-sided Laplace transform of the density in terms of a hypergeometric +function. Differential equations (2.0.3), (2.0.4) with β = 1, 4 were derived in [319] using +a method based on known evaluations of the eigenvalue densities in terms of Hermite +polynomials (recall the discussion in §1.2.3, particularly that which concerns [4]). Our +approach, using Selberg correlation integrals and duality formulae, is different to those just +recounted, and moreover unifies all the classical cases. +2.1 +Selberg Correlation Integrals +As mentioned above, our initial interest lies in the Selberg correlation integral that is specified +as the average of ∏N +i=1 |λ − λi|β (0 < λ < 1, β > 0) against the j.p.d.f. of the Jacobi β ensemble, +which we recall from equations (1.2.81) and (1.2.9) as +p(J)(λ1, . . . , λN; β) = +1 +N (J) +N,β +N +∏ +i=1 +λa +i (1 − λi)b |∆N(λ)|β, +0 < λ1, . . . , λN < 1, β > 0. +(2.1.1) +The normalisation constant N (J) +N,β is known as the Selberg integral, which, in turn, explains +why the average (2.0.1) is referred to as a Selberg correlation integral. +2.1.1 +The Selberg and Dixon–Anderson integrals +Highlighting the dependence on the parameters a, b, we denote the Selberg integral by +SN(a, b, κ) := N (J) +N,β = +� +[0,1]N +N +∏ +i=1 +λa +i (1 − λi)b |∆N(λ)|2κ dλ1 · · · dλN, +(2.1.2) +where we now allow a, b ∈ C with Re(a), Re(b) > −1. This integral was first computed by +Selberg [285] to be given by +SN(a, b, κ) = +N−1 +∏ +j=0 +Γ(a + 1 + κj)Γ(b + 1 + κj)Γ(1 + κ(j + 1)) +Γ(a + b + 2 + κ(N + j − 1))Γ(1 + κ) +, +(2.1.3) +68 + +2.1. Selberg Correlation Integrals +with the requirements Re(a), Re(b) > −1 now seen to be necessary for avoiding the poles in +the numerator of the right-hand side of equation (2.1.3). Since the work of Selberg, there +have been a number of proofs produced for the above evaluation [88], [22], [20] (see [119, +Ch. 4] for a review). To make connection with the recursive theme of this thesis, we present +below the method of Anderson, which consists of setting up a recurrence in N. +Following [119, Sec. 4.2], we fix parameters s1, . . . , sN+1 > 0 and refer to the quantity +D(s1,...,sN+1) +N +(a1, . . . , aN+1) := +� +aN+1<λN<···<λ1 0 for all i = 1, . . . , N + 1 +� +. +Now, introduce the random rational function +R(λ) := +N+1 +∑ +i=1 +wi +ai − λ. +(2.1.6) +In a similar fashion to the calculation surrounding the ratio of characteristic polynomials +(1.2.87) of the matrix model MN (1.2.86) for the Gaussian β ensemble that was outlined in +§1.2.4, taking the residues of both sides of equation (2.1.6) at λ = ai expresses the wi in +terms of the {ai}N+1 +i=1 and the zeroes {λi}N +i=1 of R(λ). Substituting these expressions into the +j.p.d.f. (2.1.5) and including the Jacobian yields the Dixon–Anderson j.p.d.f. +p(s1,...,sN+1)(λ1, . . . , λN; a1, . . . , aN+1) := Γ(s1 + · · · + sN+1) +Γ(s1) · · · Γ(sN+1) +N +∏ +i=1 +N+1 +∏ +p=1 +|λi − ap|sp−1 +× +∏ +1⩽j N. It is known [119, Ch. 4] that the above +differential-difference equation (2.1.17) is satisfied by a broader class of functions +˜J(N) +n,p,q(λ) := +1 +NN,p +� +n +∏ +i=1 +|λi − λ|N χλ1,...,λN−q 0, and +N (sJ) +N,β = 2N(κ(N−1)+a+b+1)N (J) +N,β = 2N(κ(N−1)+a+b+1)SN(a, b, κ), +(2.2.2) +with the value of SN(a, b, κ) given in equation (2.1.3). Denoting an average with respect to +this j.p.d.f. by the subscript sJEN,β(a, b), in keeping with the notation in equation (2.1.15), +the spectral moments (1.1.15) of the shifted and un-shifted Jacobi β ensembles are related as +follows: +m(sJ) +k += +� +N +∑ +i=1 +λk +i +� +sJEN,β(a,b) += +� +N +∑ +i=1 +(1 − 2λi)k +� +JEN,β(a,b) += +k +∑ +s=0 +�k +s +� +(−2)s +� +N +∑ +i=1 +λs +i +� +JEN,β(a,b) += +k +∑ +s=0 +�k +s +� +(−2)sm(J) +s , +(2.2.3) +where the binomial expansion has been used in the second line. +74 + +2.2. Relating the Cauchy and Jacobi Ensembles Through Analytic Continuation +Having established the mechanisms for moving between shifted and un-shifted Jacobi +ensembles, the rest of this section is devoted to relating the Cauchy ensembles to the shifted +Jacobi ensembles. We first discuss the symmetric case (recall from §1.2.1 that this corresponds +to taking a = b and α ∈ R), where the necessary arguments are easier to follow, before +outlining the (non-symmetric) general case. +2.2.1 +Relating the symmetric Cauchy and shifted Jacobi ensembles +When a = b and α ∈ R, the Cauchy and shifted Jacobi weights w(Cy)(λ) = (1 + λ2)η and +w(sJ)(λ) = (1 − λ2)aχ−1<λ<1 are even functions of λ (if α := −κ(N − 1) − 1 − η is real, then +so too is η), i.e., symmetric about λ = 0. Integrations of multivariable symmetric functions +against these weights can be related via the following proposition. +Proposition 2.2. Let f (x1, . . . , xN) be a multivariable symmetric polynomial of degree d in each xi. +For 2η < −(d + 1), define +I(Cy) +N,η [ f (x1, . . . , xN)] := +� ∞ +−∞ dx1 (1 + x2 +1)η · · · +� ∞ +−∞ dxN (1 + x2 +N)η f (x1, . . . , xN), +(2.2.4) +and for η outside of this range, define I(Cy) +N,η [ f (x1, . . . , xN)] by its analytic continuation. Also, in +relation to the shifted Jacobi weight with a = b > −1, define +I(sJ) +N,a [ f (x1, . . . , xN)] := +� 1 +−1 dx1 (1 − x2 +1)a · · · +� 1 +−1 dxN (1 − x2 +N)a f (x1, . . . , xN), +(2.2.5) +and for a outside of this range, define I(J) +N,a[ f (x1, . . . , xN)] by its analytic continuation. We have +I(Cy) +N,η [ f (ix1, . . . ixN)] = (tan πη)NI(sJ) +N,η [ f (x1, . . . xN)]. +(2.2.6) +Proof. Let p ∈ Z≥0. Suppose 2η < −(p + 1) and a > −1. A simple change of variables (for +k even) and use of the Euler beta integral evaluation (see [119, Exercises 5.4, q.2]) shows that +� ∞ +−∞(1 + x2)ηxk dx = +� +� +� +� +� +0, +k odd, +(−1)k/2 tan πη Γ(1+η)Γ((k+1)/2) +Γ((k+3)/2+η) +, +k even, +(2.2.7) +and +� 1 +−1(1 − x2)axk dx = +� +� +� +� +� +0, +k odd, +Γ(1+a)Γ((k+1)/2) +Γ((k+3)/2+a) +, +k even. +(2.2.8) +75 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +The functions f (x1, . . . , xN) and f (ix1, . . . , ixN) are polynomials, so the computation of +I(Cy) +N,η [ f (ix1, . . . , ixN)] and I(sJ) +N,a [ f (x1, . . . , xN)] reduces to the above one-dimensional integrals. +Since, as analytic functions of η, we read off from the respective evaluations that +� ∞ +−∞(1 + x2)η(ix)2k dx = tan πη +� 1 +−1(1 − x2)ηx2k dx, +the stated result (2.2.6) follows +One immediate consequence is a proof for the value of N (Cy) +N,β given in equation (2.1.14), +in the symmetric case with κ = β/2 ∈ N. +Corollary 2.1. Let α be real and let η = −κ(N − 1) − 1 − α. For β even, we have +(−1)κN(N−1)/2N (Cy) +N,β = (− tan πα)NN (sJ) +N,β +��� +a=b=−κ(N−1)−1−α, +(2.2.9) +where both sides are to be interpreted as analytic functions in α. +Proof. With α and thus η real, we have +N (Cy) +N,β = +� ∞ +−∞ dx1(1 + x2 +1)η · · · +� ∞ +−∞ dxN(1 + x2 +N)η +∏ +1⩽j 0; Carlson’s theorem states +that if a function f (κ − 1) is analytic for Re(κ) ⩾ 1, has the bound | f (κ − 1)| = O(eτ|κ−1|) +for some τ < π, and vanishes at all κ ∈ N, then f ≡ 0. +For the particular values of β even, β = 2, 4, we will use the identity (2.2.11) relating the +eigenvalue density of the Cauchy ensemble for α real to (an analytic continuation of) the +eigenvalue density of the shifted Jacobi ensemble supported on (−1, 1) with a = b and the +same value of β to study properties of the former — extension to β = 1 is enabled through +the above remark or Lemma 1.4. This is presented in §2.3.2, but first we outline a relationship +between the Cauchy ensemble when Im(α) ̸= 0 (also known as the non-symmetric case or +the generalised Cauchy ensemble) and the shifted Jacobi ensemble now requiring a = b. +2.2.2 +Relating the non-symmetric Cauchy and shifted Jacobi ensembles +A classical result of Cauchy [59] gives +� ∞ +−∞ +dt +(1 − it)γ(1 + it)δ = 22−γ−δπ Γ(γ + δ − 1) +Γ(γ)Γ(δ) +(2.2.13) +subject to the requirement that Re(γ + δ) > 1; outside of this range we consider the integral +as defined by the analytic continuation given by the right-hand side. Use of the reflection +77 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +equation for the gamma function allows this to be rewritten +� ∞ +−∞(1 − it)γ(1 + it)δ dt = 2γ+δ+2 sin πγ sin πδ +sin π(γ + δ) +Γ(γ + 1)Γ(δ + 1) +Γ(γ + δ + 2) +, +(2.2.14) +subject now to the requirement Re(γ + δ) < −1 on the left-hand side. +The form (2.2.14) is to be compared against the Euler beta function evaluation +� 1 +0 tc(1 − t)d dt = Γ(c + 1)Γ(d + 1) +Γ(c + d + 2) +(2.2.15) +or, equivalently, +� 1 +−1(1 − t)c(1 + t)d dt = 2c+d+1 Γ(c + 1)Γ(d + 1) +Γ(c + d + 2) +, +(2.2.16) +where on the left-hand side, it is required that Re(c), Re(d) > −1. The agreement in the +gamma function dependence of both integrals allows for a relation between multiple integrals +analogous to that in Proposition 2.2 to be derived. +Proposition 2.4. Let f (x1, . . . , xN) be a multivariable symmetric polynomial of degree ˜d in each xi. +For Re(γ + δ) < − ˜d − 1, define +˜I(Cy) +N,γ,δ[ f (x1, . . . , xN)] := +� ∞ +−∞ dx1 (1 − ix1)γ(1 + ix1)δ · · · +· · · +� ∞ +−∞ dxN (1 − ixN)γ(1 + ixN)δ f (x1, . . . , xN), +(2.2.17) +and for γ, δ outside of this range, define ˜I(Cy) +N,γ,δ[ f (x1, . . . , xN)] by its analytic continuation. Also, in +relation to the shifted Jacobi weight with Re(c), Re(d) > −1, define +˜I(sJ) +N,c,d[ f (x1, . . . , xN)] := +� 1 +−1 dx1 (1 − x1)c(1 + x1)d · · · +· · · +� 1 +−1 dxN (1 − xN)c(1 + xN)d f (x1, . . . , xN), +(2.2.18) +and for c, d outside of this range, define ˜I(sJ) +N,c,d[ f (x1, . . . , xN)] by its analytic continuation. We have +˜I(Cy) +N,γ,δ[ f (1 − ix1, . . . , 1 − ixN)] = +� +2sin πγ sin πδ +sin π(γ + δ) +�N +˜I(sJ) +N,γ,δ[ f (1 − x1, . . . 1 − xN)]. +(2.2.19) +Proof. For p ∈ Z≥0, it follows from equations (2.2.14) and (2.2.16) upon setting c = γ and +d = δ that +� ∞ +−∞(1 − it)γ+p(1 + it)δ dt = 2γ+δ+p+2 sin πγ sin πδ +sin π(γ + δ) +Γ(γ + p + 1)Γ(δ + 1) +Γ(γ + δ + p + 2) +, +(2.2.20) +78 + +2.2. Relating the Cauchy and Jacobi Ensembles Through Analytic Continuation +and +� 1 +−1(1 − t)γ+p(1 + t)δ dt = 2γ+δ+p+1 Γ(γ + p + 1)Γ(δ + 1) +Γ(γ + δ + p + 2) +. +(2.2.21) +Hence, in the sense of analytic continuation, +� ∞ +−∞(1 − it)γ+p(1 + it)δ dt = 2sin πγ sin πδ +sin π(γ + δ) +� 1 +−1(1 − t)γ+p(1 + t)δ dt. +(2.2.22) +The stated result now follows from the assumption that f (x1, . . . , xN) in (2.2.17) and +(2.2.18) is a polynomial and so the evaluation of the multiple integrals reduces to the +one-dimensional integrals (2.2.20) and (2.2.21), which are related by equation (2.2.22). +We can use Proposition 2.4 to extend Corollary 2.1 to the case that α, hence η, are complex. +Corollary 2.2. Let α be, in general, complex and recall that we choose η = −κ(N − 1) − 1 − α. For +κ = β/2 ∈ N, +(−1)κN(N−1)/2N (Cy) +N,β = +� +−2sin πα sin πα +sin π(α + α) +�N +N (sJ) +N,β +��� +a=b=−κ(N−1)−1−α, +(2.2.23) +where the right-hand side is to be regarded as defined by its analytic continuation. +Proof. We observe that for β even, the product of differences |∆N(λ)|β in the definition of +the normalisation constants, +NN,β = +� +RN p(λ1, . . . , λN) dλ1 · · · dλN, +is a polynomial, and moreover, +(λk − λj)β = (−1)κ((1 − iλk) − (1 − iλj))β. +The result now follows from the definition of the normalisation constants and the identity +(2.2.19). +Remark 2.6. +1. Analogous to equation (2.2.10), the identity (2.2.23) can be rewritten as +SN(−κ(N − 1) − 1 − α, −κ(N − 1) − 1 − α, κ) += (−1)κN(N−1)/2 +� +−π sin π(α + α) +sin πα sin πα +�N N−1 +∏ +j=0 +Γ(2Re(α) + 1 + κj)Γ(1 + κ(j + 1)) +Γ(α + 1 + κj)Γ(α + 1 + κj)Γ(1 + κ). +(2.2.24) +79 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +This can be verified using the same steps as for equation (2.2.10), and is known to be +true for β > 0 due yet again to Carlson’s theorem (cf. Remark 2.5). +2. In the case α is real, identity (2.2.23) reduces to (2.2.9). +We can make use of Corollary 2.2 and a further application of Proposition 2.4 in the +specification (2.0.1) of the eigenvalue densities ρ(Cy)(λ) and ρ(sJ)(λ) to deduce the analogue +of Proposition 2.3 in the non-symmetric case. +Proposition 2.5. In the setting of Corollary 2.2, +ρ(Cy)(iλ; N, β) = − sin π(α + α) +2 sin πα sin πα ρ(sJ)(λ; N, β) +��� +a=b=−κ(N−1)−1−α. +(2.2.25) +Remark 2.7. +1. In the case α is real, equation (2.2.25) reduces to (2.2.11). +2. As discussed in §1.2.3, there exist expressions in terms of orthogonal polynomials for +both sides of equation (2.2.25) when β = 1, 2, or 4. The expressions for the right-hand +side can be found in [4], and the left-hand side in [145]. These expressions can be +checked to be consistent with equation (2.2.25), using the fact that the orthogonal poly- +nomials associated with the shifted Jacobi weight are the Jacobi polynomials P(a,b) +N +(λ), +while those associated with the Cauchy weight are the scaled Jacobi polynomials +i−NP(η,η) +N +(iλ). +2.3 +Linear Differential Equations for the Eigenvalue Densities and +Resolvents +The Selberg integral theory in §2.1.2 allows us to derive order β + 1 linear differential +equations satisfied by the eigenvalue densities (2.0.1) of the Jacobi β ensembles with β = 2, 4; +the β ↔ 4/β duality given in Lemma 1.4 then extends our results to β = 1. Analogous to +Proposition 2.1, these differential equations are homogeneous, and they have inhomogeneous +counterparts that are solved by the corresponding resolvents (recall that the resolvents have +multiple valid definitions, as introduced in §1.1.1). +After presenting the aforementioned differential equations for the Jacobi ensembles in +§2.3.1, we use the theory of Section 2.2 to translate our results to the shifted Jacobi and +80 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +Cauchy ensembles. We treat the β = 2 case in full generality and report on the β = 1, 4 cases +only in the symmetric case, since the non-symmetric β = 1, 4 differential equations have +cumbersome presentations with little expository benefit. The differential equations are all of +the same orders and simplify greatly in the symmetric case. They are given in §2.3.2, along +with brief discussions regarding connections to the circular Jacobi ensemble. +Extending the limiting procedures in Lemma 1.1, in a similar fashion to what was seen for +the normalisation constants at the end of §2.1.1, the Jacobi ensembles’ differential equations +transform into differential equations satisfied by the Laguerre ensembles’ eigenvalue densities +and resolvents. We again treat β = 1, 2, and 4, and the differential equations are of the same +orders, but with simpler coefficients due to the loss of parameter b. Repeating this exercise +to scale out both the parameters a and b results in the even simpler differential equations for +the Gaussian ensembles displayed in Proposition 2.1. Rather than outlining this calculation, +we demonstrate an alternate extension of Lemma 1.1 that allows us to derive differential +equations for the Gaussian ensembles without needing to first make the Jacobi ensemble +analogues explicit. Indeed, in §2.3.4, we derive seventh order linear differential equations +that characterise the eigenvalue densities and resolvents of the β = 2/3 and β = 6 Gaussian +β ensembles, without any knowledge of the Jacobi β ensembles for these values of β. These +differential equations serve as a proof of concept: Our method applies for positive integer β +outside of the classical regime β = 1, 2, 4, and also allows us to access non-integer β values. +As discussed in Section 1.4, the interest in the upcoming differential equations is that +comparable and/or equivalent (in the GOE, GSE, and classical unitary cases) results in the +works [221], [164], [151], [2], [319], [260], [23] have been shown therein and also in, e.g., +[223], [213] to be amenable to various forms of analysis. The differential equations of this +section will be given two applications in this thesis: In Section 2.4, we apply global and +edge scalings to the differential equations to give characterisations of the relevant ensembles +at these scaling limits, while in Section 3.1, we present linear recurrences for the spectral +moments mk of the classical matrix ensembles, supplementing the works [174], [224], [72], +among others. Another appeal of our derivations below is that they treat all of the classical +matrix ensembles through a unified method, in contrast to the aforementioned works whose +results we recover. +81 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +2.3.1 +Differential equations for the Jacobi ensembles +Recall from §2.1.2 that the eigenvalue density ρ(J)(λ) is a scalar multiple of w(J)(λ) times +the function I(J) +N,β(λ) defined in equation (2.1.15). When β is an even integer, this latter +function is moreover related, via equation (2.1.24), to the auxiliary function J(N) +β,0 (λ) defined +in equation (2.1.16). Using the differential-difference equation (2.1.17) for the J(N) +β,p (λ), now +with p = 0, 1, . . . , β, we derive a differential equation for J(N) +β,0 (λ). Applying straightforward +manipulations according to equations (2.1.15) and (2.1.17) then yields the sought differential +equations for ρ(J)(λ; N, β) with β ∈ 2N. +Lemma 2.1. The function J(N) +2,0 (x) (2.1.16) satisfies the differential equation +0 = x2(x − 1)2 d3 +dx3 J(N) +2,0 (x) − [3C2(x) − 2(1 − 2x)] x(x − 1) d2 +dx2 J(N) +2,0 (x) ++ [((a + N)(1 + 4N) + 4 + N) x(x − 1) + (2C2(x) − 3(1 − 2x)) C2(x)] d +dx J(N) +2,0 (x) +− 2N(a + N) [2C2(x) − 3(1 − 2x)] J(N) +2,0 (x), +(2.3.1) +where C2(x) = (a + 2N)(x − 1) − b − 2x. +Proof. Setting n = 2 and taking p = 0, 1, 2 in equation (2.1.17), we obtain the matrix differen- +tial equation +d +dx +� +���� +J(N) +2,0 (x) +J(N) +2,1 (x) +J(N) +2,2 (x) +� +���� = +� +���� +A0x+B0 +x(x−1) +2E0 +x(x−1) +0 +−D1 +A1x+B1 +x(x−1) +E1 +x(x−1) +0 +−D2 +0 +� +���� +� +���� +J(N) +2,0 (x) +J(N) +2,1 (x) +J(N) +2,2 (x) +� +���� . +(2.3.2) +The second row gives an expression for J(N) +2,2 (x) which transforms the third row into a +differential equation involving only J(N) +2,0 (x) and J(N) +2,1 (x). This equation further transforms +into an equation for just J(N) +2,0 (x) upon substitution of the expression for J(N) +2,1 (x) drawn from +the first row: +0 = x2(x − 1)2 d3 +dx3 J(N) +2,0 (x) − +� ˜C2,0(x) + ˜C2,1(x) + 1 − 2x +� +x(x − 1) d2 +dx2 J(N) +2,0 (x) ++ +�(D2E1 + 2D1E0 − A1 − 2A0 + 2)x(x − 1) + ˜C2,0(x) ˜C2,1(x) +� d +dx J(N) +2,0 (x) ++ +� +A0 ˜C2,1(x) + (A1 − D2E1)(A0x + B0) − 2D1E0(1 − 2x) +� +J(N) +2,0 (x), +(2.3.3) +where ˜C2,p(x) = (Ap − 2)x + Bp + 1 with n = β = 2. Substituting the appropriate values for +the constants Ap, Bp, Dp and Ep gives the claimed result. +82 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +Lemma 2.2. Now taking n = β = 4, the function J(N) +4,0 (x) satisfies the differential equation with +polynomial coefficients +0 = 4x4(x − 1)4 d5 +dx5 J(N) +4,0 (x) − 20 [(a + 4N)(x − 1) − b − 2x] x3(x − 1)3 d4 +dx4 J(N) +4,0 (x) ++ +� +5(a + 4N)2 − 5a(a − 2) − 12 +� +x3(x − 1)3 d3 +dx3 J(N) +4,0 (x) + · · · +(2.3.4) +where the (lengthier) specific forms of the coefficients of the lower order derivatives have been sup- +pressed. (One may obtain the full expression by inverting the upcoming proof of Theorem 2.2. In fact, +this is the most efficient method of obtaining the full expression using computer algebra.) +Proof. Like the preceding proof, setting n = 4 in equation (2.1.17) and taking p = 0, 1, . . . , 4 +yields the matrix differential equation +d +dx +� +���������� +J(N) +4,0 (x) +J(N) +4,1 (x) +J(N) +4,2 (x) +J(N) +4,3 (x) +J(N) +4,4 (x) +� +���������� += +� +���������� +A0x+B0 +x(x−1) +4E0 +x(x−1) +0 +0 +0 +−D1 +A1x+B1 +x(x−1) +3E1 +x(x−1) +0 +0 +0 +−D2 +A2x+B2 +x(x−1) +2E2 +x(x−1) +0 +0 +0 +−D3 +A3x+B3 +x(x−1) +E3 +x(x−1) +0 +0 +0 +−D4 +0 +� +���������� +� +���������� +J(N) +4,0 (x) +J(N) +4,1 (x) +J(N) +4,2 (x) +J(N) +4,3 (x) +J(N) +4,4 (x) +� +���������� +. +(2.3.5) +For 1 ⩽ p ⩽ 4, the pth row gives an expression for J(N) +4,p (x) in terms of d +dx J(N) +4,p−1(x) and J(N) +4,k (x) +with k < p. Substituting these expressions (in the order of decreasing p) into the differential +equation corresponding to the fifth row yields a fifth order differential equation for J(N) +4,0 (x) +similar to that of Lemma 2.1. +Now, we may easily obtain differential equations for I(J) +N,β(λ) for β = 2 and 4, which, in +turn, give us differential equations for ρ(J)(λ; N, β) for β = 1, 2, 4. +Theorem 2.1. Define +D(J) +N,2 = x3(1 − x)3 d3 +dx3 + 4(1 − 2x)x2(1 − x)2 d2 +dx2 ++ +�(a + b + 2N)2 − 14 +� +x2(1 − x)2 d +dx − +� +a2(1 − x) + b2x − 2 +� +x(1 − x) d +dx ++ 1 +2 +�(a + b + 2N)2 − 4 +� (1 − 2x)x(1 − x) + 3 +2 +� +a2 − b2� +x(1 − x) +− a2(1 − x) + b2x. +(2.3.6) +83 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +Then, +D(J) +N,2 ρ(J)(x; N, 2) = 0 +(2.3.7) +and +D(J) +N,2 +1 +NW(J) +1 (x; N, 2) = (a + b + N)(a(1 − x) + bx). +(2.3.8) +Proof. We change variables x �→ 1/x in equation (2.3.1) to obtain a differential equation for +J(N) +2,0 (1/x) using the fact that +d +d(1/x) = −x2 d +dx. Since this differential equation is homogeneous, +we can ignore constants of proportionality and substitute in +J(N) +2,0 (1/x) = x−a−2N(1 − x)−bρ(J)(x; N + 1, 2) +according to equations (2.1.15) and (2.1.24). Repeatedly applying the product rule and then +replacing N + 1 by N gives equation (2.3.7). Applying the Stieltjes transform to this equation +term by term (see Appendix B) and substituting in the values of the spectral moments m(J) +1 +and m(J) +2 +with β = 2 [242] yields equation (2.3.8). +Differential equation (2.3.7) lowers the order by one relative to the fourth order differential +equation for ρ(J)(λ; N, 2) given in the recent work [72, Prop. 6.7]. +Remark 2.8. With 1 ⩽ k ⩽ N, let ρ(w) +k +(λ1, . . . , λN) denote the k-point correlation function of +the random matrix ensemble corresponding to the classical weight w(λ) (1.2.9), as specified +by equation (1.2.72). It is well known (see, e.g., [119, Ch. 9]) that the generating function +E(w) +N ((s, ∞); ξ) for the probabilities {E(w) +N,k(s, ∞)}N +k=0 of there being exactly k eigenvalues in +the interval (s, ∞) can be written in terms of the correlation functions according to +E(w) +N ((s, ∞); ξ) = 1 + +N +∑ +k=1 +(−ξ)k +k! +� ∞ +s +dx1 · · · +� ∞ +s +dxk ρ(w) +k +(x1, . . . , xk). +(2.3.9) +The logarithmic derivative of these generating functions multiplied by simple polynomials +dependent on the weight w(λ) are known to be characterised by particular σ Painlev´e +equations [298], [136], [318], [137], [138]. It has been observed in [132, §3.3] that the third +order linear differential equations (2.0.3) and (2.3.43) for the eigenvalue densities of the +Gaussian and Laguerre unitary ensembles are equivalent to related σ Painlev´e equations +(see, e.g., [119, Ch. 8]). An analogous result holds true for the differential equation (2.3.7) +84 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +for ρ(J)(x; N, 2). Thus, with v1 = v3 = N + (a + b)/2, v2 = (a + b)/2, v4 = (b − a)/2, in +studying the gap for the interval (0, s) one encounters the nonlinear equation [119, Eq. (8.76)] +(t(1 − t) f ′′)2 − 4t(1 − t)( f ′)3 + 4(1 − 2t)( f ′)2 f + 4f ′ f 2 − 4f 2v2 +1 ++( f ′)2� +4tv2 +1(1 − t) − (v2 − v4)2 − 4tv2v4 +� ++ 4f f ′(−v2 +1 + 2tv2 +1 + v2v4) = 0, +subject to the boundary condition f (t) +∼ +t→0+ −ξt(1 − t)ρ(J)(t; N, 2). Substituting this bound- +ary condition for f and equating terms of order ξ2 shows that u(t) := t(1 − t)ρ(J)(t; N, 2) +satisfies the second order nonlinear differential equation +(t(1 − t)u′′(t))2 − 4(u(t))2v2 +1 + (u′(t))2(4tv2 +1(1 − t) − (v2 − v4)2 − 4tv2v4) ++ 4u(t)u′(t)(−v2 +1 + 2tv2 +1 + v2v4) = 0. +(2.3.10) +Differentiating this and simplifying gives a third order linear differential equation that agrees +with equation (2.3.7). +Theorem 2.2. Recalling that κ := β/2, let +aβ := +a +κ − 1, +bβ := +b +κ − 1, +Nβ := (κ − 1)N +so that (a4, b4, N4) = (a, b, N) and (a1, b1, N1) = (−2a, −2b, −N/2). For β = 1 or 4, define +D(J) +N,β = 4x5(1 − x)5 d5 +dx5 + 40(1 − 2x)x4(1 − x)4 d4 +dx4 + +� +5˜c2 − 493 +� +x4(1 − x)4 d3 +dx3 +− +� +5f+(x; ˜a, ˜b) − 88 +� +x3(1 − x)3 d3 +dx3 + 41 +�˜a − ˜b +� +x3(1 − x)3 d2 +dx2 ++ +� +19˜c2 − 539 +� (1 − 2x)x3(1 − x)3 d2 +dx2 − 22f−(x; ˜a, ˜b)x2(1 − x)2 d2 +dx2 ++ 16(1 − 2x)x2(1 − x)2 d2 +dx2 + +� +˜c4 − 64˜c2 + 719 +� +x3(1 − x)3 d +dx +− +�(˜c2 − 45)(˜a + ˜b − 6) + (˜a − ˜b)2 − 248 +� +x2(1 − x)2 d +dx +− +�(˜c2 − 37)(˜a − ˜b) +� (1 − 2x)x2(1 − x)2 d +dx ++ +� +f+(x; ˜a2, ˜b2) − 14f+(x; ˜a, ˜b) − 16 +� +x(1 − x) d +dx ++ 1 +2 +� +5(˜c2 − 9)(˜a − ˜b) +� +x2(1 − x)2 + 1 +2(˜c2 − 9)2(1 − 2x)x2(1 − x)2 +− 1 +2 +�(3˜c2 − 35) f−(x; ˜a, ˜b) + 7 +2(˜a2 − ˜b2) + 4(˜a − ˜b) +� +x(1 − x) +− 1 +2 +� +4˜c2 − 36 + 3 +2(˜a − ˜b)2� (1 − 2x)x(1 − x) + f−(x; ˜a2, ˜b2), +(2.3.11) +85 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +where +˜a = aβ(aβ − 2), +˜b = bβ(bβ − 2), +˜c = aβ + bβ + 4Nβ − 1, +f±(x; ˜a, ˜b) = ˜a(1 − x) ± ˜bx. +(2.3.12) +Then, for β = 1 and 4, +D(J) +N,β ρ(J)(x; N, β) = 0 +(2.3.13) +and +D(J) +N,β +1 +NW(J) +1 (x; N, β) = (˜c − 2Nβ)x(1 − x) +� +(aβ + bβ)(aβ + bβ − 2)(aβ(1 − x) + bβx) ++ 4Nβ(˜c − 2Nβ)(2aβ(1 − x) + 2bβx − 1) − aβbβ(aβ + bβ − 6) − 4(aβ + bβ − 1) +� +− (˜c − 2Nβ) +� +aβ(aβ − 2)2(1 − x)2 + bβ(bβ − 2)2x2� +. +(2.3.14) +Proof. The proof is done in four steps. First, to see that equation (2.3.13) holds for β = 4, one +undertakes the same steps as in the proof of Theorem 2.1. That is, change variables x �→ 1/x +in equation (2.3.4), substitute in +J(N) +4,0 (1/x) = x−a−4N(1 − x)−bρ(J)(x; N + 1, 4), +and then replace N + 1 by N. For the second step, apply the Stieltjes transform according +to Appendix B and substitute in the values of the spectral moments m(J) +1 +to m(J) +4 +[96], [242] +to obtain (2.3.14) for β = 4. The third step proves equation (2.3.14) for β = 1 by employing +the β ↔ 4/β duality relation (1.2.98), which leads us to formulate equation (2.3.14) in terms +of the (aβ, bβ, Nβ) parameters. Since this step essentially redefines constants, applying the +inverse Stieltjes transform to this result returns equation (2.3.13) with the new constants. +In the way that equation (2.3.11) is presented, all of its dependencies on a, b and N are +captured by ˜a, ˜b and ˜c. Moreover, it can be seen that this operator is invariant under the +symmetry (x, a, b) ↔ (1 − x, b, a), which is a property of ρ(J)(x; N, β). Equations (2.3.7) and +(2.3.13) have been checked for N = 1, 2 and to hold in the large N limit, while (2.3.8) and +(2.3.14) have been checked to be consistent with expressions for the resolvent expansion +coefficients W(J),0 +1 +(x; β), . . . , W(J),4 +1 +(x; β) (recall Theorem 1.2 in §1.1.1) generated via the loop +equation analysis presented in [142] (see also §4.1.1 for a summary of the methods used in +86 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +this work). It should be noted that while the moments m(J) +1 +to m(J) +4 +of the Jacobi β ensemble’s +eigenvalue density are complicated rational functions of a, b, and N, the right-hand sides of +equations (2.3.8) and (2.3.14) are relatively simple polynomials, even though they are linear +combinations of these moments. +2.3.2 +Differential equations for the shifted Jacobi and Cauchy ensembles +Upon making the simple change of variables x �→ (1 − x)/2 in Theorems 2.1 and 2.2, +analogous differential equations for the shifted Jacobi ensembles can be obtained. The +outcomes are not different in any essential way, so no insight is gained from explicitly +presenting them. +Instead, we further enforce the restriction b = a so that significant +simplification occurs. Then, the differential equations are for the eigenvalue densities and +resolvents of the symmetric shifted Jacobi ensembles. +Proposition 2.6. Define +D(sJ) +N,2 = (1 − x2)3 d3 +dx3 − 8x(1 − x2)2 d2 +dx2 +− 2(1 − x2)[3 − 2N2 − 4aN + (2(a + N)2 − 7)x2] d +dx ++ 4x(a2 + 1 − N2 − 2aN + (a + N)2x2 − x2) +(2.3.15) +and for β = 1 and 4, +D(sJ) +N,β = 4(1 − x2)5 d5 +dx5 − 80x(1 − x2)4 d4 +dx4 + (5˜c2 − 493)(1 − x2)4 d3 +dx3 +− 4(5˜a − 88)(1 − x2)3 d3 +dx3 + 16(11˜a − 8)x(1 − x2)2 d2 +dx2 − 2(19˜c2 − 539)x(1 − x2)3 d2 +dx2 ++ (˜c4 − 64˜c2 + 719)(1 − x2)3 d +dx − 8 +�(˜c2 − 45)(˜a − 3) − 124 +� (1 − x2)2 d +dx ++ 16 +�(˜a − 7)2 − 65 +� (1 − x2) d +dx − (˜c2 − 9)2x(1 − x2)2 ++ 4 +� +4(˜c2 − 9) + (3˜c2 − 35)˜a +� +x(1 − x2) − 32˜a2x, +(2.3.16) +taking ˜a, ˜c as defined in Theorem 2.2, with b = a. Then, for β = 1, 2, and 4, we have +D(sJ) +N,β ρ(J)(x; N, β)|a=b = 0 +(2.3.17) +87 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +and +D(sJ) +N,β +1 +NW(sJ) +1 +(x; N, β)|a=b += +� +� +� +� +� +� +� +� +� +� +� +� +� +4a(2a + N), +β = 2, +8(2Nβ + 2aβ − 1) +� +2a3 +βx2 − 2(2Nβ + 1)(Nβ − 1)(x2 − 1) ++a2 +β((8Nβ − 3)x2 − 8Nβ − 5) + 8aβ(Nβ(Nβ − 1)(x2 − 1) + 1) +� +, +β = 1, 4, +(2.3.18) +where aβ, Nβ are also as in Theorem 2.2. +Differential equation (2.3.17) with β = 2 is a generalisation of that given in [151, Sec. 4] for +the β = 2 Legendre ensemble, which is the sJUE with a = b = 0. According to Proposition 2.3 +and calculations similar to those seen in Appendix B, the differential equations satisfied by +ρ(Cy)(x) and W(Cy) +1 +(x) for β even and α > −1/2 real can be obtained from those satisfied by +ρ(sJ)(x)|a=b and W(sJ) +1 +(x)|a=b, respectively. This is done by setting a = −κ(N − 1) − 1 − α and +replacing x by ix. Doing so in Proposition 2.6 gives third and fifth order linear differential +equations for β = 2 and 4, respectively. Furthermore, Remark 2.5 tells us that the latter +differential equations are valid for β = 1 after reparametrising properly. +Proposition 2.7. Let α > −1/2 be real so that we are restricted to the symmetric Cauchy case. +Recalling the definition of Nβ in Theorem 2.2, let +αβ := +α +κ − 1, +˜α := αβ(αβ − 1), +˜N := (2Nβ + αβ)2. +Define +D(Cy) +N,2 = (1 + x2)3 d3 +dx3 + 8x(1 + x2)2 d2 +dx2 + 2(1 + x2)[3 + 2N(N + 2α) + (7 − 2α2)x2] d +dx ++ 4x(1 + α2 + 2N(N + 2α) + (1 − α2)x2) +(2.3.19) +and for β = 1 and 4, +D(Cy) +N,β = 4(1 + x2)5 d5 +dx5 + 80x(1 + x2)4 d4 +dx4 − 4(5˜α − 122)(1 + x2)4 d3 +dx3 ++ 4(5 ˜N − 93)(1 + x2)3 d3 +dx3 − 8(19˜α − 130)x(1 + x2)3 d2 +d2x + 16(11 ˜N − 19)x(1 + x2)2 d2 +d2x ++ 8(2˜α2 − 31˜α + 82)(1 + x2)3 d +dx − 32 +�(˜α − 11)( ˜N − 4) − 31 +� (1 + x2)2 d +dx ++ 16 +�( ˜N − 8)2 − 65 +� (1 + x2) d +dx + 16(˜α − 2)2x(1 + x2)2 +− 16 +�(3˜α − 8) ˜N + ˜α +� +x(1 + x2) + 32( ˜N − 1)2x. +(2.3.20) +88 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +Then, for β = 1, 2, and 4, with α > −1/2 real, we have +D(Cy) +β,N ρ(Cy)(x; N, β) = 0 +(2.3.21) +and +D(Cy) +N,β +1 +NW(Cy) +1 +(x; N, β) += +� +� +� +� +� +� +� +� +� +� +� +� +� +4(N + α)(N + 2α), +β = 2, +8(2Nβ + 2αβ − 1) +� +(2Nβ + 1)(8N2 +β + x2 − 1) − 2αβ(1 + x2α2 +β) ++4Nβαβ(6Nβ + x2 + 3) + α2 +β(3x2 − 4Nβ(x2 − 2) + 5) +� +, +β = 1, 4. +(2.3.22) +Equation (2.3.21) with β = 2 was recently derived in [23] using a method of Ledoux [223]. +Remark 2.9. Recall from Remark 2.8 the relationship between the generating function +E(w) +N ((s, ∞); ξ) and k-point correlation functions ρ(w) +k +recounted therein. We now supplement +Remark 2.8 with its counterpart for the Cauchy case. In the case β = 2, it is known [318] that +σ(s) := (1 + s2) d +ds log E(Cy) +N +((s, ∞); ξ) +(2.3.23) +satisfies the nonlinear equation (which can be identified in terms of the σ-PVI equation [138]; +see also equation (2.3.35) below) +(1 + s2)2(σ′′)2 + 4(1 + s2)(σ′)3 − 8sσ(σ′)2 ++ 4σ2(σ′ − α2) + 8α2sσσ′ + 4[N(N + 2α) − α2s2](σ′)2 = 0. +(2.3.24) +Note that equation (2.3.24) is independent of the parameter ξ in equation (2.3.9). +According to equation (2.3.9), to leading order in ξ, +σ(s) = ξr(s) + O(ξ2), +r(s) := (1 + s2)ρ(Cy) +1 +(s) = (1 + s2)ρ(Cy)(s). +(2.3.25) +Substituting in (2.3.24) and equating terms to the leading order in ξ (which is O(ξ2)) shows +(1 + s2)2(r′′(s))2 − 4α2(r(s))2 + 8α2sr(s)r′(s) + 4[N(N + 2α) − α2s2](r′(s))2 = 0. +(2.3.26) +Upon differentiating with respect to s, a factor of r′′(s) can be cancelled and a third order +linear differential equation results, +(1 + s2)2r′′′(s) + 2s(1 + s2)r′′(s) + 4[N(N + 2α) − α2s2]r′(s) + 4α2sr(s) = 0. +(2.3.27) +89 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +Recalling the definition of r(s) in terms of ρ(Cy)(s) (2.3.25), we see that equation (2.3.27) is +equivalent to the third order differential equation given in Proposition 2.7. +The β = 1, 4 analogue of equation (2.3.27), obtained by taking ρ(Cy)(s) = r(s)/(1 + s2) in +equation (2.3.21), is +(1 + s2)4r(5)(s) + 10s(1 + s2)3r(4)(s) − (5˜α − 22)(1 + s2)3r′′′(s) ++ (5 ˜N − 13)(1 + s2)2r′′′(s) − 8(˜α − 1)s(1 + s2)2r′′(s) ++ 2(7 ˜N + 1)s(1 + s2)r′′(s) + 4˜α2(1 + s2)2r′(s) − 2 +�(4˜α − 1) ˜N + 1 +� (1 + s2)r′(s) ++ 4( ˜N − 1)2r′(s) − 4˜α2s(1 + s2)r(s) + 4˜α( ˜N − 1)sr(s) = 0, +(2.3.28) +where ˜α and ˜N are as given in Proposition 2.7. In comparing equations (2.3.27) and (2.3.28) +to their counterparts in Proposition 2.7, we see that degree of each of the coefficients (which +alternate between being even and odd in s) has been reduced by 2. +Remark 2.10. Recall from §1.2.1 that the the Cauchy and circular Jacobi ensembles are related +by the Cayley transformation (1.2.25) and stereographic projection λi = cot(θi/2) (1.2.26). In +particular, the eigenvalue densities of the two ensembles are related by +ρ(Cy)(λ) = 2 sin2(θ/2) ρ(cJ)(eiθ). +(2.3.29) +Equivalently, in the notation r(s) of equation (2.3.25) with s = cot(θ/2), +r(s) = 2ρ(cJ)(eiθ). +(2.3.30) +In the case α = 0, the circular Jacobi weight w(cJ)(eiθ) (1.2.28) is a constant and so according +to equation (2.3.30), r(s) is then also a constant. We can see immediately that this is consistent +with equations (2.3.27) and (2.3.28). +Suppose now θ is scaled by writing θ = 2X/N. Then, in the scaling limit N → ∞, +the circular Jacobi ensemble exhibits a spectrum singularity at θ = 0, in keeping with the +discussion in §1.2.1. In view of equation (2.3.30), in the cases β = 1, 2, and 4, differential +equations for the corresponding density ρ(s.s.)(X) can be obtained by setting s = N/X and +r(s) = ρ(s.s.)(X) in equations (2.3.27) and (2.3.28), and then equating terms at leading order +in N. Specifically for β = 2, we therefore have that R(X) = ρ(s.s.)(X) satisfies the third order +linear differential equation +X2R′′′(X) + 4XR′′(X) + (2 − 4α2 + 4X2)R′(X) − 4α2 +X R(X) = 0. +(2.3.31) +90 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +This is consistent with the known exact formula [257], [119, Eq. (7.49) with πρ = 1] +ρ(s.s.)(x) = x +2 +� +(Jα−1/2(x))2 + (Jα+1/2(x))2 − 2α +x Jα−1/2(x)Jα+1/2(x) +� +, +(2.3.32) +as can be checked using computer algebra. +Though there is no present benefit in presenting the analogue of Proposition 2.6 for the +non-symmetric shifted Jacobi ensemble, there is merit in deriving the differential equation +satisfied by the eigenvalue density of the non-symmetric CyUE, especially considering its +relatively compact form. Repeating the process outlined above, we first make the change +of variables x �→ (1 − x)/2 in equation (2.3.7) to obtain a differential equation for the +non-symmetric shifted Jacobi unitary ensemble. Setting a = −N − α, b = −N − α, and +replacing x by ix in this differential equation, it follows from Proposition 2.5 that we obtain +the differential equation satisfied by ρ(Cy)(x; N, 2). +Proposition 2.8. Define the third order differential operator +˜D(Cy) +N,2 = (1 + x2)3 d3 +dx3 + 8x(1 + x2)2 d2 +dx2 ++ (1 + x2)[6 + 4N(N + α + α) + (α − α)2 + 2i(α − α)(2N + α + α)x + (14 − (α + α)2)x2] d +dx ++ [4 + 8N(N + α + α) + 3α2 − 2αα + 3¯α2 + (4 − (α + α)2)x2]x ++ i(α − α)(2N + α + α)(3x2 − 1). +(2.3.33) +For β = 2 and α ∈ C with constraint Re(α) > −1/2, we have +˜D(Cy) +N,2 ρ(Cy)(x) = 0. +(2.3.34) +Remark 2.11. After a simple change of variables, the Jimbo–Miwa–Okamoto σ-form of the +Painlev´e differential equation reads [138, Eq. (1.32)] +h′� +(1 + t2)h′′�2 ++ 4 +� +h′(h − th′) − ib1b2b3b4 +�2 ++ 4 +4 +∏ +k=1 +(h′ + b2 +k) = 0. +(2.3.35) +Let b = (b1, b2, b3, b4) and define e′ +2[b], e2[b] as the elementary symmetric polynomials of +degree two in {b1, b3, b4} and {b1, b2, b3, b4}, respectively. Set +U(Cy) +N +(t; (α1, α2); ξ) = (t2 + 1) d +dt log +� +(it − 1)e′ +2[b]−e2[b]/2(it + 1)e2[b]/2E(Cy) +N +((t, ∞); ξ)) +� +, +(2.3.36) +91 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +where E(Cy) +N +is specified by equation (2.3.9) in the non-symmetric case with α = α1 + iα2 +(α2 ̸= 0). We have from [138, Prop. 15] that U(Cy) +N +satisfies the transformed σ-PVI equation +(2.3.35) with parameters +b = (−α1, −iα2, N + α1, α1). +(2.3.37) +Analogous to equation (2.3.25), we have from equations (2.3.9), (2.3.36), and (2.3.37) that +U(Cy) +N +(t; (α1, α2); ξ) = (N + α1)α2 − α2 +1t + ξr(t) + O(ξ2), +r(t) := (1 + t2)ρ(Cy)(t). (2.3.38) +Substituting (2.3.38) in equation (2.3.35) and equating terms of leading order in ξ, which +as for equation (2.3.25) occurs at order ξ2, then differentiating and cancelling a factor of r′′ +shows +(1 + t2)2r′′′ + 2t(1 + t2)r′′ + 4[((N + α1)α2 + α2 +1t)(r − tr′) ++ ((N + α1)2 − (N + α1)α2t − α2 +1 − α2 +2)r′] = 0. +(2.3.39) +In the case α2 = 0, this agrees with equation (2.3.27). Now, substituting for r(t) in terms of +ρ(Cy)(t) as specified in equation (2.3.38) reclaims equation (2.3.34). +2.3.3 +Differential equations for the Laguerre ensembles +As mentioned at the beginning of this section, the eigenvalue density ρ(L)(x) of the Laguerre +β ensemble can be obtained from that of the Jacobi β ensemble via a limiting procedure: +From equation (2.0.1), upon extending the idea of Lemma 1.1, we see (cf. equation (2.1.12)) +ρ(L)(x; N + 1, β) = lim +b→∞ b(N+3)Nκ+(N+2)a+N+1 ρ(J)� x +b ; N + 1, β +� +. +(2.3.40) +This fact allows one to transform differential equations satisfied by ρ(J)(x) into analogues +satisfied by ρ(L)(x), which will be presented in a moment. +As an aside, suppose that one would like to obtain differential equations for ρ(L)(x) +without prior knowledge of the analogous differential equations for ρ(J)(x), i.e., if the results +of §2.3.1 were not available, or if one were interested in ensembles with β /∈ {1, 2, 4}. Then, it +is actually more efficient to circumvent computation of the differential equations for ρ(J)(x) +and use the aforementioned limiting procedure indirectly. That is, let +L(N) +β,p (x) = lim +b→∞ +�− x +b +�βN+p J(N) +β,p (b/x), +˜L(N) +β,p (x) = xae−x L(N) +β,p (x) +92 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +so that by equation (2.0.1), ρ(L)(x; N + 1, β) is proportional to ˜L(N) +β,0 (x) when β is a positive +even integer. Equation (2.1.17) gives a differential-difference equation for +�− x +b +�βN+p J(N) +β,p (b/x) +and taking the limit b → ∞ thus gives a differential-difference equation for L(N) +β,p (x). Sub- +stituting L(N) +β,p (x) = x−aex ˜L(N) +β,p (x) then gives a differential-difference equation for ˜L(N) +β,p (x). +These equations with p = 0, 1, . . . , β are equivalent to matrix differential equations not unlike +(2.3.2) and (2.3.5). However, having taken the limit b → ∞, we ensure that these new matrix +differential equations are simpler, and have the added benefit of simplifying down to scalar +differential equations for ρ(L)(x; N + 1, β) rather than for auxiliary functions. One may then +apply the Stieltjes transform to obtain differential equations for the resolvents, and use the +β ↔ 4/β dualities given in Lemma 1.4 to obtain mirror differential equations like those seen +in Theorem 2.2. +Since we are interested in ρ(L)(x) with β ∈ {1, 2, 4} and we have differential equations for +ρ(J)(x) for these β values, we use the more direct approach to give the following proposition. +Proposition 2.9. Retaining the definitions of aβ, Nβ, and ˜a from Theorem 2.2, define +D(L) +2,N = x3 d3 +dx3 + 4x2 d2 +dx2 − +� +x2 − 2(a + 2N)x + a2 − 2 +� +x d +dx + +�(a + 2N)x − a2� +(2.3.41) +and for β = 1 or 4, +D(L) +N,β = 4x5 d5 +dx5 + 40x4 d4 +dx4 − +� +5 +� +x +κ − 1 +�2 +− 10 +� +aβ + 4Nβ +� +x +κ − 1 + 5˜a − 88 +� +x3 d3 +dx3 +− +� +16 +� +x +κ − 1 +�2 +− 38 +� +aβ + 4Nβ +� +x +κ − 1 + 22˜a − 16 +� +x2 d2 +dx2 ++ +�� +x +κ − 1 +�2 +− 4 +� +aβ + 4Nβ +� � +x +κ − 1 +� ++ 2 +� +2 +� +aβ + 4Nβ +�2 + ˜a − 2 +�� +x3 +(κ − 1)2 +d +dx +− +� +4(˜a − 3) +� +aβ + 4Nβ +� +x +κ − 1 − ˜a2 + 14˜a + 16 +� +x d +dx − +� +aβ + 4Nβ +� � +x +κ − 1 +�3 ++ +� +2 +� +aβ + 4Nβ +�2 + ˜a +� � +x +κ − 1 +�2 +− (3˜a + 4) +� +aβ + 4Nβ +� +x +κ − 1 + ˜a2. +(2.3.42) +Then, for β = 1, 2, and 4, +D(L) +N,β ρ(L)(x; N, β) = 0 +(2.3.43) +93 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +and +D(L) +N,β +1 +NW(L) +1 +(x; N, β) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(x + a), +β = 2, +4 +κ−1 +� +2 +� +x +κ−1 +�2 + (2aβ − 1) +x +κ−1 +� +Nβ +− +1 +κ−1 +�� +x +κ−1 +�3 − (aβ + 2) +� +x +κ−1 +�2� ++ +1 +κ−1 +� +(a2 +β + 4aβ − 4) +x +κ−1 − aβ(aβ − 2)2� +, +β = 1, 4. +(2.3.44) +Proof. To obtain equation (2.3.43) from (2.3.7) and (2.3.13), change variables x �→ x/b, mul- +tiply both sides by b(N+2)(N−1)κ+(N+1)a+N according to equation (2.3.40), and then take the +limit b → ∞. This is equivalent to changing variables x �→ x/b, extracting terms of leading +order in b, then rewriting ρ(J) as ρ(L). +To obtain equation (2.3.44), apply the same prescription to equations (2.3.8) and (2.3.14). +Alternatively, one may apply the Stieltjes transform to equation (2.3.43). This is considerably +easier than in the Jacobi case (see Appendix B) since +� ∞ +0 +xn +s − x +dn +dxn ρ(L)(x) dx = sn dn +dsn W(L) +1 +(s) +can be computed through repeated integration by parts using the fact that the boundary +terms vanish at all stages: For a > −1 real and m > n non-negative integers, +xm +s−x +dn +dxn ρ(L)(x) +has a factor of xa+m−n which dominates at x = 0 and a factor of e−x which dominates as +x → ∞. +Equation (2.3.43) with β = 2 is equivalent to that seen in [164], [2]. When β = 1, 4, it has +been checked for N = 1, 2 using computer algebra. From inspection, it seems that the natural +variable of equations (2.3.43) and (2.3.44) is x/(κ − 1), which is the limit limb→∞ bβ +� x +b +� +. This +is in keeping with the duality relation on the resolvent presented in Lemma 1.4. Strictly +speaking, changing variables to y = x/(κ − 1) is not natural and would be counterproductive +since the corresponding weight [(κ − 1)y]a e(1−κ)y vanishes at κ = 1 and has different support +depending on whether κ < 1 or κ > 1. +2.3.4 +Differential equations for the Gaussian ensembles +The previous subsection contains a discussion on how one would obtain differential equations +for the densities and resolvents of Laguerre ensembles with even β without knowledge of +94 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +differential equations for the corresponding Jacobi β ensembles’ densities and resolvents. +We now elucidate those ideas by explicitly applying them to the study of the Gaussian +ensembles with β = 6 and consequently, by duality, β = 2/3. Indeed, since we do not have +differential equations for the β = 6 Jacobi ensemble at hand, we cannot immediately apply +the direct limiting approach used in the proof of Proposition 2.9. Thus, for our purposes, it +is in fact more efficient to replicate the proofs of §2.3.1 while taking limits at an earlier stage +so as to circumvent the need for investigating the β = 6 Jacobi ensemble altogether. +Like in the Jacobi and Laguerre cases, our initial focus is the average +I(G) +N,β(λ) := +� +N +∏ +i=1 +|λ − λi|β +� +GEN,β +, +(2.3.45) +where the subscript GEN,β indicates that the average is with respect to the eigenvalue +j.p.d.f. (1.2.7) with weight w(G)(λ) = e−λ2. The Gaussian analogue of the duality relation +(2.1.21) is [27] +� +N +∏ +i=1 +� +λ − κ−1/2λi +�n +� +GEN,β += +� +n +∏ +i=1 +(λ − iλi)N +� +GEn,4/β +. +(2.3.46) +Replacing λ by κ−1/2λ in the above duality relation and then factoring (−i)nN from the +right-hand side shows that for even β, I(G) +N,β(λ) is proportional to G(N) +β,0 (λ), where +G(N) +n,p (λ) := (−i)nN+p +� +n +∏ +i=1 +� +λi + iκ−1/2λ +�N+χi⩽p +� +GEn,4/β +, +0 ⩽ p ⩽ n. +(2.3.47) +Setting a′ = b′ = L and changing variables λi �→ 1 +2 +� +1 + λi +√ +L +� +in J(N) +n,p (λ) (2.1.16), we see that +G(N) +n,p (λ) = lim +L→∞ 4nL(−2i +√ +L)nN+p(2 +√ +L)n(n−1)/κ+n J(N) +n,p +� +1 +2 +� +1 − i +� +2 +βLλ +����� +a′=b′=L . +(2.3.48) +Thus, equation (2.1.17) simplifies to a differential-difference equation for G(N) +n,p (λ), +(n − p)G(N) +n,p+1(λ) = (n−p) +√κ λG(N) +n,p (λ) − +√κ +2 +d +dλG(N) +n,p (x) ++ p +2 +� 1 +κ(n − p) + N + 1 +� +G(N) +n,p−1(λ) +(2.3.49) +(cf. [133, Eq. (5.5)]). Taking the L → ∞ limit early has already yielded a simpler equation +than (2.1.17). However, we can take this one step further by defining +˜G(N) +n,p (λ) = e−λ2G(N) +n,p (λ) +(2.3.50) +95 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +so that ρ(G)(λ; N + 1, β) is proportional to ˜G(N) +β,0 (λ). It is then easy to obtain the differential- +difference equation +d +dλ +˜G(N) +n,p (λ) = +p +√κ +� 1 +κ(n − p) + N + 1 +� ˜G(N) +n,p−1(λ) ++ 2 +� 1 +κ(n − p) − 1 +� +λ ˜G(N) +n,p (λ) − +2 +√κ(n − p) ˜G(N) +n,p+1(λ). +(2.3.51) +With n = β ∈ 2N and p = 0, 1, . . . , n, this is equivalent to a matrix differential equation on +the vector v = +� +˜G(N) +n,0 (x) · · · ˜G(N) +n,n (x) +�T +, which is moreover equivalent to a scalar differential +equation for ρ(G)(λ; N + 1, β). +Proposition 2.10. For β = 2/3 or 6, define +D(G) +N,β = 81(κ − 1)7/2 d7 +dx7 + 1008 +� +3Nβ − 2x2 +κ − 1 + 2 +� +(κ − 1)5/2 d5 +dx5 ++ 2016x(κ − 1)3/2 d4 +dx4 ++ 64 +� +21Nβ − 14x2 +κ − 1 + 5 +� � +21Nβ − 14x2 +κ − 1 + 23 +� +(κ − 1)3/2 d3 +dx3 ++ 9984 +� +3Nβ − 2x2 +κ − 1 + 2 +� +x(κ − 1)1/2 d2 +dx2 ++ 256 +� +54Nβ +� +4N2 +β + 8Nβ + 3 +� +− (432N2 +β + 576Nβ + 57) x2 +κ − 1 ++ 96(3Nβ + 2) +x4 +(κ − 1)2 − +64x6 +(κ − 1)3 − 20 +� +(κ − 1)1/2 d +dx ++ 256 +� +144N2 +β + 192Nβ − 64(3Nβ + 2) x2 +κ − 1 + +64x4 +(κ − 1)2 + 25 +� +x +(κ − 1)1/2 , +(2.3.52) +where we retain the definition Nβ = (κ − 1)N. Then, for these same β values, +D(G) +N,β ρ(G)(x; N, β) = 0 +(2.3.53) +and +D(G) +N,β +1 +NW(G) +1 +(x; N, β) = +211 +√ +κ − 1 +� 4x2 +κ − 1 − 6Nβ − 7 +�2 +− 3 +212 +√ +κ − 1 +. +(2.3.54) +Proof. Take equation (2.3.51) with n = β = 6 and N replaced by N − 1 to obtain the matrix +96 + +2.3. Linear Differential Equations for the Eigenvalue Densities and Resolvents +differential equation +dv +dx = +� +����������������� +2x +−4 +√ +3 +0 +0 +0 +0 +0 +3N+5 +3 +√ +3 +4x +3 +− 10 +√ +3 +0 +0 +0 +0 +0 +6N+8 +3 +√ +3 +2x +3 +− 8 +√ +3 +0 +0 +0 +0 +0 +√ +3(N + 1) +0 +−2 +√ +3 +0 +0 +0 +0 +0 +12N+8 +3 +√ +3 +− 2x +3 +− 4 +√ +3 +0 +0 +0 +0 +0 +15N+5 +3 +√ +3 +− 4x +3 +− 2 +√ +3 +0 +0 +0 +0 +0 +2 +√ +3N +−2x +� +����������������� +v +(2.3.55) +for v = +� +˜G(N−1) +6,0 +(x) · · · ˜G(N−1) +6,6 +(x) +�T +. Like in the proofs of Lemmas 2.1 and 2.2, for 1 ⩽ p ⩽ 6, +the pth row of the above matrix differential equation gives an expression for ˜G(N−1) +6,p +(x) in +terms of +d +dx ˜G(N−1) +6,p−1 (x) and ˜G(N−1) +6,k +(x) with k < p. Substituting these expressions into the +equation corresponding to the last row in the order of decreasing p then yields a seventh- +order differential equation satisfied by ˜G(N−1) +6,0 +(x). Since ρ(G)(x; N, 6) is proportional to +˜G(N−1) +6,0 +(x), this equation is equivalent to (2.3.53) for β = 6. Taking the Stieltjes transform of +this result and substituting in the spectral moments m(G) +2 +and m(G) +4 +from [319] then yields +equation (2.3.54) for β = 6. Employing the duality relation given in Lemma 1.4 for the +resolvent then shows that equation (2.3.54) also holds for β = 2/3. Finally, taking the inverse +Stieltjes transform of this result shows that equation (2.3.53) holds for β = 2/3 as well. +Equation (2.3.53) has been checked for N = 1, 2 using computer algebra. Similar to +the Laguerre case, it seems like x/√ +κ − 1 is the natural variable in Proposition 2.10. This +is evidently due to the duality relation used in the proof of this proposition. Like in the +Laguerre case, there is presently no benefit in changing variables to x/√ +κ − 1. +It has been mentioned that equation (2.3.51) leads to a matrix differential equation which +is equivalent to a scalar differential equation for ρ(G)(x; N + 1, β) when β is even. For β = 2 +and 4, these differential equations are respectively +d +dx +� +���� +˜G(N−1) +2,0 +(x) +˜G(N−1) +2,1 +(x) +˜G(N−1) +2,2 +(x) +� +���� = +� +���� +2x +−4 +0 +N + 1 +0 +−2 +0 +2N +−2x +� +���� +� +���� +˜G(N−1) +2,0 +(x) +˜G(N−1) +2,1 +(x) +˜G(N−1) +2,2 +(x) +� +���� +(2.3.56) +97 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +and +d +dx +� +���������� +˜G(N−1) +4,0 +(x) +˜G(N−1) +4,1 +(x) +˜G(N−1) +4,2 +(x) +˜G(N−1) +4,3 +(x) +˜G(N−1) +4,4 +(x) +� +���������� += +� +���������� +2x +−4 +√ +2 +0 +0 +0 +2N+3 +2 +√ +2 +x +−3 +√ +2 +0 +0 +0 +√ +2(N + 1) +0 +−2 +√ +2 +0 +0 +0 +6N+3 +2 +√ +2 +−x +− +√ +2 +0 +0 +0 +2 +√ +2N +−2x +� +���������� +� +���������� +˜G(N−1) +4,0 +(x) +˜G(N−1) +4,1 +(x) +˜G(N−1) +4,2 +(x) +˜G(N−1) +4,3 +(x) +˜G(N−1) +4,4 +(x) +� +���������� +. +(2.3.57) +The corresponding scalar differential equations for the respective eigenvalue densities agree +with those of Proposition 2.1 after scaling x �→ +� +Nκ +2g x. So too does the differential equation +for ρ(G)(x; N, 1) obtained through the β ↔ 4/β duality given in Lemma 1.4. +2.4 +Scalings of the Differential Equations +The differential equations of the previous section hold true not only for finite integers N, +but also in the large N limit, upon appropriate scaling. In this section we consider two +scaling regimes: In §2.4.1, we study the classical matrix ensembles (and also the Gaussian +β ensembles with β = 2/3, 6) under the global scalings introduced in Definition 1.6, while +in §2.4.2, we look at the soft and hard edge scaling regimes (recall the discussion following +Proposition 1.2). +In the global scaling regime, we use Theorem 1.2 to expand the scaled resolvents ˜W1(x), +specified by equation (1.1.24), in 1/N and derive first order differential equations which +characterise the expansion coefficients. Being first order, these differential equations are +straightforward to solve, and using the Sokhotski–Plemelj inversion formula (1.1.25), we +are able to compute expressions for the large N limiting forms ρ0(λ) of the smoothed +eigenvalue densities and their 1/N corrections ρ1(λ). As expected, we recover the Wigner +semi-circle, Marˇcenko–Pastur, and Wachter laws seen in Proposition 1.2, along with the +expression for ρ(Cy),0(λ) presented therein. Moreover, in keeping with Note 1.1 following +this proposition, we observe that while the large N limiting forms ρ0(λ) are β-independent, +the 1/N corrections ρ1(λ) are different for each β that we consider. This is in keeping with +the general-β results known from loop equation analysis (see, e.g., [319], [142] and references +therein). +98 + +2.4. Scalings of the Differential Equations +In contrast to the β-universality seen for the large N limiting forms of the global scaled +eigenvalue densities, we see a different type of universality at the soft and hard edges. +Namely, it is known [115], [256] that for fixed β, a soft edge of a classical β ensemble behaves +statistically as that of any other classical β ensemble, in the large N limit — likewise for a +hard edge. Thus, we may denote the large N limiting forms of the soft and hard edge scaled +eigenvalue densities of the classical β ensembles as ρ(so f t)(λ; β) and ρ(hard)(λ; β), respectively. +In §2.4.2, we apply these scalings to the differential equations of Section 2.3 to derive linear +differential equations satisfied by ρ(so f t)(λ; β) for β ∈ {2/3, 1, 2, 4, 6} and ρ(hard)(λ; β) for +β ∈ {1, 2, 4}. +2.4.1 +Global scaled differential equations +Global scaling of the eigenvalue density refers to a choice of scaling which ensures that, in +the large N limit, the scaled density is supported on a finite interval on which it integrates to +one. Such a transformation is not unique, due to the length of the interval of support being +unspecified; for convenience, we follow the conventions outlined in Definition 1.6, which +are chosen to be consistent with the earlier works [319], [142]. We recall then that the global +scaled eigenvalue densities of the classical β ensembles are +˜ρ(G)(λ) = +� +κ +N ρ(G)( +√ +κNλ), +(2.4.1) +˜ρ(L)(λ) = κ ρ(L)(κNλ), +(2.4.2) +˜ρ(J)(λ) = 1 +N ρ(J)(λ), +(2.4.3) +˜ρ(Cy)(λ) = 1 +N ρ(Cy)(λ) +��� +α=ˆακN, +(2.4.4) +with corresponding scaled resolvents (1.1.24), +˜W(G) +1 +(x) := +√ +κN W(G) +1 +( +√ +κNx), +(2.4.5) +˜W(L) +1 +(x) := κN W(L) +1 +(κNx), +(2.4.6) +˜W(Cy) +1 +(x) := W(Cy) +1 +(x) +��� +α=ˆακN, +(2.4.7) +where ˆα is constant in N; since the spectrum of the Jacobi β ensemble is already supported +on [0, 1], there is no need to scale its resolvent. In §1.1.1, it was mentioned that the scaled +resolvents are more robust than the global scaled eigenvalue densities in the sense that they +99 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +admit well-defined large N expansions due to Theorem 1.2. Thus, following [319], [142] +again for consistency, we consider the expansions +˜W(G) +1 +(x; N) = 2N +∞ +∑ +l=0 +W(G),l +1 +(x) +(2√κN)l , +(2.4.8) +˜W(L) +1 +(x; N) = N +∞ +∑ +l=0 +W(L),l +1 +(x) +(√κN)l , +(2.4.9) +W(J) +1 (x; N) = N +∞ +∑ +l=0 +W(J),l +1 +(x) +(κN)l +, +(2.4.10) +˜W(Cy) +1 +(x; N) = N +∞ +∑ +l=0 +W(Cy),l +1 +(x) +(κN)l +. +(2.4.11) +Substituting these expansions into the differential equations of Section 2.3 (after scaling +according to (2.4.5)–(2.4.7)) and equating terms of like order in N results in linear differential +equations for the expansion coefficients Wl +1(x). The differential equations for Wl +1(x) are +first order with inhomogeneous terms dependent on {Wk +1(x)}l−1 +k=0. Hence, they constitute +a recursive process for computing the scaled resolvents ˜W1(x) up to any desired order in +1/N. The topological recursion or, equivalently, loop equation formalism [110] (see also the +review given in Section 4.1) accomplishes the same task for general β > 0, but our approach, +valid for the specific β values considered, is more efficient since there is no need to consider +the n-point correlators Un, Wn for n > 1. As a consequence, our work is able to isolate extra +structures, such as those studied by Haagerup and Thorbjørnsen [170]: In [170], the authors +showed in the GUE case that W(G),l +1 +(x) is a polynomial in (x2 − 2)−1/2, with the polynomial +coefficients satisfying a certain three-term recurrence. +Differential equations for W(G),l +1 +(x) +We begin with the Gaussian β ensembles with β ∈ {2/3, 1, 4, 6} and refer to [170] for the +β = 2 case. +Proposition 2.11. We reuse the notation of Proposition 2.1 with g = 1/2 so that h = √κ − 1/√κ +and y(G) := +√ +x2 − 2. Then, for β = 1 and 4, the expansion coefficients of ˜W(G) +1 +(x; N, β) (2.4.8) +satisfy the differential equations +y2 +(G) +d +dxW(G),0 +1 +(x) − xW(G),0 +1 +(x) = −1, +(2.4.12) +100 + +2.4. Scalings of the Differential Equations +y2 +(G) +d +dxW(G),1 +1 +(x) − xW(G),1 +1 +(x) = 4h d +dxW(G),0 +1 +(x) − +h +y2 +(G) +� +2xW(G),0 +1 +(x) + 5 +� +, +(2.4.13) +and for l ⩾ 2, the general differential equation +y2 +(G) +d +dxW(G),l +1 +(x) − xW(G),l +1 +(x) = 4h d +dxW(G),l−1 +1 +(x) − 2hx +y2 +(G) +W(G),l−1 +1 +(x) ++ +1 +y2 +(G) +�5y2 +(G) +2 +d3 +dx3 − 3x d2 +dx2 + d +dx +� +W(G),l−2 +1 +(x) +− 5h +y2 +(G) +d3 +dx3W(G),l−3 +1 +(x) + +1 +y2 +(G) +d5 +dx5W(G),l−4 +1 +(x), +(2.4.14) +where we set W(G),k +1 +:= 0 for k < 0. +Proof. Beginning with the β = 1, 4 case of equation (2.0.4), set g = 1/2 and map x �→ +√ +κNx +to obtain a differential equation for ˜W(G) +1 +(x; N, β). Substitute in the expansion (2.4.8) and +equate terms of like order in N. The terms of leading order in N correspond to equation +(2.4.12), the next to leading order terms correspond to equation (2.4.13), and so on. +Let us recall from §1.1.1 the relationship between the resolvent expansion coefficients +{Wl +1(x)}∞ +l=0 and the coefficients { ˜Mk,l}∞ +l=0 of the 1/N expansions of the spectral moments ˜mk +of ˜ρ(λ). The facts that ˜m0 = m0/N = 1 and Wl +1(x) has asymptotic expansion ∑∞ +k=0 ˜Mk,l/xk+1 +tell us that the solutions of the differential equations (2.4.12)–(2.4.14) are fully determined by +the boundary conditions at large x, +W0 +1(x) = 1 +x + O +� 1 +x2 +� +, +(2.4.15) +Wl +1(x) = O +� 1 +x2 +� +, +l > 1. +(2.4.16) +This is in fact true for all of the first order differential equations given in this subsection. +Now, applying the above proof to differential equation (2.3.54) gives analogous differential +equations for the expansion coefficients W(G),l +1 +(x) when β = 2/3, 6. +Proposition 2.12. Retain the definition of y(G) from Proposition 2.11. Then, for β = 2/3 and 6, the +expansion coefficients of ˜W(G) +1 +(x; N, β) satisfy the differential equations +y2 +(G) +d +dxW(G),0 +1 +(x) − xW(G),0 +1 +(x) = −1, +(2.4.17) +101 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +y2 +(G) +d +dxW(G),1 +1 +(x) − xW(G),1 +1 +(x) = 6h d +dxW(G),0 +1 +(x) − +h +y2 +(G) +� +4xW(G),0 +1 +(x) − 7 +� +, +(2.4.18) +y2 +(G) +d +dxW(G),2 +1 +(x) − xW(G),2 +1 +(x) = 6h d +dxW(G),1 +1 +(x) − 4hx +y2 +(G) +W(G),1 +1 +(x) − +43 +3y4 +(G) ++ +1 +12y4 +(G) +� +49y4 +(G) +d3 +dx3 − 78xy2 +(G) +d2 +dx2 + 3(72 − 19x2) d +dx + 25x +� +W(G),0 +1 +(x), +(2.4.19) +and for l ⩾ 3, the general differential equation +y2 +(G) +d +dxW(G),l +1 +(x) − xW(G),l +1 +(x) = 6h d +dxW(G),l−1 +1 +(x) − 4hx +y2 +(G) +W(G),l−1 +1 +(x) ++ +1 +12y4 +(G) +� +49y4 +(G) +d3 +dx3 − 78xy2 +(G) +d2 +dx2 + 3(72 − 19x2) d +dx + 25x +� +W(G),l−2 +1 +(x) ++ +h +3y4 +(G) +� +49y2 +(G) +d3 +dx3 + 39x d2 +dx2 − 10 d +dx +� +W(G),l−3 +1 ++ 7h +y4 +(G) +d5 +dx5W(G),l−5 +1 ++ +1 +18y4 +(G) +� +63y2 +(G) +d5 +dx5 + 63x d4 +dx4 + 230 d3 +dx3 +� +W(G),l−4 +1 ++ +3h +4y4 +(G) +d7 +dx7W(G),l−6 +1 +, +(2.4.20) +where we again set W(G),k +1 +:= 0 for k < 0. +This proposition has been checked against [319] up to l = 6, and thus also serves as a +check for differential equation (2.3.54) up to order six in 1/N. Similar checks for consistency, +against [142], have been carried out for Propositions 2.13 to 2.15 below. +Differential equations for W(L),l +1 +(x) and W(J),l +1 +(x) +Since the differential equations for the Laguerre and Jacobi ensembles’ resolvents involve the +parameters a, b, equating terms of equal order in N requires us to choose how these parame- +ters depend on N. To make our results suitable for most known or potential applications +[33], [56], [305], [266], [229], [240], [241], [70], [71] and to improve clarity, we take a = ˆaκN +and b = ˆbκN with ˆa, ˆb constant in N. Note that our method easily accommodates for more +general N-expansions of a and b. +Proposition 2.13. Set a = ˆaκN with ˆa constant in N, and let y(L) := +� +(ˆa − x)2 − 4x. The +expansion coefficients of the LUE scaled resolvent ˜W(L) +1 +(x; N, 2) (2.4.9) satisfy the differential equation +− xy2 +(L) +d +dxW(L),0 +1 +(x) + +�(ˆa + 2)x − ˆa2� +W(L),0 +1 +(x) = x + ˆa, +(2.4.21) +102 + +2.4. Scalings of the Differential Equations +and for l ⩾ 2, the general differential equation +xy2 +(L) +d +dxW(L),l +1 +(x) − +�(ˆa + 2)x − ˆa2� +W(L),l +1 +(x) += +� +x3 d3 +dx3 + 4x2 d2 +dx2 + 2x d +dx +� +W(L),l−2 +1 +(x). +(2.4.22) +Proof. Consider equation (2.3.44) with β = 2. In it, set a = ˆaN and map x �→ κNx to obtain a +differential equation for ˜W(L) +1 +(x) +�� +a=ˆaN. Substitute in the expansion (2.4.9) and equate terms +of equal order in N to extract differential equations (2.4.21), (2.4.22). +In the κ = β/2 = 1, a = ˆaN setting considered here, the operator D(L) +N,2 +�� +x�→κNx (2.3.41) +is even in N while the right-hand side of differential equation (2.3.44) under the mapping +x �→ κNx is odd in N. It follows that ˜W(L) +1 +(x) (2.4.6) is an odd function of N, so W(L),k +1 +(x) = 0 +when k is odd. Thus, differential-difference equation (2.4.22) holds vacuously for odd l, and +should otherwise be interpreted as a recursion over even l. Note also that the vanishing +of W(L),k +1 +(x) for k odd implies that the spectral moments ˜m(L) +k +(recall §1.1.1) of the scaled +eigenvalue density ˜ρ(L)(λ; N, 2) are hence even functions of N, in keeping with the second +point of Remark 2.1. In the β = 1, 4 analogue of Proposition 2.13 given below, which is +obtained by repeating the above proof starting with the β = 1, 4 cases of equation (2.3.44) +and taking a = ˆaκN, such phenomena are not seen since W(L),k +1 +(x) is non-zero for all k ∈ N. +Proposition 2.14. Retain the definitions of h, ˆa and y(L) from Propositions 2.1 and 2.13. For β = 1 +and 4, the expansion coefficients of the scaled resolvent ˜W(L) +1 +(x; N, β) satisfy the differential equations +− xy2 +(L) +d +dxW(L),0 +1 +(x) + +�(ˆa + 2)x − ˆa2� +W(L),0 +1 +(x) = x + ˆa, +(2.4.23) +xy4 +(L) +d +dxW(L),1 +1 +(x) − y2 +(L) +�(ˆa + 2)x − ˆa2� +W(L),1 +1 +(x) += 4hˆaxy2 +(L) +d +dxW(L),0 +1 +(x) + 2hˆa +� +x2 − 3(ˆa + 2)x + 2ˆa2� +W(L),0 +1 +(x) ++ 2h +� +x(x − 1) + 2ˆax + 2ˆa2� +, +(2.4.24) +103 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +xy4 +(L) +d +dxW(L),2 +1 +(x) − y2 +(L) +�(ˆa + 2)x − ˆa2� +W(L),2 +1 +(x) += 4hˆaxy2 +(L) +d +dxW(L),1 +1 +(x) + 2hˆa +� +x2 − 3(ˆa + 2)x + 2ˆa2� +W(L),1 +1 +(x) ++ 5 +2x3y2 +(L) +d3 +dx3W(L),0 +1 +(x) + x2 � +8x2 − 19(ˆa + 2)x + 11ˆa2� d2 +dx2W(L),0 +1 +(x) ++ x +� +2x2 − 6(ˆa + 2)x + 5ˆa2� d +dxW(L),0 +1 +(x) ++ 2 +�(ˆa + 2)x − ˆa2� +W(L),0 +1 +(x) − 2(ˆa + x), +(2.4.25) +and for l ⩾ 3, the general differential equation +xy4 +(L) +d +dxW(L),l +1 +(x) − y2 +(L) +�(ˆa + 2)x − ˆa2� +W(L),l +1 +(x) += 4hˆaxy2 +(L) +d +dxW(L),l−1 +1 +(x) + 2hˆa +� +x2 − 3(ˆa + 2)x + 2ˆa2� +W(L),l−1 +1 +(x) ++ 5 +2x3y2 +(L) +d3 +dx3W(L),l−2 +1 +(x) + x2 � +8x2 − 19(ˆa + 2)x + 11ˆa2� d2 +dx2W(L),l−2 +1 +(x) ++ x +� +2x2 − 6(ˆa + 2)x + 5ˆa2� d +dxW(L),l−2 +1 +(x) + 2 +�(ˆa + 2)x − ˆa2� +W(L),l−2 +1 +(x) +− hˆax +� +5x2 d3 +dx3 + 22x d2 +dx2 + 14 d +dx +� +W(L),l−3 +1 +(x) +− +� +x5 d5 +dx5 + 10x4 d4 +dx4 + 22x3 d3 +dx3 + 4x2 d2 +dx2 − 4x d +dx +� +W(L),l−4 +1 +(x), +(2.4.26) +where we set W(L),−1 +1 +:= 0. +Before moving onto the Cauchy ensembles, we present the differential equations char- +acterising the expansion (2.4.10) in the JUE case with a = ˆaκN, b = ˆbκN as discussed +immediately prior to Proposition 2.13. We do not display the analogous results in the JOE +and JSE cases due to their (relatively speaking) unwieldy forms, but note that they can be +derived from equation (2.3.14) following the same proof as below. +Proposition 2.15. Let y(J) := +� +(ˆa + ˆb + 2)2x2 − 2(ˆa + 2)(ˆa + ˆb)x + ˆa2 with ˆa, ˆb constant in N. +The expansion coefficients specified by equation (2.4.10) in the JUE case with a = ˆaκN and b = ˆbκN +satisfy the differential equation +x(x − 1)y2 +(J) +d +dxW(J),0 +1 +(x) +− +� +ˆa2(1 − x)3 − ˆa(ˆb + 2)x(1 − x)(1 − 2x) − (ˆb + 2)2x3 + 2(ˆb + 1)(3x2 + x) +� +W(J),0 +1 +(x) += (ˆa + ˆb + 1)(ˆa(1 − x) + ˆbx), +(2.4.27) +104 + +2.4. Scalings of the Differential Equations +and for l ⩾ 2, the general differential equation +x(1 − x)y2 +(J) +d +dxW(J),l +1 +(x) ++ +� +ˆa2(1 − x)3 − ˆa(ˆb + 2)x(1 − x)(1 − 2x) − (ˆb + 2)2x3 + 2(ˆb + 1)(3x2 + x) +� +W(J),l +1 +(x) += x3(1 − x)3 d3 +dx3W(J),l−2 +1 +(x) + 4x2(1 − x)2(1 − 2x) d2 +dx2W(J),l−2 +1 +(x) +− 2x(1 − x)(7x(1 − x) − 1) d +dxW(J),l−2 +1 +(x) − 2x(1 − x)(1 − 2x)W(J),l−2 +1 +(x). +(2.4.28) +Proof. Substitute expansion (2.4.10) into equation (2.3.8) and set a = ˆaN, b = ˆbN. Equating +terms of equal order in N then produces the sought differential equations. +Similar to Proposition 2.13, in the β = 2, a = ˆaN, b = ˆbN setting, W(J),k +1 +(x) vanishes for +odd k, and the differential equations of Proposition 2.15 constitute a recursion over even l. +Differential equations for W(Cy),l +1 +(x) in the symmetric case α real +The procedure outlined above extends to the Cauchy ensembles in the expected manner. +Having already set α = ˆακN (recall that η = −κ(N − 1) − 1 − α) to ensure that the global +scaled eigenvalue density ˜ρ(Cy)(λ) has the necessary properties, we simplify further by +constraining ˆα to be a real and positive constant. This decision to consider only the symmetric +Cauchy ensembles is simply due to the fact that the equivalent results in the non-symmetric +case are cumbersome to display; they can however be derived in the same way. Thus, setting +α = ˆακN in equation (2.3.22) with ˆα = O(1) in N, substituting in the expansion (2.4.11) +for ˜W(Cy) +1 +(x), and then collecting terms of like order in N, we obtain the following two +propositions. +Proposition 2.16. In the Cauchy weight, set η = −κ(N − 1) − 1 − α, α = ˆακN with ˆα positive +real and constant in N. Moreover, let y(Cy) := +√ +ˆα2x2 − 2ˆα − 1. Then, the expansion coefficients of +the symmetric CyUE scaled resolvent ˜W(Cy) +1 +(x; N, 2) +�� +ˆα∈R+ (2.4.11) satisfy the differential equation +− (1 + x2)y2 +(Cy) +d +dxW(Cy),0 +1 +(x) − +�ˆα2x2 − (ˆα + 2)2 + 2 +� +xW(Cy),0 +1 +(x) = (ˆα + 1)(2ˆα + 1), +(2.4.29) +105 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +and for l ⩾ 2, the general differential equation +4(1 + x2)y2 +(Cy) +d +dxW(Cy),l +1 +(x) + 4 +�ˆα2x2 − (ˆα + 2)2 + 2 +� +xW(Cy),l +1 +(x) += (1 + x2)3 d3 +dx3W(Cy),l−2 +1 +(x) + 8x(1 + x2)2 d2 +dx2W(Cy),l−2 +1 +(x) ++ 2(7x2 + 3)(1 + x2) d +dxW(Cy),l−2 +1 +(x) + 4x(1 + x2)W(Cy),l−2 +1 +(x). +(2.4.30) +As with the other classical unitary ensembles, ˜W(Cy) +1 +(x) is an odd function of N when +β = 2, so W(Cy),k +1 +(x) = 0 when k is odd. It can thus be easily checked that differential- +difference equation (2.4.30) trivially holds true for odd values of l and is otherwise to be +interpreted as a recursion over even l. +Proposition 2.17. Retain the notation of Proposition 2.16. For β = 1 and 4, the expansion coefficients +of the scaled resolvent ˜W(Cy) +1 +(x; N, β) +�� +ˆα∈R+ satisfy the differential equations +− (1 + x2)y2 +(Cy) +d +dxW(Cy),0 +1 +(x) − +�ˆα2x2 − (ˆα + 2)2 + 2 +� +xW(Cy),0 +1 +(x) = (ˆα + 1)(2ˆα + 1), +(2.4.31) +2(1 + x2)y4 +(Cy) +d +dxW(Cy),1 +1 +(x) + 2y2 +(Cy) +�ˆα2x2 − (ˆα + 2)2 + 2 +� +xW(Cy),1 +1 +(x) += 4(κ − 1)ˆαy2 +(Cy)(1+ x2)2 d +dxW(Cy),0 +1 +(x) + 2(κ − 1)ˆα +� +2ˆα2x2 − (ˆα + 3)2 + 6 +� +x(1+ x2)W(Cy),0 +1 +(x) ++ (κ − 1)ˆα +�(8ˆα2 + 9ˆα + 2)x2 + (2ˆα + 1)(5ˆα + 4) +� +, +(2.4.32) +8(1 + x2)y4 +(Cy) +d +dxW(Cy),2 +1 +(x) + 8y2 +(Cy) +�ˆα2x2 − (ˆα + 2)2 + 2 +� +xW(Cy),2 +1 +(x) += 16(κ − 1)ˆαy2 +(Cy)(1 + x2)2 d +dxW(Cy),1 +1 +(x) ++ 8(κ − 1)ˆα +� +2ˆα2x2 − (ˆα + 3)2 + 6 +� +x(1+ x2)W(Cy),1 +1 +(x) + 10(κ − 1)2y2 +(Cy)(1+ x2)3 d3 +dx3W(Cy),0 +1 +(x) ++ 4(κ − 1)2 � +19ˆα2x2 − 3ˆα2 − 22(2ˆα + 1) +� +x(1 + x2)2 d2 +dx2W(Cy),0 +1 +(x) ++ 4(κ − 1)2 � +29ˆα2x4 − 2ˆα2x2 − 44(2ˆα + 1)x2 + ˆα2 − 12(2ˆα + 1) +� +(1 + x2) d +dxW(Cy),0 +1 +(x) ++ 8(κ − 1)2 � +3ˆα2x4 − (ˆα + 8)2x2 − 4(2ˆα + 1) +� +W(Cy),0 +1 +(x) +− 4(κ − 1)2 �(3ˆα2 − 1)x2 + 9ˆα2 + 10ˆα + 3 +� +, +(2.4.33) +106 + +2.4. Scalings of the Differential Equations +and for l ⩾ 3, the general differential equation +4(1 + x2)y4 +(Cy) +d +dxW(Cy),l +1 +(x) + 4y2 +(Cy) +�ˆα2x2 − (ˆα + 2)2 + 2 +� +xW(Cy),l +1 +(x) += 8(κ − 1)ˆαy2 +(Cy)(1 + x2)2 d +dxW(Cy),l−1 +1 +(x) ++ 4(κ − 1)ˆα +� +2ˆα2x2 − (ˆα + 3)2 + 6 +� +x(1 + x2)W(Cy),l−1 +1 +(x) ++ 5(κ − 1)2y2 +(Cy)(1 + x2)3 d3 +dx3W(Cy),l−2 +1 +(x) ++ 2(κ − 1)2 � +19ˆα2x2 − 3ˆα2 − 22(2ˆα + 1) +� +x(1 + x2)2 d2 +dx2W(Cy),l−2 +1 +(x) ++ 2(κ − 1)2 � +29ˆα2x4 − 2ˆα2x2 − 44(2ˆα + 1)x2 + ˆα2 − 12(2ˆα + 1) +� +(1 + x2) d +dxW(Cy),l−2 +1 +(x) ++ 4(κ − 1)2 � +3ˆα2x4 − (ˆα + 8)2x2 − 4(2ˆα + 1) +� +W(Cy),l−2 +1 +(x) +− (κ − 1)3ˆα +� +5(1 + x2)4 d3 +dx3 + 38x(1 + x2)3 d2 +dx2 ++2(31x2 + 15)(1 + x2)2 d +dx + 4x(4x4 + 9x2 + 5) +� +W(Cy),l−3 +1 +(x) +− (κ − 1)4 +� +(1 + x2)5 d5 +dx5 + 20x(1 + x2)4 d4 +dx4 + (122x2 + 29)(1 + x2)3 d3 +dx3 ++ 4x(65x2 + 46)(1 + x2)2 d2 +dx2 + 4(41x4 + 56x2 + 14)(1 + x2) d +dx ++ 8x(2x4 + 4x2 + 3) +� +W(Cy),l−4 +1 +(x) + 4(κ − 1)3ˆαx2χl=3 + 2(κ − 1)4(1 − x2)χl=4, +(2.4.34) +where we have set W(Cy),−1 +1 +:= 0. +Remark 2.12. +1. It can be observed that throughout this subsection, the differential equa- +tions characterising the leading order terms W0 +1(x) depend on the classical weight +w(λ), but are otherwise the same for all β. Thus, W0 +1(x) is β-independent and use +of the Sokhotski–Plemelj inversion formula (1.1.25) tells us that the large N limiting +form ρ0(λ) of ˜ρ(λ) also has this property. On the other hand, with a, b = O(N), +W1 +1(x) vanishes for β = 2 but can be seen to be non-zero for β ̸= 2, in keeping with +Note 1.1 following Proposition 1.2. In fact, solving the differential equations of Proposi- +tions 2.11–2.15 with the boundary conditions (2.4.15), (2.4.16) recovers expressions for +W0 +1(x) known from the general-β works [319], [142]. Applying the Sokhotski–Plemelj +formula (1.1.25) then recovers the expressions for ρ0(λ) given in Proposition 1.2 and +its 1/N, 1/N2 corrections seen in [319], [142] (see also the earlier work [17]). +107 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +2. According to the second point of Remark 1.9 (more specifically equation (1.2.88)), +taking a = ˆaκN and b = ˆbκN with ˆa, ˆb constant in N is equivalent to setting γ1 = 1 + ˆa +and γ2 = 1 + ˆb. Taking this into account, we see that the auxiliary functions y(w) used +in Propositions 2.11 to 2.17 are related to the large N limiting forms given in Propo- +sition 1.2. To be precise, y(G), y(L), y(J), and y(Cy) are equal to the Stieltjes transform +of ρ(G),0(λ), 2λρ(L),0(λ), 2λ(1 − λ)ρ(J),0(λ), and (1 + λ2)ρ(Cy),0(λ), respectively. The +significance of their presence in the differential equations of this subsection is that for +l > 0, the singularities of Wl +1(x) lie exactly at the zeroes of y(w), a fact well known from +the viewpoint of topological recursion [110]. +We do not make explicit here the expressions for Wl +1(x) nor ρl(x) in the Gaussian, +Laguerre, or Jacobi cases, since these can be found in the literature [17], [319], [142]. Instead, +we demonstrate the structures discussed in the above remark in the Cauchy case. +Proposition 2.18. Retain the notation of Proposition 2.16, with ˆα positive real and constant in N. +For β = 1, 2, and 4, we have +W(Cy),0 +1 +(x) = (ˆα + 1)x − y(Cy) +1 + x2 +, +(2.4.35) +W(Cy),1 +1 +(x) = (κ − 1)ˆα +2 +� +1 +y(Cy) +− +ˆαx +y2 +(Cy) +� +, +(2.4.36) +W(Cy),2 +1 +(x) = (κ − 1)2ˆα +2 +� +ˆα2((4ˆα2 + 10ˆα + 5)x2 + 2ˆα + 1) +4y5 +(Cy) +− ˆα(ˆα2 + 2ˆα + 1)x +y4 +(Cy) +� ++ κ +8 +ˆα2(2ˆα + 1)(1 + x2) +y5 +(Cy) +, +(2.4.37) +along with the large N asymptotic for the smoothed eigenvalue density (2.4.4) +˜ρ(Cy)(λ) +��� +ˆα∈R+ += ρ(Cy),0(λ) + κ − 1 +κN +� +ˆα +2π +√ +1 + 2ˆα − ˆα2λ2 χ|λ|<√ +2ˆα+1/ˆα +−1 +4δ(λ − +√ +2ˆα + 1/ˆα) − 1 +4δ(λ + +√ +2ˆα + 1/ˆα) +� ++ O +� 1 +N2 +� +, +(2.4.38) +where δ(λ) is the Dirac delta and ρ(Cy),0(λ) is specified by equation (1.2.19) with ˆα = ˆα1 and ˆα2 = 0. +Proof. The general solution of equation (2.4.29), equivalently (2.4.31), is +W(Cy),0 +1 +(x) = (ˆα + 1)x + Cy(Cy) +1 + x2 +, +108 + +2.4. Scalings of the Differential Equations +with the integration constant C forced to equal −1 due to the boundary condition (2.4.15). +Similarly, the integration constants present in the general solutions of equations (2.4.30), +(2.4.32), and (2.4.33) must be zero due to the boundary condition (2.4.16). Equation (2.4.38) is +obtained by applying the Sokhotski–Plemelj formula (1.1.19) to the series (2.4.11) truncated +to have two terms; note that the Stieltjes transform (1.1.18) of δ(λ − λ0) is 1/(x − λ0). +Before moving on to the soft and hard edge scaling regimes, let us remark that the results +contained in Proposition 2.18 are expected to hold for general real β > 0, in line with what +is known about the other classical β ensembles [319], [142]. +2.4.2 +Soft and hard edge scaled differential equations +In this subsection we study the eigenvalue densities of the classical matrix ensembles after +they have been shifted to be centred on either the largest or smallest eigenvalue and scaled +such that the mean spacing between this eigenvalue and its neighbour is order unity. The +large N limits of the eigenvalue densities recentred and scaled in this manner have one of +two universal forms, depending on whether the centring is performed at a soft or hard edge; +we denote these large N limiting forms ρ(so f t)(λ; β) and ρ(hard)(λ; β), respectively. Recalling +from the discussion following Proposition 1.2, centring on the largest (smallest) eigenvalue +corresponds to a soft edge if ρ0(λ) exhibits a square root profile at the upper (lower) endpoint +of its support, and a hard edge if it instead has an inverse square root profile. Thus, some +observations can be made from the contents of Proposition 1.2 using equation (1.2.88): When +the parameters a, b, α are proportional to N, i.e., upon setting a = ˆaκN, b = ˆbκN, and +α = ˆακN with ˆa, ˆb, ˆα constant in N, all of the edges of the classical β ensembles are of the soft +type. If one instead takes a = O(1), the lower edge of the Laguerre and Jacobi β ensembles +is of the hard type. Likewise, if b = O(1), ρ(J),0(λ) has a hard edge at the upper endpoint +of its support. A point of interest we will observe is that while edge scaling is a procedure +local to the endpoint of support being centred on, it depends on both parameters a, b where +applicable, even though a (b) mainly governs the behaviour of the lower (upper) edge. +The particular scalings required to ensure that the large N limits of the eigenvalue +densities of the classical matrix ensembles share the same universal forms can be a little +complicated, so we (re)introduce some notation. Recall from the definitions of a, b, γ1, γ2 in +109 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +Propositions 1.1 and 1.2 and equation (1.2.88) contained in Remark 1.9 that +γ1 = lim +N→∞ +M1 +N = 1 + lim +N→∞ +a − κ + 1 +κN +, +(2.4.39) +γ2 = lim +N→∞ +M2 +N = 1 + lim +N→∞ +b − κ + 1 +κN +, +(2.4.40) +where M1, M2 are now interpreted as continuous parameters dependent on a, b. Thus in +the large N limit, γ1, γ2 = 1 when a, b = O(1), while if we set a = ˆaκN and b = ˆbκN +with ˆa, ˆb = O(1), we have γ1 = 1 + ˆa and γ2 = 1 + ˆb. Recall too from Proposition 1.2 +that λ(MP) +± +, λ(Wac) +± +, λ(Cy) +± +denote the endpoints of the supports of ˜ρ(L)(λ), ˜ρ(J)(λ), ˜ρ(Cy)(λ), +respectively; in the following, we only consider the symmetric Cauchy ensembles, so we set +ˆα ∈ R and λ(Cy) +± += ±√ +2ˆα + 1/ˆα. Let us also define +γ3 := (ˆα + 1)2 +ˆα3 +(2.4.41) +for convenience and +q2 := +M1 +M1 + M2 +, +r2 := +M2 +M1 + M2 +, +(2.4.42) +˜q2 := +N +M1 + M2 +, +˜r2 := 1 − ˜q2, +(2.4.43) +uN := +� +qr ˜q˜r√M1 + M2 +˜q˜r(q2 − r2) + qr( ˜q2 − ˜r2) +�4/3 +(2.4.44) +following [178]. +Differential equations characterising the soft edge +We now present differential equations satisfied by ρ(so f t)(λ) for β ∈ {2/3, 1, 2, 4, 6}. +Theorem 2.3. Define the soft edge limiting forms of the differential operators DN,β introduced in +Section 2.3 as +D(so f t) +β += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +d3 +dx3 − 4x d +dx + 2, +β = 2, +d5 +dx5 − 10κx d3 +dx3 + 6κ d2 +dx2 + 16κ2x2 d +dx − 8κ2x, +β = 1, 4, +3 d7 +dx7 − 56κx d5 +dx5 + 28κ d4 +dx4 + 784 +3 κ2x2 d3 +dx3 +−208κ2x d2 +dx2 − 4κ2(64κx3 − 17) d +dx + 128κ3x2, +β = 2/3, 6, +(2.4.45) +110 + +2.4. Scalings of the Differential Equations +where we recall that κ = β/2. Then, for β ∈ {2/3, 1, 2, 4, 6}, the Gaussian, Laguerre, Jacobi, and +symmetric Cauchy β ensembles’ eigenvalue densities satisfy the following differential equation in the +soft edge limit: +D(so f t) +β +ρ(so f t)(x; β) = 0. +(2.4.46) +Proof. Make the change of variables [115] x �→ √κ +�√ +2N + +x +√ +2N1/6 +� +in equations (2.0.3) and +(2.3.53). Then, multiply through by N−1/2 for β = 2, N−5/6 for β = 1 and 4, or N−7/6 for +β = 2/3 and 6. Equating terms of order one then yields equation (2.4.46) above, while all +other terms vanish in the N → ∞ limit. +In the case β = 2, the differential equation (2.4.46) satisfied by the soft edge density has +been isolated in the earlier works of Brack et al. [50, Eq. (C.2) with ¯h2/m = 1, λM − V(r) = +− 1 +2r, D = 1] and of Dean et al. [77, Eq. (207) with d = 1]. For β = 1, 2, and 4, the above proof +can be replicated by instead considering the differential equations (2.3.7), (2.3.13), (2.3.21) +and (2.3.43) for the eigenvalue densities of the Jacobi, symmetric Cauchy, and Laguerre +ensembles, due to universality. Explicitly, differential equation (2.4.46) is satisfied by the +leading order term of the following scaled densities in the large N limit: +• In the regime of the largest eigenvalue [115], [28], [132], [133], +ρ(G) +�√ +κ +�√ +2N + δ(G) + +x +√ +2N1/6 +� +; N, β +� +, +(2.4.47) +where δ(G) = o(N−1/6) is an arbitrary parameter (the regime of the smallest eigenvalue +can be treated by exploiting the symmetry x ↔ −x); +• In the regime of the largest eigenvalue [115], [196], [121], [132], [133], +ρ(L) +� +κ +� +λ(MP) ++ +N + δ(L,so f t) ++ ++ (√γ1 + 1)4/3 +√γ1 +N1/3x +� +; N, β +� +, +(2.4.48) +where δ(L,so f t) +± += o(N1/3) is henceforth an arbitrary parameter. Note that this choice of +scaling is effective whether we set a = ˆaκN or take a to be constant in N. In the latter +case, the argument of (2.4.48) simplifies to κ(4N + δ(L,so f t) ++ ++ 24/3N1/3x); +• In the regime of the smallest eigenvalue, having set a = ˆaκN with ˆa = O(1) [28], [121], +ρ(L) +� +κ +� +λ(MP) +− +N − δ(L,so f t) +− +− (√γ1 − 1)4/3 +� N +√γ1 +�1/3 +x +� +; N, β +� +; +(2.4.49) +111 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +• In the regime of the largest eigenvalue, with b = ˆbκN and ˆb = O(1) [197], [121], [178], +ρ(J) +� +λ(Wac) ++ ++ δ(J,so f t) + qr ˜q˜r +uN +x; N, β +� +, +(2.4.50) +where δ(J,so f t) = o(N−2/3) is arbitrary and q, r, ˜q, ˜r, uN are defined in (2.4.42)–(2.4.44). +Like with (2.4.48), this scaling is effective regardless of whether a is constant or linear +in N (we must take b = O(N) to ensure that this is actually a soft edge). The regime of +the smallest eigenvalue can be treated by using the symmetry (x, a, b) ↔ (1 − x, b, a); +• In the regime of the largest eigenvalue, having set α = ˆακN with ˆα positive real and +constant in N, +ρ(Cy) +� +�λ(Cy) ++ ++ δ(Cy) + +� +γ2 +3 +2λ(Cy) ++ +N2 +�1/3 +x; N, β +� +� , +(2.4.51) +where δ(Cy) = o(N−2/3) is an arbitrary parameter and γ3 = (ˆα + 1)2/ˆα3 (2.4.41). +Remark 2.13. For even β, ρ(so f t)(x) has an explicit representation as a β-dimensional integral +due to [81], while [140] provides alternate forms for β = 1, 2, and 4. To date, there is no +explicit functional form for ρ(so f t)(x) when β = 2/3. On the other hand, [90] shows for all +κ = β/2 > 0 that for the first two leading orders as x → ∞, +ρ(so f t)(x) ∝ exp +�−4κx3/2/3 +� +x3κ/2 +, +which is an extension of the even-β result of [120], +ρ(so f t)(x) ∼ +x→∞ +1 +π +Γ(1 + κ) +(8κ)κ +exp +�−4κx3/2/3 +� +x3κ/2 +. +(2.4.52) +This result is consistent with the differential equations of Theorem 2.3. So too is the result +[81], +ρ(so f t)(x) +∼ +x→−∞ +� +|x| +π +, +β ∈ 2N. +(2.4.53) +Likewise, Theorem 2.4 below is consistent with the result [116], +ρ(hard)(x) ∼ +x→∞ +1 +2π√x, +β ∈ 2N. +(2.4.54) +Note that the asymptotic forms (2.4.52), (2.4.53), and (2.4.54) respectively capture the facts +that moving past the soft edge results in exponential decay, moving from the soft edge into +the bulk results in a square root profile, and moving from the hard edge into the bulk shows +a square root singularity. +112 + +2.4. Scalings of the Differential Equations +Differential equations characterising the hard edge +We now give the hard edge analogue of Theorem 2.3 for β ∈ {1, 2, 4}. For uniformity across +the Laguerre and Jacobi ensembles, we study the lower edge centred at x = 0 (the hard edge +at x = 1 for the Jacobi ensemble can be studied by interchanging x with 1 − x and a with b). +In contrast to the soft edge scaling, the differential equations characterising the hard edge +are seen to depend on the a paramater, which is taken to be constant in N. +Theorem 2.4. Define the hard edge limiting forms of the differential operators DN,β specified in +Section 2.3 as +D(hard) +β += +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +x3 d3 +dx3 + 4x2 d2 +dx2 + +� +x − a2 + 2 +� +x d +dx + 1 +2x − a2, +β = 2, +4x5 d5 +dx5 + 40x4 d4 +dx4 + [10κx − 5˜a + 88] x3 d3 +dx3 ++ [38κx − 22˜a + 16] x2 d2 +dx2 ++ +� +(2κx − ˜a)2 + 12κx − 14˜a − 16 +� +x d +dx ++(2κx − ˜a)(κx − ˜a) − 4κx, +β = 1, 4, +(2.4.55) +where we retain the definition of ˜a given in Theorem 2.2. Then, for β ∈ {1, 2, 4}, the hard edge scaled +eigenvalue densities of the Laguerre and Jacobi β ensembles satisfy the differential equation +D(hard) +β +ρ(hard)(x; β) = 0. +(2.4.56) +Proof. Since the Gaussian ensembles do not exhibit a hard edge, we turn to the Laguerre +ensembles as they are the next simplest to work with. Thus, we begin by changing variables +[115], [256] x �→ κx/(4N) in equation (2.3.43). Equating terms of order one yields the +differential equation (2.4.56) above, and we note that all other terms are O( 1 +N). +As in the soft edge case, there is a little bit of freedom in the choice of scaling used in +the above proof, and the proof itself can be formulated in terms of the differential equations +(2.3.7) and (2.3.13) characterising the eigenvalue densities of the Jacobi ensembles. To be +precise, the differential equation (2.4.56) above is satisfied by the leading order term of the +following scaled densities in the large N limit: +• With δ(L,hard) = o(N) an arbitrary parameter [115], [256], [134], +ρ(L) +� +κx +4N + δ(L,hard) ; β +� +; +(2.4.57) +113 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +• With δ(J,hard) = o(N) also an arbitrary parameter, +ρ(J) +� +x +N(4γ2N + δ(J,hard)); β +� +. +(2.4.58) +While a must be constant in N, applying this scaling to the differential equations of +Section 2.3 recovers Theorem 2.4 regardless of whether we take b to be proportional to +N or constant. +Optimal scaling at the soft and hard edges +It has previously been observed in [132] that the differential equation characterisation of the +soft edge scaled eigenvalue density can be extended to similarly characterise the optimal +leading order correction term. The latter is obtained by a tuning of the δ( · ) parameters in +equations (2.4.47)–(2.4.51) so as to obtain the fastest possible decay in N of the leading order +correction, and thus the fastest possible convergence to the limit. On this latter point, and +considering the Gaussian case for definiteness, we know from [133] that for β even (at least), +ˆρN,β(x) := +√κ +√ +2N7/6 ρ(G) +�√ +κ +�√ +2N + δ(G) + +x +√ +2N1/6 +� +; N, β +� +(2.4.59) += ρ(so f t)(x; β) + +�√ +2N1/6δ(G) − (1 − 1/κ)/(2N1/3) +� d +dxρ(so f t)(x; β) ++ O(N−2/3), +(2.4.60) +where the O(N−2/3) terms do not depend as simply on ρ(so f t)(x), d +dxρ(so f t)(x) as those made +explicit above. Thus, choosing δ(G) = (1 − 1/κ)/(2 +√ +2N) gives the fastest convergence to +the limit, +ˆρN,β(x) = ρ(so f t)(x; β) + +1 +N2/3 ζβ(x) + o(N−2/3) +for some ζβ(x) which, for the values of β permitting a differential equation characterisation +of ρ(G)(x) and ρ(so f t)(x), can itself be characterised as the solution of a differential equation. +The simplest case is β = 2, when +ζ′′′ +2 (x) − 4xζ′ +2(x) + 2ζ2(x) = x2 d +dxρ(so f t)(x; 2) − xρ(so f t)(x; 2), +(2.4.61) +which is an inhomogeneous generalisation of equation (2.4.46) for β = 2. For the particular +Laguerre soft edge scaling (2.4.48), again considering β = 2, the analogue of equation (2.4.61) +is given in [132, Eq. (4.19)], with a = ˆaκN and δ(L,so f t) ++ += 0. +114 + +2.4. Scalings of the Differential Equations +More generally, multiplying the functions (2.4.47)–(2.4.51) by appropriate scalars and +powers of N results in a function whose large N asymptotic expansion has leading order +term given by ρ(so f t)(x) and next to leading order correction some function generically +O(N−1/3) (when β = 2 and not all relevant parameters are constant, the correction term +is actually of order N−2/3). Likewise, multiplying the functions (2.4.57) and (2.4.58) by +the correct pre-factors produces a function whose large N limit is given by ρ(hard)(x) with +correction term of generic order 1/N. In a similar fashion to the computation (2.4.60) above, +proper tuning of the δ( · ) parameters seen in (2.4.47)–(2.4.51), (2.4.57), (2.4.58) ensures that +the correction terms are O(N−2/3) in the soft edge case, and O(1/N2) in the hard edge +case, which corresponds to the scaled eigenvalue densities experiencing the optimal rate of +convergence to the limiting forms ρ(so f t)(x) and ρ(hard)(x), respectively. +For β an even integer, [133] shows that the scaling (2.4.48) becomes optimal when we set +δ(L,so f t) ++ += +� +� +� +� +� +2a +κ , +a = O(1), +� +1 − 1 +κ +� γ1−1 +2√γ1 , +a = O(N). +(2.4.62) +Likewise, [134] shows for general real β > 0 and a ∈ N that the scaling (2.4.57) is optimal +upon setting δ(L,hard) = 2a/κ, which coincides with the value of δ(L,so f t) ++ +given above for the +a = O(1) setting. We are able to extend these results by utilising Theorems 2.3 and 2.4 +as follows: Letting ˆρN,β(x) denote the analogue of (2.4.59) constructed from either of the +functions (2.4.47)–(2.4.51), (2.4.57), or (2.4.58), we have that ˆρN,β(x) is, at leading order, +equal to either ρ(so f t)(x; β) or ρ(hard)(x; β). Then, ˆρN,β(x) satisfies an edge-scaled version of +one of the differential equations of Section 2.3, while ρ(so f t)(x; β) and ρ(hard)(x; β) satisfy +the differential equations of Theorems 2.3 and 2.4. Hence, the correction term ˆρN,β(x) − +ρ(so f t)(x; β), respectively ˆρN,β(x) − ρ(hard)(x; β), can be seen to satisfy a certain differential +equation. Analysis of this latter differential equation reveals the order in N of the correction +term and its dependence on the δ( · ) parameters. Thus, for β = 1, 2, and 4, we are able to +identify the values of δ( · ) which optimally tune the scalings (2.4.47)–(2.4.49), (2.4.51), (2.4.57), +and (2.4.58). We confirm that the values δ(G) = (1 − 1/κ)/(2 +√ +2N), δ(L,so f t) ++ +(2.4.62), and +δ(L,hard) = 2a/κ recounted above from the works [132], [133], [134] remain optimal when +β = 1 and/or a > −1 is allowed to be general real. Moreover, we find that the scalings +115 + +CHAPTER 2. Differential Equations for the Classical Matrix Ensembles +(2.4.49), (2.4.51) and (2.4.58) are optimal when taking +δ(L,so f t) +− += +� +1 − 1 +κ +� γ1 − 1 +2√γ1 +, +(2.4.63) +δ(Cy) = +� +1 − 1 +κ +� +γ3 +2λ(Cy) ++ +N +, +(2.4.64) +δ(J,hard) = +� +� +� +� +� +4(a+b) +κ +− 4 +� +1 − 1 +κ +� +, +b = O(1), +2a +κ (γ2 + 1) − 4 +� +1 − 1 +κ +� +, +b = O(N). +(2.4.65) +Note that the value of δ(L,so f t) +− +given above is equal to that of δ(L,so f t) ++ +given in equation (2.4.62) +in the a = O(N) setting. Moreover, the value of δ(J,hard) given above for the b = O(1) case +agrees with computations seen in the recent study [126, App. A]. Due to computational +limitations, we have not been able to identify the optimal value of δ(J,so f t) for β = 1, 4, but +note that its value in the β = 1 case can be surmised from [197] (which presumably extends +to the β = 4 case through the duality principles of Lemma 1.4), while for β = 2, it should +reduce to zero in the same fashion as δ(G), δ(L,so f t) +− +, δ(Cy). +116 + +Chapter 3 +Characterisations of the Moments and +Cumulants +It was mentioned in §1.1.1 that the eigenvalue densities ρ(λ) of the random matrix ensembles +studied in this thesis are fully characterised by their spectral moments (though, one must +keep in mind the caveat regarding the Cauchy ensembles) +mk = +� +R λkρ(λ) dλ = +� +N +∑ +i=1 +λk +i +� +, +k ∈ N. +(3.0.1) +In the above, the latter presentation of the spectral moments is to be read as an average with +respect to the eigenvalue j.p.d.f. of the matrix ensemble of interest. We begin this chapter +with an application of the differential equations given in Section 2.3 to the derivation of +linear recurrences characterising the spectral moments of the classical matrix ensembles (and +the Gaussian β ensembles with β = 2/3, 6). We recall from Definition 1.5 that the classical +matrix ensembles are exactly those whose eigenvalue j.p.d.f.s are of the form +p(w)(λ1, . . . , λN; β) = +1 +N (w) +N,β +N +∏ +i=1 +w(λi) |∆N(λ)|β, +(3.0.2) +where β = 1, 2, or 4, +∆N(λ) = +∏ +1⩽i0, +(3.0.4) +w(J)(λ) = λa(1 − λ)bχ0<λ<1, +(3.0.5) +w(sJ)(λ) +�� +a=b = (1 − λ2)aχ−1<λ<1, +(3.0.6) +w(Cy)(λ) +�� +α∈R = (1 + λ2)−κ(N−1)−1−α; +(3.0.7) +we recall that κ := β/2 and the indicator function χA is defined to equal one when A is true +and zero otherwise. The parameters a, b, α are taken to be real throughout this chapter and, +for convergence issues, are further constrained such that a, b > −1 and α > −1/2. +The aforementioned linear recurrences on the spectral moments are given in §3.1.1, where +we also show that these recurrence relations hold true for the negative-integer moments (m−k +with k ∈ N), which have gained interest primarily due to their connection to problems of +quantum transport [56], [240], [241], [71] and more recently due to duality principles relating +the negative-integer moments m−k to their counterparts mk−1 [72]; in fact, we show moreover +that our moment recurrences are satisfied by a further generalisation (see equation (3.1.1) +forthcoming) of the mk concerning complex values of k, which have recently been studied +in [72], [23]. Having linear recurrence relations on the spectral moments of the classical +matrix ensembles at hand means that we are able to derive so-called 1-point recursions [61] +characterising the moment expansion coefficients defined in Lemma 1.3. To be precise, we +proceed in §3.1.2 by substituting the expansions +m(G) +2k = +k +∑ +l=0 +M(G) +k,l N1+k−l, +(3.0.8) +m(L) +k += +k +∑ +l=0 +M(L) +k,l N1+k−l, +(3.0.9) +m(J) +k += +∞ +∑ +l=0 +M(J) +k,l N1−l +(3.0.10) +into the moment recurrences of §3.1.1 and equating terms of like order in N to obtain 1-point +recursions on the M(w) +k,l . Following [61, Defn. 1], this means that we find for each ensemble +some integers imax, jmax and a non-trivial set of polynomials {q(w) +ij (k)}0⩽i⩽imax,0⩽j⩽jmax such +118 + +that whenever the involved terms are well-defined, we have the equality +imax +∑ +i=0 +jmax +∑ +j=0 +q(w) +ij (k)M(w) +k−i,l−j = 0. +(3.0.11) +We present recursions of this type for the moment expansion coefficients of the Gaussian β +ensembles with β = 2/3, 6, the (classical) Laguerre ensembles, and the (symmetric shifted) +Jacobi unitary ensemble, choosing to forgo consideration of the remaining classical matrix +ensembles in the interest of brevity. However, we confirm here that applying our method to +the GUE recovers the celebrated Harer–Zagier recursion [174] +4(k + 1)M(GUE) +k,l += 4(2k − 1)M(GUE) +k−1,l + (k − 1)(2k − 1)(2k − 3)M(GUE) +k−2,l−2 +(3.0.12) +subject to the initial conditions M(GUE) +0,0 += 2M(GUE) +1,0 += 1, M(GUE) +1,1 += 0, and M(GUE) +k,l += 0 for all +k, l < 0, whereas treating instead the GOE and GSE recovers the corresponding recursions of +Ledoux given in [224]. +Evidently, the problem of computing the spectral moments of the classical matrix en- +sembles is not a new one. Aside from recurrences on the moments mk and their expansion +coefficients Mk,l known from the works [174], [169], [223], [224], [319], [71], [72], [23] (which +the computations of Section 3.1 recover and/or supplement), the literature contains a range +of different approaches for studying the spectral moments of interest. In Section 3.2, we +review some of these alternative characterisations of the spectral moments of the classical +matrix ensembles; in short, we survey parts of the literature that show how the study of +these spectral moments relates to skew-orthogonal polynomial theory, symmetric function +theory, and hypergeometric orthogonal polynomials from the Askey scheme. +After the review of Section 3.2, the remainder of this chapter focuses on elucidating how +the spectral moments mk, mixed moments (1.1.27) +mk1,...,kn = +� +n +∏ +i=1 +Tr Xki +� +P(X) +, +k1, . . . , kn ∈ N, +(3.0.13) +and mixed cumulants ck1,...,kn of particular random matrix ensembles relate to the enumeration +of ribbon graphs satisfying certain constraints. Here, the term ‘ribbon graph’ needs to be +formally introduced and it is helpful to reiterate the definition of ‘mixed cumulants’. The +mixed cumulants cκi are defined implicitly by the moment-cumulants relation (1.1.28) +mk1,...,kn = +∑ +K⊢{k1,...,kn} ∏ +κi∈K +cκi, +(3.0.14) +119 + +CHAPTER 3. Characterisations of the Moments and Cumulants +where we recall that K ⊢ {k1, . . . , kn} is read as “K is a partition of {k1, . . . , kn}”, which means +that K = {κi}m +i=1 for some 1 ⩽ m ⩽ n such that the disjoint union κ1 ⊔ · · · ⊔ κm is equal to +{k1, . . . , kn}. For example, K ⊢ {4, 6} if and only if K is equal to { {4, 6} } or { {4}, {6} }, so +m4,6 = c4,6 + c4 c6. +As for ribbon graphs, we give here a pictorial definition that is well suited to our purposes +(cf. [211], [250], [219] for alternative but near-equivalent definitions). +Definition 3.1. Letting k ∈ N, a k-ribbon graph is a collection of n polygons (0 < n ⩽ 2k) with +labelled vertices and k rectangles, referred to as ribbons, satisfying the following properties: +1. The number of edges of the polygons must sum to 2k. +2. For each ribbon, designate a pair of opposite edges to be referred to as ends of the +ribbon. Then, each of the 2k polygon edges must be identified, in a topological sense, +with exactly one ribbon-end, so that each ribbon connects two polygon edges together. +The Euler genus of an orientable (non-orientable) connected ribbon graph is the smallest +˜g ∈ N such that the ribbon graph can be drawn on a sphere with ˜g/2 handles ( ˜g cross-caps) +without intersecting with itself. In the orientable case, it is double the usual genus g. +Note 3.1. Since there are two ways to identify a pair of lines, depending on the relative +orientation of the lines, we allow our ribbons to have at most one M¨obius half-twist. We +refer to the class of ribbon graphs with untwisted ribbons as ‘(globally) orientable’, while +the term ‘locally orientable’ refers to the scenario where ribbons are allowed to be M¨obius +half-twisted. In the literature, ribbon graphs are usually restricted to be of the first type, +while the latter type are referred to as M¨obius graphs [251], [201], [57]. +Figure 3.1: A 3-ribbon graph constructed from a monogon, bigon, and triangle, with a grey +ribbon connecting the single edge of the monogon to the left edge of the bigon, a M¨obius +half-twisted pink ribbon connecting the right edge of the bigon to an edge of the triangle, +and a blue ribbon connecting the remaining two edges of the triangle together. +120 + +The connection between ribbon graphs and random matrix theory was first explored by +Br´ezin et al. in their 1978 work [54] (see also [36]), where they were strongly influenced by +’t Hooft’s 1974 work [179] on the large N limit of gauge theories. Closely related ideas were +seen independently in [326], [304], [174] over the next decade, with Harer and Zagier’s 1986 +work [174] being of particular interest to us. Technically speaking, Harer and Zagier did +not look directly at ribbon graphs, but rather showed and utilised the fact that 2kM(GUE) +k,l +equals the number of ways of identifying pairs of edges of a 2k-gon to form a compact +orientable surface of genus l/2 — of course, this is equal to the number of orientable genus +l/2 ribbon graphs that can be built from a single 2k-gon since the ribbons can be interpreted +as identifying the polygon edges at their ends. In 1997, Goulden and Jackson [165] (see also +[166], [251], [201]) gave the GOE analogue of the above statement, which is that 4kM(GOE) +k,l +equals the number of locally orientable ribbon graphs of Euler genus l that can be built from +a single 2k-gon. Other related works from this time period include [271], [211], [190], [287]. +In §3.3.1, we give a detailed review of how the spectral moments of the GUE and GOE +relate to the ribbon graphs described above. Then, in §3.3.2, we review the analogous +statements for the LUE [82] and LOE [57]; the upshot here is that M(LUE) +k,l +, M(LOE) +k,l +count +the same ribbon graphs as M(GUE) +k,l +, M(GOE) +k,l +, respectively, except that the vertices of the +underlying 2k-gons are alternately coloured black and red, and the edge identifications +induced by the ribbons respect this bicolouring. Our arguments, which flow from the Isserlis– +Wick theorem (see Theorem 3.1 of Section 3.3), have been shown to extend to the GSE and +LSE [251], [57], but we do not explore this fact in order to keep our discussion concise; +M(GSE) +k,l +, M(LSE) +k,l +can roughly be ascribed the same combinatorial meaning as M(GOE) +k,l +, M(LOE) +k,l +due to the β ↔ 4/β duality of Lemma 1.4. Finally, in §3.3.3, we extend the arguments of +§3.3.1 and §3.3.2 to produce ribbon graph interpretations for the spectral moments of the +Hermitised and antisymmetrised matrix product ensembles defined in Section 1.3. +One of our motivations for supplying the review contained in §3.3.2 is that it allows us to +interpret the 1-point recursions of §3.1.2 pertaining to the Laguerre ensembles as solving +combinatorial problems similar to that considered by Harer and Zagier — unfortunately, we +are currently unable to find analogous applications for the other recursions given in §3.1.2 +due to complications that we outline in Section 3.3. As mentioned earlier, the discussion of +Section 3.3 does not focus solely on the spectral moments of the random matrix ensembles +121 + +CHAPTER 3. Characterisations of the Moments and Cumulants +at hand, but also their mixed moments and cumulants. Indeed, we show therein how +straightforward generalisations of the ribbon graphs described above, now constructed +from multiple polygons, relate to the mixed moments and cumulants of interest. +We +show moreover that the relevant mixed moments and cumulants are polynomial in N. +A consequence of this is that, upon appropriate scaling, we are able to assume that the +connected n-point correlators Wn (1.1.29) of the Hermitised and antisymmetrised matrix +product ensembles have large N (actually N0) expansions of the form (1.1.21) without +needing to check the hypotheses of Theorem 1.2. The existence of these large N expansions +is significant because they are crucial to the loop equation analysis presented in Chapter 4. +3.1 +Recurrence Relations for the Moments of the Classical Matrix +Ensembles +We now derive the aforementioned linear recurrence relations on the spectral moments mk +(3.0.1), and explain why they hold true when the mk are instead taken to be the complex-k +generalised moments (following Cunden et al. [72]) +mk := +� +R |λ|kρ(λ) dλ, +k ∈ {k′ ∈ C | mk′ < ∞}. +(3.1.1) +Note that these generalised moments agree with the spectral moments defined by equation +(3.0.1) for k ∈ N in the Jacobi and Laguerre cases, and for k ∈ 2N in the Gaussian, symmetric +shifted Jacobi, and symmetric Cauchy cases — in the latter set of cases, mk = 0 for k odd +according to equation (3.0.1), but not equation (3.1.1). Since the eigenvalue densities studied +in this section are either symmetric about the origin or have support contained within +the positive real line, the generalised moments (3.1.1) can be interpreted as the spectral +moments (we reserve the adjective ‘spectral’ for the moments defined by equation (3.0.1)) of +the constrained density ρ(λ)χλ>0, with a possible factor of two. +The study of the former type of spectral moments is motivated by the fact that they +fully characterise the eigenvalue densities of the classical matrix ensembles, as discussed in +§1.1.1. Generalisation to negative integers k was first considered by Brouwer et al. [56] in the +Laguerre case, where they showed that the reciprocated eigenvalues of the Wigner–Smith +time delay matrix Q are distributed according to the Laguerre ensemble with a = κN so that +122 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +� +Tr Qk� = m(L) +−k |a=κN [71]. The significance of the Wigner–Smith matrix is that it relates to +the study of ballistic chaotic scattering, whereby its eigenvalues, referred to as proper delay +times, describe the amount of time incident particles spend in the ballistic cavity during a +scattering event, among other observables [33], [56], [70], [71], [229], [240], [241]. The more +recent extension to the complex-k moments (3.1.1) considered by Cunden et al. [72] relates +to the study of so-called spectral zeta functions of random matrices X, ζX(k) := Tr (X†X)−k +(note that if we take X to be the infinite-dimensional deterministic matrix diag(1, +√ +2, +√ +3, . . .), +ζX(k) is then the Riemann zeta function). It is shown in [72] that spectral zeta functions +satisfy a reflection symmetry across the line Re(k) = 1/2 (much like the Riemann zeta +function), which implies reciprocity laws between the moments m−k and mk−1 of the LUE +and JUE, for k ∈ N; see Remark 3.1 within §3.2.3. +In §3.1.2, we return to the study of spectral moments (3.0.1) with k ∈ N since then, it is +known (recall Lemma 1.3) that m(G) +k +, m(L) +k +are polynomial in N and m(J) +k +is a rational function +in N. Thus, the expansions (3.0.8)–(3.0.10) are valid and we are able to derive 1-point recur- +sions (3.0.11) characterising the moment expansion coefficients Mk,l. As mentioned earlier, +1-point recursions satisfied by these moment expansion coefficients have been studied in spe- +cial cases, with particular attention paid to their topological or combinatorial interpretations +[174], [224], [86]. These aspects are briefly discussed in §3.1.2, with the highlighted point +being that even though the moment expansion coefficients Mk,l may have combinatorial +interpretations in certain cases (fully detailed in Section 3.3), there are presently no such +interpretations for the 1-point recursions satisfied by them. +3.1.1 +Recurrences for the spectral moments +One may obtain recurrences for the spectral moments mk of the classical matrix ensembles by +recalling from §1.1.1 that the resolvent W1(x) acts as a generating function for these moments, +presuming that the moments are well-defined for all k ∈ N or otherwise interpreting the +large x expansion of W1(x) in a formal sense: +W1(x) = +∞ +∑ +k=0 +mk +xk+1 . +(3.1.2) +Substituting this series into the differential equations for W1(x) given in Section 2.3 and +then equating terms of equal order in x gives relations between the spectral moments. The +123 + +CHAPTER 3. Characterisations of the Moments and Cumulants +equations obtained from terms of negative order in x give the upcoming recurrences on the +spectral moments, while the terms of order one and positive order in x give the first few +moments required to run the recursions (the latter are also available in earlier literature; see, +e.g., [142] and references therein). +The recurrence relations derived from the above procedure enable the iterative com- +putation of the spectral moment mk for any positive integer k starting with knowledge of +m0, . . . , mp for some small integer p. However, these recurrences are actually valid for the +complex-k generalisations defined by equation (3.1.1), so long as all involved moments mk are +such that in the Gaussian and symmetric shifted Jacobi cases, Re(k) > −1; in the Jacobi and +Laguerre cases, Re(k) > −a − 1; and in the symmetric Cauchy case, −1 < Re(k) < 2α + 1 +(this is a refinement of the constraint −1 < k < 2α1 + 1 discussed in §1.2.1). To ascertain +the validity of our moment recurrences in these more general settings, we must provide an +alternative proof to that outlined in the previous paragraph. Thus, rather than the differential +equations for W1(x), we begin at the differential equations for the eigenvalue density ρ(x) +given in Section 2.3. Multiplying both sides of these differential equations by |x|k and then +integrating over the support of the eigenvalue density produces a relation between terms of +the form � +supp ρ xp|x|k dn +dxn ρ(x) dx for positive integers p, n. The goal then is to use integration +by parts to reduce these terms to the form (3.1.1). This is done in a similar fashion to what +is shown in Appendix B, with the notable exception being that the domain of integration +must be split at the origin due to the absolute value sign in the factor |x|k — in any case, all +boundary terms arising from integration by parts vanish, owing to our restrictions on k. +Proposition 3.1. Recall the definitions of ˜a, ˜b, and ˜c given in Theorem 2.2. For β = 2 and k ∈ C +such that Re(k) > 2 − a, the moments (3.1.1) of the eigenvalue density of the Jacobi ensemble satisfy +the third order linear recurrence +3 +∑ +l=0 +d(J) +2,l m(J) +k−l = 0, +(3.1.3) +where +d(J) +2,0 = k +�(a + b + 2N)2 − (k − 1)2� +, +d(J) +2,1 = 3k3 − 11k2 − k +� +2(a + b + 2N)2 + a2 − b2 − 14 +� ++ 3(a + b)(a + 2N) + 6(N2 − 1), +124 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +d(J) +2,2 = (2k − 3) [2N(a + b + N) + ab] − (k − 2) +� +3k2 − 10k − 3a2 + 9 +� +, +d(J) +2,3 = (k − 3) +�(k − 2)2 − a2� +. +For β = 1, 4 and k ∈ C such that Re(k) > 4 − a, the moments of the Jacobi ensemble’s eigenvalue +density satisfy the fifth order linear recurrence +5 +∑ +l=0 +d(J) +4,l m(J) +k−l = 0, +(3.1.4) +where +d(J) +4,0 = k(˜c2 − (k − 2)2)(˜c2 − (2k − 1)2), +d(J) +4,1 = 1 +2(˜c2 − 9)2(5 − 6k) + 1 +2(˜a − ˜b) +�(˜c2 − 9)(5 − 4k) + 2k(5(k − 1)(k − 5) + 4k) +� ++ (˜c2 − 9)k [5(4k − 3)(k − 3) + 2k] − 4k2(k − 5) [5(k − 2)(k − 1) − 2] , +d(J) +4,2 = ˜c4(3k − 5) + ˜c2 � 1 +2(˜a + ˜b)(2k − 5) + 5(˜a − ˜b)(k − 2) +� +− ˜c2 � +30k3 − 171k2 + 339k − 230 +� − 1 +2(˜a + ˜b) +� +5k3 − 44k2 + 129k − 125 +� +− 1 +2(˜a − ˜b) +� +35k3 − 246k2 + 581k − 460 +� + 1 +2(2k − 5)(˜a − ˜b)2 ++ 40k4(k − 11) + 1966k3 − 4443k2 + 5056k − 2305, +d(J) +4,3 = 1 +2 ˜c4(5 − 2k) + ˜c2 � 1 +4(˜a + ˜b)(25 − 8k) + 1 +4(˜a − ˜b)(45 − 16k) +� ++ ˜c2 � +20k3 − 155k2 + 401k − 345 +� + 5 +4(˜a + ˜b) +� +6k3 − 62k2 + 216k − 253 +� ++ 1 +4(˜a − ˜b) +� +90k3 − 806k2 + 2436k − 2485 +� + 1 +4(˜a2 − ˜b2)(15 − 4k) ++ 1 +4(˜a − ˜b)2(25 − 8k) − 4k3(10k2 − 140k + 789) + 8923k2 − 12600k + 14125 +2 +, +d(J) +4,4 = (k − 4) +� +k3 + k2 − 18k − +�˜c2 − ˜b − 4k2 + 29k − 51 +� � +5k2 − 29k + 40 +�� ++ 1 +2 ˜a2(6k − 25) + 1 +2 ˜a +�(4k − 15) +�˜c2 − ˜b − 10k2 + 76k − 147 +� − 2(k − 5) +� +, +d(J) +4,5 = (k − 5) [4(k − 5)(k − 4) − ˜a] [˜a − (k − 4)(k − 2)] . +The sequences of spectral moments {m(J) +k }k∈Z, k>−a−1 are fully determined for β = 1, 2, 4 by the above +recurrence relations and the initial terms m(J) +0 , m(J) +1 , . . . , m(J) +4 , which can be computed through MOPS +[96] or via the methods presented in [242], [142] — according to the discussion at the beginning of +this subsection, the required initial terms are also encoded within the right-hand sides of differential +equations (2.3.8) and (2.3.14). +125 + +CHAPTER 3. Characterisations of the Moments and Cumulants +The recurrence (3.1.3) for the moments m(J) +k +of the JUE eigenvalue density was recently +given in [72, Prop. 4.8], wherein it is formulated as a recurrence on the differences of moments +∆m(J) +k +:= m(J) +k +− m(J) +k+1. The fact that this recurrence relation can be naturally formulated +in terms of the differences ∆m(J) +k +is equivalent to the observation that ∑3 +l=0 d(J) +2,l = 0. Since +∑5 +l=0 d(J) +4,l = 0, we see that it is also reasonable to rewrite the recurrence relation (3.1.4) in +terms of the ∆m(J) +k . It turns out that the analogous recurrences for the symmetric shifted +Jacobi and symmetric Cauchy ensembles are most compactly presented when written in +terms of the differences and sums, respectively, of even moments +µ(sJ) +k +:= m(sJ) +2k+2 − m(sJ) +2k , +(3.1.5) +µ(Cy) +k +:= m(Cy) +2k+2 + m(Cy) +2k +. +(3.1.6) +Here, the moments mk are again defined by equation (3.1.1) with ρ(λ) being the eigenvalue +density of the classical matrix ensemble specified by either the constrained weight (3.0.6) or +(3.0.7), respectively. The fact that the µ(sJ) +k +, µ(Cy) +k +recurrences are simpler than the correspond- +ing recurrences on the m(sJ) +2k , m(Cy) +2k +is in keeping with the fact that the former are, respectively, +the moments of r(sJ)(x) := (1 − x2)ρ(sJ)(x)|a=b and r(Cy)(x) := (1 + x2)ρ(Cy)(x)|α∈R, which +are characterised by differential equations whose coefficients are of degree two less than +the corresponding coefficients of the differential equations (2.3.17), (2.3.21) for ρ(sJ)(x) and +ρ(Cy)(x); the differential equations satisfied by r(sJ)(x) can be surmised from those satisfied +by r(Cy)(x), which are themselves given in §2.3.2 as equations (2.3.27), (2.3.28). +Proposition 3.2. Define µ(sJ) +k +by equation (3.1.5) and retain the definitions of aβ, ˜a, and ˜c given +in Theorem 2.2. For β = 2 and k ∈ C such that Re(k) > 1/2, we have the second order linear +recurrence +(2k + 4) +�(2k + 3)2 − 4(a + N)2� +µ(sJ) +k+1 − 2(2k + 1) +�(2k + 2)2 − 2N(N + 2a) +� +µ(sJ) +k ++ (2k + 1)(2k)(2k − 1)µ(sJ) +k−1 = 0. +(3.1.7) +The initial condition determining the sequence {µ(sJ) +k +|β=2}k∈N is +µ(sJ) +0 += 2N(a + N)(2a + N) +1 − 4(a + N)2 +. +(3.1.8) +For β = 1, 4 and k ∈ C such that Re(k) > 3/2, we have the fourth order linear recurrence +2 +∑ +l=−2 +d(sJ) +4,l µ(sJ) +k−l = 0, +(3.1.9) +126 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +where +d(sJ) +4,−2 := −4(2k + 1)(2k)(2k − 1)(2k − 2)(2k − 3), +d(sJ) +4,−1 := (2k + 1)(2k)(2k − 1) +� +20˜a − 5˜c2 + 16k(4k + 5) + 77 +� +, +d(sJ) +4,0 := (2k + 1) +� +8˜a(5k + 8)(2k + 3) − +�˜c2 − 4˜a − 30k2 − 67k − 42 +�2 ++ 516k4 + 2292k3 + 3653k2 + 2534k + 673 +� +, +d(sJ) +4,1 := ˜c2 �(˜c2 − 4˜a)(4k + 7) − 120k3 − 656k2 − 1210k − 746 +� ++ ˜a +� +160k3 + 736k2 + 1128k + 580 +� ++ 512k5 + 4480k4 + 16056k3 + 29360k2 + 27270k + 10243, +d(sJ) +4,2 := −2(k + 3)(˜c + 4k + 11)(˜c + 2k + 4)(˜c − 2k − 4)(˜c − 4k − 11). +The initial conditions determining the sequences {µ(sJ) +k +|β=1,4}k∈N are +µ(sJ) +0 += (˜c − 1) +�˜c + 2aβ − 3 +� �˜c − 2aβ + 1 +� +8˜c(˜c − 3)(1 − κ) +, +(3.1.10) +µ(sJ) +1 += (˜c2 − 5)(˜c − 7) − 4˜a(˜c − 1) +4(˜c2 − 4)(˜c − 7) +µ(sJ) +0 +. +(3.1.11) +We note that the generalisation of recurrence (3.1.7) concerning the non-symmetric shifted +JUE corresponding to the weight w(sJ)(λ) = (1 − λ)a(1 + λ)bχ−1<λ<1 and β = 2 was first +given by Ledoux in [223], albeit consideration was given only to the µ(sJ) +k +with k ∈ N. Moving +on, let us observe that the contents of Section 2.2, particularly Proposition 2.2, imply the +relation +µ(sJ) +k +��� +a�→−κ(N−1)−1−α = (−1)k−1µ(Cy) +k +, +(3.1.12) +where we recall that κ = β/2. A simple consequence of this relation is that Proposition 3.2 +leads to the following recurrence relations for the sums of moments µ(Cy) +k +. +Corollary 3.1. Define µ(Cy) +k +by equation (3.1.6) and retain the definitions of d(sJ) +4,−2, d(sJ) +4,−1, . . . , d(sJ) +4,2 +listed in Proposition 3.2. For β = 2 and k ∈ C such that 1/2 < Re(k) < α − 3/2, we have the +second order linear recurrence (recently given in [23]) +(2k + 4) +�(2k + 3)2 − 4α2� +µ(Cy) +k+1 + 2(2k + 1) +�(2k + 2)2 + 2N(N + 2α) +� +µ(Cy) +k ++ (2k + 1)(2k)(2k − 1)µ(Cy) +k−1 = 0. +(3.1.13) +127 + +CHAPTER 3. Characterisations of the Moments and Cumulants +The initial condition determining the sequence {µ(Cy) +k +|β=2}k∈N is +µ(Cy) +0 += +2Nα(N + 2α) +(2α − 1)(2α + 1), +α > 1/2. +(3.1.14) +For β = 1, 4 and k ∈ C such that 3/2 < Re(k) < α − 5/2, we have the fourth order linear recurrence +2 +∑ +l=−2 +d(Cy) +4,l +µ(Cy) +k+l = 0, +d(Cy) +4,l +:= (−1)l−1d(sJ) +4,l +��� +a�→−κ(N−1)−1−α. +(3.1.15) +The initial conditions required to fully characterise the sequences {µ(Cy) +k +|β=1,4}k∈N are obtained by +parsing equations (3.1.10) and (3.1.11) through the relation (3.1.12). +In a similar vein to how the arguments of Section 2.2 enable us to simply translate the +recurrences of Proposition 3.2 for the µ(sJ) +k +into the analogous recurrences on the µ(Cy) +k +given +in Corollary 3.1, the limiting procedure (1.2.23) described in Lemma 1.1 can be used to obtain +recurrences on the (generalised) spectral moments m(L) +k +of the Laguerre ensembles from the +recurrences presented in Proposition 3.1. Indeed, to derive the following recurrences on the +m(L) +k , one may circumvent the strategy outlined at the beginning of this subsection by setting +m(J) +k += b−km(L) +k +in Proposition 3.1 and then taking the limit b → ∞ (cf. equation (2.3.40), in +addition to Lemma 1.1). +Corollary 3.2. Recall the definitions of aβ, Nβ, and ˜a given in Theorem 2.2. For β = 2 and k ∈ C +such that Re(k) > 1 − a, the moments of the LUE eigenvalue density satisfy the second order linear +recurrence +(k + 1)m(L) +k += (2k − 1)(a + 2N)m(L) +k−1 + (k − 2) +�(k − 1)2 − a2� +m(L) +k−2. +(3.1.16) +For β = 1, 4 and k ∈ C such that Re(k) > 3 − a, we have the fourth order linear recurrence +4 +∑ +l=0 +d(L) +4,l (κ − 1)lm(L) +k−l = 0, +(3.1.17) +where +d(L) +4,0 = k + 1, +d(L) +4,1 = (1 − 4k)(aβ + 4Nβ), +d(L) +4,2 = (1 − k)(5k2 − 11k + 4) + (2k − 3) +�˜a + 2(aβ + 4Nβ)2� +, +d(L) +4,3 = (aβ + 4Nβ) +�(11 − 4k)˜a + 10k3 − 68k2 + 146k − 96 +� +, +d(L) +4,4 = (k − 4) [˜a − 4(k − 4)(k − 3)] [˜a − (k − 3)(k − 1)] . +128 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +The initial terms m(L) +0 , m(L) +1 , . . . , m(L) +3 +required to fully determine the sequences {m(L) +k }k∈Z, k>−a−1 +in the orthogonal, unitary, and symplectic cases can be found in [242], [142]. +It should come as no surprise that recurrence (3.1.16) on the moments of the LUE +eigenvalue density has already been given in the literature, seeing as this is true of its JUE +analogue (3.1.3), which is more involved. Indeed, it was first obtained by Haagerup and +Thorbjørnsen [169], where they considered only the case a, k ∈ N. A bit over a decade later, +Cunden et al. [71] showed that the LUE moment recurrence (3.1.16) holds for all integers +k > −a − 1 with a > −1 a continuous real parameter. On the other hand, recurrence +(3.1.17) pertaining to the moments of the LOE and LSE can be seen to be a homogeneous +simplification of [71, Eq. (43)], wherein the inhomogeneous terms depend on the moments +of the LUE. +Another simple observation is that for each of β = 1, 2, and 4, the m(L) +k +recurrences +displayed in Corollary 3.2 involve one less term than the corresponding m(J) +k +recurrences +given in Proposition 3.1 — the terms in the recurrences are themselves drastically simpler, as +well. This is in keeping with the loss of parameter b. A similar simplification of terms can +be observed in the moment recurrences in the Gaussian case, again to be understood as a +consequence of the hierarchy implied by Lemma 1.1. However, we do not see a reduction in +the order of the moment recurrences when moving from the Laguerre to Gaussian ensembles. +Thus, the spectral (and generalised) moments of the GUE satisfy a second order linear +recurrence first derived by Harer and Zagier in [174], while the spectral moments of the GOE +and GSE are characterised by fourth order linear recurrences given by Ledoux in [224]. We +do not present them here, but note (as was mentioned earlier) that the strategies discussed +in this section recover the recurrences of [174], [224], with the added insight that they hold +for complex k. Furthermore, our methods allow us to derive linear recurrence relations for +the β = 2/3 and β = 6 Gaussian ensembles’ spectral moments, which we now give. +Proposition 3.3. For β = 2/3, 6 and k ∈ C such that Re(k) > 11, the moments of the Gaussian β +ensemble’s eigenvalue density satisfy the sixth order linear recurrence +6 +∑ +l=0 +d(G) +6,l +�κ − 1 +4 +�l +m(G) +k−2l = 0, +(3.1.18) +129 + +CHAPTER 3. Characterisations of the Moments and Cumulants +where +d(G) +6,0 = −4(k + 2), +d(G) +6,1 = 8(3k − 1)(3Nβ + 2), +d(G) +6,2 = 48(8 − 3k)Nβ(3Nβ + 4) + 49k3 − 216k2 + 92k + 320, +d(G) +6,3 = 4(k − 5)(3Nβ + 2) +� +24Nβ(3Nβ + 4) − 49k2 + 304k − 442 +� +, +d(G) +6,4 = 2(k − 5)3 +� +294Nβ(3Nβ + 4) − 63k(k − 6) − 274 +� +, +d(G) +6,5 = 252(k − 5)5(3Nβ + 2), +d(G) +6,6 = 81(k − 5)7, +and we retain the definition Nβ = (κ − 1)N. Here, (x)n = x(x − 1) · · · (x − n + 1) is the falling +Pochhammer symbol. The initial terms m(G) +0 +, m(G) +2 +, . . . , m(G) +10 +needed to determine the sequences +{m(G) +k +|β=2/3,6}k∈N are listed in [319]. +As we proceed to our discussion on 1-point recursions, let us conclude this subsection +with a couple of general remarks. The recurrences given above describe relations between +the complex-k generalised moments mk defined by equation (3.1.1), so long as all moments +involved are well-defined. However, the recurrences hold true when the mk are taken to +be (integer-k) spectral moments as defined by equation (3.0.1), even though these spectral +moments fail to coincide with the generalised moments when considering the Gaussian, +symmetric Jacobi, or symmetric Cauchy ensembles, with k an odd integer. In these cases, the +definitions (3.0.1) and (3.1.1) disagree only in that, according to the first definition, mk = 0 +when k is odd — the relevant recurrences are then seen to hold trivially for odd integers k +due to their homogeneous structure and the fact that they run over every second integer-k +moment (e.g., recurrence (3.1.15) involves m(Cy) +k+6 , m(Cy) +k+4 , . . . , m(Cy) +k−4 ). +The methods employed in [224] manifest a coupling between the GOE and GUE moments, +which does not arise naturally from our viewpoint. Likewise for the coupling between the +LOE and LUE moments implied by the recurrence [71, Eq. (43)]. As made clear in [71], +both couplings can be traced back to a structural formula expressing the β = 1 eigenvalue +densities in terms of their β = 2 counterparts plus what can be regarded as rank one +corrections. This inter-relation was discussed briefly in §1.2.3, where we saw that the context +underlying it is that the skew-orthogonal polynomials pertaining to the GOE and LOE can +130 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +be constructed from the Hermite and Laguerre orthogonal polynomials used to study the +GUE and LUE, respectively. At present, there is no evidence implying a relation connecting +the moments of the β = 6 Gaussian ensemble to the moments of the GOE and/or GUE. +3.1.2 +1-point recursions for the moment expansion coefficients +Following on from the preceding subsection, it is a simple matter to obtain 1-point recursions +of the form (3.0.11) for the coefficients of the moment expansions (3.0.8)–(3.0.10). In fact, +all one needs to do is substitute the moment expansions (3.0.8)–(3.0.10) into the recurrence +relations of §3.1.1, now considering k large enough integers such that all involved moments +are well-defined according to definition (3.0.1), and then equate terms of equal order in +N. There is however another (instructive) avenue towards the sought 1-point recursions +beginning at the differential-difference equations of §2.4.1: In the large N limit, the spectral +moments m(G) +2k , m(L) +k , and m(J) +k +do not converge, so the moment expansions (3.0.8)–(3.0.10) +may only be understood in a formal sense. Rectifying this issue by introducing the scalings +˜m(G) +2k +:= κ−kN−k−1 m(G) +2k , +(3.1.19) +˜m(L) +k +:= κ−kN−k−1 m(L) +k , +(3.1.20) +˜m(J) +k +:= m(J) +k /N, +(3.1.21) +as is compliant with Definition 1.6 (that is, ˜mk = � +R λk ˜ρ(λ) dλ), we observe that the (scaled) +resolvents +˜W(G) +1 +(x) (2.4.5), +˜W(L) +1 +(x) (2.4.6), and W(J) +1 (x) act as generating functions for +{ ˜m(G) +2k }k∈N, { ˜m(L) +k }k∈N, and { ˜m(J) +k }k∈N, respectively. Moreover, the scaled moments have +convergent large N expansions (finite sums in the Gaussian and Laguerre cases) of the +form ˜mk = ∑∞ +l=0 ˜Mk,l N−l (1.1.32), with the expansion coefficients ˜Mk,l of the scaled moments +themselves generated by the resolvent expansion coefficients Wl +1(x) defined implicitly in +equations (2.4.8)–(2.4.10). Thus, substituting the large x expansions +W(G),l +1 +(x) = 2l+1κl/2 +∞ +∑ +k=0 +˜M(G) +k,l x−k−1, +W(L),l +1 +(x) = κl/2 +∞ +∑ +k=0 +˜M(L) +k,l x−k−1, +W(J),l +1 +(x) = +∞ +∑ +k=0 +˜M(J) +k,l x−k−1 +131 + +CHAPTER 3. Characterisations of the Moments and Cumulants +into the differential-difference equations of §2.4.1 and then comparing terms of like order in +x yields 1-point recursions on the ˜Mk,l (recall the diagram displayed at the end of §1.1.1). +To obtain equivalent recursions for the unscaled moment expansion coefficients Mk,l, it is +enough to note from equations (3.1.19)–(3.1.21) that +M(G) +k,l = κk ˜M(G) +2k,l, +M(L) +k,l = κk ˜M(L) +k,l , +M(J) +k,l = ˜M(J) +k,l . +Confirming agreement between the two derivations outlined above, flowing separately +from the results of §3.1.1 and §2.4.1, respectively, serves as a consistency check. That aside, +the upcoming 1-point recursions are themselves motivated by the fact that they enable +efficient computation of the expansion coefficients Mk,l, assuming knowledge of sufficiently +many of them for small k, l ∈ N — we do not provide the necessary initial conditions here, +but note that they can be computed through the strategy discussed at the beginning of +§3.1.1. Compared to the recurrences of §3.1.1, which run over the spectral moments mk, the +1-point recursions are faster at isolating the leading order behaviour of the mk, in the large +N limit. On the other hand, changing viewpoint to that of Section 3.3, the 1-point recursions +are interesting because they relate enumerations of ribbon graphs in a manner that has no +obvious combinatorial interpretation; we make only brief comments on the relevant ribbon +graphs here, deferring full discussion to Section 3.3. +As mentioned earlier, our 1-point recursions agree with that of Harer and Zagier (3.0.12) +in the GUE case and of Ledoux [224] in the GOE and GSE cases. Let us recall that the +motivation for studying the former came from the interpretation of 2kM(GUE) +k,l +as the number +of ways of gluing the edges of a 2k-gon to form a compact orientable surface of genus l/2 +(cf. Figure 3.5 of §3.3.1), while 4kM(GOE) +k,l +is known [165], [166], [251], [201] to be equal to the +number of such gluings that form a compact locally orientable surface of Euler characteristic +2 − l (see Figure 3.9); M(GSE) +k,l +has been given equivalent interpretations in [251], [57], which +can be understood through the β ↔ 4/β duality of Lemma 1.4. We do not recount the +GOE, GUE, and GSE 1-point recursions here, for brevity. Likewise, we do not report on +the symmetric Cauchy ensembles, nor the (symmetric shifted) Jacobi ensemble when β ̸= 2; +though these have not been given in the literature before, they are relatively cumbersome to +display while being as equally easy to derive from the recurrences of §3.1.1 as the recursions +given below. Our first result is hence on the Gaussian β ensembles with β = 2/3 and 6. +132 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +Proposition 3.4. Expand the spectral moments of the Gaussian ensemble according to (3.0.8). Then, +for β = 2/3, 6 and k ⩾ 6, +(k + 1)M(G) +k,l = +6 +∑ +i=1 +i +∑ +j=0 +(κ − 1)2i−j +2i +fi,jM(G) +k−i,l−j, +(3.1.22) +where +f1,0 = 3(6k − 1), +f1,1 = 2(6k − 1), +f2,0 = 36(4 − 3k), +f2,1 = 48(4 − 3k), +f2,2 = 49k3 − 108k2 + 23k + 40, +f3,0 = 108(2k − 5), +f3,1 = 216(2k − 5), +f3,2 = 3(5 − 2k)(98k2 − 304k + 189), +f3,3 = 2(5 − 2k)(98k2 − 304k + 221), +f4,2 = 441 +2 (2k − 5)3, +f4,3 = 294(2k − 5)3, +f4,4 = 1 +2(2k − 5)3 (126k(3 − k) − 137) , +f5,4 = 189 +2 (2k − 5)5, +f5,5 = 63(2k − 5)5, +f6,6 = 81 +8 (2k − 5)7, +and all other fi,j are zero. We also set M(G) +k,l = 0 if l < 0 or l > k. +When equation (3.1.22) is used to compute M(G) +k,0 , it reduces to +16(k + 1)M(G) +k,0 = 12(6k − 1)(κ − 1)2M(G) +k−1,0 − 36(3k − 4)(κ − 1)4M(G) +k−2,0 ++ 27(2k − 5)(κ − 1)6M(G) +k−3,0, +κ = 1/3 or 3. +(3.1.23) +This is in keeping with the limiting scaled eigenvalue density ρ(G),0(λ) of the Gaussian +ensembles equalling the Wigner semi-circle law specified by the density +√ +2 − λ2/π (1.2.14) +supported on |λ| < +√ +2: up to a scale factor the Catalan numbers are the even moments. +Thus, equation (3.1.23) can essentially be seen to be a four term linear recurrence for the +Catalan numbers. To the best of the author’s knowledge, it does not have a combinatorial +interpretation comparable to what is known for the traditional two term recurrence given in +equation (3.1.25) below. However, we note that equation (3.1.23) and its three term analogue +(3.1.27) are implied by repeated applications of equation (3.1.25) with appropriate scaling. +To expand the Laguerre and Jacobi ensembles’ spectral moments in N, we need to decide +how the parameters a and b scale with N. We were faced with this same decision in §2.4.1, +where we opted to set a = ˆaκN and b = ˆbκN with ˆa, ˆb constant in N. We continue with +this choice for consistency, with one caveat: Since the 1-point recursion in the LUE case +is relatively simple, we consider the slightly more general parametrisation a = ˆaκN + δa, +133 + +CHAPTER 3. Characterisations of the Moments and Cumulants +where ˆa, δa = O(1) — this gives us an opportunity to demonstrate the insight gained, at +the cost of complexity, from implementing this generalisation. Substituting this generalised +parametrisation together with the expansion (3.0.9) into recurrence (3.1.16) yields the LUE +1-point recursion. +Proposition 3.5. For β = 2, a = ˆaN + δa with ˆa, δa constant in N, and k ⩾ 2, +(k + 1)M(LUE) +k,l += (2k − 1) +� +(2 + ˆa)M(LUE) +k−1,l + δaM(LUE) +k−1,l−1 +� +− ˆa2(k − 2)M(LUE) +k−2,l + (k − 2) +�(k − 1)2 − δ2 +a +� +M(LUE) +k−2,l−2, +(3.1.24) +where we set M(LUE) +k,l += 0 if l < 0 or l > k. +When this is used to compute M(LUE) +k,0 +in the case a = δa = O(1) and thus ˆa = 0, one +recovers the familiar Catalan recursion +(k + 1)M(LUE) +k,0 += 2(2k − 1)M(LUE) +k−1,0 . +(3.1.25) +This recurrence is well known (see, e.g., [290]) and can be seen to be consistent with +Proposition 1.2: When a = O(1), we have from equation (1.2.88) that γ1 = 1 so that the +Marˇcenko–Pastur density ρ(L),0(λ) (1.2.15) simplifies to √ +4/λ − 1/(2π)χ0<λ<4, whose kth +spectral moment is exactly the kth Catalan number. Observe also that the Harer–Zagier +recursion (3.0.12) reduces to (3.1.25) upon setting l = 0 and noting that M(LUE) +k,0 += 2kM(GUE) +k,0 +, +this equality being implied by the fact that ρ(G),0(λ) = 2|λ| ρ(L),0(2λ2). +In contrast to the four term recurrence (3.1.23), equation (3.1.25) has a famous combina- +torial interpretation, which we now present in the language of Section 3.3. For k ∈ N and +β = 2, the expansion coefficients M(LUE) +k,0 +count the number of unique planar ribbon graphs +that can be constructed from a 2k-gon1 [82]. To see that they are equal to the kth Catalan +number, observe that the set of planar ribbon graphs that can be constructed from a 2k-gon +are in bijection with the set of balanced parenthesisations involving k pairs of parentheses, +which are well known to be enumerated by the Catalan numbers [290]; we make the bijection +explicit by moving clockwise around the 2k-gon, starting at the top edge, assigning the label +1In §3.3.2, we will see that the ribbon graphs pertaining to the Laguerre ensembles must obey a certain +bicolouring constraint. This constraint is automatically satisfied when the ribbon graphs are planar, so we ignore +it for now. +134 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +‘1’ (open parenthesis) to the first free edge and ‘−1’ (corresponding closed parenthesis) to the +edge connected to it by a ribbon, ‘2’ and ‘−2’ to the next free edge and its ribbon-connected +counterpart, and so on. As exemplified in Figure 3.2 below, the edges are then labelled — in +clockwise order starting at the top edge — by the sequence (i1, i2, . . . , i2k) with i1 = 1 and +j < l whenever 0 < ij < il or 0 < ij = −il. +Figure 3.2: The first image relates a planar ribbon graph to the balanced parenthesisation +()(()) by labelling the top edge ‘1’ and then, moving clockwise, labelling the tail of the +ribbon connected to the top edge ‘−1’, the head of the next ribbon ‘2’, the head of the final +ribbon ‘3’, and then the tails of these ribbons ‘−3’ and ‘−2’. In the second illustration, we +highlight the edges of the hexagon which fit the edge-marking criteria below; in addition to +the sixth edge in the clockwise ordering, which is special, an edge is eligible if it precedes +an edge that is the tail of some ribbon. In the final image, we show how to split the vertex +preceding the top edge in the clockwise ordering so that there are seven vertices available +for the vertex-marking procedure outlined below. +To elucidate the combinatorics underlying equation (3.1.25), we take the ribbon graphs +labelled in the manner described above and assign them markings in one of two ways. The +first marking convention requires that one edge of the 2k-gon be marked, with eligible edges +being those labelled by ij with either j = 2k or j such that ij+1 < 0; that is, we mark either the +final edge in the clockwise ordering, or one that immediately precedes a negatively-labelled +edge (see the second illustration of Figure 3.2 above). There are (k + 1) such eligible edges, so +that the set of k-ribbon graphs edge-marked in this fashion are enumerated by the left-hand +side of equation (3.1.25). For the second marking convention, we split the vertex connecting +the first and last edges so that, in clockwise ordering, we have a vertex immediately preceding +135 + +1 +3 +2 +3CHAPTER 3. Characterisations of the Moments and Cumulants +the first edge, a vertex immediately following the last edge, and 2k − 1 vertices in between. +Labelling one of these 2k + 1 vertices by either of ±0 results in a set of 2(2k + 1)M(LUE) +k,0 +vertex-marked ribbon graphs. Since this number is just the right-hand side of equation +(3.1.25) with k replaced by k + 1, it is unsurprising that there is a bijection between the set of +k-ribbon graphs edge-marked in the first described convention and the set of (k − 1)-ribbon +graphs vertex-marked in the second convention. To map from the first set to the second, +delete the marked edge, the ribbon connected to it, and the edge at the other end of said +ribbon, so that each deleted edge collapses to a vertex. Of these two edge-turned-vertices, +mark the one corresponding to the deleted edge that was positively-labelled; label it ‘−0’ +if the positively-labelled edge was marked, or ‘0’ otherwise. Finally, we split the vertex +connecting the first and last edges. For the reverse mapping, split the marked vertex into +two and connect them with an edge. If the vertex was marked with a ‘−0’, insert a marked +edge immediately anticlockwise to the new edge, and connect the two with a ribbon. If +the vertex was instead marked with a ‘0’, we need to travel clockwise to a second vertex, +turn it into a marked edge, and connect said edge to the first through a ribbon; to uniquely +determine the second vertex, we simply pick the furthest one in the clockwise ordering +(stopping before reaching the first vertex and edge) that allows us to perform this procedure +whilst still producing a planar ribbon graph. +Example 3.1. The planar ribbon graphs displayed in Figure 3.3 below map to each other via +the bijection detailed above. On the left, we have a planar 3-ribbon graph labelled by the +sequence (1, −1, 2, 3, −3, −2), with a marking on the final edge ‘−2’. On the right, we have a +planar 2-ribbon graph labelled by the sequence (1, −1, 2, −2), with the vertex between edges +‘−1’ and ‘2’ marked with a ‘+0’. To map the left image to the right, we delete the marked +edge, together with its partner ‘2’ and the ribbon connecting them, then relabel the edges +3, −3 as 2, −2 and split the vertex connecting edges ‘1’ and ‘−2’. Since the original edge ‘2’ +(we focus on the positively-labelled deleted edge) was nestled between edges ‘−1’ and ‘3’ +and was unmarked, we mark the corresponding vertex with a ‘+0’ (if the edge marking in +the left image was at the edge ’2’, this vertex-marking would have been a ‘−0’). To obtain the +left illustration from the right, we expand the vertex labelled ‘+0’ into an edge, the vertex d +into a marked edge, and connect the two with a ribbon. Then, we join up the new marked +136 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +edge and the top edge ‘1’ and relabel the edges in the appropriate manner. The new marked +edge has to be at vertex d because it is the furthest along in the clockwise ordering since +vertices a, b appear earlier than vertex ‘+0’ in said ordering; note that picking vertex c would +result in a non-planar (genus one) ribbon graph. +Figure 3.3: Representatives of the sets of edge-marked planar 3-ribbon graphs with a +hexagon at their core and vertex-marked planar 2-ribbon graphs stemming from a square. +These sets are in bijection and have cardinality given manifestly by, respectively, the left- +and right-hand sides of equation (3.1.25) with k = 3. +When we remove the constraint l = 0, M(LUE) +k,l +retains a combinatorial interpretation if +we fix a = δa = 0. Then, as discussed in §3.3.2, it enumerates ribbon graphs that are now +bicoloured and of genus l/2 [82] (when a = 0, M(LUE) +k,l +vanishes for all odd l). Equivalently, +M(LUE) +k,l +|a=0 counts the number of ways of forming a compact orientable genus l/2 surface +by gluing together the edges of a 2k-gon whose vertices are alternately coloured black and +red, with the gluing respecting this bicolouring. When a = 0, the general-l recursion (3.1.24) +reduces to a recursion that runs over even l, which was recently derived in [86, Thrm. 4.1] +as a means of enumerating the aforementioned gluings. Keeping δa = 0, but letting ˆa ⩾ 0 +be free, M(LUE) +k,l +|a=ˆaN counts the same ribbon graphs, with the red vertices now weighted +by (ˆa + 1). Thus, M(LUE) +k,l +|a=ˆaN is a polynomial in (ˆa + 1) with the coefficient of (ˆa + 1)p +equalling the number of genus l/2 bicoloured gluings of the aforementioned 2k-gon that have +p distinct red vertices surviving the gluing procedure (again, we refer to §3.3.2 for details). +In this a = ˆaN setting, recursion (3.1.24) continues to run over even l so that m(LUE) +k +|a=ˆaN is +manifestly an odd function in N, in keeping with the second point of Remark 2.1. +137 + +1 +a +-2 +-1 +b +a +3 +2 +-2 +-1 +3 +2 +0 +C ++0CHAPTER 3. Characterisations of the Moments and Cumulants +A curious point to note is that even though the moment expansion coefficients M(LUE) +k,l +have combinatorial interpretations for k, l ∈ N and a = ˆaN with ˆa ⩾ 0, the recursion (3.1.24) +relating them does not presently have such an interpretation (at least not one comparable to +the bijection demonstrated in Example 3.1 above). Indeed, it may be the case that this 1-point +recursion describes a relation between the M(LUE) +k,l +|a=ˆaN which simply does not correspond +to a topological procedure at the level of the ribbon graphs themselves — ascertaining the +validity of this intriguing proposition remains an open problem. Similar to the LUE case, +the moment expansion coefficients Mk,l of the GUE, GOE, GSE, LOE, and LSE also have +combinatorial meaning, to be detailed in Section 3.3, but the 1-point recursions characterising +them are yet to be understood in a combinatorial sense. With the GUE, GOE, and GSE 1-point +recursions known from the works [174], [224], we now supply the analogous recursions for +the LOE and LSE. +Proposition 3.6. For β = 1 and 4, a = ˆaκN with ˆa constant in N, and k ⩾ 4, +(k + 1)M(L) +k,l = +4 +∑ +i=1 +i +∑ +j=0 +(κ − 1)jgi,jM(L) +k−i,l−j, +(3.1.26) +where +g1,0 = (4k − 1) (ˆaκ + 4(κ − 1)) (κ − 1), +g2,0 = (3 − 2k) +�ˆa2κ2 + 2(ˆaκ + 4(κ − 1))2(κ − 1)2� +, +g2,1 = 2(2k − 3)ˆaκ, +g2,2 = (k − 1) +� +5k2 − 11k + 4 +� +, +g3,0 = (4k − 11)(ˆaκ + 4(κ − 1))ˆa2κ2(κ − 1), +g3,1 = 2(11 − 4k)(ˆaκ + 4(κ − 1))ˆaκ(κ − 1), +g3,2 = 2(3 − k) +� +5k2 − 19k + 16 +� (ˆaκ + 4(κ − 1))(κ − 1), +g4,0 = (4 − k)ˆa4κ4, +g4,1 = 4(k − 4)ˆa3κ3, +g4,2 = (k − 4) +� +5k2 − 32k + 47 +� ˆa2κ2, +g4,3 = 2(4 − k)(k − 3)(5k − 17)ˆaκ, +g4,4 = 4(1 − k) [(k − 4)(k − 3)]2 , +and all other gi,j are zero. We also set M(L) +k,l = 0 if l < 0 or l > k. +When a = 0, M(LOE) +k,l +is shown in §3.3.2 to count the same bicoloured ribbon graphs +as M(LUE) +k,l +except that the ribbons are now allowed to have M¨obius half-twists so that the +ribbon graphs are no longer guaranteed to be globally orientable, though they still have +Euler genus l (recall Definition 3.1 and see Figures 3.15 and 3.19 for examples). Thus, in +138 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +keeping with the discussion comparing equations (3.1.23) and (3.1.25), we see that the three +term analogue of these equations obtained by setting a = l = 0 in Proposition 3.6, +(k + 1)M(L) +k,0 = 4(4k − 1)(κ − 1)2M(L) +k−1,0 − 32(2k − 3)(κ − 1)4M(L) +k−2,0, +(3.1.27) +is, up to scaling, satisfied by the Catalan numbers when κ = 1/2 or 2. Indeed, this recursion +can be used to verify that 2kM(LOE) +k,0 +|a=0, equivalently 2−kM(LSE) +k,0 +|a=0, is equal to the kth +Catalan number. Moreover, we observe that equation (3.1.27) is equivalent to that obtained +by setting l = 0 in the GOE 1-point recursion given in [224, Cor. 7]. +Now returning to the general-l case, let us compare the moment expansion coefficients +M(L) +k,l and M(G) +k,l . We immediately observe that the former is somehow more complicated than +the latter: M(L) +k,l enumerates the same ribbon graphs as M(G) +k,l +but with the extra bicolouring +constraint described earlier and in §3.3.2, so that M(L) +k,l |a=0 ⩽ M(G) +k,l and when a = ˆaκN ̸= 0, +(ˆa + 1) acts as a weight that keeps track of the ratio of red to black vertices in the ribbon +graphs. Note also that the presence of the parameter a results in the expansion coefficients +M(L) +k,l and the 1-point recursions (3.1.24), (3.1.26) being more complex than their Gaussian +counterparts. This increase in complexity is linked to the fact that the Laguerre ensembles +sit above the Gaussian ensembles in the hierarchy implied by Lemma 1.1. We posit that +since the Jacobi ensembles inhabit the tier immediately above the Laguerre ensembles in +the aforementioned hierarchy, it is not unreasonable to expect that the M(J) +k,l (3.0.10) count +some type of ribbon graphs that include, as special cases, the ribbon graphs enumerated by +the M(G) +k,l +or M(L) +k,l . However, there is no such construction currently known in the literature, +with attempts at replicating the constructions of Section 3.3 thwarted by the fact that the +Jacobi matrices of Definition 1.4 involve inverse matrices whose entries are not compatible +with the Isserlis–Wick theorem (see the discussion following Theorem 3.1 of Section 3.3). It +is nonetheless the author’s belief, which we justify below (see also the discussion in §4.4.3 +regarding the works [326], [25], [73], [159], [160]), that the M(J) +k,l should have combinatorial +interpretations in line with what is known for the Gaussian and Laguerre ensembles. +To motivate the study of the Jacobi ensembles’ moment expansion coefficients M(J) +k,l with +a combinatorial flavour, let us review some properties of the classical matrix ensembles that +are relevant in the combinatorial setting. To begin, recall from the paragraph preceding +Corollary 3.2 that m(J) +k , consequently M(J) +k,l , reduces to m(L) +k , respectively M(L) +k,l , when taking +139 + +CHAPTER 3. Characterisations of the Moments and Cumulants +b → ∞ (possibly by setting b = ˆbκN and working in the large N limit), so that if M(J) +k,l +were to enumerate a class of ribbon graphs, these ribbon graphs would degenerate to those +enumerated by M(L) +k,l in the large b limit. This is understood to be a simple consequence of +the limit (1.2.23) presented in Lemma 1.1, which is key to comparing the Laguerre and Jacobi +ensembles. The other limit given in Lemma 1.1, relating the Jacobi and Gaussian ensembles, +has a cleaner form when written in terms of the symmetric shifted Jacobi ensemble: +lim +a→∞ w(sJ)(λ/√ +a) +�� +a=b = w(G)(λ). +A similar observation can be made by comparing the identities +ρ(G),0(λ) = +√ +2(1 − λ2/2)ρ(sJ),0(λ/ +√ +2) +�� +a=b=0, +(3.1.28) +ρ(G),0(λ) = |λ|(1 − λ2/2)ρ(J),0(λ2/2) +�� +a=b=0. +(3.1.29) +Recalling the discussion following equation (3.1.25), equation (3.1.29) is reminiscent of the +analogous relation between ρ(G),0(λ) and ρ(L),0(λ), suggesting that while the GUE can be +compared at leading order to the sJUE by a linear change of variables, the LUE is more +closely related to the JUE. If we expand µ(sJ) +k +(3.1.5) as +µ(sJ) +k += +∞ +∑ +l=0 +∆M(sJ) +k,l N1−l +(3.1.30) +and define ∆M(J) +k,l := M(J) +k,l − M(J) +k+1,l, equations (3.1.28) and (3.1.29) imply that +2k∆M(J) +k,0 +�� +a=b=0 = −2k+1∆M(sJ) +k,0 +�� +a=0 = M(G) +k,0 , +(3.1.31) +which links the differences of the moments of ρ(J),0(λ)|a=b=0 and ρ(sJ),0(λ)|a=b=0 to the +Catalan numbers. Combining these observations together, it seems that there is merit in +studying the moment expansion coefficients of both the un-shifted and symmetric shifted +Jacobi ensembles. At this point in time, a good approach for progressing this line of study +is to find more ways of comparing the moment expansion coefficients Mk,l of the classical +matrix ensembles. To this end, we now derive the 1-point recursions satisfied by M(JUE) +k,l +, +∆M(JUE) +k,l +, and ∆M(sJUE) +k,l +, fixing β = 2 for clarity. +Proposition 3.7. Fix β = 2, a = ˆaN, and b = ˆbN, with ˆa, ˆb = O(1). Then, for k ⩾ 3, +k(ˆa + ˆb + 2)2M(JUE) +k,l += k(k − 1)2M(JUE) +k,l−2 + +3 +∑ +i=1 +� +hi,0M(JUE) +k−i,l + hi,1M(JUE) +k−i,l−2 +� +, +(3.1.32) +140 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +where +h1,0 = 2(4k − 3)(ˆa + ˆb + 1) + +� +3ˆa(k − 1) + ˆbk +� +(ˆa + ˆb), +h1,1 = (1 − k)(3k2 − 8k + 6), +h2,0 = 3ˆa2(2 − k) + (3 − 2k) +� +(ˆa + 2)(ˆb + 2) − 2 +� +, +h2,1 = (k − 2)(3k2 − 10k + 9), +h3,0 = ˆa2(k − 3), +h3,1 = (3 − k)(k − 2)2, +and we set M(JUE) +k,l += 0 if l < 0. +Proof. The derivation of this recursion is slightly different to that described earlier, so we +supply the details. Substitute expansion (3.0.10) into the recurrence (3.1.3), along with the +substitutions a = ˆaN and b = ˆbN, and then equate terms of order 3 − l in N. Note that +with our choice of a and b, the coefficients d(J) +2,0, . . . , d(J) +2,3 all contain a term of order two and +a term of order one in N, and no other terms. The O(N2) terms of d(J) +2,i correspond to the +coefficients hi,0 of M(JUE) +k−i,l +in equation (3.1.32), while the terms of order one in N correspond +to the coefficients hi,1 of M(JUE) +k−i,l−2. +Proposition 3.7 leads directly to the 1-point recursion on the ∆M(JUE) +k,l +|a=ˆaN, b=ˆbN = +M(JUE) +k,l +|a=ˆaN, b=ˆbN − M(JUE) +k+1,l |a=ˆaN, b=ˆbN, while applying the inverse of equation (2.2.3) yields +the equivalent relation on the ∆M(sJUE) +k,l +|a=ˆaN (3.1.30). Alternatively, the latter can be derived +from equation (3.1.7) by following the prescription at the beginning of this subsection. +Corollary 3.3. In the setting of Proposition 3.7 with ∆M(JUE) +k,l += M(JUE) +k,l +− M(JUE) +k+1,l and ∆M(sJUE) +k,l +defined implicitly by equation (3.1.30), we have for k ⩾ 2 the 1-point recursions +(k + 1)(ˆa + ˆb + 2)2∆M(JUE) +k,l += (2k − 1) +� +ˆa(ˆa + ˆb + 2) + 2(ˆb + 1) +� +∆M(JUE) +k−1,l ++ (2 − k)ˆa2∆M(JUE) +k−2,l + k2(k + 1)∆M(JUE) +k,l−2 +− k(k − 1)(2k − 1)∆M(JUE) +k−1,l−2 + (k − 2)(k − 1)2∆M(JUE) +k−2,l−2 +(3.1.33) +and +4(k + 1)(ˆa + 1)2∆M(sJUE) +k,l += 2(2k − 1)(2ˆa + 1)∆M(sJUE) +k−1,l ++ (k + 1)(2k + 1)2∆M(sJUE) +k,l−2 ++ 4k2(2k − 1)∆M(sJUE) +k−1,l−2 + (k − 1)(2k − 1)(2k − 3)∆M(sJUE) +k−2,l−2, +(3.1.34) +with ∆M(JUE) +k,l +, ∆M(sJUE) +k,l +set to zero for all l < 0. +141 + +CHAPTER 3. Characterisations of the Moments and Cumulants +There are three immediate observations relating to Proposition 3.7 and Corollary 3.3. +Firstly, the recursions therein run over even l, similar to Proposition 3.5. Indeed, when +the first few moments m(JUE) +k +, m(sJUE) +k +with k = 0, . . . , 3 (known from [96], [242], [142]) are +expanded as series in 1/N, they do not contain terms of even powers in N, so the recursions +(3.1.32)–(3.1.34) are trivially satisfied when l is odd. As with Proposition 3.5, this feature +is due to our choice of taking β = 2 and a, b ∝ N. Our second observation is that, unlike +the analogous 1-point recursions for the GUE and LUE moment expansion coefficients, +the recursions of Proposition 3.7 and Corollary 3.3 show that M(JUE) +k,l +, ∆M(JUE) +k,l +, ∆M(sJUE) +k,l +depend, respectively, on M(JUE) +k,l−2 , ∆M(JUE) +k,l−2 , ∆M(sJUE) +k,l−2 . Thus, in contrast to the Gaussian and +Laguerre cases, knowledge of m(JUE) +k′ +, ∆m(JUE) +k′ +, µ(sJUE) +k′ +with k′ < k is not enough to determine +M(JUE) +k,l +, ∆M(JUE) +k,l +, ∆M(sJUE) +k,l +for l > 0. Our final observation is that of the three recursions +(3.1.32)–(3.1.34), the final one is the simplest. +If we are to assume that the moment expansion coefficients of the Jacobi ensembles +count some type of surface similar to those counted by the GUE and LUE spectral moments, +comparing the corresponding 1-point recursions may shed light on what these surfaces +actually are. For example, the fact that the recursions of Proposition 3.7 and Corollary 3.3 +run over even l suggests that l/2 might play the role of genus in the Jacobi case, as it does in +the Gaussian and Laguerre cases. On the other hand, the second observation given above +could give a hint as to what sets the JUE apart from the GUE and LUE. Since setting a = 0 +in Proposition 3.5 yields a simpler recursion that retains an interpretation in terms of ribbon +graphs, we set a = b = 0 in Proposition 3.7 and Corollary 3.3 for comparison: +Corollary 3.4. In the context of Proposition 3.7 and Corollary 3.3, setting a = b = 0 reduces +recursion (3.1.32) to +4kM(JUE) +k,l += k(k − 1)2M(JUE) +k,l−2 + 2(4k − 3)M(JUE) +k−1,l ++ (1 − k)(3k2 − 8k + 6)M(JUE) +k−1,l−2 + 2(3 − 2k)M(JUE) +k−2,l ++ (k − 2)(3k2 − 10k + 9)M(JUE) +k−2,l−2 + (3 − k)(k − 2)2M(JUE) +k−3,l−2, +(3.1.35) +recursion (3.1.33) to +4(k + 1)∆M(JUE) +k,l += 2(2k − 1)∆M(JUE) +k−1,l + k2(k + 1)∆M(JUE) +k,l−2 +− k(k − 1)(2k − 1)∆M(JUE) +k−1,l−2 + (k − 2)(k − 1)2∆M(JUE) +k−2,l−2, +(3.1.36) +142 + +3.1. Recurrence Relations for the Moments of the Classical Matrix Ensembles +and recursion (3.1.34) to +4(k + 1)∆M(sJUE) +k,l += 2(2k − 1)∆M(sJUE) +k−1,l ++ (k + 1)(2k + 1)2∆M(sJUE) +k,l−2 ++ 4k2(2k − 1)∆M(sJUE) +k−1,l−2 + (k − 1)(2k − 1)(2k − 3)∆M(sJUE) +k−2,l−2. +(3.1.37) +Comparing these forms to the GUE analogue (3.0.12) and the result [86, Thrm. 4.1] of +setting a = 0 in Proposition 3.5, +(k + 1)M(LUE) +k,l += 2(2k − 1)M(LUE) +k−1,l + (k − 2)(k − 1)2M(LUE) +k−2,l−2, +(3.1.38) +we note that equations (3.1.36) and (3.1.37) contain familiar terms, whereas the same is not +true for equation (3.1.35). Setting equation (3.1.35) aside, we see that upon ignoring the terms +with indices (k, l − 2) and (k − 1, l − 2), equation (3.1.36) has the same structure as equation +(3.1.38), while equation (3.1.37) is almost equivalent to the Harer–Zagier recursion (3.0.12). +Hence, it seems that the JUE should be compared to the LUE and the sJUE should be com- +pared to the GUE (cf. the discussion following equations (3.1.28) and (3.1.29)). In addition, +the coefficients of the ignored terms contain, respectively, the factors (k + 1) and (2k − 1), +which suggests that we should group them with the terms with indices (k, l) and (k − 1, l). +As expected, setting l = 0 in equations (3.1.36) and (3.1.37) shows that 2k∆M(JUE) +k,0 +|a=b=0 +and −2k+1∆M(sJUE) +k,0 +|a=0 satisfy the two term Catalan recursion (3.1.25), which is in keeping +with equation (3.1.31). Moving past these observations, the challenge now is to extend the +combinatorial interpretation of equation (3.1.25) to the general-l recursions (3.0.12), (3.1.36), +(3.1.37), and (3.1.38). As mentioned earlier, this remains an open problem that we are unable +to address here. Nonetheless, some comments on promising directions are given in §4.4.2. As +we move on to the next section, let us make a final remark: Experimentation for 0 ⩽ k ⩽ 10, +0 ⩽ l ⩽ 5 suggests that for each k, l ⩾ 0, −22(k+l)+1∆M(sJUE) +k,l +|a=0 is a positive integer, which +supports the idea that these scaled differences of moments are equal to the cardinality of +some class of combinatorial objects. Moreover, it can be seen that these positive integers +grow extremely quickly as k, l are increased, relative to what is seen for their GUE and LUE +counterparts; to be more concrete, comparing equations (3.0.12) and (3.1.37) shows that for +all k, l ⩾ 0, +−22(k+l)+1∆M(sJUE) +k,l +|a=0 ⩾ M(GUE) +k,l +⩾ M(LUE) +k,l +|a=0. +143 + +CHAPTER 3. Characterisations of the Moments and Cumulants +3.2 +Established Results on the Moments of the Classical Matrix +Ensembles +In the previous section, we derived recursions characterising the spectral moments of the +classical matrix ensembles, along with their complex-k generalisations (3.1.1) and their +expansion coefficients Mk,l. To summarise Sections 2.3 and 3.1, our methodology consisted +of using the theory of Selberg correlation integrals to obtain differential equations satisfied +by the eigenvalue densities ρ(x) and resolvents W1(x) of the classical matrix ensembles — +applying integration by parts on the former set of differential equations or inserting the +expansion W1(x) = ∑∞ +k=0 mk/xk+1 in the latter set resulted in the sought recursions. Some +of the recursions derived in Section 3.1 have already been given in the literature, but our +approach involving Selberg correlation integrals is distinct and is able to treat all of the +classical matrix ensembles in a unified manner. In this section, we give a select review on +recent studies that provide characterisations of the moments of the classical matrix ensembles, +focusing particularly on those that are different to the results of Section 3.1 (classical results +on this topic are reviewed in a plethora of texts, including [238], [119]). Seeing as this +gives us an opportunity to showcase connections to various fields of mathematics, we +compartmentalise our review by methodology rather than outcome. +3.2.1 +Results from skew-orthogonal polynomial theory +A straightforward approach to computing the spectral moments is to simply integrate +monomials against closed form expressions for the corresponding eigenvalue densities. +This approach was taken by Livan and Vivo in 2011 [229], with their starting point being +expressions for the classical Laguerre and Jacobi ensembles’ eigenvalue densities known +from (skew-)orthogonal polynomial theory. Recalling the discussion of §1.2.3, the idea then +is to take ρ(λ) to be given by equation (1.2.67) with x = y = λ when β = 2, and by this +plus a correction term when β = 1, 4. Integrating these expressions against λk for k ∈ N, +Livan and Vivo gave exact expressions for the moments of the classical Laguerre and Jacobi +ensembles in terms of sums of ratios of gamma functions. At this same time, in a parallel +but independent work, Mezzadri and Simm [240] derived equivalent expressions for the +144 + +3.2. Established Results on the Moments of the Classical Matrix Ensembles +spectral moments mk of the classical Gaussian, Laguerre, and Jacobi ensembles, moreover +showing that their expressions hold true for all k ∈ C such that the mk are well-defined. +In order to ease the required integrations against the aforementioned expressions for the +eigenvalue densities ρ(w)(λ), a key idea of Mezzadri and Simm was to use polynomials that +are orthogonal with respect to the perturbations λkw(λ) of the classical weights w(λ) (1.2.9). +The moment recurrences given in [169], [223], [224], [71] were also obtained through +consideration of (skew-)orthogonal polynomial theory. In the 2003 work [169], Haagerup and +Thorbjørnsen used elementary identities satisfied by the Hermite and Laguerre orthogonal +polynomials to express the exponential moment generating function (cf. equation (1.1.18)) +R1(x) := +� +supp ρ exp(xλ)ρ(λ) dλ, +for the GUE and LUE cases in terms of (confluent) hypergeometric functions. They then +transformed known differential equations satisfied by these hypergeometric functions into +differential equations characterising R1(x). From there, Haagerup and Thorbjørnsen obtained +recurrences for the GUE and LUE (positive-integer) spectral moments through essentially +the same strategy as used in §3.1.1, thereby recovering the recursion of Harer and Zagier +[174], and finding its LUE analogue. Ledoux [223] extended these ideas to the JUE case +a year later, and to the GOE case [224] in 2009 — clever manipulations were needed to +handle the correction term that must be added to the GUE eigenvalue density to obtain +its GOE analogue. In 2016, Cunden et al. [71] extended the work of Ledoux to derive an +inhomogeneous moment recurrence for the LOE, where the inhomogeneous terms were given +in terms of the LUE moments. This recurrence, together with the LUE moment recurrence +given in [169], was shown to hold for negative-integer moments (m−k with 0 < k < a + 1). +3.2.2 +Results from symmetric function theory +Of the classical β ensembles (recall §1.2.4), only the β = 1, 2, and 4 regimes can be studied +through (skew-)orthogonal polynomial theory. To compute the spectral moments in the +general β case, other approaches must be taken. One option is to use loop equation analysis +to iteratively compute the resolvent expansion coefficients W0 +1(x), W1 +1(x), . . . (1.1.21) and +then interpret them as generating functions for the moment expansion coefficients Mk,l. +In the Gaussian and Laguerre cases, the spectral moments are polynomials in N, so this +145 + +CHAPTER 3. Characterisations of the Moments and Cumulants +procedure can be used to compute the spectral moment mk for any given k, assuming enough +computation power is available; in the case of the Jacobi and Cauchy β ensembles, the best +one can hope for is a truncation of the large N expansion of the spectral moments. We do not +review the loop equation formalism any further here since it will be revisited in Chapter 4, +but note that the Gaussian, Laguerre, and Jacobi β ensembles were studied in this manner in +the 2014 and 2017 works [319], [142]. +While the loop equation formalism can be used to compute (the leading order behaviour +of) spectral moments of the classical β ensembles, it does not provide any closed form +expressions for these moments. Such closed form expressions can however be obtained in +the general β setting through symmetric function theory. The relevance of said theory should +not be surprising since equation (3.0.1) shows that the kth spectral moment is given by the +integral of the symmetric polynomial ∑N +i=1 λk +i multiplied by the appropriate eigenvalue j.p.d.f. +Some well known bases for the ring of symmetric polynomials are the sets of monomial +symmetric polynomials, Schur polynomials, zonal polynomials, and the Jack polynomials. +As an algebra, it is also generated by the sets of power-sum symmetric polynomials and +elementary symmetric polynomials. We refer the reader to [232] for definitions of these +polynomials, but note that the elementary symmetric polynomials were mentioned earlier in +Remark 2.2. Of particular interest to us is the set of Jack polynomials of type “C”, which differ +from those of type “J” and “P” by a choice of normalisation (see, e.g., [92]) and are written as +C(α) +σ (λ1, . . . , λN). +They are indexed by partitions σ of integers (i.e., σ = (σ1, . . . , σk) with k, σ1, . . . , σk ∈ N +and σ1 ⩾ σ2 ⩾ · · · ⩾ σk > 0) and a positive real parameter α not to be confused with that +associated with the Cauchy ensemble; setting α = 1 recovers the Schur polynomials while +α = 2 corresponds to the zonal polynomials. +In the 1997 work on Selberg correlation integrals [199], Kadell gave an explicit closed +form expression, in terms of ratios of gamma functions, for the integral +� +[0,1]N C(2/β) +σ +(λ1, . . . , λN) p(J)(λ1, . . . , λN; β) dλ1 · · · dλN, +(3.2.1) +where p(J)(λ1, . . . , λN; β) is the eigenvalue j.p.d.f. (1.2.81) of the Jacobi β ensemble and +σ is any given partition. In the same year, Baker and Forrester [27] showed that when +146 + +3.2. Established Results on the Moments of the Classical Matrix Ensembles +w(λ) is either the Gaussian or Laguerre weight (1.2.9), the above integral (3.2.1) with p(J) +replaced by p(w) is equal to the generalised Hermite (up to a minus sign), respectively +Laguerre, polynomial of index σ evaluated at zero, these generalised polynomials being the +multivariable symmetric polynomials that are orthogonal with respect to the inner products +⟨ f, g⟩(w) := +� +RN f (λ1, . . . , λN)g(λ1, . . . , λN) p(w)(λ1, . . . , λN; β) dλ1 · · · dλN. +In the Jacobi case, one can surmise from [27] a result analogous to that of Kadell [199]. +Around 2003 [92], Dumitriu et al. used the results of [199], [27] (among others) to devise +algorithms, implemented in the MOPS package [96], for computing the spectral moments +of the Gaussian, Laguerre, and Jacobi β ensembles. These algorithms essentially boil down +to expressing the polynomial ∑N +i=1 λk +i as a linear combination of Jack “C” polynomials so +that the problem of computing the spectral moments (3.0.1) reduces to the computation +of integrals of the form (3.2.1). One should note that these algorithms are not equivalent +to closed form expressions for the spectral moments. However, in the 2017 work [242], +Mezzadri et al. provide such closed form expressions in the Jacobi and Laguerre cases by +way of giving explicit formulae for the coefficients in the aforementioned linear combination +of Jack “C” polynomials that is equal to ∑N +i=1 λk +i . Thus, they express m(J) +k +and m(L) +k +as +sums over partitions of k with each summand being an explicit ratio of gamma functions. +Prompted by Fyodorov and Le Doussal, they also show how to use a relation between +� +C(α) +σ (1/λ1, . . . , 1/λN) +� +σ and +� +C(α) +σ′ (λ1, . . . , λN) +� +σ′ to extend their expressions for m(J) +k , m(L) +k +to the case that k is a negative integer. In fact, these expressions for the integer moments of +the Jacobi and Laguerre β ensembles as sums of ratios of gamma functions were given by +Fyodorov and Le Doussal in the independent work [149]. Moreover, this latter work contains +contour integral representations for m(J) +k , m(L) +k +(k ∈ Z sufficiently large) due to Borodin and +Gorin. Before moving on, let us finally mention that Novaes also studied the negative-integer +moments of the LUE using Schur polynomials in the 2015 work [267]. +3.2.3 +Results in terms of hypergeometric orthogonal polynomials +In the recent work [72], Cunden at al. made the novel observation that in the classical +Gaussian, Laguerre, and Jacobi cases, simple linear combinations of the complex-k moments +mk (3.1.1), interpreted as functions of k and renormalised by factors dependent on N, are +147 + +CHAPTER 3. Characterisations of the Moments and Cumulants +hypergeometric orthogonal polynomials in k. To be precise, they showed that when β = 2 and +N is fixed, m(G) +k +is essentially a Meixner–Pollaczek polynomial in k, m(L) +k +is a continuous dual +Hahn polynomial, and ∆m(J) +k += m(J) +k +− m(J) +k+1 is a Wilson polynomial. These three polynomials +belong to the Askey scheme of hypergeometric orthogonal polynomials, with the first two +being degenerations of the latter. Cunden et al. prove this observation in the Gaussian +and Laguerre cases by seeing agreement in the hypergeometric function representations +of the moments and the corresponding polynomials when k ∈ N, and then extending this +agreement to k ∈ C through an application of Carlson’s theorem (cf. Remark 2.5); in the +Jacobi case, lacking such a representation in terms of hypergeometric functions, they derive +the analogue of recurrence (3.1.3) for ∆m(J) +k +and show that this recurrence, after appropriate +scaling, also characterises the Wilson polynomials in a unique sense. By noting that certain +linear combinations of the β = 1, 4 moments are known to be given by linear combinations +of their β = 2 counterparts, Cunden et al. further show within [72] how the polynomial +characterisations of the β = 2 moments extends to the linear combinations considered in the +β = 1, 4 cases. +Remark 3.1. It was shown in [72] that the LUE moments and JUE moment differences satisfy +the following reciprocity laws: +m(L) +−k−1 = +� +k +∏ +j=−k +1 +a − j +� +m(L) +k , +(3.2.2) +∆m(J) +−k−1 = +� +k +∏ +j=−k +a + b + 2N − j +a − j +� +∆m(J) +k . +(3.2.3) +It is not immediately obvious that there exist similar laws for the moments of the orthogonal +or symplectic ensembles; one can experiment using data computable from Proposition 3.1 +and Corollary 3.2. It is believed that this is a consequence of three term recurrences, as seen +in the β = 2 cases, being simpler than their five term (β = 1, 4) analogues in an integrable +sense. Indeed, this is the reason that Cunden et al. were able to place the β = 2 moments +m(G) +k +, m(L) +k , m(J) +k +in the Askey scheme of hypergeometric orthogonal polynomials, but were +only able to do so for linear combinations of the moments when taking β = 1, 4. +Soon after the work [72], Assiotis et al. showed [23] that the complex-k sum of moments +µ(CyUE) +k +(3.1.6) of the symmetric Cauchy unitary ensemble, again understood as a function of +148 + +3.2. Established Results on the Moments of the Classical Matrix Ensembles +k with fixed N and properly renormalised, is also a hypergeometric orthogonal polynomial. +This time, the relevant polynomials are the continuous Hahn polynomials, which are also +degenerations of the Wilson polynomials. Recalling that the hypergeometric functions are +defined by +pFq +�a1, . . . , ap +b1, . . . , bq +���� z +� +:= +∞ +∑ +n=0 +(a1)(n) · · · (ap)(n) +(b1)(n) · · · (bq)(n) +zn +n!, +with (x)(n) = x(x + 1) · · · (x + n − 1) denoting the rising Pochhammer symbol, the continuous +Hahn polynomials are defined as [207] +Sn(x; a, b, c, d) := in (a + c)(n)(a + d)(n) +n! +3F2 +�−n, n + a + b + c + d − 1, a + ix +a + c, a + d +���� 1 +� +. +(3.2.4) +Assiotis et al. proceed by deriving a differential equation characterising the eigenvalue +density ρ(Cy)(λ; N, 2) of the CyUE, using this differential equation to obtain recurrence +(3.1.13) on the sums of moments µ(Cy) +k +(3.1.6), and then confirming that this recurrence also +characterises the continuous Hahn polynomials. Thus, they show that +µ(CyUE) +k += +Γ (k + 1/2) Γ (α − k − 1/2) +Γ (α + 3/2) Γ (−α − 1/2) √π +i1−N +2 +Γ (1/2 − α − N) α(2α + N) +× SN−1 +� +−i(k + 1); 1, 1 +2 + α, 1, 1 +2 + α +� +, +(3.2.5) +with SN−1 defined by equation (3.2.4). +We provide now the analogue of the above result of Assiotis et al. in the case of the +symmetric shifted Jacobi unitary ensemble corresponding to the weight w(sJ)(λ) +�� +a=b (3.0.6). +Let us first recall from the third point of Remark 2.1 that ρ(sJUE)(λ) +�� +a=b is equal to (1 − λ2)a +multiplied by a polynomial of order N − 1. Thus, we see from equations (2.2.8) and (3.1.5) +that +˜µk := Γ(k + N + a + 3/2) +Γ(k + 1/2) +µ(sJUE) +k +��� +a=b +(3.2.6) +is a polynomial in k of order N − 1. Substituting this scaling into recurrence (3.1.7) results in +the simpler recurrence +(2k + 4)[(2k + 3) − 2(a + N)] ˜µk+1 + 2[(2k + 2)2 − 2N(N + 2a)] ˜µk ++ (2k)(2(N + a + k) + 1) ˜µk−1 = 0 +(3.2.7) +149 + +CHAPTER 3. Characterisations of the Moments and Cumulants +valid for k ∈ C with Re(k) > 1/2. Now, let us constrain the continuous Hahn polynomials +(3.2.4) by defining +sn(x; a, b) := Sn(x; a, b, a, b) +(3.2.8) +and observe that this polynomial satisfies the recurrence [1, 18.22.13–18.22.15] +A(x)sn(x + i; a, b) − [A(x) + C(x) − n(n + 2Re(a + b) − 1)] sn(x; a, b) ++ C(x)sn(x − i; a, b) = 0, +(3.2.9) +with +A(x) = (x + ia)(x + ib), +C(x) = (x − ia)(x − ib). +Proposition 3.8. For k ∈ C with Re(k) > −1/2 and ˜µk defined by equation (3.2.6), we have +˜µk = +˜µ0i1−N +N(3/2 − (a + N))(N−1) sN−1 +� +i(k + 1); 1, 1 +2 − (a + N) +� +, +(3.2.10) +with +˜µ0 = Γ(N + a + 3/2) +Γ(1/2) +�2N(a + N)(2a + N) +1 − 4(a + N)2 +� +. +(3.2.11) +Thus, the difference of moments µ(sJUE) +k +of the symmetric shifted Jacobi unitary ensemble is given in +terms of the continuous Hahn polynomials according to +µ(sJUE) +k += +˜µ0i1−NΓ(k + 1/2) +NΓ(k + N + a + 3/2)(3/2 − (a + N))(N−1) sN−1 +� +i(k + 1); 1, 1 +2 − (a + N) +� +. +(3.2.12) +Proof. Equation (3.2.11) is obtained by combining equations (3.1.8) and (3.2.6). Comparing +equations (3.2.7) and (3.2.9) shows that +q(k; a, N) := CN,a sN−1 +� +i(k + 1); 1, 1 +2 − (a + N) +� +, +with CN,a independent of k, satisfies the recurrence (3.1.8). To obtain agreement with ˜µk at +k = 0, we set +CN,a = +˜µ0 +sN−1 +� +i; 1, 1 +2 − (a + N) +� = +˜µ0 i1−N +N(3/2 − (a + N))(N−1) , +where the second equality follows from definition (3.2.4) of the continuous Hahn polynomials. +With this choice, both q(k; a, N) and ˜µk are polynomials of order N − 1 in k that coincide at +k = 0. As they both satisfy recurrence (3.1.8), they also coincide at k ∈ Z. This agreement +is extended to k ∈ C via Carlson’s theorem (recall Remark 2.5). Finally, equation (3.2.12) +follows from substituting equation (3.2.6) into equation (3.2.10). +150 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +3.3 +The Isserlis–Wick Theorem and Ribbon Graphs +A major motivation for studying ribbon graphs (recall Definition 3.1) is their relation to +topological (hyper)maps, which we briefly discuss throughout this section. For now, we note +that the term ‘map’ is not synonymous with ‘function’ or ‘mapping’. Instead, topological +maps are so-called because they are reminiscent of the geographic maps of countries that +one would find in an atlas — they are graphs embedded in surfaces. These topological +maps have been studied by combinatorialists and geometers since the 1960s [302], [198], +but became prominent in algebraic geometry around 1984 due to Grothendieck’s work on +dessins d’enfants (children’s drawings) [284] — technically, a topological map is equivalent to +a ‘clean’ dessin (see Remark 3.6 at the end of §3.3.1). +The connection between ribbon graphs (equivalently, topological maps) and random +matrix theory is that, through consequences of the Isserlis–Wick theorem that we will soon +expound, random matrix theory provides a neat method for enumerating ribbon graphs; the +upshot here is that the pairwise polygon-edge identifications represented by the ribbons of a +ribbon graph can be interpreted as pairings of random matrix entries. Algebraic geometers +and mathematical physicists have used this relationship in a number of intriguing ways, of +which we mention a select few: +1. Let Ms +g denote the moduli space of genus g compact Riemann surfaces with s marked +points (i.e., the (6g − 6 + 2s)-dimensional space of parameters that determine the +complex structure of such surfaces). Using the uniqueness of the Jenkins–Strebel +quadratic differential on compact Riemann surfaces [193], [291], [292], Harer [173] +showed how to canonically assign a genus g, orientable, connected, metrised ribbon +graph built from s polygons to each point of Ms +g (here, metrised means that each +ribbon is labelled with a length). This had the consequence of endowing Ms +g with the +structure of a simplicial complex (each k-simplex corresponding to a k-ribbon graph +representative of all metrised ribbon graphs that reduce to it upon ‘forgetting’ ribbon +lengths), which enabled Harer and Zagier [174] to reduce the problem of computing +the (orbifold) Euler characteristic of Ms +g to that of enumerating ribbon graphs. This +treatment was extended to the moduli space of real algebraic curves in [165], [166], +with the relevant ribbon graphs now allowed to be non-orientable. +151 + +CHAPTER 3. Characterisations of the Moments and Cumulants +2. In string theory (see, e.g., [274] for an introduction), one replaces particles with strings, +which are topologically lines or circles. The trajectory of a string is called a worldsheet, +which is a surface with boundaries. Considering only closed (circular) strings, a +worldsheet is a surface of, say, genus g with s punctures — the handles counted by +the genus arise from strings splitting and combining, while the punctures are simply +contractions of the boundaries that correspond to the initial and final states of strings. In +the theory of 2D quantum gravity, one considers a generic worldsheet with its structure, +that is, a choice of metric, left undetermined. The partition function is formulated as an +integral over the space of all possible worldsheet metrics, with the integrand given in +terms of aspects of the worldsheet that depend on the choice of metric. This integral is +difficult to make sense of because the space of worldsheet metrics is unwieldy for many +reasons. In 1990, progress was made by the independent works [52], [89], [168], where +the idea was to discretise the worldsheet and approximate it by triangulations (i.e., +topological maps with trivalent vertices or ribbon graphs built purely from triangles). +When considering the limit of infinitely many triangles, these discrete approximations +improve to the point that each choice of triangulation corresponds to a worldsheet +metric. Consequently, the relevant partition function can essentially be written as a +sum over orientable, connected ribbon graphs. +3. Another approach to 2D quantum gravity, called topological gravity, has partition func- +tion equal to a generating function for certain intersection numbers. These intersection +numbers are given by integrals over the compactifications M +s +g of the moduli spaces +defined in point 1 above. A 1991 conjecture of Witten [322] is that topological gravity +should be equivalent to the formalism outlined in point 2 above. This conjecture was +famously proven by Kontsevich the following year [211] using the ideas of Harer that +we reviewed in point 1. In short, Kontsevich used the structure of Ms +g as a simplicial +complex (with each simplex corresponding to a specific ribbon graph) in order to relate +the aforementioned intersection numbers to explicit sums over ribbon graphs. +For detailed surveys of these topics, see [83], [250], [220], [248]. +The studies listed above share the philosophy of reducing a difficult problem to that +of enumerating ribbon graphs, and then using random matrix theory to perform said +152 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +enumerations. In contrast to this philosophy, the literature also contains many examples +of using ribbon graphs (and similar objects) to diagrammatically encode calculations of +the (mixed) moments and cumulants defined through equations (3.0.1), (3.0.13), (3.0.14). +Naturally, such examples were first seen in the physics literature [304], [53], [287], [192], +since both random matrix theory and diagrammatic theories [179], [54] were prevalent in +the field during the mid-twentieth century. In particular, Wick’s theorem [312] on reducing +products of creation and annihilation operators to sums of products of creation-annihilation +operator pairings was well known to quantum field theorists and easily extends to treatment +of similar such products of random matrix entries. In fact, the required generalisation +pertaining to products of normally distributed random variables was actually known much +earlier to probability theorists as Isserlis’ theorem [186]. Let us now state a version of this +theorem that is suitable for our purposes. +Theorem 3.1 (Isserlis–Wick). For k ∈ N, let x1, . . . , xk be centred normal variables. Then, if k is +odd, the expectation ⟨x1 · · · xk⟩ is equal to zero, while for k even, it decomposes as +⟨x1 · · · xk⟩ = ∑ +σ∈Pk +� +xσ(1)xσ(2) +� +· · · +� +xσ(k−1)xσ(k) +� +, +(3.3.1) +where Pk is the set of all permutations of {1, . . . , k} such that for all odd integers i, j with i < j, +σ(i) < σ(i + 1) and σ(i) < σ(j). +The key idea of this section, in conjunction with the Isserlis–Wick theorem, is that for +random matrices X that can be expressed as sums and products of Ginibre matrices (recall +Definition 1.3), Tr Xk is given by a sum of products of normally distributed random variables. +Thus, recalling Definition 1.4, the (mixed) moments and cumulants of the GOE and LOE can +be simplified using the Isserlis–Wick theorem; graphically, the covariances ⟨xσ(j)xσ(j+1)⟩ in +the summand of equation (3.3.1) are to be represented by ribbons connecting edges labelled +xσ(j) and xσ(j+1). By linearity of the expectation, the Isserlis–Wick theorem also holds when +the variables x1, . . . , xk are drawn from mean-zero complex normal distributions, so it can +be used to study the GUE and LUE, as well. Similar reasoning applies in the GSE and LSE +cases, too [251], [57], though we do not discuss these cases for brevity (instead, we appeal to +the β ↔ 4/β duality of Lemma 1.4). +Unfortunately, the Isserlis–Wick theorem cannot be used to relate ribbon graphs to any of +the other ensembles studied thus far in this thesis. Indeed, the Gaussian β ensembles with +153 + +CHAPTER 3. Characterisations of the Moments and Cumulants +β = 2/3, 6 do not have matrix realisations in terms of Ginibre matrices and must instead +be understood through the general-β constructions of §1.2.4. Likewise, there is no relation +between the Cauchy and Ginibre ensembles, while matrix realisations of the Jacobi ensembles +involve inverse Ginibre matrices (see Definition 1.4 and Remark 1.4). However, it should be +noted that the combinatorial approach of [326] was used in [25] to express m(JOE) +k +in terms +of the negative-integer moments of the LOE; the latter do not have any known ribbon graph +interpretations (see §4.4.3 for some further discussion on this point). +In §3.3.1, we show how the Isserlis–Wick theorem leads to interpretations of the (mixed) +moments and cumulants of the GUE in terms of (globally) orientable ribbon graphs, and +then explain how the GOE analogues are similarly related to locally orientable ribbon +graphs. In §3.3.2, we repeat these exercises for the LUE and LOE, detailing a necessary +bicolouring constraint on the vertices of the polygons that are present in the relevant +ribbon graphs. +Finally, in §3.3.3, we combine the ideas of §3.3.1 and §3.3.2 to derive +ribbon graph representations for the mixed moments and cumulants of the Hermitised and +antisymmetrised matrix products introduced in Section 1.3, thereby justifying the existence +of the large N expansions (1.1.21) for the corresponding connected n-point correlators Wn +(this latter point being significant in the context of Chapter 4). We will be relaxed with our +referencing since our discussion will be an amalgamation of new results, results already +cited, and results that are most likely known to the community but unpublished in full detail. +That said, let us note that the exposition of the first half of §3.3.1 is equivalent to that of +[36], [329], while the first half of §3.3.2 contains statements that loosely follow from [82]. +The second half of both of these subsections discuss concepts that have been studied and +reviewed in [57], [219] and references therein. The contents of §3.3.3 are original and based +on the ongoing work [75]. +3.3.1 +Moments of the Gaussian unitary and orthogonal ensembles +Let us now demonstrate how the Isserlis–Wick theorem can be used to express the spectral +moments m(GUE) +k +as sums over certain (k/2)-ribbon graphs. Afterwards, we illustrate how +these ideas extend to the mixed moments and cumulants of the GUE and then show how to +tweak our arguments to make them suitable for the GOE case. At the end of this subsection, +we also discuss connections to topological and combinatorial (hyper)maps. +154 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +Ribbon graphs for the GUE spectral moments +Let H be drawn from the N × N Gaussian unitary ensemble. According to the convention +established in Section 1.2, this means that the entries {Hij}N +i,j=1 of the complex Hermitian +matrix H are centred normal variables with covariances given by +� +HijHkl +� = 1 +2χk=jχl=i, +(3.3.2) +where we recall that the indicator function χA equals one when A is true and zero otherwise. +Here and throughout the remainder of this subsection, we suppress the fact that our averages +are with respect to the p.d.f. P(G)(H) given in equation (1.2.1). Our first order of business is +to use the formula (3.3.2) to give a multi-sum expression for the GUE spectral moments. +Lemma 3.1. Let k ∈ N, recall the definition of Pk from Theorem 3.1, and set ik+1 := i1. The spectral +moment m(GUE) +k +is zero if k is odd, while for even k, +m(GUE) +k += 2−k/2 ∑ +σ∈Pk +N +∑ +i1,...,ik=1 +(χiσ(1)=iσ(2)+1χiσ(2)=iσ(1)+1) · · · (χiσ(k−1)=iσ(k)+1χiσ(k)=iσ(k−1)+1). +(3.3.3) +Proof. Taking equation (1.1.15) as our definition for m(GUE) +k +, it is well known that +m(GUE) +k += +� +Tr Hk� += +� +N +∑ +i1,...,ik=1 +Hi1i2Hi2i3 · · · Hik−1ik Hiki1 +� += +N +∑ +i1,...,ik=1 +� +Hi1i2Hi2i3 · · · Hik−1ik Hiki1 +� +; +(3.3.4) +the second line can be checked via straightforward induction, while the third line follows +from linearity of the expectation. The summand on the right-hand side of equation (3.3.4) is +the expected value of a product of normally distributed random variables, so the Isserlis–Wick +theorem applies. Hence, it vanishes for odd k, and for even k, we have +� +Hi1i2Hi2i3 · · · Hik−1ik Hiki1 +� += ∑ +σ∈Pk +� +Hiσ(1)iσ(1)+1Hiσ(2)iσ(2)+1 +� +· · · +� +Hiσ(k−1)iσ(k−1)+1Hiσ(k)iσ(k)+1 +� +. +(3.3.5) +Substituting this expression into equation (3.3.4), interchanging the order of summation, and +using equation (3.3.2) then produces the desired result. +155 + +CHAPTER 3. Characterisations of the Moments and Cumulants +The inner sum in equation (3.3.3) reduces to ∑N +j1,...,jl=1 1 = Nl for some 1 ⩽ l ⩽ k/2 + 1 +since many of the indices i1, . . . , ik must be identified to ensure that the product of indicator +functions is non-zero. To better understand this statement, it is convenient to recast it in +terms of ribbon graphs. This can be done in a natural way by recognising that each choice +of σ ∈ Pk determines a partitioning of the set {Hi1i2, Hi2i3, . . . , Hiki1} into k disjoint pairs: To +begin, we encode the fact that the first index of each element of {Hi1i2, Hi2i3, . . . , Hiki1} is equal +to the second index of one other element of the same set by drawing a k-gon whose vertices +are labelled i1, i2, . . . , ik in clockwise order; each edge of the k-gon can be oriented in two +ways, with (il → il+1) representing the matrix entry Hilil+1 and (il+1 → il) representing Hilil+1. +Next, for each l = 1, . . . , k/2, we represent the identification of the oriented polygon edges +(iσ(2l−1) → iσ(2l−1)+1) and (iσ(2l)+1 → iσ(2l)), implied by the factor χiσ(2l−1)=iσ(2l)+1χiσ(2l)=iσ(2l−1)+1 +in the right-hand side of equation (3.3.3), by connecting them with an untwisted ribbon. +Figure 3.4: These are the ribbon graph representations of the terms given in equations +(3.3.7)–(3.3.9) below. The purple (grey) ribbons represent the first (second) factor in the +summands of the corresponding equations. +Example 3.2. Let us illustrate the above formalism in the k = 4 case. The set P4 contains +three permutations: the identity, (2, 4, 3), and (2, 3). Hence, equation (3.3.3) tells us that +m(GUE) +4 += 1 +4 +� +Sid + S(2,4,3) + S(2,3) +� +(3.3.6) +with +Sid := 4 +N +∑ +i1,...,i4=1 +⟨Hi1i2Hi2i3⟩ ⟨Hi3i4Hi4i1⟩ = +N +∑ +i1,...,i4=1 +(χi1=i3χi2=i2)(χi3=i1χi4=i4), +(3.3.7) +S(2,4,3) := 4 +N +∑ +i1,...,i4=1 +⟨Hi1i2Hi4i1⟩ ⟨Hi2i3Hi3i4⟩ = +N +∑ +i1,...,i4=1 +(χi1=i1χi4=i2)(χi2=i4χi3=i3), +(3.3.8) +S(2,3) := 4 +N +∑ +i1,...,i4=1 +⟨Hi1i2Hi3i4⟩ ⟨Hi2i3Hi4i1⟩ = +N +∑ +i1,...,i4=1 +(χi1=i4χi3=i2)(χi2=i1χi4=i3). +(3.3.9) +156 + +i2 +11 +i1 +i2 +1 +i2 +14 +14 +13 +i4 +i3 +(2,3)3.3. The Isserlis–Wick Theorem and Ribbon Graphs +These sums are then represented by the ribbon graphs in Figure 3.4. Seen as two-dimensional +surfaces, they have, respectively, three, three, and one boundaries. This reflects the fact that +i3 ≡ i1 in the first ribbon graph, i4 ≡ i2 in the second ribbon graph, and i4 ≡ i3 ≡ i2 ≡ i1 in +the third ribbon graph. Hence, +Sid = +N +∑ +i1,i2,i4=1 +1 = N3, +(3.3.10) +S(2,4,3) = +N +∑ +i1,i2,i3=1 +1 = N3, +(3.3.11) +S(2,3) = +N +∑ +i1=1 +1 = N, +(3.3.12) +so that by equation (3.3.6), m(GUE) +4 += N3/2 + N/4. +The computation of the above example extends quite simply to the case of k being a +general positive even integer: One need only interpret Pk within Lemma 3.1 as the set of all +orientable (k/2)-ribbon graphs that can be built from a single k-gon. +Lemma 3.2. Let k ∈ 2N and Pk be as in Theorem 3.1. Then, we have +m(GUE) +k += 2−k/2 ∑ +σ∈Pk +NV(σ), +(3.3.13) +where V(σ) is equal to the number of boundaries of the ribbon graph constructed from σ according to +the prescription above Figure 3.4. +Proof. Inspection of the ribbon graph corresponding to σ shows that each indicator function +in the summand of equation (3.3.3) is represented by the side of a ribbon identifying two +vertices of the k-gon at the centre of the ribbon graph. Thus, the number of unique summation +indices i1, . . . , ik is equal to V(σ), the number of boundaries of said ribbon graph. Hence, we +see the reduction +N +∑ +i1,...,ik=1 +(χiσ(1)=iσ(2)+1χiσ(2)=iσ(1)+1) · · · (χiσ(k−1)=iσ(k)+1χiσ(k)=iσ(k−1)+1) = +N +∑ +j1,...,jV(σ)=1 +1 = NV(σ). +Substituting this into equation (3.3.3) produces the sought result. +Remark 3.2. Recalling that ribbons represent edge identifications of the underlying k-gons, +V(σ) is also equal to the number of distinct vertices retained by a k-gon after identifying +edges according to the ribbon graph corresponding to σ; see Figure 3.5 below. +157 + +CHAPTER 3. Characterisations of the Moments and Cumulants +Figure 3.5: These compact orientable surfaces (sphere, sphere, and torus) are obtained by +enlargening the white squares in Figure 3.4 while collapsing the ribbons therein to their +now-identified ends. Three, three, and one distinct vertices survive the edge identification +procedure, respectively. +It is evident from equation (3.3.13) that m(GUE) +k +is a polynomial in N. In fact, according +to the statement (which we are yet to fully justify) following the proof of Lemma 3.1, it is a +polynomial of degree k/2 + 1. This means that equation (3.3.13) can be rewritten as +m(GUE) +k += 2−k/2 +k/2+1 +∑ +V=1 +NV #{σ ∈ Pk | V(σ) = V}, +k ∈ 2N, +(3.3.14) +where #S denotes the number of elements in the set S. With our current definitions of the +function V(σ), computing the coefficient of NV on the right-hand side of equation (3.3.14) +requires us to generate all orientable (k/2)-ribbon graphs that can be built from a k-gon +and then count how many of them have exactly V boundaries. However, we are able to +give a much neater interpretation of these polynomial coefficients, while also justifying our +assumption that the degree of m(GUE) +k +in N is k/2 + 1, by classifying our ribbon graphs with +respect to the genera of the corresponding compact surfaces. +Let us proceed by first observing that rather than collapsing the ribbons of a ribbon graph +(as described in Figure 3.5), another way of obtaining the compact surface represented by a +connected ribbon graph is to glue open disks along the boundaries of said ribbon graph. This +is seen to be true by noting that contracting the disks in the resulting polygonised surface to +points while also collapsing the ribbons to edges is equivalent to just collapsing the ribbons +in the original ribbon graph (see Figure 3.6 below). Hence, connected ribbon graphs can +be drawn on the compact surfaces that they represent and, moreover, these surfaces are of +minimal (Euler) genus such that this is possible without self-intersections (if one were able +to draw a ribbon graph on a surface of lower (Euler) genus without self-intersections, then +158 + +13 +S(2,3) +Sid3.3. The Isserlis–Wick Theorem and Ribbon Graphs +shrinking the ribbons and disks bounded by the ribbons in the described way could not +produce the compact surface that the ribbon graph represents). As one would expect, this +means that the (Euler) genus of a ribbon graph (recall Definition 3.1) is equal to that of the +compact surface it represents. +Figure 3.6: Gluing a pink, green, and blue disk to the boundaries of the ribbon graph +illustrated on the left produces the polygonised sphere in the middle. Shrinking the ribbons +to edges and (deformed) disks to vertices while enlargening the white square produces the +illustration on the right, which is exactly the first surface presented in Figure 3.5. +Recall that the Euler characteristic of a compact surface is given by the classical formula +χ = F − E + V, +where F, E, V are respectively the number of faces, edges, and vertices of a chosen poly- +gonisation of the surface. In our case, the surfaces of interest are constructed by gluing +together pairs of edges of a k-gon, so an obvious choice of polygonisation for a given surface +is the edge-identified k-gon that it is equivalent to. Thus, we have F = 1 and E = k/2 since +our k-gons have one face and k edges which are identified pairwise into k/2 distinct edges +(cf. Figure 3.5). Combining this with Euler’s formula and noting that the Euler characteristic +of a compact orientable surface is also given by χ = 2 − 2g, where g is the genus of the +surface, we have +V = 1 + k/2 − 2g. +(3.3.15) +Thus, we may reformulate equation (3.3.14) as a genus expansion. +Proposition 3.9. Let k ∈ N and let Pk be as in Theorem 3.1. If k is odd, the spectral moment +m(GUE) +k +is zero, while if k is even, we have that +m(GUE) +k += 2−k/2 +k/2 +∑ +l=0 +N1+k/2−l #{σ ∈ Pk | g(σ) = l/2}, +(3.3.16) +where g(σ) is equal to the genus of the ribbon graph labelled by σ. +159 + +CHAPTER 3. Characterisations of the Moments and Cumulants +Proof. Lemma 3.2 implies equation (3.3.14) with the possible requirement that the upper +terminal of the sum be changed. Equation (3.3.15) and the discussion above Figure 3.6 tells +us that the coefficient of NV on the right-hand side of equation (3.3.14) is equal to the number +of genus (1 + k/2 − V)/2, connected, orientable ribbon graphs that consist of a single k-gon +and k/2 ribbons. Moreover, since g ⩾ 0, we see that V ⩽ k/2 + 1, which establishes that +the upper limit of the sum in equation (3.3.14) is indeed k/2 + 1; it is a simple exercise to +check that the identity permutation σ = id ∈ Pk corresponds to a (genus zero) sphere, so +that V(id) = k/2 + 1. Letting l = 2g and changing the summation index in equation (3.3.14) +according to equation (3.3.15) gives equation (3.3.16), as desired. +Replacing k by 2k in equation (3.3.16) gives a proof of equation (3.0.8), i.e., equation +(1.2.89) of Lemma 1.3, in the GUE case and confirms our claim following Figure 3.1 that +2kM(GUE) +k,l +counts the number of orientable genus l/2 ribbon graphs that can be constructed +from a 2k-gon. Furthermore, since g ∈ N and l = 2g, M(GUE) +k,l += 0 for l odd and m(GUE) +2k +is consequently an odd polynomial in N, in keeping with the second point of Remark 2.1. +Finally, let us point out that the interpretation of M(GUE) +k,l +in terms of ribbon graphs extends +to a representation of the generating functions W(GUE),l +1 +(x) (recall the discussion at the +beginning of §3.1.2) as genus l/2 compact orientable surfaces with one hole — if one can +replace the hole of such a representation of W(GUE),l +1 +(x) by a 2k-gon for some positive integer +k and connect the edges of the 2k-gon with ribbons drawn on the surface without self- +intersections such that excising the resulting ribbon graph yields a disjoint union of sets +homeomorphic to open disks, then that ribbon graph contributes to the value of M(GUE) +k,l +. +Figure 3.7: The sphere with a hole on the left represents W(GUE),0 +1 +(x). On the right, we have +a collection of ribbon graphs drawn on this sphere with the hole replaced by a suitable +2k-gon. The illustrations on the right contribute to the value of M(GUE) +k,0 +with k = 1, 2, 2, 3, +respectively. Therefore, they also contribute to the coefficients of the 1/x expansion of +W(GUE),0 +1 +(x) since the latter is essentially a generating function for the M(GUE) +k,0 +. +160 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +Remark 3.3. +1. Recalling definition (3.1.19) of ˜m(G) +2k , Proposition 3.9 tells us that +˜m(GUE) +2k += 2−k +k +∑ +l=0 +N−l #{σ ∈ P2k | g(σ) = l/2}. +Thus, from equation (1.1.32), +˜M(GUE) +2k,0 += limN→∞ ˜m(GUE) +2k += M(GUE) +k,0 +is equal to 2−k +multiplied by the number of genus zero (i.e., planar) ribbon graphs that can be built +from a 2k-gon. Hence, from the discussion contained in the caption of Figure 3.2 and +the paragraph preceding it, 2k ˜M(GUE) +2k,0 +is equal to the kth Catalan number. Consequently, +W(GUE),0 +1 +(x), as described below equation (3.1.19), is essentially a generating function +for the Catalan numbers, which is well known [290] to be given by +W(GUE),0 +1 +(x) = 1 +2(x − +� +x2 − 2) +after appropriate scaling (see also differential equation (2.4.17)). Of course, applying +the Sokhotski–Plemelj inversion formula (1.1.25) yields the Wigner semi-circle law +(1.2.14). This proof of the Wigner semi-circle law is a rewording of Wigner’s original +proof [313], which used the so-called method of moments. +2. The above proof can be extended to show that the Wigner semi-circle law holds for the +Hermitian Wigner matrix ensemble, which is represented by the same matrices as the +GUE except that the moments ⟨Hp1+1 +ij +Hp2+1 +ji +⟩ (p1, p2 ∈ N) no longer need to be zero +whenever p1, p2 > 0. Setting rigour aside (see, e.g., [21, Ch. 2] for a detailed treatment), +the idea is that when H is a Hermitian Wigner matrix, the right-hand side of equation +(3.3.5) needs to be replaced by a sum over all partitions of k, rather than just pair +partitions. As an example, the contribution of the terms ⟨Hi1i2Hi2i3⟩ ⟨Hi3i4Hi4i5Hi5i6Hi6i1⟩ +and ⟨Hi1i2Hi2i3Hi3i4⟩ ⟨Hi4i5Hi5i6Hi6i1⟩ to the value of ⟨Hi1i2Hi2i3 · · · Hi5i6Hi6i1⟩ must now +be taken into account. However, since these terms are interpreted as products of +indicator functions which constrain our summation indices i1, . . . , i6, only pair partitions +contribute at leading order — ⟨Hi1i2Hi2i3Hi3i4Hi4i5⟩ is non-zero only if i3, i4, i5 depend +on i1, i2 while ⟨Hi1i2Hi2i3⟩ ⟨Hi3i4Hi4i5⟩ being non-zero merely constrains i3, i5 to depend +on i1, i2, i4. Thus, at leading order in N, the spectral moments of the Hermitian Wigner +matrix H are given by the Catalan numbers up to some normalisation factor, and the +argument given above follows through. +161 + +CHAPTER 3. Characterisations of the Moments and Cumulants +Ribbon graphs for the GUE mixed moments and cumulants +Letting H be again drawn from the N × N GUE, the mixed moments of the GUE (3.0.13) are +given by +m(GUE) +k1,...,kn = +� +n +∏ +i=1 +Tr Hki +� +, +k1, . . . , kn ∈ N. +The analogue of equation (3.3.4) is then +m(GUE) +k1,...,kn = +N +∑ +i(1) +1 ,...,i(1) +k1 =1 +· · · +N +∑ +i(n) +1 ,...,i(n) +kn =1 +�� +Hi(1) +1 i(1) +2 Hi(1) +2 i(1) +3 · · · Hi(1) +k1 i(1) +1 +� +· · · +· · · +� +Hi(n) +1 i(n) +2 Hi(n) +2 i(n) +3 · · · Hi(n) +kn i(n) +1 +�� +. +(3.3.17) +The summand on the right-hand side of this equation is the expected value of a product +of centred normal variables, so it can be simplified using the Isserlis–Wick theorem, in a +similar manner to equation (3.3.5). Thus, m(GUE) +k1,...,kn vanishes if k1 + · · · + kn is an odd integer, +whereas for k1 + · · · + kn an even integer, we can express m(GUE) +k1,...,kn in terms of ribbon graphs: +For each j = 1, . . . , n, the set {Hi(j) +1 i(j) +2 , Hi(j) +2 i(j) +3 , . . . , Hi(j) +kj i(j) +1 } is represented by a kj-gon with +vertices labelled i(j) +1 , i(j) +2 , . . . , i(j) +kj in clockwise order, while the factors ⟨Hi(j) +p i(j) +p+1Hi(l) +q i(l) +q+1⟩ (setting +i(j) +kj+1 := i(j) +1 ) in the sum of products of covariances produced by the Isserlis–Wick theorem +are represented by untwisted ribbons connecting the appropriate polygon edges. +Figure 3.8: In computing m(GUE) +4,3,3 += +� +Tr(H4) Tr(H3) Tr(H3) +� +, one first draws a square and +two triangles with vertices labelled as shown. Then, one must list out all possible ways of +pairing the polygon edges using untwisted ribbons. The illustrated planar ribbon graph +represents ⟨Hi(1) +1 i(1) +2 +Hi(1) +2 i(1) +3 ⟩⟨Hi(1) +3 i(1) +4 +Hi(1) +4 i(1) +1 ⟩⟨Hi(2) +2 i(2) +3 +Hi(2) +3 i(2) +1 ⟩⟨Hi(2) +1 i(2) +2 +Hi(3) +3 i(3) +1 ⟩⟨Hi(3) +1 i(3) +2 +Hi(3) +2 i(3) +3 ⟩, +which, when summed over the indices, contributes a value of N6 to m(GUE) +4,3,3 +. +As a slight abuse of notation, let us now define Pk1,...,kn, for positive integers k1, . . . , kn +such that k1 + · · · + kn is even, to be the set of all 1 +2(k1 + · · · + kn)-ribbon graphs that can be +162 + +z(1 +2 +1 +1 +(3)3.3. The Isserlis–Wick Theorem and Ribbon Graphs +constructed by drawing n polygons with respectively k1, k2, . . . , kn edges and then connecting +said edges with untwisted ribbons. Then, equation (3.3.14) extends to the form +m(GUE) +k1,...,kn = 2− 1 +2 (k1+···+kn) +1 +2 (k1+···+kn)+n +∑ +V=1 +NV #{Γ ∈ Pk1,...,kn | V(Γ) = V}, +(3.3.18) +where V(Γ) is equal to the number of boundaries of Γ, which is equivalent to the number +of distinct polygon vertices retained by Γ upon collapsing its ribbons in order to identify +their ends (e.g., in Figure 3.8 above, collapsing the ribbons `a la Figure 3.5 results in there +being six distinct vertices labelled by i(1) +1 +≡ i(1) +3 , i(1) +2 , i(1) +4 , i(2) +1 +≡ i(2) +2 +≡ i(3) +1 +≡ i(3) +3 , i(2) +3 +and i(3) +2 ). +When we discussed the n = 1 case earlier, we saw that the equivalent function V(σ) could +be expressed in terms of topological invariants of the ribbon graph corresponding to σ. We +would now like to see if such relations can be replicated in the general-n case and moreover +justify the upper limit of the sum in equation (3.3.18). +Let Γ ∈ Pk1,...,kn and let Σ denote the surface obtained by collapsing the ribbons of Γ such +that the ribbon-ends are identified. As in the n = 1 case, the Euler characteristic of Σ is given +by the classical formula χ = F − E + V, where F, E, V are again the number of faces, edges, +and vertices of any polygonisation of Σ. Polygonising Σ by the polygons of Γ with their +edges identified according to the ribbon data of Γ, we have F = n, the number of polygons +in Γ, and E = (k1 + · · · + kn)/2, half the total number of polygon edges in Γ. Hence, the +number of vertices, V = V(Γ), is given by +V(Γ) = χ + 1 +2(k1 + · · · + kn) − n. +(3.3.19) +Writing C(Σ) for the number of connected components of Σ and gi ⩾ 0 for the genus of +the ith such component, we have by the additive property of the Euler characteristic that +χ = ∑ +C(Σ) +i=1 (2 − 2gi). It follows that χ ⩽ 2n since 1 ⩽ C(Σ) ⩽ n. Combined with the above +equation, this means that V(Γ) ⩽ 1 +2(k1 + · · · + kn) + n, which is precisely the upper limit +of the sum in equation (3.3.18). However, unlike in the n = 1 case, this upper limit cannot +always be attained. Indeed, if any of the ki are odd, the corresponding polygon must be +ribbon-connected to at least one other polygon, so Γ, equivalently Σ, cannot have n connected +components. This slight complication in determining the degree of m(GUE) +k1,...,kn as a polynomial +in N, in addition to the reasons listed below, suggests that we should somehow focus on +studying connected ribbon graphs. +163 + +CHAPTER 3. Characterisations of the Moments and Cumulants +1. If we define +Pc +k1,...,kn := {Γ ∈ Pk1,...,kn | Γ is connected}, +we have by the arguments above that for all Γ ∈ Pc +k1,...,kn, +V(Γ) ⩽ 1 +2(k1 + · · · + kn) + 2 − n. +Furthermore, this relation is a strict equality if Γ is planar (genus zero) — it is easy to +convince oneself that there exists at least one such Γ in each Pc +k1,...,kn. +2. Connected surfaces have well-defined genera, so we can rewrite the analogue of +equation (3.3.18) with Pk1,...,kn replaced by Pc +k1,...,kn as a genus expansion, in a similar +fashion to equation (3.3.16). +3. No generality is lost by studying connected ribbon graphs as every ribbon graph is a +countable disjoint union of its connected components, which is easy to construct. +To expand on the third point given above, it is well known to combinatorialists [290, Ch. 5] +that if the mixed moment mk1,...,kn on the left-hand side of the moment-cumulants relation +(3.0.14) is the weighted count of some structures of a given type (graphs, marked surfaces, +etc.) built from n objects, then the corresponding cumulants ck1,...,kn are the analogous counts +of the subsets of such structures that are connected. This can be understood in the context +of ribbon graphs from the following inductive argument: If we assume for all p < n and +k1, . . . , kp ∈ N such that k1 + · · · + kp is an even integer that the mixed cumulants ck1,...,kp +enumerate (with suitable weights) the number of connected ribbon graphs that can be built +from p polygons with respectively k1, . . . , kp sides, then the product ck1,...,kpcl1,...,lq (q < n) +counts the number of ribbon graphs consisting of two connected components with the first +being a ribbon graph built from p polygons with k1, . . . , kp sides and the second component +being a ribbon graph built from q polygons with l1, . . . , lq sides. Extending this reasoning to +general products, we see that the sum +mk1,...,kn − ck1,...,kn = +∑ +K⊢{k1,...,kn} +K̸={{k1,...,kn}} +∏ +κi∈K +cκi +counts the number of disconnected ribbon graphs that can be constructed from n polygons +with k1, . . . , kn sides. Since mk1,...,kn enumerates both the disconnected and connected ribbon +164 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +graphs that can be built from n polygons with k1, . . . , kn sides, it follows that ck1,...,kn must +enumerate all of the connected ribbon graphs that can be constructed in said manner. +Remark 3.4. If mk1,...,kn vanishes whenever k1 + · · · + kn is odd, so too does ck1,...,kn. This can +be seen from the inverse of the moment-cumulants relation (cf. equation (1.1.30)), +ck1,...,kn = +∑ +K⊢{k1,...,kn} +(−1)#K−1(#K − 1)! ∏ +κi∈K +mκi. +(3.3.20) +Having established that we are interested in enumerating connected, orientable ribbon +graphs and that these enumerations are given by the mixed cumulants of the GUE, let us +now give the general-n analogue of Proposition 3.9. +Proposition 3.10. Let n, k1, . . . , kn ∈ N be such that k1 + · · · + kn is even and let Pc +k1,...,kn be the +set of connected, orientable 1 +2(k1 + · · · + kn)-ribbon graphs built from n polygons with respectively +k1, . . . , kn sides. The mixed cumulant c(GUE) +k1,...,kn is given by +c(GUE) +k1,...,kn = +� N +2 +� 1 +2 (k1+···+kn) 1 +2 (k1+···+kn)+1−n +∑ +l=0 +N2−n−l #{Γ ∈ Pc +k1,...,kn | g(Γ) = l/2}, +(3.3.21) +where g(Γ) is the genus of Γ. +Proof. First, note that if Γ ∈ Pc +k1,...,kn, it is connected with Euler characteristic χ = 2 − 2g, so +equation (3.3.19) reads +V(Γ) = 1 +2(k1 + · · · + kn) + 2 − n − 2g ⩽ 1 +2(k1 + · · · + kn) + 2 − n. +(3.3.22) +Thus, the analogue of equation (3.3.18) corresponding to connected ribbon graphs is +c(GUE) +k1,...,kn = 2− 1 +2 (k1+···+kn) +1 +2 (k1+···+kn)+2−n +∑ +V=1 +NV #{Γ ∈ Pc +k1,...,kn | V(Γ) = V}. +(3.3.23) +Setting l = 2g and changing summation index according to equation (3.3.22) in equation +(3.3.23) concludes the proof. +In comparing equations (3.3.18) and (3.3.21), we see that a benefit of studying the mixed +cumulants is that upon scaling them by a factor of N−(k1+···+kn)/2 (equivalent to considering +the cumulants of the GUE scaled by replacing the matrix H in P(G)(H) (1.2.1) by +√ +NH as is +consistent with Definition 1.6), they are O(N2−n), while the mixed moments are O(Nn) when +165 + +CHAPTER 3. Characterisations of the Moments and Cumulants +scaled in this manner. Thus, the connected n-point correlators ˜W(GUE) +n +of the global scaled +GUE, which are generating functions for the scaled cumulants, have large N expansions of +the form given in Theorem 1.2, whereas the corresponding unconnected n-point correlators +˜U(GUE) +n +(1.1.26) do not. In keeping with the discussion surrounding Figure 3.7, let us also +mention that it is convenient to represent the correlator expansion coefficients W(GUE),l +n +defined in Theorem 1.2 as genus l/2 compact, connected, orientable surfaces with n holes. +Ribbon graphs in the GOE case +The calculations pertaining to the GUE that have been outlined thus far in this subsection +follow through in much the same way when H is drawn from the N × N Gaussian orthogonal +ensemble. Indeed, when H is an N × N GOE matrix, the right-hand side of equation (3.3.4) +now equals m(GOE) +k +with the summand still simplifying to the form given in equation (3.3.5) +when k is a positive even integer (and vanishing if k is odd). The key difference in studying +the GOE is that equation (3.3.2) must be replaced with +� +HijHkl +� = 1 +4 +� +χk=jχl=i + χk=iχl=j +� +, +(3.3.24) +which reflects the fact that H is now a real symmetric matrix with off-diagonal entries having +variance 1/4 (when i ̸= j, ⟨H2 +ij⟩ = +� +HijHji +� = 1/4) and diagonal entries having variance 1/2. +To ease notation, let us define +ξkl +ij (t) := (1 − t)χk=jχl=i + tχk=iχl=j +(3.3.25) +so that the right-hand side of equation (3.3.24) is equal to 1 +4[ξkl +ij (0) + ξkl +ij (1)] (the parameter +t can be thought of as keeping track of orientability; cf. the role of γ in [166]). Then, +substituting equation (3.3.24) into equation (3.3.5) shows that for k a positive even integer, +� +Hi1i2Hi2i3 · · · Hik−1ik Hiki1 +� = 2−k ∑ +σ∈Pk +1 +∑ +t1,...,tk/2=0 +ξ +iσ(2)iσ(2)+1 +iσ(1)iσ(1)+1(t1) · · · ξ +iσ(k)iσ(k)+1 +iσ(k−1)iσ(k−1)+1(tk/2). +(3.3.26) +As in the GUE case, we now encode the index-matching of the set {Hi1i2, Hi2i3, . . . , Hiki1} by +drawing a k-gon with vertices labelled i1, i2, . . . , ik in clockwise order, and then represent +the edge-matching implied by the summand of equation (3.3.26) by drawing k/2 ribbons +connecting the edges of said k-gon. The factors ξ +iσ(2l)iσ(2l)+1 +iσ(2l−1)iσ(2l−1)+1(tl) (1 ⩽ l ⩽ k/2) in the +166 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +right-hand side of equation (3.3.26) are equivalent to the products χiσ(2l−1)=iσ(2l)+1χiσ(2l)=iσ(2l−1)+1 +seen in equation (3.3.3) when we set tl = 0. Thus, the term ξ +iσ(2l)iσ(2l)+1 +iσ(2l−1)iσ(2l−1)+1(0) is represented +by an untwisted ribbon connecting the oriented polygon edges (iσ(2l−1) → iσ(2l−1)+1) and +(iσ(2l)+1 → iσ(2l)). On the other hand, the term ξ +iσ(2l)iσ(2l)+1 +iσ(2l−1)iσ(2l−1)+1(1) is non-zero if the edge +(iσ(2l−1) → iσ(2l−1)+1) is identified to (iσ(2l) → iσ(2l)+1), so we represent it by a M¨obius +half-twisted ribbon connecting these two edges. +Figure 3.9: These ribbon graphs represent the term ξi2i3 +i1i2(t1)ξi4i1 +i3i4(t2) for various choices of +t1, t2 ∈ {0, 1}, with the purple (grey) ribbons representing the first (second) factor. In order, +the graphs correspond to setting (t1, t2) = (0, 1), (1, 0), and (1, 1); they can be obtained by +twisting either or both of the ribbons of the left image of Figure 3.4, which incidentally +corresponds to setting (t1, t2) = (0, 0). +Extending the above reasoning to products of the form presented in the summand of +the right-hand side of equation (3.3.17) shows that computing the mixed moments and +cumulants of the GOE amounts to enumerating locally orientable ribbon graphs of specific +types. The necessary arguments for obtaining explicit expressions for these moments and +cumulants are the same as in the GUE case, with the key numeric still being the number of +boundaries of each ribbon graph. Without reiterating any further details, let us simply state +the GOE analogue of Proposition 3.10. +Proposition 3.11. Let n, k1, . . . , kn ∈ N be such that k1 + · · · + kn is even and let ˜Pc +k1,...,kn be the set +of all connected, locally orientable 1 +2(k1 + · · · + kn)-ribbon graphs that can be built from n polygons +with respectively k1, . . . , kn edges. The mixed cumulant c(GOE) +k1,...,kn is a degree 1 +2(k1 + · · · + kn) + 2 − n +polynomial in N given by the formula +c(GOE) +k1,...,kn = +� N +4 +� 1 +2 (k1+···+kn) 1 +2 (k1+···+kn)+1−n +∑ +l=0 +N2−n−l #{Γ ∈ ˜Pc +k1,...,kn | ˜g(Γ) = l}, +(3.3.27) +where ˜g(Γ) is the Euler genus of Γ (recall Definition 3.1). +167 + +i1 +i1 +i2 +i4 3 +i3 +i4 +i3 +i4CHAPTER 3. Characterisations of the Moments and Cumulants +Remark 3.5. The set ˜Pc +k1,...,kn defined above is in bijection with Pc +k1,...,kn × {0, 1}(k1+···+kn)/2 +since Γ ∈ ˜Pc +k1,...,kn if and only if it is the result of twisting the ribbons of some ribbon graph +drawn from Pc +k1,...,kn; the tuple (t1, t2, . . . , t(k1+···+kn)/2) ∈ {0, 1}(k1+···+kn)/2 determines which +ribbons are to be twisted. +Upon setting n = 1, comparing equation (3.3.27) to equation (3.0.8) confirms that M(GOE) +k,l +has the combinatorial interpretation described in the paragraph below Figure 3.1 (as an +aside, observe that the ribbon graph in Figure 3.1 is genus zero and thus contributes a value +of N2 to the computation of c(GOE) +1,2,3 +). Furthermore, the equality +{Γ ∈ ˜Pc +k | ˜g(Γ) = 0} = {Γ ∈ Pc +k | g(Γ) = 0} +implies that the moment expansion coefficients M(GUE) +k,0 +and M(GOE) +k,0 +differ only by a factor +of 2k/2, so the proof of Remark 3.3 can be used to show that the global scaled eigenvalue +density of the GOE is given by the Wigner semi-circle law (1.2.14). +A key difference between c(GUE) +k1,...,kn and c(GOE) +k1,...,kn is that, since the (Euler) genus is a non- +negative integer, the summand of the expression (3.3.21) for c(GUE) +k1,...,kn is identically zero for +odd values of l, but this is not true for the corresponding expression (3.3.27) for c(GOE) +k1,...,kn. +Hence, unlike in the GUE case, the correlator expansion coefficients W(GOE),l +n +do not vanish +for odd values of l. Another difference between W(GUE),l +n +and W(GOE),l +n +is that it does not +make sense to represent the latter as a compact surface on which to draw ribbon graphs (like +in Figure 3.7) since it is a generating function for cumulants that count both orientable and +non-orientable ribbon graphs. +Relations to topological and combinatorial maps and hypermaps +It is oftentimes beneficial, either for visualisation or computational purposes, to consider +alternative representations of ribbon graphs. We now outline the connection between ribbon +graphs and two such representations: topological and combinatorial maps [303], [220], [219]. +Definition 3.2. A topological map is a graph embedded in a compact surface such that the +edges of the graph do not intersect and excising the graph from the surface results in a +disjoint union of open sets that are homeomorphic to open disks — these open sets are +referred to as the faces of the topological map. +168 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +Note 3.2. The second condition in the above definition requires that for a given embedding to +be a valid topological map, each connected component of the graph must live on a separate +connected component of the surface. Most authors require topological maps, hence the +involved graphs, to be connected. In such settings, it suffices to define a topological map as +the embedding of a graph in a connected compact surface of minimal (Euler) genus such +that the edges of the graph do not intersect. +There are two natural ways of transforming a ribbon graph into a topological map. In +the first formalism, one glues open disks to the boundaries of a given ribbon graph and +then contracts the polygons of the ribbon graph to points which become the vertices of a +topological map. Shrinking the polygons to vertices also has the consequence of thinning the +ribbons to edges connecting said vertices. The disks that were initially glued to the ribbon +graph become the faces of the topological map. +Figure 3.10: As in Figure 3.6, we glue three open disks (pink, green, and blue) to the +boundaries of the ribbon graph on the left in order to embed it into a sphere (middle image). +Shrinking the white square to a vertex and the ribbons to edges incident to said vertex +produces the topological map on the right. Our usual convention, which we break here, +will be to colour the vertices black and the faces white. +In the second formalism, one also glues disks to the boundaries of the given ribbon graph, +but then contracts these disks to vertices. The ribbons then collapse to their ends, which +become the edges of the topological map, and the polygons of the ribbon graph become +the faces of the topological map. This construction was already exemplified in Figure 3.6. +The two formalisms just described lead to topological maps that are dual to each other in +the sense that canonically follows from the duality principle well known in standard graph +theory: one can be obtained from the other by placing a vertex at the centre of every face, +drawing an edge between each pair of new vertices whenever the corresponding faces of +the original topological map share an edge, and then deleting the edges and vertices of the +169 + +CHAPTER 3. Characterisations of the Moments and Cumulants +original topological map. For example, the topological maps of Figures 3.6 and 3.10 are dual. +For the remainder of this subsection, when we refer to the topological map corresponding to +a ribbon graph, we will mean the one constructed via the first formalism (i.e., that illustrated +in Figure 3.10). +It is important to observe at this point that neither of the above constructions are injective. +For example, the first two ribbon graphs displayed in Figure 3.4, which can be distinguished +due to the labelling of the polygon vertices (that we often leave implicit), produce the same +topological map. In the case of orientable ribbon graphs, we are able to turn the first +construction into a bijection by additionally marking a half-edge of each of the n vertices in +a systematic manner, thereby forming a so-called n-rooted topological map [163]. One possible +convention is to mark the half-edges corresponding to the ribbon-ends (i(l) +1 +→ i(l) +2 ) and label +them by l; see [219], [277] for more comprehensive reviews of (1-)rooted topological maps. +Figure 3.11: These rooted topological maps are constructed from the ribbon graphs of +Figure 3.4 using the first prescribed formalism. The half-edges that are root-marked red +correspond to the ribbon ends that are glued to the polygon edges (i1 → i2). +In the case of locally orientable ribbon graphs, one also needs to keep track of local +orientation at the vertices and which edges of a given topological map correspond to twisted +ribbons. This can be done by assigning orientation to the half-edge root-markings and +labelling each edge by, say, ±1, with +1 indicating that the edge represents an untwisted +ribbon and −1 instead referring to a M¨obius half-twisted ribbon (cf. [252]). Seeing as how +introducing markings and labellings to give a combinatorial flavour to our topological +maps makes the theory more tractable, one might wonder if much more can be gained +by introducing even more labels. It turns out that it is possible to transition to a fully +combinatorial theory, removing the need for explicitly constructing graphs or surfaces +altogether. Thus, we are led to combinatorial maps, for which there are multiple definitions. +170 + +1 +03.3. The Isserlis–Wick Theorem and Ribbon Graphs +Definition 3.3 (Tutte ’84). Following [303], [161, §17.10], a combinatorial map is a quadruple +(EQ, τ0, τ1, τ2) where +1. EQ is a finite set with the number of elements being divisible by four, +2. τ0, τ1, τ2 are fixed-point free involutions on EQ, +3. τ0τ1 = τ1τ0 and τ0τ1 is also fixed-point free, +4. the group ⟨τ0, τ1, τ2⟩ generated by τ0, τ1, τ2 is transitive, meaning that it has exactly one +orbit, that being all of EQ. +The elements of EQ are called quarter-edges. The orbits of τ0, ⟨τ0, τ1⟩, ⟨τ0, τ2⟩, and ⟨τ1, τ2⟩ are +respectively referred to as the half-edges, edges, vertices, and faces of the combinatorial map. +Combinatorial maps are in one-to-one correspondence with connected, n-rooted topologi- +cal maps that have been edge-labelled ±1 in the aforementioned manner (the connectedness +being enforced by the fourth condition in the above definition). This correspondence follows +from the suggestive names given to the elements of EQ and the various orbits highlighted +in Definition 3.3: Given a suitable topological map, one must first divide each edge both +width- and length-wise to form quarter-edges and then assign these quarter-edges distinct +labels. Then, the corresponding combinatorial map has for EQ the set of said quarter-edges, +while the involutions τ0, τ1, τ2 on EQ are such that τ0 interchanges two quarter-edges if and +only if they belong to the same half-edge, τ1 interchanges two quarter-edges if and only if +they belong to the same length-wise half-edge, and τ2 interchanges two quarter-edges if and +only if they are incident to the same vertex without belonging to the same half-edge and are +consecutive to each other in the cyclic ordering around that vertex. The two possible choices +of (local) orientation at the vertices of the topological map are encoded in the cycles of τ0τ2. +Example 3.3. The combinatorial map corresponding to Figure 3.12 below is (EQ, τ0, τ1, τ2) +with +EQ = {1, 1′, 2, 2′, 3, 3′, 4, 4′}, +τ0 = (1, 1′)(2, 2′)(3, 3′)(4, 4′), +τ1 = (1, 2′)(1′, 2)(3, 4′)(3′, 4), +τ2 = (1, 4′)(1′, 2)(2′, 3)(3′, 4). +Observe that the first cycle of τ0τ2 = (1, 4, 3, 2)(1′, 2′, 3′, 4′) describes an anticlockwise order- +ing of the half-edges incident to the single vertex of the topological map, while the second +171 + +CHAPTER 3. Characterisations of the Moments and Cumulants +cycle of τ0τ2 describes the opposite ordering. These cyclic orderings each prescribe a choice +of orientation for the topological map. +Figure 3.12: We divide the ribbons of a ribbon graph (left) and edges of the corresponding +topological map (right) into quarters that are labelled 1, 1′, 2, 2′, 3, 3′, 4, 4′ in clockwise order. +The half-edge marked red and oriented clockwise corresponds to the ribbon-end (i1 → i2), +while the blue plus signs signify that these edges represent untwisted ribbons. +A combinatorial map is (globally) orientable if and only if ⟨τ0τ1, τ0τ2⟩ has two distinct +orbits [303]. If this is the case, one may remove the redundancy of τ0 by identifying quarter- +edges whenever they belong to the same half-edge. In the context of Figure 3.12, this amounts +to setting j′ ≡ j for j = 1, . . . , 4. Identifying quarter-edges according to τ0 induces a choice of +orientation and moreover leads to a simpler combinatorial theory on the set of half-edges +EH ≃ EQ/τ0, which has half as many elements as that considered in Definition 3.3. This +simplification is to be expected since every compact surface has an orientable double cover +[176], so a theory allowing for non-orientable objects should intuitively require twice as +many labels as an equivalent theory focusing solely on orientable structures. +Definition 3.4 (Edmonds ’60). Following [101], [220], an oriented combinatorial map is a triple +(EH, τe, τv) where +1. EH is a finite set with an even number of elements, +2. τv is a permutation of EH and τe is a fixed-point free involution on EH, +3. the group ⟨τe, τv⟩ is transitive. +The elements of EH are called half-edges. The cycles of τe, τv, and τ−1 +v τ−1 +e +are respectively +called the edges, vertices, and faces of the oriented combinatorial map. +172 + +:1 +i2 +2 +4' +4 +2' +2 +i4 +4 +i3 +3' +33.3. The Isserlis–Wick Theorem and Ribbon Graphs +This definition is preferred to Definition 3.3 in the orientable case because the correspon- +dence between oriented combinatorial maps and orientable, connected, n-rooted topological +maps is somehow more transparent. Indeed, if one labels the half-edges of such a topological +map and settles on a choice of orientation, then the corresponding oriented combinatorial +map has EH being the set of labelled half-edges, τe being the involution that interchanges +half-edges that belong to the same edge, and τv being a product of disjoint cycles where each +cycle describes the cyclic ordering of the half-edges incident to a vertex that is induced by +the orientation of the topological map. Each cycle of the face permutation τ−1 +v τ−1 +e +describes +a face of the topological map by listing out the first half of each edge that borders the face +when traversing in the direction determined by the choice of orientation. +Figure 3.13: Illustrated is an oriented planar graph with labelled half-edges. It should be +thought of as being embedded in the sphere and thus equivalent to an oriented topological +map. The corresponding oriented combinatorial map is (EH, τe, τv) with EH = {1, 2, . . . , 8}, +τe = (1, 2)(3, 5)(4, 6)(7, 8), and τv = (1, 8)(2, 3, 4)(5, 7, 6). Note that the cycles of τv comply +with the orientation of the topological map. Likewise, the first, second, and third cycle +of the face permutation τ−1 +v τ−1 +e += (1, 4, 7)(2, 8, 5)(3, 6) respectively list out, in appropriate +cyclic order, the blue, black, and yellow halves of the edges bordering the triangular face, +the ‘external face’, and the bigonal face. +Let us now briefly review the connection between the structures discussed above and +their hypermap analogues. A topological hypermap is simply a connected topological map +whose vertices are coloured black and white (later, we use red) in such a way that each +edge connects a black vertex to a white one [220]. In that vein, a combinatorial hypermap is +the triple of Definition 3.4 except that τe is now free to be any permutation of EH; EH is +then the set of edges of the hypermap, while the cycles of τv (τe) are interpreted as black +vertices (white vertices or so-called hyperedges). Thus, an oriented combinatorial map is a +173 + +3 +2 +4 +1 +8 +6 +7 +5CHAPTER 3. Characterisations of the Moments and Cumulants +combinatorial hypermap with τe constrained to be a fixed-point free involution. Similarly, a +topological hypermap with all white vertices constrained to be bivalent is equivalent to a +connected topological map — the edges of the hypermap are interpreted as half-edges of the +topological map so that each white vertex sits at the middle of a map-edge. +Remark 3.6. In the literature surrounding Grothendieck’s dessins d’enfants [284], a topological +hypermap is referred to as a dessin d’enfant, while a topological map — seen as a topological +hypermap with bivalent white vertices — is referred to as a clean dessin d’enfant. +3.3.2 +Moments of the Laguerre unitary and orthogonal ensembles +Since the Laguerre unitary and orthogonal ensembles are represented by matrices that +can be expressed as products of Ginibre matrices, whose entries are normally distributed, +their (mixed) moments and cumulants can also be studied using the Isserlis–Wick theorem. +Assuming mastery over the Gaussian case, as outlined in the previous subsection, we first +show how the spectral moments m(LUE) +k +can be written as sums over certain bicoloured +ribbon graphs. Then, as in the Gaussian case, we discuss how the presented ideas extend to +the mixed cumulants of the LUE and also the LOE. +Ribbon graphs for the LUE spectral moments +Let G be drawn from the M × N complex Ginibre ensemble and throughout this subsection, +let ⟨ · ⟩ denote averages with respect to the p.d.f. P(Gin)(G) given in Definition 1.3. According +to Definition 1.4, the N × N Wishart–Laguerre matrix W = G†G then represents the (M, N) +Laguerre unitary ensemble and the spectral moments of this ensemble are given by (1.1.15) +m(LUE) +k += +� +Tr Wk� +, +k ∈ N += +� +Tr (G†G)k� += +N +∑ +i1,...,ik=1 +M +∑ +j1,...,jk=1 +� +(G†)i1j1Gj1i2(G†)i2j2Gj2i3 · · · (G†)ikjkGjki1 +� +, +(3.3.28) +in analogy with equation (3.3.4). Since the real components of the entries of G are indepen- +dent, centred, normal variables such that +� +(G†)ijGkl +� += χi=lχj=k, +(3.3.29) +174 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +the Isserlis–Wick theorem then shows that +� +(G†)i1j1Gj1i2 · · · (G†)ikjkGjki1 +� += ∑ +σ∈Sk +k +∏ +l=1 +� +(G†)iljlGjσ(l)iσ(l)+1 +� += ∑ +σ∈Sk +k +∏ +l=1 +(χil=iσ(l)+1χjl=jσ(l)), +(3.3.30) +where Sk is the set of all permutations of {1, . . . , k} and we have defined ik+1 := i1. Hence, +combining equations (3.3.28) and (3.3.30) gives the analogue of Lemma 3.1 for the LUE. +Lemma 3.3. Fix k ∈ N, let Sk be the set of permutations of {1, . . . , k} and set ik+1 := i1. We have +that the spectral moments of the LUE are given by +m(LUE) +k += ∑ +σ∈Sk +N +∑ +i1,...,ik=1 +M +∑ +j1,...,jk=1 +k +∏ +l=1 +(χil=iσ(l)+1χjl=jσ(l)). +(3.3.31) +As one might expect, the summand of the right-hand side of equation (3.3.31) can be +represented by ribbon graphs. To do so, we first draw a 2k-gon and label its vertices +i1, j1, i2, j2, . . . , ik, jk in clockwise order. Next, to distinguish the two index sets, we colour +the vertices labelled by the i1, . . . , ik black and those labelled by the j1, . . . , jk red. Then, for +1 ⩽ l ⩽ k, edges of the form (il → jl) represent (G†)iljl, while those of the form (jl → il+1) +represent Gjlil+1. The factors in the summand of the right-hand side of equation (3.3.31) are +thus represented by untwisted ribbons connecting these two types of edges. The condition +that the ribbons cannot join edges of the same type is equivalent to that of only allowing +ribbon graphs that respect the colouring of the vertices. +Figure 3.14: The illustrated ribbon graphs represent the terms given in equations (3.3.33) +and (3.3.34) below. Having alternately coloured the vertices of the squares black and red +according to the formalism described above, we see that the vertex colourings induce +colours on the boundaries of the ribbon graph. The number of black (red) boundaries is +equal to the exponent of N (M) in the evaluations of S′ +id and S′ +(1,2). +175 + +11 +J1 +12 +i2 +12 +i2CHAPTER 3. Characterisations of the Moments and Cumulants +Example 3.4. Let us now give the LUE analogue of Example 3.2, wherein we saw that m(GUE) +4 +is equal to (N3 + N3 + N)/4, with each of the terms N3, N corresponding to a ribbon graph +of Figure 3.4. In the LUE case, we compute the second spectral moment m(LUE) +2 +. By equation +(3.3.31), it is given by +m(LUE) +2 += S′ +id + S′ +(1,2) +(3.3.32) +with +S′ +id := +N +∑ +i1,i2=1 +M +∑ +j1,j2=1 +(χi1=i2χj1=j1)(χi2=i1χj2=j2) = M2N, +(3.3.33) +S′ +(1,2) := +N +∑ +i1,i2=1 +M +∑ +j1,j2=1 +(χi1=i1χj1=j2)(χi2=i2χj2=j1) = MN2. +(3.3.34) +The evaluation S′ +id = M2N follows from the fact that the product of indicator functions +in equation (3.3.33) enforces the equivalence i1 ≡ i2, but otherwise does not constrain the +indices j1, j2 — similarly for S′ +(1,2). These equivalences of summation indices can be read off +from the ribbon graphs of Figure 3.14 above. +In general, the kth spectral moment for the LUE is given by +m(LUE) +k += ∑ +σ∈Sk +MVr(σ)NVb(σ), +(3.3.35) +where Vr(σ) and Vb(σ) are equal to the number of red, respectively black, boundaries of the +ribbon graph corresponding to σ — the proof of this fact is the same as for Lemma 3.2. As +with the GUE case, we are able to reformulate equation (3.3.35) as a genus expansion, so +long as the red and black vertices of the involved ribbon graphs are placed on equal footing. +Proposition 3.12. Let k ∈ N and Sk be as in Lemma 3.3 Setting the Laguerre parameter a = ˆaN +with ˆa = O(1) so that M = (ˆa + 1)N, as per the convention set out in §2.4.1 and §3.1.2, we have +m(LUE) +k += +k +∑ +l=0 +N1+k−l +k +∑ +p=1 +(ˆa + 1)p #{σ ∈ Sk | g(σ) = l/2 and Vr(σ) = p}, +(3.3.36) +where g(σ) is again the genus of the ribbon graph corresponding to σ — equivalently, g(σ) is +the minimal genus of the surface on which this ribbon graph can be embedded into without self- +intersections and it is also the genus of the surface obtained by pairwise identifying the edges of a +2k-gon according to the factors (χil=iσ(l)+1χjl=jσ(l)) in the right-hand side of equation (3.3.31). +176 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +Proof. Observe that the construction outlined above Example 3.4 describes a bijection between +Sk and the set of orientable k-ribbon graphs built from 2k-gons with well-defined bicoloured +boundaries (i.e., each ribbon has both a black and red side, and black (red) vertices lie on +black (red) boundaries). Thus, interpreting Sk as this latter set, equation (3.3.35) can be +rewritten as +m(LUE) +k += +k +∑ +Vr,Vb=1 +MVr NVb #{σ ∈ Sk | Vr(σ) = Vr and Vb(σ) = Vb}, +(3.3.37) +where #S has the same meaning as in equation (3.3.14). Substituting in M = (ˆa + 1)N and +using equation (3.1.15) to relate V(σ) = Vr(σ) + Vb(σ), the total number of boundaries of +the ribbon graph labelled by σ, to g(σ) yields equation (3.3.36). +Comparing equation (3.3.36) to equation (3.0.9) shows that M(LUE) +k,l +has precisely the inter- +pretation described below Figure 3.3. Each ribbon graph that contributes to the computation +of m(LUE) +k +also contributes to the value of m(GUE) +2k +upon ‘forgetting’ the bicolouring; compare +Figures 3.4 and 3.14. However, not every LUE ribbon graph arises in this way, since some +orientable ribbon graphs have no valid bicolouring. For example, colouring any vertex of +the third ribbon graph displayed in Figure 3.4 red forces the single boundary and hence all +vertices to also be coloured red, which violates the bicolouring requirement. Nonetheless, +every planar ribbon graph has a valid bicolouring, so that the genus zero ribbon graphs +pertaining to the calculation of m(GUE) +2k +are in one-to-one correspondence with the genus zero +bicoloured ribbon graphs contributing to the value of m(LUE) +k +. In keeping with the discussion +following Proposition 3.5, this implies the very simple relationship M(LUE) +k,0 += 2kM(GUE) +k,0 +when +ˆa = 0. In combination with the proof of the Wigner semi-circle law given in Remark 3.3, we +thus have a combinatorial proof of the Marˇcenko–Pastur law (1.2.15) in the regime a = O(1). +Ribbon graphs for the LUE and LOE mixed cumulants +The mixed moments m(LUE) +k1,...,kn and cumulants c(LUE) +k1,...,kn of the LUE are respectively defined +through the formulae (3.0.13), (3.0.14) +m(LUE) +k1,...,kn = +� +n +∏ +i=1 +Tr Wki +� +, +k1, . . . , kn ∈ N +(3.3.38) += +∑ +K⊢{k1,...,kn} ∏ +κi∈K +c(LUE) +κi +. +(3.3.39) +177 + +CHAPTER 3. Characterisations of the Moments and Cumulants +The proofs of equations (3.3.18) and (3.3.21) can be extended to give their LUE analogues. +Proposition 3.13. Let n, k1, . . . , kn ∈ N and define Sk1,...,kn to be the set of (k1 + · · · + kn)-ribbon +graphs that consist of n polygons with respectively 2k1, 2k2, . . . , 2kn edges and vertices alternately +coloured black and red, whose edges are connected by untwisted ribbons that respect the bicolouring of +the polygon vertices — let Sc +k1,...,kn be the subset of connected ribbon graphs in Sk1,...,kn. Then, taking +a = ˆaN with ˆa = O(1), the mixed moments and cumulants of the LUE are given by +m(LUE) +k1,...,kn = +k1+···+kn+n +∑ +V=1 +NV +× +k1+···+kn +∑ +p=1 +(ˆa + 1)p #{Γ ∈ Sk1,...,kn | Vr(Γ) + Vb(Γ) = V and Vr(Γ) = p}, +(3.3.40) +c(LUE) +k1,...,kn = +k1+···+kn+1−n +∑ +l=0 +Nk1+···+kn+2−n−l +× +k1+···+kn+1−n +∑ +p=1 +(ˆa + 1)p #{Γ ∈ Sc +k1,...,kn | g(Γ) = l/2 and Vr(Γ) = p}, +(3.3.41) +where Vr(Γ) and Vb(Γ) are respectively the number of red and black boundaries of the ribbon graph Γ +and g(Γ) is the genus of Γ. +In contrast to what is seen in the GUE case, m(LUE) +k1,...,kn is always guaranteed to be a +polynomial of degree k1 + · · · + kn + n in N since there is at least one Γ ∈ Sk1,...,kn such that +Vr(Γ) + Vb(Γ) equals this degree. This Γ, which has n connected components that are each +of genus zero, exists because the polygons present in the ribbon graphs of Sk1,...,kn each have +an even number of sides, which allows for polygon edges to be paired without need for any +ribbons connecting two separate polygons — the GUE analogue of Sk1,...,kn is the set Pk1,...,kn, +which contains no ribbon graphs with n connected components if any of the ki are odd. +The (M, N) Laguerre orthogonal ensemble is represented by the N × N Wishart–Laguerre +matrix W = GTG, where G is an element of the M × N real Ginibre ensemble. Its (mixed) +moments and cumulants (which are specified by equations (3.3.38), (3.3.39) with our new +definition of W) can be computed in the same way as in the LUE case, except that the involved +ribbon graphs are now allowed to have M¨obius half-twisted ribbons. This is because G being +real means that (GT)ij = Gji, so equation (3.3.29) should be rewritten as +� +(GT)ijGkl +� += +� +(GT)ij(GT)lk +� += +� +GjiGkl +� = 1 +2χi=lχj=k; +(3.3.42) +178 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +recalling that our ribbon graphs are built from polygons whose edges represent entries of +either GT or G, the first average in equation (3.3.42) is represented by an untwisted ribbon +connecting these two types of edges, while the second (third) average therein is represented +by a M¨obius half-twisted ribbon connecting together two edges of the GT (G) type. +Proposition 3.14. Let n, k1, . . . , kn ∈ N and define ˜Sc +k1,...,kn to be the set of locally orientable, +connected (k1 + · · · + kn)-ribbon graphs built from n polygons with respectively 2k1, 2k2, . . . , 2kn +edges and vertices alternately coloured black and red, whose boundaries have well-defined colours +induced by the vertex colouring. Then, in the setting a = ˆaN with ˆa = O(1), we have +c(LOE) +k1,...,kn = +k1+···+kn+1−n +∑ +l=0 +� N +2 +�k1+···+kn +N2−n−l +× +k1+···+kn+1−n +∑ +p=1 +(ˆa + 1)p #{Γ ∈ ˜Sc +k1,...,kn | ˜g(Γ) = l and Vr(Γ) = p}, +(3.3.43) +where Vr(Γ) is the number of red boudnaries of Γ and ˜g(Γ) is the Euler genus of Γ. +Note that the expression (3.3.43) is very similar to that for c(LUE) +k1,...,kn given in equation +(3.3.41), with a key difference being that the summand of equation (3.3.43) may be non-zero +for odd values of l, in contrast to what is seen in equation (3.3.41). Likewise, the mixed +moments of the LOE are given by a perturbation of equation (3.3.40) involving locally +orientable ribbon graphs, though we do not display it here. +Figure 3.15: The illustrated bicoloured, locally orientable ribbon graph represents the +term ⟨(GT)i(1) +1 j(1) +1 (GT)i(2) +1 j(2) +1 ⟩⟨Gj(1) +1 i(1) +2 +Gj(2) +2 i(2) +1 ⟩⟨(GT)i(1) +2 j(1) +2 +Gj(1) +2 i(1) +1 ⟩⟨Gj(2) +1 i(2) +2 (GT)i(2) +2 j(2) +2 ⟩, where +we have adopted the vertex labelling convention of Figure 3.8. Interpreting the ribbons +as identifying the polygon edges, we have two faces, four edges, and four vertices (equiv- +alently, boundaries). Thus, by the classical formula χ = F − E + V = 2 − ˜g, this ribbon +graph is an element of ˜Sc +2,2 with two red boundaries and Euler genus zero. Hence, by +equation (3.3.43), it contributes a value of (ˆa + 1)2N4 to c(LOE) +2,2 +. +179 + +R +i(1) +ji +i(2) +i2 +i2 +i记 +12CHAPTER 3. Characterisations of the Moments and Cumulants +Like in the Gaussian case, the generating functions ˜W(LUE) +n +, ˜W(LOE) +n +of the corresponding +mixed cumulants scaled according to Definition 1.6 are O(N2−n), so they have large N +expansions of the form given in Theorem 1.2. Another similarity between the GUE and LUE +is that the correlator expansion coefficients W(LUE),l +n +(recall their definition in Theorem 1.2 +and cf. the discussion following equation (3.3.21)) can be visualised as genus l/2 compact, +connected, orientable surfaces with n holes. This represents the fact that for given (n, l), +W(LUE),l +n +is a generating function for all of the ribbon graphs that can be drawn on such a +surface by replacing the holes with even-sided polygons and connecting the edges of these +polygons with ribbons such that the resulting (embedded) ribbon graph satisfies the criteria +given above Figure 3.7 (the ribbon graph does not self-intersect and excising it from the +surface results in a disjoint union of sets homeomorphic to open disks), along with the +condition that the disks bounded by the ribbons can be coherently bicoloured black and red +such that each ribbon borders both a black and red face — in the LOE case, one needs to be +careful and categorise the relevant ribbon graphs by their orientability. +Relations to topological and combinatorial maps and hypermaps +As was discussed in §3.3.1, alternative representations of ribbon graphs are interesting for +various reasons. The bicoloured ribbon graphs constructed above can be represented by +the topological maps introduced in Definition 3.2 in a similar fashion to that discussed +below said definition, with a few differences: Recall that a true bijection between sets of +ribbon graphs and their topological map counterparts requires that the latter be rooted (see +Figure 3.11) and that, in the locally orientable (i.e., LOE) case, each vertex be assigned a local +orientation. However, in contrast to the GOE case, we no longer need to keep track of which +map edges correspond to twisted ribbons since this data can be recovered by checking if +the ends of a map edge correspond to polygon edges of the same or different type (recall +that the polygon edges of an LOE ribbon graph represent entries of either GT or G) — this is +assuming that the relevant topological maps are constructed using the formalism illustrated +in Figure 3.10. (As an aside, observe that there is a bijection between ˜Sc +k1,...,kn and the set +Pc +2k1,...,2kn of connected, orientable GUE ribbon graphs: Alternately labelling the edges of +each polygon of Γ ∈ Pc +2k1,...,2kn by GT and G, with the ‘first’ or rooted edge of each polygon +180 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +labelled GT, and then twisting any ribbons connecting polygon edges of the same type results +in a ribbon graph that can be uniquely bicoloured in the necessary way.) When constructing +a topological map from a bicoloured ribbon graph, the bicolouring of the boundaries of the +ribbon graph extends naturally to a face bicolouring of the resulting topological map. +In the combinatorial setting of Definition 3.3, the necessary bicolouring constraint is +enforced by requiring that EQ admit a partitioning EQ = Er ∪ Eb into sets Er ≃ EQ/τ0 and +Eb = EQ \ Er of red and black quarter-edges, respectively, such that τ0(Ex) = EQ \ Ex and +τ1(Ex) = τ2(Ex) = Ex for x = r, b. The equivalent condition for the oriented combinatorial +maps of Definition 3.4 is that EH = E′ +r ∪ E′ +b with E′ +b = EH \ E′ +r and τe(E′ +x) = τv(E′ +x) = EH \ E′ +x, +where we again take x = r, b. Thus, writing Er = {1′, 2, 3′, 4} shows that the topological map +of Figure 3.12 can be bicoloured in a way that relates it to the first ribbon graph displayed +in Figure 3.14, while it can be checked that the oriented combinatorial map of Figure 3.13 +cannot be bicoloured in any valid way. +Figure 3.16: Pictured is the bicoloured 2-rooted topological map corresponding to the ribbon +graph of Figure 3.15. It has two black (depicted as dark grey) and red faces. Since it is +embedded in the sphere, it is orientable. Nonetheless, the notches on the blue root-markings +indicate that the local orientations of the vertices are inverse to each other. +The topological maps discussed above can be simplified in an interesting way by trans- +forming them into topological hypermaps (whose definition we recall from the end of §3.3.1). +In the LUE case, this is done by contracting red faces to red vertices while identifying half- +edges as follows: Traverse clockwise around each vertex, starting at the rooted half-edge, and +identify the rooted half-edge with the second visited half-edge (retaining the root-marking), +the third visited half-edge with the fourth, and so on. This procedure results in a topological +181 + +CHAPTER 3. Characterisations of the Moments and Cumulants +hypermap consisting of red and black vertices that are connected by (possibly rooted) edges. +The number of faces and red vertices of the resulting topological hypermap is respectively +equal to the number of black and red faces of the original topological map. Moreover, the +act of contracting faces to vertices does not affect the genus of the relevant surface, so the +topological hypermap obtained through the above procedure has the same genus as the +bicoloured topological map (and ribbon graph) that it corresponds to. Thus, we may rewrite +equation (3.3.41) in terms of topological hypermaps by interpreting the coefficient +#{Γ ∈ Sc +k1,...,kn | g(Γ) = l/2 and Vr(Γ) = p} +as the number of genus l/2 n-rooted topological hypermaps with p red vertices and n +black vertices of valency k1, . . . , kn (it is straightforward to see that the above construction +is invertible and thus bijective). In terms of combinatorial hypermaps, this is equivalent +to counting the number of triples (EH, τe, τv) such that EH has k1 + . . . + kn edges, τe has p +cycles, τv has n cycles of length k1, . . . , kn, and τ−1 +v τ−1 +e +has k1 + · · · + kn + 2 − n − p − l cycles +(this ensures that the number of faces is consistent with the genus equalling l/2). +Figure 3.17: The topological hypermap illustrated on the right is obtained from the topo- +logical map on the left by contracting the red faces to vertices while making the half-edge +identifications 2 ≡ 1, 4 ≡ 3, 2′ ≡ 1′, and 4′ ≡ 3′ — note that we retain the blue root- +markings. The corresponding combinatorial map is (EH, τe, τv) with EH = {1, 3, 1′, 3′}, +τe = (1, 3′, 1′)(3), and τv = (1, 3)(1′, 3′). Observe that τ−1 +v τ−1 +e += (1, 3′, 3)(1′). +The procedure described above is also valid in the LOE case, but one needs to decorate +the resulting locally orientable topological hypermaps in an appropriate manner to ensure +that the mapping is injective. To understand this requirement, let us first recall that each +half-edge of a topological map is either of type GT or G, as discussed below equation (3.3.42) +182 + +3 +3 +2 +23.3. The Isserlis–Wick Theorem and Ribbon Graphs +(e.g., in the left image of Figure 3.17, the half-edges 1, 3, 1′, 3′ are of type GT, while 2, 4, 2′, 4′ +are of type G). In the case of LUE topological maps, traversing the boundary of a red face +shows that such a boundary is a chain of half-edges with no two consecutive half-edges being +of the same type. This is not the case when dealing with LOE topological maps since, e.g., the +boundary of the large red face in Figure 3.16 has two rooted half-edges (automatically of type +GT) meeting each other. To account for this subtlety, we assign ±1 ‘twist’ labellings (cf. the +discussion below Figure 3.11) to the edges of our locally orientable topological hypermaps +in such a way that interchanging the type GT ↔ G of map half-edges whenever they are +represented by twisted hypermap edges (labelled −1) results in topological maps whose red +faces are bounded by a chain of half-edges of alternating type, as seen in the LUE case. Note +that the edges labelled −1 are truly twisted in the sense that the faces glued to such edges +must switch sides somewhere along them; see Figure 3.18 below. +Figure 3.18: The left image is the result of gluing a white open disk to a (−1-labelled) +twisted edge. To make this possible, we have folded a portion of the disk over the grey +length of the edge; the green (blue) lines are attached to the boundary of the white disk at +points that are relatively close to each other and one should be able to imagine a continuum +of similar lines in between them. Folding the disk over the grey length of the edge results +in a peak (thin black line) and trough (dotted orange line), which are better visualised +through the depicted cross sections — the first (second) cross section refers to the leftmost +(rightmost) green and blue lines in the illustration on the left. In this case, flattening the +peak and trough (i.e., untwisting the edge) shows that we simply have a sphere. +The alert reader will have noticed that there are multiple ways of assigning twists to +the edges of a topological hypermap so that it correctly corresponds to a given locally +orientable, bicoloured topological map (e.g., changing the −1 label in Figure 3.18 to +1 is +183 + +>Z +X 1 cases, but that +such cases can be treated combinatorially by introducing the theory of constellations. We do +not discuss constellations here and instead refer the reader to the textbook [220]. +Let us first consider the (N, N) Hermitised Laguerre ensemble represented by the N × N +matrix H1 = G†HG with G drawn from the N × N complex Ginibre ensemble and H +drawn from the N × N GUE. From the discussion preceding Figure 3.20, we know that +the mixed cumulants c(Hm) +k1,...,kn (3.0.13) of this ensemble can be computed by enumerating +connected, orientable ribbon graphs constructed in the following manner: Draw n polygons +with respectively 3k1, 3k2, . . . , 3kn sides, pick one vertex of each polygon to be the ‘first’ or +rooted vertex of that polygon and then, moving clockwise around the polygon while starting +198 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +at the first vertex, colour each sequence of three vertices red, yellow, and yellow, in that +order (taking φ1, φ2 to be yellow and red, respectively). Next, connect polygon edges with +untwisted ribbons whose sides identify vertices of like colour in a way that results in a +connected ribbon graph. For visual clarity, we also colour the ribbons connecting edges +representing entries of G† (H) to those representing G (H) black (purple); see Figure 3.22 +below for an example. These ribbon graphs can immediately be reformulated as n-rooted +topological maps by gluing yellow (red) faces to the yellow (red) boundaries, shrinking +the polygons to vertices, thinning the ribbons to edges connecting said vertices, and root- +marking the half-edges appearing immediately clockwise to rooted vertices (cf. Figures 3.10 +and 3.11). To simplify even further to n-rooted topological hypermaps (which we recall were +introduced at the end of §3.3.1), we contract the red faces to red vertices while identifying +black half-edges in the manner shown in Figure 3.17, and then split the purple edges into +two by inserting green vertices in the middle of them; again, see Figure 3.22 below. +Figure 3.22: The ribbon graph on the left contributes to the computation of c(H1) +2,1,1 (obtained +by removing grey ribbons from Figure 3.20). The blue arrows point to the rooted vertices +and the rooted polygon edges are likewise coloured blue. Gluing faces to the boundaries +and shrinking the ribbons and polygons results in the toroidal (genus one) topological map +depicted on the top right. Inserting green vertices in the middle of the purple edges and +shrinking the red faces in the aforedescribed manner then yields the topological hypermap +on the bottom right. The blue edges are actually root-marked black edges. +We now give a reformulation of the m = 1, ν1 = ν0 = 0 case of Proposition 3.16 in terms +199 + +CHAPTER 3. Characterisations of the Moments and Cumulants +of these particular topological hypermaps. +Proposition 3.18. For n, k1, . . . , kn ∈ N with k1 + · · · + kn even, let Hk1,...,kn be the set of orientable +n-rooted topological hypermaps consisting of k1 + · · · + kn black edges and as many purple edges, +1 +2(k1 + · · · + kn) green vertices, any number of red vertices, and n black vertices such that +1. the green vertices are bivalent with only purple edges attached to them, +2. the red vertices only have black edges attached to them, +3. for i = 1, . . . , n, the ith black vertex has ki purple and black edges attached to it in alternating +cyclic order, +4. for i = 1, . . . , n, one black edge incident to the ith black vertex is root-marked and labelled by i. +Then, appropriately simplifying equation (3.3.50) shows that the mixed cumulants of the global +scaled (N, N) Hermitised Laguerre ensemble (represented by ˜H1 := ˜G† ˜H ˜G with ˜G := G/ +√ +N and +˜H := √ +2/NH) is given by +˜c(H1) +k1,...,kn = N2−n +3 +2 (k1+···+kn)+1−n +∑ +l=0 +1 +Nl #{Ω ∈ Hk1,...,kn | g(Ω) = l/2}. +(3.3.63) +Moving on, consider now the (N, N) antisymmetrised Laguerre ensemble represented by +the N × N matrix J1 = GTJNG, where N is an even integer, G is drawn from the N × N real +Ginibre ensemble, and JN is the N × N elementary antisymmetric matrix defned in equation +(1.1.5). From the discussion leading to Lemma 3.8, we know that the mixed cumulants c(J1) +k1,...,kn +can be computed by counting LOE ribbon graphs with appropriate weights — the weight of +each ribbon graph is the product of the weights of its boundaries, with black boundaries +being weighted by N and red boundaries being weighted by the trace of a suitable monomial +(3.3.59) in JN and JT +N. Recall that these monomials can be determined by replacing each red +vertex of a given LOE ribbon graph Γ by a new polygon edge representing an entry of JN and +then traversing the red boundaries while constructing the strings ω1, . . . , ωVr(Γ) described +above Lemma 3.8, where Vr(Γ) is the number of red boundaries of Γ. Translating this idea +to the setting concerning the associated topological maps, which are obtained by gluing +red (black) faces to the red (black) boundaries and then shrinking polygons and ribbons to +vertices and edges, we need to compute a string ωj for each red face of the topological map. +200 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +Now, recall from the discussion between Figures 3.17 and 3.18 that the LOE topolog- +ical maps have half-edges either of type GT or G and further observe that half-edges of +type GT (G) correspond to ribbon ends that make contact with vertices with labels of +the form i(s;1) +t +(i(s;−1) +t +); here, we remember that we are working with ribbon graphs that +have been obtained by replacing the red vertices of LOE ribbon graphs with polygon +edges representing JN and we have adopted the vertex labelling convention given above +Figure 3.21. +Thus, an edge of type GT − GT (G − G) signifies an identification of the +form i(s;1) +t +≡ i(s′;1) +t′ +(i(s;−1) +t +≡ i(s′;−1) +t′ +) so that in the relevant string ωj, we have either of the +substrings (JT +N)i(s;−1) +t +i(s;1) +t +(JN)i(s′;1) +t′ +i(s′;−1) +t′ +or (JT +N)i(s′;−1) +t′ +i(s′;1) +t′ +(JN)i(s;1) +t +i(s;−1) +t +((JN)i(s;1) +t +i(s;−1) +t +(JT +N)i(s′;−1) +t′ +i(s′;1) +t′ +or +(JN)i(s′;1) +t′ +i(s′;−1) +t′ +(JT +N)i(s;−1) +t +i(s;1) +t +) appearing — edges of type GT − GT and G − G correspond to sub- +strings of the form (JN)i(s;1) +t +i(s;−1) +t +(JN)i(s′;1) +t′ +i(s′;−1) +t′ +or (JT +N)i(s;−1) +t +i(s;1) +t +(JT +N)i(s′;−1) +t′ +i(s′;1) +t′ +. Hence, rewriting a +term (JN)i(s;1) +t +i(s;−1) +t +as (JT +N)i(s;−1) +t +i(s;1) +t +or vice versa in a given string ωj is equivalent to interchang- +ing the types GT ↔ G of the half-edges representing the ribbon ends that interact with the +vertices labelled i(s;±1) +t +. Consequently, the number of JT +N seen in the monomial q(j) +Γ (JN, JT +N) +induced by ωj is equal to the minimum number of times such GT ↔ G interchanges would +need to be applied to the jth red boundary to yield a red boundary made up of a chain of +half-edges of alternating type. However, these are the same GT ↔ G interchanges discussed +between Figures 3.17 and 3.18: They can be kept track of by transforming our topological +maps into hypermaps through the manner shown in Figure 3.17 and then applying a ±1 +labelling to the hypermap edges following the prescription given above Figure 3.18. Then, +the monomial trace Tr q(j) +Γ (JN, JT +N) weighting the jth red map face is equal to Tr J +a+ +j +N (JT +N)a− +j , +where a± +j is equal to the number of edges of type ±1 incident to the red hypermap vertex +corresponding to the jth red map face — consequently, we have that +sgn(ωj) = (−1)a− +j , +|ωj| = a− +j + a+ +j . +(3.3.64) +Example 3.8. Although the above prescription is quite longwinded, it is much easier to +compute the weight of a given topological hypermap than the ribbon graph it represents. For +example, if we want to compute the contribution of the ribbon graph displayed on the top +left of Figure 3.19 to the value of c(J1) +2 +, we first need to replace the red vertices with diagonal +edges so that the white square becomes a hexagon. The new edge in the top right represents +(JN)i(1;1) +1 +i(1;−1) +1 +, while the new edge in the bottom left represents (JN)i(1;1) +2 +i(2;−1) +1 +. Following the +201 + +CHAPTER 3. Characterisations of the Moments and Cumulants +red ribbon border starting at the bottom vertex of the edge representing (JN)i(1;1) +1 +i(1;−1) +1 +shows +that i(1;−1) +1 +≡ i(1;−1) +2 +and continuing to traverse this red boundary by following the other red +ribbon border confirms that i(1;1) +2 +≡ i(1;1) +1 +. Thus, we have ω1 = (JN)i(1;1) +1 +i(1;−1) +1 +(JT +N)i(1;−1) +1 +i(1;1) +1 +and +so this boundary is weighted Tr JNJT +N = N sgn(ω1) Re(i|ω1|) = N. Since we also have one +black boundary, which is automatically weighted N, this ribbon graph contributes a value of +N2/4 to c(J1) +2 +in our ν1 = ν0 = 0 setting. Now, if we look instead at either of the topological +hypermaps given in the bottom of Figure 3.19, we immediately see that the red vertex has +one twisted and one untwisted edge adjacent to it, so it is weighted by N(−1)1Re(i2) = N; +there being one black vertex means that the hypermap is weighted N2/4. +Working with topological hypermaps rather than ribbon graphs allow us to derive the +following simplification of the m = 1, ν1 = ν0 = 0 case of Proposition 3.17. +Proposition 3.19. For n ∈ N and k1, . . . , kn ∈ 2N, let ˜Ak1,...,kn be the subset of ±1-labelled locally +orientable n-rooted topological hypermaps in the set ˜T∗ +k1,...,kn, defined in Proposition 3.15, whose +vertices all have even valency. Furthermore, let ˜A ˜g,± +k1,...,kn ⊆ ˜Ak1,...,kn be such that for each Ω ∈ ˜A± +k1,...,kn, +Ω has Euler genus ˜g and the product of the signs of the edges of Ω is ±1. Then, the mixed cumulants +of the global scaled (N, N) antisymmetrised Laguerre ensemble (represented by ˜J1 := ˜GTJN ˜G with +˜G = √ +2/NG) is given by +˜c(J1) +k1,...,kn = (−1) +1 +2 (k1+···+kn)N2−n +k1+···+kn+1−n +∑ +l=0 +1 +Nl +� +# ˜Al,+ +k1,...,kn − # ˜Al,− +k1,...,kn +� +, +(3.3.65) +where we recall that #S denotes the number of elements in S. +Proof. Having converted the ribbon graphs of Proposition 3.17 (with m = 1) into topological +hypermaps in the manner prescribed above, we first note that hypermaps with black vertices +of odd valency have vanishing weight: Recall from Proposition 3.17 that the cumulants +c(Jm) +k1,...,kn vanish if any of the ki are odd — the black vertices of our topological hypermaps +have valency k1, . . . , kn. +Next, note that the jth red vertex (which corresponds to the jth red boundary of the +original ribbon graph) has weight sgn(ωj) Re(i|ωj|) according to Proposition 3.17, which +simplifies to +sgn(ωj) Re(i|ωj|) = (−1)a− +j Re(ia− +j +a+ +j ) +(3.3.66) +202 + +3.3. The Isserlis–Wick Theorem and Ribbon Graphs +by equation (3.3.64). This weight also vanishes if a− +j + a+ +j , the valency of the jth red vertex, is +odd. Thus, only hypermaps with all vertices of even valency contribute to the value of the +mixed cumulants. +Now, using equation (3.3.66) in the third line of equation (3.3.61) shows that the weight +of a topological hypermap due to the weights of the red vertices is the product +V0(Γ) +∏ +j=1 +sgn(ωj) Re(i|ωj|) = +V0(Γ) +∏ +j=1 +(−1)a− +j Re(ia− +j +a+ +j ), +where, following Lemma 3.8 and Proposition 3.17, V0(Γ) is the number of red boundaries +of Γ and is thus the number of red vertices of the hypermap at hand. As our weight is +non-vanishing only when a− +j + a+ +j is even for all j, we see that +V0(Γ) +∏ +j=1 +Re(ia− +j +a+ +j ) = +V0(Γ) +∏ +j=1 +(−1) +1 +2 (a− +j +a+ +j ) = (−1) +1 +2 (a− +1 +···+a− +V0(Γ)+a+ +1 +···+a+ +V0(Γ)) = (−1) +1 +2 (k1+···+kn), +where the last line follows from the fact that the sum of the valencies a− +j + a+ +j of the red +vertices is equal to the total number of edges in the hypermap, which is k1 + · · · + kn. This +is the source of the factor seen at the front of the right-hand side of equation (3.3.65). On the +other hand, since each edge is incident to exactly one red vertex, ∏ +V0(Γ) +j=1 (−1)a− +j is simply the +product of the signs of all of the edges present in the hypermap, which leads to the definition +of ˜A± +k1,...,kn. Substituting m = 1, ν1 = ν0 = 0 into equation (3.3.61) while incorporating the +simplifications just discussed produces the sought formula (3.3.65). +As we move on to the next chapter, let us mention that, although we have not discussed +them in full generality, many of the arguments leading to Proposition 3.19 can be extended to +the general m ∈ N setting of Proposition 3.17. Our reason for not presenting these arguments +in this general setting is simply that in the related ribbon graphs, the extra ribbons that +interact with polygon edges representing entries of GT +2 , G2, . . . , GT +m, Gm are distracting while +being completely irrelevant to our discussion, as they do not interact with the polygon edges +representing entries of JN0. +203 + +Chapter 4 +Loop Equations for the Matrix Product +Ensembles +In this chapter, we take the first step in exploring the feasibility of studying the Hermitised +and antisymmetrised matrix product ensembles introduced in Section 1.3 using the loop +equation formalism. Recall from Definition 1.10 that for m ∈ N and N0, . . . , Nm ∈ N such +that N1, . . . , Nm ⩾ N0, the (N0, . . . , Nm−1, N) (writing N := Nm) Hermitised matrix product +ensemble is represented by the N × N random matrix product +Hm := G† +m · · · G† +1HG1 · · · Gm, +(4.0.1) +where H is drawn from the N0 × N0 GUE and for 1 ⩽ i ⩽ m, each Gi is drawn independently +from the Ni−1 × Ni complex Ginibre ensemble. If we further take N0, . . . , Nm to be even and +now draw each Gi from the Ni−1 × Ni real Ginibre ensemble, we have by Definition 1.11 that +the (N0, . . . , Nm−1, N) antisymmetrised matrix product ensemble is represented by +Jm := GT +m · · · GT +1 JN0G1 · · · Gm, +(4.0.2) +where JN0 is the elementary antisymmetric matrix defined in equation (1.1.5). Equivalently, +in the notation of Definition 1.1, the (N0, . . . , Nm−1, N) Hermitised matrix product ensemble +E (Hm) = (S(Hm), P(Hm)) is the set S(Hm) := S(G)�� +F=C (1.2.4) of N × N complex Hermitian +matrices with the p.d.f. +P(Hm)(X) := δ(X − Hm) P(G)(H) +��� +β=2,N�→N0 +m +∏ +i=1 +P(Gin)(Gi) +��� +β=2,(M,N)�→(Ni−1,Ni), +(4.0.3) +204 + +where δ is the Dirac delta and the p.d.f.s P(Gin)(G) of the Ginibre ensemble and P(G)(H) of +the Gaussian ensemble are respectively specified in Definition 1.3 and Proposition 1.1, while +the (N0, . . . , Nm−1, N) antisymmetrised matrix product ensemble E (Jm) = (S(Jm), P(Jm)) is +the set S(Jm) of N × N antisymmetric real matrices with the p.d.f. +P(Jm)(X) := δ(X − Jm) +m +∏ +i=1 +P(Gin)(Gi) +��� +β=1,(M,N)�→(Ni−1,Ni), +(4.0.4) +where the parameters N0, . . . , Nm and matrices G1, . . . , Gm have been appropriately redefined. +What we mean by ‘taking the first step’ is that we will be studying the simplest examples of +these ensembles, which correspond to setting m = 1 and N1 = N0 = N — as we will soon +see, taking m to be any greater would result in loop equations too unwieldy to display in the +present setting, while many of the interesting features of such loop equations are already +present in the m = 1 case. In the language of Definitions 1.10 and 1.11, we will thus be +deriving loop equations for the Hermitised and antisymmetrised Laguerre ensembles. +One of our motivations for initiating the aforementioned study is that while the eigenvalue +j.p.d.f.s of the Hermitised and antisymmetrised matrix product ensembles have recently +been made explicit in the works [143], [144], as reviewed in §1.3.1 (technically, we discuss +the non-zero real eigenvalues of the matrix Hm and the positive real eigenvalues of the +matrix iJm), their moments and associated moment generating functions present challenges +relative to the setting of the classical matrix ensembles. Thus, as with our studies of the +classical matrix ensembles presented in the earlier parts of this thesis, we would like to +derive recursive characterisations of, say, the spectral moments mk (1.1.13) and resolvents +W1(x) (1.1.17) of the Hermitised and antisymmetrised matrix product ensembles. One +might then suggest that we study the ensembles at hand using the methodology laid out +in Chapters 2 and 3 to obtain linear differential equations on the resolvent expansion +coefficients Wl +1(x) and 1-point recursions on the moment expansion coefficients Mk,l defined +implicitly through the expansions (1.1.21) and (1.1.32). Recall, though, that the development +of these chapters is based on the Selberg correlation integral theory reviewed in Section 2.1, +which can only be used to study eigenvalue j.p.d.f.s that are (tractably) expressible in terms +of the j.p.d.f. (2.1.1) seen in the Selberg integral (2.1.2). This is actually the case for the +antisymmetrised Laguerre ensemble, since it is an example of a Laguerre Muttalib–Borodin +ensemble (recall Proposition 1.11 and Remark 1.14) and the work [125] relates such Muttalib– +205 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +Borodin ensembles to the Selberg integral, so it may therefore be possible to apply the +techniques of Chapters 2 and 3 to the antisymmetrised Laguerre ensemble, but we do not +pursue this line of research since it is quite unlikely that this approach could be extended +effectively to the general m > 1 ensembles E (Jm) and E (Hm), as we currently have no way of +relating the relevant eigenvalue j.p.d.f.s (recall their specifications in terms of the relatively +complicated Meijer G-functions given in Proposition 1.7) to the Selberg integral. Instead, we +are led to considering the loop equation formalism, which is well known to be applicable to +a large variety of random matrix ensembles. +The (final sets of) loop equations derived in this chapter constitute triangular recursive +systems on the coefficients Wl +n of the large N expansions (1.1.21) +Wn(x1, . . . , xn) = N2−n +∞ +∑ +l=0 +Wl +n(x1, . . . , xn) +Nl +(4.0.5) +of the connected n-point correlators Wn (1.1.29), meaning that they facilitate the systematic, +recursive computation of said coefficients. Although their forming triangular recursive +systems means that these loop equations are not as efficient as corresponding 1-point +recursions (if they exist) for computing the spectral moments m(H1) +k +and m(J1) +k +, they have +the added benefit of producing the coefficients W(H1),l +n +and W(J1),l +n +for n ⩾ 1 and l ⩾ 0, +from which we can extract coefficients of the genus expansions (3.3.63) and (3.3.65) of the +corresponding mixed cumulants ˜ck1,...,kn, thereby giving us a method of enumerating the +ribbon graphs and topological hypermaps constructed in §3.3.3. +Another motivation for deriving loop equations for the Hermitised and antisymmetrised +Laguerre ensembles is that they are of higher order than the equivalent loop equations for the +random matrix ensembles that are more commonly studied in the literature (see, e.g., §4.1.1 +forthcoming). Indeed, we will see in §4.2.2 and §4.3.2 that the loop equations specifying +W0 +1(x), also know as the spectral curves [109], for the antisymmetrised (Hermitised) Laguerre +ensembles are third (fourth) order polynomials in W(J1),0 +1 +(x) (W(H1),0 +1 +(x)), whereas the +spectral curves of the classical matrix ensembles reviewed in §4.1.1 are quadratic polynomials +in W0 +1(x). Spectral curves of order higher than two have garnered interest in the abstract +setting [110], [45], [46], [268], but there have not been many concrete examples of such +higher order spectral curves and, more broadly, loop equations (here, higher order means +higher than usual) arising from random matrix theory (see, however, the studies on so-called +206 + +4.1. A Brief Introduction to Loop Equations +multi-matrix models [110]). As mentioned above, we produce such examples through the +analysis carried out in Sections 4.2 and 4.3 below, complementing the higher order loop +equations given in the recent work [74] on the (N, N, N) complex Wishart product ensemble +(see Definition 1.12). Note that even though the Hermitised and antisymmetrised Laguerre +ensembles are closely related to the complex Wishart product ensemble studied in [74], the +associated loop equations are structurally quite different (bar the similarities one expects +from universality principles). These differences can be intuitively understood from the +discussion of §3.3.3: At the level of ribbon graphs and topological hypermaps, all three of +these ensembles are distinct variations of the Laguerre ensemble. +In Section 4.1, we introduce the loop equation formalism by means of giving a general +outline of the key ideas behind it. We cannot hope to do much better than this since there +are a variety of techniques for obtaining loop equations, as evidenced by the term ‘loop +equations’ being somewhat interchangeable with ‘Ward identities’, ‘Virasoro constraints’, +‘Tutte’s equations’ and ‘Schwinger–Dyson equations’ (see, e.g., [243] and Chapters 8, 10, 16, +17, 26, and 30 of the handbook [14]), as well as the method of constructing and recursively +solving loop equations being related to the topological recursion [110]. This relation to the +topological recursion is briefly reviewed in §4.1.2, while §4.1.1 contains a survey of how +loop equations for the Gaussian, Laguerre, and Jacobi β ensembles (defined in §1.2.4) were +derived in [108], [142]. Following the introductory content of Section 4.1, we demonstrate +the loop equation formalism in more detail, using an alternative approach to that discussed +in §4.1.1, by applying it to the (N, N) antisymmetrised and Hermitised Laguerre ensembles +in Sections 4.2 and 4.3, respectively. +4.1 +A Brief Introduction to Loop Equations +As with many tools in mathematics, the loop equation formalism is essentially a sophisticated +application of integration by parts. Following the notation of Section 1.1, let X be an N × N +random matrix drawn from the ensemble E = (S, P), let p(λ1, . . . , λN) be the j.p.d.f. of +the eigenvalues {λi}N +i=1 of X, let ⟨ · ⟩ denote an average with respect to this eigenvalue +j.p.d.f., and let ⟨ · ⟩P(X) denote an average with respect to the matrix p.d.f. P(X). The main +idea behind the loop equation formalism [243], [19] is to integrate the total derivative of +207 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +the product of P(X) and a well-chosen product of functions, F(X) = ∏i Fi(X) say, in two +different ways: the first is to observe that the integral vanishes by the fundamental theorem +of calculus due to the function F(X) having been chosen such that its product with P(X) is +zero on the boundary ∂S of the domain of integration; the second is to expand the derivative +under the integral sign using the Leibniz product rule and then integrate each of the terms +separately so as to obtain the identity +0 = +� +S +� d +dX P(X) +� +F(X) dX + ∑ +i +�� d +dX Fi(X) +� F(X) +Fi(X) +� +P(X) +. +(4.1.1) +For this identity to be relevant to our goal, one of the integrals on the right-hand side must +be equal to either the mixed moment (1.1.27) +mk1,...,kn := +� +N +∏ +i=1 +Tr Xki +� +P(X) += +� +N +∑ +i1,...,in=1 +λk1 +i1 λk2 +i2 · · · λkn +in +� +, +(4.1.2) +with n, k1, . . . , kn ∈ N generic, or the unconnected n-point correlator (1.1.26) +Un(x1, . . . , xn) := +� +N +∏ +i=1 +Tr +1 +xi − X +� +P(X) += +� +N +∑ +i1,...,in=1 +1 +(x1 − λi1) · · · (xn − λin) +� +, +(4.1.3) +with generic n ∈ N. +It turns out that good choices of F(X) are those such that the average +� +d +dX F(X) +� +P(X) +(recall from §1.1.1 that the average of an operator with respect to a p.d.f. P(X) is the integral +of that operator applied to said p.d.f.) can be seen to be slight perturbations of either the +average (4.1.2) or (4.1.3). If, after choosing such an F(X), all of the integrals in the identity +(4.1.1) can be written in terms of mixed moments mk′ +1,...,k′ +n′ with at least one such integral +being mk1,...,kn exactly, alternatively unconnected correlators Un′(x1, . . . , xn′) with at least +one integral being Un(x1, . . . , xn), we say that this identity is a loop equation on the mixed +moments, respectively unconnected correlators. In practice, at least one of the integrals in +the identity (4.1.1) cannot be expressed in the necessary way, so one must repeat this exercise +to find simpler complementary identities expressing these undesirable integrals in terms of +mixed moments or unconnected correlators, as required. +Assuming we succeed in finding the identities described above, we would then have a +set of loop equations on the unconnected correlators (one for each n ∈ N); note that if one +instead has a set of loop equations on the mixed moments, multiplying the nth loop equation +208 + +4.1. A Brief Introduction to Loop Equations +by x−k1−1 +1 +· · · x−kn−1 +n +and summing over k1, . . . , kn ⩾ 0 results in the associated loop equation +on the unconnected correlators. Such a set of loop equations on the unconnected correlators +can be transformed into a corresponding set of loop equations on the connected correlators +through the identity (1.1.31), which we repeat here for convenience, +Wn(x1, . . . , xn) = Un(x1, Jn) − ∑ +∅̸=J⊆Jn +Wn−#J(x1, Jn \ J)U#J(J), +Jn = (x2, . . . , xn). +(4.1.4) +If we also have a large N expansion (henceforth also referred to as a topological or genus +expansion) of the form (4.0.5) available, we may substitute this expansion into the loop +equations for the Wn and then equate terms of like order in N to extract loop equations on +the Wl +n. As we will see in §4.1.1, §4.2.3, and §4.3.3, it is this final set of loop equations that +are exactly solvable through a suitable recursive procedure. +Up until now, we have formulated our discussion in terms of averages with respect to +the matrix p.d.f. P(X). However, we can see from equations (4.1.2) and (4.1.3) above that +we have the option to work with either the eigenvalue j.p.d.f. p(λ1, . . . , λN) or the matrix +p.d.f. P(X), which is equivalent to working with univariate p.d.f.s on matrix entries. (Note +that in the latter case, we do not necessarily mean p.d.f.s on the entries of X; if X is a product +of simpler matrices, it is often beneficial to work with the entries of the factors of X.) Both +have their advantages and disadvantages: In the first convention, one has the benefit of +only needing to work with the N variables λ1, . . . , λN, but may have to deal with unwieldy +eigenvalue j.p.d.f.s — this is certainly the case for our matrix product ensembles (at least +when taking m > 1, as we hope to eventually do); see Proposition 1.7. On the other hand, +the second convention requires us to work with many more variables, but each of these +variables may be drawn from relatively simple distributions. +Since the Hermitised and antisymmetrised Laguerre ensembles are specified in terms of +Ginibre matrices, whose entries are normally distributed, we opt to study these ensembles +using averages with respect to the p.d.f.s. (4.0.3) and (4.0.4). To supplement our development, +we review in §4.1.1 how averages with respect to the eigenvalue j.p.d.f. (1.2.81) have been +used to study the Gaussian, Laguerre, and Jacobi β ensembles — recall that these ensembles +do not have amenable matrix p.d.f.s outside of the β = 1, 2, 4 regimes. We continue this +review in §4.1.2, where we discuss how the topological recursion [110] results from using +residue calculus to simplify the GUE loop equations to their most aesthetic forms. +209 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +4.1.1 +Loop equations for the classical β ensembles +Loop equations for the general β ensembles, which correspond to the general-β eigenvalue +j.p.d.f. (1.2.81) with w(λ) =: e−κNV(λ) no longer constrained to be classical (recall that +κ = β/2), were first obtained in the 2006 work [62] of Chekhov and Eynard. Writing the +potential V(λ) as the formal power series +V(λ) = 1 + +∞ +∑ +k=1 +tkλk, +(4.1.5) +the loop equations of [62] were formulated in terms of the linear integral operator ˆK[ · ] and +functional derivative +∂ +∂V(x) defined by (see also [19]) +ˆK [ f (x)] = +� +γ +dξ +2πi +V′(ξ) +x − ξ f (ξ), +∂ +∂V(x) = − +∞ +∑ +k=1 +1 +xk+1 +∂ +∂tk +, +(4.1.6) +where x lies outside the integration contour γ, which circles the branch cuts of the resolvent +W1(ξ) in a positive direction, and the so-called loop insertion operator +∂ +∂V(x) is such that the +n-point connected correlator (4.1.4) can be (formally) written as +Wn(x1, . . . , xn) = +∂ +∂V(xn) +∂ +∂V(xn−1) · · · +∂ +∂V(x2)W1(x1). +These operators posed subtle challenges when it came to solving the relevant loop equations +in practice, so about three years later, Eynard and Marchal [108] derived a more tractable +reformulation of said loop equations. There, it was shown through a consideration of the +infinitesimal change of variables +λi �→ λi + ϵ +1 +x1 − λi ++ O(ϵ2) +(4.1.7) +in the average (4.1.3) that the relevant connected correlators (4.1.4) satisfy, for each n ⩾ 1, +the loop equation (see also [55], [43], [319]) +κ ∑ +J⊆Jn +W#J+1(x1, J)Wn−#J(x1, Jn \ J) + κWn+1(x1, x1, Jn) + (κ − 1) ∂ +∂x1 +Wn(x1, Jn) += κN +� +V′(x1)Wn(x1, Jn) − Pn(x1, Jn) +� − +n +∑ +i=2 +∂ +∂xi +�Wn−1(x1, Jn \ {xi}) − Wn−1(Jn) +x1 − xi +� +, +(4.1.8) +where Jn is as in equation (4.1.4), #S denotes the size of S, and +Pn(x1, Jn) := +� +N +∑ +i1=1 +V′(x1) − V′(λi1) +x1 − λi1 +N +∑ +i2,...,in=1 +1 +(x2 − λi2) · · · (xn − λin) +� +conn +. +(4.1.9) +210 + +4.1. A Brief Introduction to Loop Equations +Here, the connected average ⟨ · ⟩conn relates to the usual average ⟨ · ⟩ in the same way that the +connected and unconnected correlators relate to each other via equation (4.1.4). +Now, it is straightforward to specialise the loop equation (4.1.8) to the case of the Gaussian +β ensemble: one need only observe that taking V(λ) to be the quadratic potential λ2 in +equation (4.1.9) shows that (see, e.g., [319, Thrm. 1]) +P(G) +n +(x1, Jn) = +� +2N +N +∑ +i2,...,in=1 +1 +(x2 − λi2) · · · (xn − λin) +� +conn += 2Nχn=1, +(4.1.10) +where we recall that the indicator function χA equals one when A is true and is otherwise +zero. Hence, if we let ˜W(G) +n +(x1, . . . , xn) denote the connected n-point correlator of the global +scaled Gaussian β ensemble with eigenvalue j.p.d.f. (apply the scaling λi �→ +√ +κNλi, in +keeping with Definition 1.6, to the j.p.d.f. (1.2.81) with w(λ) = e−λ2) +˜p(G)(λ1, . . . , λN; β) = (κN) +N +2 (κ(N−1)+1) +N (G) +N,β +N +∏ +i=1 +e−κNλ2 +i |∆N(λ)|β, +(4.1.11) +substituting equation (4.1.10) into the loop equation (4.1.8) shows that +κ ∑ +J⊆Jn +˜W(G) +#J+1(x1, J) ˜W(G) +n−#J(x1, Jn \ J) + κ ˜W(G) +n+1(x1, x1, Jn) + (κ − 1) ∂ +∂x1 +˜W(G) +n +(x1, Jn) += 2κN +� +x1 ˜W(G) +n +(x1, Jn) − Nχn=1 +� +− +n +∑ +i=2 +∂ +∂xi +� ˜W(G) +n−1(x1, Jn \ {xi}) − ˜W(G) +n−1(Jn) +x1 − xi +� +. +(4.1.12) +Moreover, invoking Theorem 1.2 to further substitute the expansion (4.0.5) +˜W(G) +n +(x1, . . . , xn) = N2−n +∞ +∑ +l=0 +W(G),l +n +(x1, . . . , xn) +Nl +(4.1.13) +into equation (4.1.12) and equating terms of equal order in N yields, for n ⩾ 1 and l ⩾ 0, the +(n, l) loop equation on the correlator expansion coefficients, +κ ∑ +J⊆Jn +l +∑ +k=0 +W(G),k +#J+1 (x1, J)W(G),l−k +n−#J +(x1, Jn \ J) + κW(G),l−2 +n+1 +(x1, x1, Jn) += (1 − κ) ∂ +∂x1 +W(G),l−1 +n +(x1, Jn) + 2κx1W(G),l +n +(x1, Jn) − 2κχn=1,l=0 +− +n +∑ +i=2 +∂ +∂xi +� +W(G),l +n−1 (x1, Jn \ {xi}) − W(G),l +n−1 (Jn) +x1 − xi +� +, +(4.1.14) +with W(G),−2 +n +, W(G),−1 +n +:= 0 for all n ⩾ 1. It can readily be seen that for any given m ∈ N, the +set of loop equations obtained by setting n = 1, . . . , m in equation (4.1.12) involves more than +211 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +m correlators and is thus unsolvable, but the same is not true for equation (4.1.14). Indeed, +setting (n, l) = (1, 0) in equation (4.1.14) produces the so-called spectral curve [109] +� +W(G),0 +1 +(x1) +�2 +− 2x1W(G),0 +1 +(x1) + 2 = 0, +(4.1.15) +which can be easily solved to recover the Stieltjes transform W(G),0 +1 +(x1) = x1 − +� +x2 +1 − 2 of +the Wigner semi-circle law (1.2.14). Having this expression at hand then enables one to +solve equation (4.1.14) with (n, l) = (1, 1) to obtain W(G),1 +1 +(x1), which then allows for the +computation of W(G),0 +2 +(x1, x2) through the (2, 0) loop equation (4.1.14), and so on. In fact, as +mentioned in §1.1.1, the (n, l) loop equations (4.1.14) can be solved in a triangular recursive +manner and this is moreover true for all analogous Wl +n loop equations discussed in this +chapter. The entries of the table below specify the order in which said loop equations must +be solved to eventually compute W10 +1 +— given enough computation power and time, Wl +n can, +in theory, be computed for any n ⩾ 1 and l ⩾ 0. +l +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +n +Wl +n +1 +1 +2 +4 +6 +9 +12 +16 +20 +25 +30 +36 +2 +3 +5 +8 +11 +15 +19 +24 +29 +35 +3 +7 +10 +14 +18 +23 +28 +34 +4 +13 +17 +22 +27 +33 +5 +21 +26 +32 +6 +31 +It turns out that obtaining Wl +n loop equations of the form (4.1.14) for the Laguerre and +Jacobi β ensembles is more involved than in the Gaussian case, especially if one wishes to +allow the Laguerre and Jacobi parameters a, b to vary with N. This is because the potentials +V(λ) corresponding to the global scaled Laguerre and Jacobi β ensembles (compare the +weight w(λ) = e−κNV(λ) to the results of applying the scalings of Definition 1.6 to the weights +given in equation (1.2.9)) are respectively +V(L)(λ) = λ − a +κN log(λ), +V(J)(λ) = − a +κN log(λ) − b +κN log(1 − λ), +(4.1.16) +which means that the auxiliary function Pn(x1, Jn) specified by equation (4.1.9) is noticeably +more complicated than seen in equation (4.1.10). Thus, loop equations for the Laguerre +212 + +4.1. A Brief Introduction to Loop Equations +and Jacobi β ensembles were first made explicit in the 2017 work [142] of Forrester, the +present author, and Witte. The proofs therein used an adaptation of Aomoto’s method [22] +for proving the Selberg integral formula (2.1.3), which differs slightly in spirit from the +derivation of [108] reviewed above. (Note that the same adaptation of Aomoto’s method was +also used by Witte and Forrester in 2015 to obtain loop equations for the circular β ensembles +[320].) Let us now review how Aomoto’s method was used in [142] to tackle the problem of +specifying Pn(x1, Jn). +In keeping with the general formalism prescribed at the beginning of this section, we +begin by considering the average +� +N +∑ +i1=1 +∂ +∂λi1 +1 +x1 − λi1 +N +∑ +i2,...,in=1 +1 +(x2 − λi2) · · · (xn − λin) +� +(4.1.17) +with respect to the global scaled eigenvalue j.p.d.f. ˜p(λ1, . . . , λN) of either the Gaussian, +Laguerre, or Jacobi β ensemble; recall that the differential operator +∂ +∂λi1 acts on the product +of all terms on its right in the integral formulation of this average, including ˜p(λ1, . . . , λN). +The equivalent of making the change of variables (4.1.7) in the definition of the unconnected +correlators is to apply integration by parts to the above average in order to obtain an identity +of the form (4.1.1). Temporarily taking a, b > 0 so that the weights w(L)(λ), w(J)(λ) equal +(or limit to) zero at the boundary of the domain of integration (and later relaxing this +requirement to a, b > −1 using analytic continuation), we see that this average must vanish +by the fundamental theorem of calculus. On the other hand, using the Leibniz product rule +to expand the integrand shows that +0 = +� +N +∑ +i1=1 +∂ +∂λi1 +1 +x1 − λi1 +N +∑ +i2,...,in=1 +1 +(x2 − λi2) · · · (xn − λin) +� += κ ˜Un+1(x1, x1, Jn) + (κ − 1) ∂ +∂x1 +˜Un(x1, Jn) − +n +∑ +i=2 +∂ +∂xi +� ˜Un−1(x1, Jn \ {xi}) − ˜Un−1(Jn) +xi − x1 +� +− κN +� +N +∑ +i1,...,in=1 +V′(λi1) +(x1 − λi1) · · · (xn − λin) +� +, +(4.1.18) +where ˜Un denotes the average (4.1.3) with respect to the eigenvalue j.p.d.f. ˜p(λ1, . . . , λN) of +the global scaled classical β ensemble at hand. In the Gaussian case, V′(λi1) = 2λi1 = +2x1 − 2(x1 − λi1), so the average on the bottom line of equation (4.1.18) simplifies to +2x1 ˜U(G) +n +(x1, Jn) − 2N ˜U(G) +n−1(Jn) and thus equation (4.1.18) can be seen to be a loop equa- +213 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +tion on the ˜U(G) +n +which, upon the use of an inductive argument based on equation (4.1.4) +(see [142, App. A]), is equivalent to the loop equation (4.1.12) on the ˜W(G) +n +. In the Laguerre +case, we have from equation (4.1.16) that V′(λi1) = 1 − a/(κNλi1), so the aforementioned +average on the bottom line of equation (4.1.18) instead simplifies to +� +1 − +a +κNx1 +� +˜U(L) +n (x1, Jn) − +a +κNx1 +� +N +∑ +i1,...,in=1 +1 +λi1(x2 − λi2) · · · (xn − λin) +� +, +(4.1.19) +which is not immediately expressible in terms of the ˜U(L) +n . Thus, we search for a companion +to the average (4.1.17) whose expansion via Aomoto’s method produces an identity of the +form (4.1.1) that is simpler than equation (4.1.18), but contains the troublesome average seen +in equation (4.1.19) above. It was shown in [142] that it is sufficient to consider the average +� +N +∑ +i1=1 +∂ +∂λi1 +N +∑ +i2,...,in=1 +1 +(x2 − λi2) · · · (xn − λin) +� +, +(4.1.20) +which, by a replication of the above arguments, is equal to both zero and +a +� +N +∑ +i1,...,in=1 +1 +λi1(x2 − λi2) · · · (xn − λin) +� +− N ˜U(L) +n−1(Jn) − +n +∑ +i=2 +∂ +∂xi +˜U(L) +n−1(Jn). +(4.1.21) +Thus, combining equation (4.1.18) with the fact that the expression (4.1.21) equals zero +results in a loop equation on the ˜U(L) +n +that further transforms via equation (4.1.4) into a loop +equation on the ˜W(L) +n +[142, Eq. (3.13)]. This loop equation is the same as equation (4.1.8), +except with each Wn′ rewritten as ˜W(L) +n′ +and with the replacement +κN +� +V′(x1)Wn(x1, Jn) − Pn(x1, Jn) +� �→ +� +κN − a +x1 +� +˜W(L) +n (x1, Jn) +− χn=1 +κN2 +x1 +− 1 +x1 +n +∑ +i=2 +∂ +∂xi +˜W(L) +n−1(Jn). +(4.1.22) +Note that the second line of the right-hand side of the above is exactly Pn(x1, Jn) (4.1.9) in +the Laguerre case. +Finally, to obtain loop equations for the Jacobi β ensemble, one repeats the above exercise +of applying integration by parts to the averages (4.1.17) and (4.1.20), now with respect to +the eigenvalue j.p.d.f. p(J)(λ1, . . . , λN) (1.2.81). However, it is once again observed [142] that +this does not result in an equation that can be written fully in terms of the U(J) +n +(we do not +214 + +4.1. A Brief Introduction to Loop Equations +write ˜U(J) +n +nor ˜W(J) +n +since no scaling is required in the Jacobi case; see the discussion below +Definition 1.6) — one must also consider the third average +� +N +∑ +i1=1 +∂ +∂λi1 +λi1 +N +∑ +i2,...,in=1 +1 +(x2 − λi2) · · · (xn − λin) +� +. +(4.1.23) +Combining the identity resulting from applying integration by parts to this average with the +Jacobi analogues of the identity (4.1.18) and that obtained by setting the expression (4.1.21) +to zero then gives a loop equation on the U(J) +n , which is again equivalent to a loop equation +on the W(J) +n +due to the aforementioned inductive argument involving the relation (4.1.4). +Like in the Gaussian and Laguerre cases, this latter loop equation has the form (4.1.8) with +each Wn′ rewritten as W(J) +n′ +and with the replacement [142, Prop. 4.4] +κN +� +V′(x1)Wn(x1, Jn) − Pn(x1, Jn) +� �→ +� +b +1 − x1 +− a +x1 +� +W(J) +n (x1, Jn) + +n − 1 +x1(1 − x1)W(J) +n−1(Jn) +− +χn=1 +x1(1 − x1)[(a + b + 1)N + κN(N − 1)] ++ +1 +x1(1 − x1) +n +∑ +i=2 +xi +∂ +∂xi +W(J) +n−1(Jn). +(4.1.24) +As we move on, let us mention that in contrast to the Gaussian case, the Wl +n loop equations +for the Laguerre and Jacobi β ensembles depend on how a, b vary with N. Nonetheless, the +spectral curves (i.e., the analogues of equation (4.1.15)) are quadratic in W0 +1(x1) [142]. +4.1.2 +From loop equations to the topological recursion +The topological recursion was first given in its present form in the 2009 work [110] of +Eynard and Orantin, where it arose as a refinement of the loop equations derived about five +years earlier [105], [109] for the 1-Hermitian matrix model with polynomial potential. This +so-called 1-Hermitian matrix model is the ensemble of eigenvalues with j.p.d.f. +p(1HMM)(λ1, . . . , λN) = +1 +N (1HMM) +N +N +∏ +i=1 +e−NV(λi)|∆N(λ)|2, +(4.1.25) +where the potential V(λ) is as in equation (4.1.5) [19]; note that this ensemble is the same as +the general β ensembles discussed in the previous subsection, but with β = 2. Let us now +sketch the derivation of the topological recursion for the 1-Hermitian matrix model (with +V(λ) constrained to be polynomial) given in the work [110]. +215 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +Setting κ = β/2 = 1 in equation (4.1.8) produces the loop equation on the connected +n-point correlators of the 1-Hermitian matrix model, +∑ +J⊆Jn +W#J+1(x1, J)Wn−#J(x1, Jn \ J) + Wn+1(x1, x1, Jn) += N +� +V′(x1)Wn(x1, Jn) − Pn(x1, Jn) +� +− +n +∑ +i=2 +∂ +∂xi +�Wn−1(x1, Jn \ {xi}) − Wn−1(Jn) +x1 − xi +� +, +(4.1.26) +where Jn is as in equation (4.1.4) and Pn(x1, Jn) is as specified by equation (4.1.9). Let us +now restrict the potential V(λ) to be independent of N so that substituting the topological +expansion (4.0.5) into the above loop equation shows that Pn(x1, Jn) has an analogous +expansion of the form +Pn(x1, Jn) = N2−n +∞ +∑ +l=0 +Pl +n(x1, Jn) +Nl +. +(4.1.27) +Thus, substituting both expansions (4.0.5), (4.1.27) into equation (4.1.26) and equating terms +of like order in N then produces the loop equation on the correlator expansion coefficients, +∑ +J⊆Jn +l +∑ +k=0 +Wk +#J+1(x1, J)Wl−k +n−#J(x1, Jn \ J) + Wl−2 +n+1(x1, x1, Jn) += V′(x1)Wl +n(x1, Jn) − Pl +n(x1, Jn) − +n +∑ +i=2 +∂ +∂xi +� +Wl +n−1(x1, Jn \ {xi}) − Wl +n−1(Jn) +x1 − xi +� +. +(4.1.28) +In particular, taking (n, l) = (1, 0) yields the spectral curve (cf. equation (4.1.15)): +� +W0 +1(x1) +�2 − V′(x1)W0 +1(x1) + P0 +1 (x1) = 0. +(4.1.29) +Having assumed V(λ) to be polynomial, equation (4.1.9) tells us that P1(x1) must also be +polynomial in x1 of degree one less than V′(x1). Thus, equation (4.1.29) can be shown (see, +e.g., [111]) to have solution of the form +W0 +1(x1) = 1 +2 +� +V′(x1) ± M(x1) +� +σ(x1) +� +, +(4.1.30) +where M(x1) and σ(x1) are polynomials such that the latter is of even degree, say 2s, with +all roots distinct and M(x1)2σ(x1) = (V′(x1))2 − 4P0 +1 (x1). From the perspective of random +matrix theory, one must take the negative sign in equation (4.1.30) so that W0 +1(x1) ∼ 1/x1 as +x1 → ∞, in line with, say, equation (2.4.15). +216 + +4.1. A Brief Introduction to Loop Equations +The first step in simplifying the above loop equations is to circumvent the multivalued +nature of W0 +1(x1) by viewing it as a single-valued holomorphic function on an appropriate +two-sheeted cover of the complex plane. This cover can be constructed by taking two copies +of the complex plane, gluing them along the s branch cuts of W0 +1(x1) (which are the s +intervals along which the degree-2s polynomial σ(x1) is negative) and then adding a point +at infinity to each of the two complex planes so that the resulting surface is a genus s − 1 +compact Riemann surface Σ [111]; see Figure 4.1 below. Such a covering space Σ comes +equipped with a holomorphic projection x : Σ → C ∪ {∞} such that for all x1 ∈ C ∪ {∞} +not a root of σ(x1), there exist two distinct points z, z′ ∈ Σ such that x(z) = x(z′) = x1. +More importantly, there exists a single-valued meromorphic function y : Σ → C such that +whenever z, z′ ∈ Σ are such that z ̸= z′, but x(z) = x(z′), y(z) and y(z′) are the two values +of W0 +1(x(z)) given by equation (4.1.30). Here on out, let us work in the one-cut case, which +corresponds to assuming that σ(x1) is quadratic with two distinct roots, u < v say, and thus +Σ is the Riemann sphere CP1 (being genus zero). +Figure 4.1: On the left, we have two copies of the complex plane with their branch cuts [u, v] +identified; note that the blue plane is obtained from the beige plane through a reflection +in the real axis to ensure that if one approaches the branch cut from the beige positive +(negative) half plane, one leaves the branch cut by entering the blue negative (positive) half +plane. In the middle image, we have opened up the branch cuts into circles and pulled +them together so as to form a tube. Compactifying by adding (orange) points at infinity to +the beige and blue planes results in the Riemann sphere on the right. Note that if there +were more cuts, there would be more tubes which would end up becoming handles in the +final surface. For each x1 ∈ C ∪ {∞} \ {u, v}, there exists a z in the beige hemisphere and +z′ in the blue hemisphere such that x(z) = x(z′) = x1 and y(z), y(z′) are the two values of +W0 +1 (x1) given in equation (4.1.30). +217 + +Im z +Im z +7 +7 +u +Re'z +Re'z +u +V +u +V +Rez' +Rez' +Im z' +Im z'CHAPTER 4. Loop Equations for the Matrix Product Ensembles +In the one-cut case, one can use the Joukowsky transform to see that when the endpoints +of our branch cut are u < v, +x(z) = v − u +4 +� +z + 1 +z +� ++ u + v +2 +(4.1.31) +has the properties required of the projection map described above: this mapping has exactly +u, v as branch points and every x1 ∈ C ∪ {∞} \ {u, v} has two preimages under x (note that +x(−1) = u and x(1) = v). Furthermore, we can choose a sign in equation (4.1.30) to write +y(z) = 1 +2 +� +V′(x(z)) − v − u +4 +M(x(z)) +� +z − 1 +z +�� +; +(4.1.32) +observe that for all z ∈ Σ = CP1, x(z) = x(1/z), so if z ̸= ±1 is a preimage of x1, y(z) +and y(1/z) are the two values of W0 +1(x1) given in equation (4.1.30). This reformulation +of (x1, W0 +1(x1)) as the coordinates (x(z), y(z)) on Σ extends naturally to the spectral curve +(4.1.29) and, indeed, to the general (n, l) loop equation (4.1.28). Moreover, since the (n, l) loop +equation specifying Wl +n(x1, . . . , xn) is linear in this correlator expansion coefficient (when +(n, l) ̸= (1, 0)), we see by induction that each of these correlators have the same branching +structure as W0 +1(x1) and so can be viewed as single-valued holomorphic functions on Σn. +Thus, we are led to rewrite equation (4.1.28) in terms of the coordinates (x(z), y(z)) on Σ. +In fact, to facilitate upcoming residue calculus, we are encouraged to introduce differential +forms on Σ, as well. Hence, let us define for z1, . . . , zn ∈ Σ and (n, l) ̸= (1, 0), (2, 0), +ω0 +1(z1) := y(z1)dx(z1), +(4.1.33) +ω0 +2(z1, z2) := +� +W0 +2(x(z1), x(z2)) + +1 +(x(z1) − x(z2))2 +� +dx(z1)dx(z2), +(4.1.34) +ωl +n(z1, . . . , zn) := Wl +n(x(z1), . . . , x(zn))dx(z1) · · · dx(zn), +(4.1.35) +where equation (4.1.33) is a specialisation of equation (4.1.35) based on the replacement of +equation (4.1.30) with equation (4.1.32), equation (4.1.34) is chosen to differ from equation +(4.1.35) in a way that ensures that ω0 +2(z1, z2) coincides with the Bergman kernel (also known +as the fundamental differential of the second kind; see, e.g., [110], [111]) dz1dz2/(z1 − z2)2 +on CP11, and the Wl +n(x(z1), . . . , x(zn)) within equation (4.1.35) is to be interpreted as the +1It was first shown in [18], [19] that for the one-cut 1-Hermitian matrix model, W0 +2 has the universal form +implied by the right-hand side of equation (4.1.34) being equal to the Bergman kernel on CP1. This was also +observed [319], [142] for the classical β ensembles discussed in §4.1.1. +218 + +4.1. A Brief Introduction to Loop Equations +single-valued function on (CP1)n obtained from the loop equation (4.1.28) after it has been +rewritten in terms of the coordinates (x(z), y(z)). +With definitions (4.1.31)–(4.1.35) in hand and noting that V′(x1) − 2W0 +1(x1) should be +replaced by y(1/z1) − y(z1) to be consistent with our choice of replacing W0 +1(x1) with y(z1), +we rewrite equation (4.1.28) for (n, l) ̸= (1, 0), (2, 0) as (cf. [107, Sec. 3.3]) +ωl +n(z1, J′ +n) = +1 +ω0 +1(1/z1) − ω0 +1(z1) +� +ωl−2 +n+1(z1, z1, J′ +n) + +◦ +∑ +J⊆J′ +n +0⩽k⩽l +ωk +#J+1(z1, J)ωl−k +n−#J(z1, J′ +n \ J) +� +− +1 +ω0 +1(1/z1) − ω0 +1(z1) +� +χn=1,l=2 lim +ξ→z1 +dx(z1)dx(ξ) +(x(z1) − x(ξ))2 ++ 2 +n +∑ +i=2 +dx(z1)dx(zi)ωl +n−1(z1, J′ +n \ {zi}) +(x(z1) − x(zi))2 +� ++ dx(z1) · · · dx(zn) +y(1/z1) − y(z1) +� +Pl +n(x(z1), Jn) + +n +∑ +i=2 +4 +v − u +z2 +i +z2 +i − 1 +× ∂ +∂zi +� +Wl +n−1(x(z1), Jn \ {x(zi)}) − Wl +n−1(Jn) +x(z1) − x(zi) +� � +, +(4.1.36) +where we now have Jn = (x(z2), . . . , x(zn)), J′ +n = (z2, . . . , zn), and the sum ∑◦ excludes the +terms corresponding to (k, J) = (0, {zi}), (l, J′ +n \ {zi}); the terms of the second and third line +arise from the fact that occurrences of ω0 +2 in the top line must be accompanied by the extra +term shown in equation (4.1.34), as compared to equation (4.1.35). Now, replace z1 by ξ in +equation (4.1.36), multiply both sides by a half times the meromorphic differential [110] +dSξ,1/ξ(z1) := +� z=ξ +z=1/ξ ω0 +2(z1, z) = +� +1 +z1 − ξ − +1 +z1 − 1/ξ +� +dz1, +ξ ∈ CP1 \ {±1}, +(4.1.37) +and take the sum of the residues at ξ = ±1 on both sides of the resulting equation. This +shows that the left-hand side simply reduces as +1 +2 ∑ +a=±1 +Res +ξ=a dSξ,1/ξ(z1)ωl +n(ξ, J′ +n) = 1 +2 ∑ +a=±1 +Res +ξ=a +� +1 +z1 − ξ − +1 +z1 − 1/ξ +� +ωl +n(ξ, J′ +n) dz1 += ∑ +a=±1 +Res +ξ=a +1 +z1 − ξ ωl +n(ξ, J′ +n) dz1 = Res +ξ=z1 +1 +ξ − z1 +ωl +n(ξ, J′ +n) dz1 = ωl +n(z1, J′ +n); +(4.1.38) +the second equality is obtained by changing variables ξ �→ 1/ξ in Res +ξ=aωl +n(ξ, J′ +n)/(z1 − 1/ξ) +and using the fact that ωl +n(1/ξ, J′ +n) = −ωl +n(ξ, J′ +n) (this can be checked inductively through +219 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +equation (4.1.36) [110], [107, Sec. 3.3]), while the third equality follows from the fact that +the sum of all residues of a meromorphic form on CP1 must vanish (again, an inductive +argument via equation (4.1.36) shows that ωl +n(ξ, J′ +n) only has residues at ξ = ±1). On the +other hand, the right-hand side simplifies remarkably to +∑ +a=±1 +Res +ξ=a K(ξ, z1) +� +ωl−2 +n+1(ξ, 1/ξ, J′ +n) + +◦ +∑ +J⊆J′ +n +0⩽k⩽l +ωk +#J+1(ξ, J)ωl−k +n−#J(1/ξ, J′ +n \ J) +� +, +(4.1.39) +where we define the recursion kernel as +K(ξ, z1) := +dSξ,1/ξ(z1) +2(ω0 +1(ξ) − ω0 +1(1/ξ)). +(4.1.40) +The simplified form (4.1.39) results from the fact that all of the terms below the first line of +equation (4.1.36) either do not have residue at z1 = ±1 or the sum of their non-zero residues +vanish by symmetry arguments [107, Sec. 3.3]. Finally, setting the left- and right-hand sides +(4.1.38) and (4.1.39) equal to each other gives us the (Eynard–Orantin) topological recursion +for the one-cut 1-Hermitian matrix model with polynomial potential: +ωl +n(z1, J′ +n) = +∑ +a=±1 +Res +ξ=a K(ξ, z1) +� +ωl−2 +n+1(ξ, 1/ξ, J′ +n) + +◦ +∑ +J⊆J′ +n +0⩽k⩽l +ωk +#J+1(ξ, J)ωl−k +n−#J(1/ξ, J′ +n \ J) +� +. +(4.1.41) +Observe that this topological recursion formula can be seen to be a simplification of the loop +equation (4.1.28) obtained by essentially replacing the right-hand side by Wl +n(x1, Jn). Note +that this means that knowledge of the Pl +n(x1, Jn) is not needed to compute the ωl +n — recall +from §4.1.1 that without the topological recursion formula available, other techniques, such +as Aomoto’s method, must be used to deal with Pl +n(x1, Jn). +Let us now explain why the topological recursion is so named. Assume for simplicity that +the potential V(λ) is the Gaussian potential λ2 (it has however been known since the 1970s +[179], [54], [36] that the following argument applies formally to more general potentials; see, +e.g., [107, Ch. 2]). Then, by the discussion of §3.3.1, the Wl +n are zero for odd values of l and +are otherwise generating functions for the compact, connected, orientable ribbon graphs of +genus l/2 that can be built from n polygons. Thus, we should take l to be even in equation +(4.1.41) and discard terms in the sum involving odd values of k. Moreover, in keeping with +220 + +4.1. A Brief Introduction to Loop Equations +the discussion involving Figure 3.7 and that following Proposition 3.10, it is convenient to +visualise ωl +n as a genus l/2 surface with n holes or punctures. We can then interpret the +topological recursion formula (4.1.41) in the following diagrammatic fashion [109], [110]: +The process of multiplying by the recursion kernel K(ξ, z1) and then taking the sum of the +residues at ξ = ±1 is represented by the gluing of a pair of pants (i.e., a sphere with three +holes) to either two holes of a genus l/2 − 1 surface with n + 1 holes to increase the genus +and lower the number of holes by one each, or to one hole each of two separate surfaces in +order to form a genus l/2 surface with n holes; see Figure 4.2 below. The recursion thus +runs over the Euler characteristic χ = 2 − l + n. +Figure 4.2: On the left, we have two homeomorphic genus 2 surfaces with 1 hole represent- +ing the term ω4 +1(z1) in the left-hand side of the topological recursion formula (4.1.41) with +(n, l) = (1, 4) (having taken n to be the number of holes and l/2 to be the genus). In the top +row, we see that gluing a pair of pants (in pink) representing the operator ∑a=±1 Res +ξ=aK(ξ, z1) +to a genus 1 surface with 2 holes representing ω2 +2(ξ, 1/ξ) produces the surface representing +ω4 +1(z1). In the bottom row, we see that gluing such a pair of pants to two genus 1 surfaces +with 1 hole each produces the surface on the bottom left — the two blue surfaces on the +bottom right represent ω2 +1(ξ) and ω2 +1(1/ξ). +The diagrammatic interpretation of the topological recursion described above can be +traced back to Tutte’s equation and recursion [302], [309] (see also [107, Sec. 1.3] for a textbook +treatment) on the coefficients of the genus expansions (3.3.21) of the GUE mixed cumulants +c(GUE) +k1,...,kn. In the language of ribbon graphs (rather than topological maps, as considered +by Tutte), the idea behind Tutte’s recursion is to simply observe that a given connected, +orientable ribbon graph with n polygons can be obtained from a similar such ribbon graph +with n + 1 polygons by shrinking a particular ribbon to identify its ends so that the two +221 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +relevant polygons merge into one — if instead deleting said ribbon results in a connected +(disconnected) ribbon graph with two unpaired polygon edges, then we have a ribbon graph +analogue of the decomposition displayed in the top (bottom) row of Figure 4.2. +The key insight of the seminal work [110] was that the mechanism underlying the +topological recursion formula (4.1.41) is common to many enumerative problems in random +matrix theory and algebraic geometry. In the algebro-geometric setting, one abstracts by +taking the spectral curve to be a triple (Σ, x, y) of a Riemann surface Σ and two meromorphic +functions x, y : Σ → C ∪ {∞}, defines ω0 +1(z1) = y(z1)dx(z1), defines ω0 +2(z1, z2) to be the +Bergman kernel on Σ2, and then recursively defines all other ωl +n through equation (4.1.41) +reformulated in terms of the genus g = l/2 (i.e., setting ωl′ +n = 0 whenever l′ is odd) and with +the sum over a = ±1 replaced by a sum over all branch points, equivalently zeroes of the +differential form dx. Thus, for particular choices of initial data (Σ, x, y, ω0 +2), the topological +recursion has been shown to govern, for example, Gromov–Witten invariants of CP1, various +Hurwitz numbers, dessins d’enfants, the intersection numbers mentioned at the beginning +of Section 3.3, and Weil–Petersson volumes; see, e.g., [110], [91], [265], [61] and references +therein. Indeed, Mirzakhani’s groundbreaking recursion [245], [246] for Weil–Petersson +volumes has been shown to be an instance of the topological recursion [106]. +Since the work of Eynard and Orantin [110], the topological recursion has been generalised +and reformulated in a variety of ways: Eynard and Marchal [108] showed that the topological +recursion formula (4.1.41) can be applied to general β ensembles upon redefining the +recursion kernel K(ξ, z1) (4.1.40) in a suitable β-dependent way; Bouchard et al. [45], [46] +studied generalisations of the topological recursion involving higher order spectral curves +by introducing terms on the right-hand side of equation (4.1.41); Do and Norbury [86] +and Chekhov and Norbury [63] have studied aspects of the topological recursion when the +spectral curve is irregular (i.e., the eigenvalue density has hard edges) — in particular, they +showed that the connected correlators of the LUE with a = 0 and the Legendre unitary +ensemble, equivalently the sJUE with a = b = 0, obey the topological recursion (like [142], +their results provide an alternative to Proposition 3.5 and Corollary 3.4 for computing the +moment expansion coefficients of the LUE with a = 0 and the Legendre unitary ensemble); +Kontsevich and Soibelman [212] have recently reformulated the topological recursion by +replacing the initial data with a certain set of tensors derived from ω0 +1 and ω0 +2. +222 + +4.2. Loop Equations for the Antisymmetrised Laguerre Ensemble +4.2 +Loop Equations for the Antisymmetrised Laguerre Ensemble +In this section, we derive loop equations for the mixed moments ˜m(J1) +k1,...,kn (4.1.2), the uncon- +nected n-point correlators ˜U(J1) +n +(x1, . . . , xn) (4.1.3), the corresponding connected correlators +˜W(J1) +n +(x1, . . . , xn) (4.1.4), and finally some correlator expansion coefficients W(J1),l +n +(x1, . . . , xn) +(4.0.5) of the global scaled (N, N) antisymmetrised Laguerre ensemble, whose definition we +recall from Proposition 3.17. To improve clarity, we begin by introducing some new notation +(local to this section). +Thus, take N to be an even integer and introduce the scaled real N × N Ginibre matrix +X ∈ MN×N(R) with p.d.f. +P(X) = +� N +2π +�N2/2 +exp +� +− N +2 Tr(XTX) +� +. +(4.2.1) +Moreover, denote the elementary antisymmetric matrix JN (1.1.5) by simply the letter J and +define B := XTJX. Then, X is statistically equal to the scaled real Ginibre matrix ˜G = ˜G1 of +Propositions 3.17 and 3.19, and therefore relates to the real Ginibre matrix G of Definition 1.3 +through the equation X = √ +2/NG. Likewise, the antisymmetric matrix product B = XTJX +equals the matrix ˜J1 (3.3.60) of Proposition 3.17 and is thus related to the unscaled matrix +J1 (4.0.2) according to the equality B = 2J1/N. +Next, let ⟨ · ⟩ denote averages with respect to the p.d.f. P(X) (4.2.1) and recall that, e.g., +⟨∂Xab f (X)⟩ should be read as “take the partial derivative of the product f (X)P(X) with +respect to Xab and integrate the result over MN×N(R)”, while ⟨{∂Xab f (X)}⟩ reads “multiply +the partial derivative of f (X) with respect to Xab by P(X) and then integrate the result over +MN×N(R)”. For k1, . . . , kn ∈ N, the mixed moments and unconnected n-point correlators of +B are respectively specified by (recall equations (4.1.2), (4.1.3) and cf. equation (1.1.17)) +mk1,...,kn = +� +n +∏ +i=1 +Tr Bki +� +, +(4.2.2) +Un(x1, . . . , xn) = +∞ +∑ +k1,...,kn=0 +mk1,...,kn +xk1+1 +1 +· · · xkn+1 +n +; +(4.2.3) +note that whenever any of the ki are odd, the trace of the antisymmetric matrix Bki is zero, +so mk1,...,kn vanishes. The connected n-point correlator Wn(x1, . . . , xn) is most easily specified +in terms of the above through equation (4.1.4), but has an alternative charcterisation as a +223 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +generating function of the mixed cumulants ck1,...,kn — which are defined implicitly via the +moment-cumulants relation (1.1.28) — through the expansion (1.1.29) +Wn(x1, . . . , xn) = +∞ +∑ +k1,...,kn=0 +ck1,...,kn +xk1+1 +1 +· · · xkn+1 +n +. +(4.2.4) +These quantities are the same as the scaled analogues discussed in §3.3.3; we have only +simplified notation by omitting the tilde and the superscript (J1) so that, e.g., ˜c(J1) +k1,...,kn is now +written simply as ck1,...,kn. If we use the superscript (J1), without a tilde, to denote statistics +of the unscaled ensemble represented by the matrix J1 (4.0.2), we have +mk1,...,kn = +� 2 +N +�k1+···+kn +m(J1) +k1,...,kn, +(4.2.5) +ck1,...,kn = +� 2 +N +�k1+···+kn +c(J1) +k1,...,kn, +(4.2.6) +Un(x1, . . . , xn) = +� N +2 +�n +U(J1) +n +(Nx1/2, . . . , Nxn/2), +(4.2.7) +Wn(x1, . . . , xn) = +� N +2 +�n +W(J1) +n +(Nx1/2, . . . , Nxn/2). +(4.2.8) +Now, inserting the genus expansion (3.3.65) for the mixed cumulants ck1,...,kn = ˜c(J1) +k1,...,kn +into equation (4.2.4) shows that Wn(x1, . . . , xn) admits a large N expansion of the form +(4.0.5) and we may thus implicitly define the correlator expansion coefficients Wl +n(x1, . . . , xn) +accordingly. +Finally, let us adopt the Einstein summation convention, meaning that repeated matrix +indices in a product of matrix entries are to be summed over 1, . . . , N. For example, AabBbc +denotes the sum over b = 1, . . . , N of the product of the (a, b) entry of A and (b, c) entry of +B. Thus, AabBbc = (AB)ac, the (a, c) entry of the matrix product AB. In particular, we must +take care to remember that Jaa = Tr(J) = 0 and ∂XabXab = ∑N +a,b=1 1 = N2. +As discussed at the beginning of Section 4.1, the first step in deriving our loop equations +is to use integration by parts on a suitable perturbation of the average (4.2.2) (recall that a +perturbation of the average (4.2.3) can also serve as an alternative starting point) in order +to produce an identity of the form (4.1.1). Let us recall from the discussion of Section 3.3 +(particularly that preceding Example 3.2) that traces Tr Bk can be conveniently represented +by k-gons formed by chaining together edges labelled by entries of B into a closed loop. The +intuition behind the first average we consider in our development is that we would like to +224 + +4.2. Loop Equations for the Antisymmetrised Laguerre Ensemble +delete a term in the average (4.2.2) specifying mk1+1,k2,...,kn to cut open the loop representing +Tr Bk1+1 and then reintroduce that same term to join the loop back up, except that the +reintroduced term is now extracted from the p.d.f. P(X) (4.2.1) by way of acting on said +p.d.f. with an appropriate differential operator2. +Lemma 4.1. Let us write Tr Bk1+1 = (Bk1)adBda (using the Einstein summation convention) and +then erase the last term so that we are left with (Bk1)ad. Moreover, let us define In := (k2, . . . , kn) +and use the shorthand Tr BIn := Tr Bk2 Tr Bk3 · · · Tr Bkn. We have that +∂(XT)ab(JT)bc∂Xcd exp +� +− N +2 Tr(XTX) +� += N2Bda exp +� +− N +2 Tr(XTX) +� +(4.2.9) +and, consequently, +� +(Bk1)ad Tr BIn∂(XT)ab(JT)bc∂Xcd +� += N2mk1+1,In. +(4.2.10) +Proof. We give a short proof as a warm-up. First note by the Leibniz product rule that +∂(XT)ab∂Xcd exp +� +− N +2 Tr(XTX) +� += −N∂(XT)abXcd exp +� +− N +2 Tr(XTX) +� += +� +N2(XT)abXcd − Nχa=d,b=c +� +exp +� +− N +2 Tr(XTX) +� +. +Multiplying this result by (JT)bc gives +� +N2(XTJTX)ad − NJbbχa=d,b=c +� +exp +� +− N +2 Tr(XTX) +� +, +which reduces to the right-hand side of equation (4.2.9) upon recalling that Jbb = Tr J = 0 +and observing that (XTJTX)T = XTJX, so (XTJTX)ad = Bda. Pre-multiplying both sides +of equation (4.2.9) by (N/2π)N2/2(Bk1)ad Tr BIn and then integrating over MN×N(R) while +keeping track of notation yields equation (4.2.10). +From the above, one might suggest that our desired starting point be the average +� +∂(XT)ab(JT)bc∂Xcd(Bk1)ad Tr BIn +� +, as starting at the average of a full derivative enables us +to use the fundamental theorem of calculus to immediately show that said average is zero — +this is indeed very similar to the initial average (4.1.17) seen in the loop equation analysis +of the Laguerre and Jacobi β ensembles given in [142], as reviewed in §4.1.1. However, one +2This intuition is essentially the spirit behind the terminology ‘loop equations’ and is also the reason why +loop equations are sometimes referred to as cut-and-join equations. +225 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +soon sees that this path requires us to consider averages of the form +� +Tr(BpXTX) Tr Bq� +, +which are not immediately expressible in terms of the mixed moments. As seen in §4.1.1, +this inevitably leads to a game of finding identities that can be combined with the initial +analogue of equation (4.1.1) in order to obtain a valid loop equation on the mixed moments. +Such games can at times prove to be a tedious matter of trial and error, especially for large +products of matrices, and if one does not have much experience in the exercise. +It turns out that, for reasons soon to be made clear, a lot of guesswork can be avoided by +beginning our analysis at the average +�� +∂(XT)ab + NXba +� +(JT)bc +� +∂Xcd + N(XT)dc +� +(Bk1)ad Tr BIn +� +. +(4.2.11) +Thus, in §4.2.1 forthcoming, we apply integration by parts to this average to obtain loop +equations on the mixed moments mk1,...,kn in a relatively straightforward manner. These loop +equations are then easily transformed into loop equations on the unconnected correlators +Un(x1, . . . , xn). Then, in §4.2.2, we use equations (4.1.4) and (4.0.5) to obtain loop equations +for the connected correlators Wn(x1, . . . , xn) and their expansion coefficients Wl +n(x1, . . . , xn) +from the Un loop equations of §4.2.1. We then conclude this section with some explicit +computations and discussions in relation to W0 +1(x1) and W0 +2(x1, x2). +4.2.1 +Loop equations on the ˜U(J1) +n +Before unpacking the average (4.2.11), let us first give the companion to Lemma 4.1. +Lemma 4.2. The partial derivatives of (Bk1)e f with respect to (XT)ab and Xcd (taking e, f to be +arbitrary and possibly equal to each other or one of a, b, c, d) are given by +∂(XT)ab(Bk1)e f = +∑ +p1+p2=k1−1 +� +(Bp1)ea(JXBp2)b f + (Bp1XTJ)eb(Bp2)a f +� +, +(4.2.12) +∂Xcd(Bk1)e f = +∑ +p1+p2=k1−1 +� +(Bp1)ed(JXBp2)c f + (Bp1XTJ)ec(Bp2)d f +� +, +(4.2.13) +where the sums over p1 + p2 = k1 − 1 are over the range p1 = 0, 1, . . . , k1 − 1 with 0 ⩽ p2 ⩽ k1 − 1 +set to k1 − p1 − 1; these sums are empty for k1 = 0. For later convenience, let us also mention that +∂(XT)ab(Bk1XT)e f = +∑ +p1+p2=k1−1 +� +(Bp1)ea(JXBp2XT)b f + (Bp1XTJ)eb(Bp2XT)a f +� ++ χ f =b(Bk1)ea. +(4.2.14) +226 + +4.2. Loop Equations for the Antisymmetrised Laguerre Ensemble +Proof. First note that by the Leibniz product rule, +∂(XT)abBgh = +� +∂(XT)ab(XT)gl +� +(JX)lh + (XTJ)gl +� +∂(XT)abXlh +� += χg=a,l=b(JX)lh + (XTJ)glχl=b,h=a = χg=a(JX)bh + χh=a(XTJ)gb. +Thus, the left-hand side of equation (4.2.13) decomposes as +∂(XT)ab(Bk1)e f = +k1−1 +∑ +p1=0 +(Bp1)eg +� +∂(XT)abBgh +� +(Bk1−p1−1)h f += +k1−1 +∑ +p1=0 +� +(Bp1)ea(JX)bh(Bk1−p1−1)h f + (Bp1)eg(XTJ)gb(Bk1−p1−1)a f +� +, +which simplifies to the right-hand side of equation (4.2.12) upon summing over the repeated +indices g, h. To obtain equation (4.2.13) from equation (4.2.12), simply observe that setting +(c, d) = (b, a) shows that Xcd = (XT)ab. For equation (4.2.14), note that +∂(XT)ab(Bk1XT)e f = +� +∂(XT)ab(Bk1)eg +� +(XT)g f + (Bk1)eg +� +∂(XT)ab(XT)g f +� += +� +∂(XT)ab(Bk1)eg +� +(XT)g f + (Bk1)egχg=a,f =b += +� +∂(XT)ab(Bk1)eg +� +(XT)g f + (Bk1)eaχ f =b; +substituting equation (4.2.12) with f �→ g into the braces then completes the proof. +With these identities in hand, we can now apply integration by parts to the average +(4.2.11) in two different ways to obtain a precursor to the loop equation on the Un. +Proposition 4.1. Recalling from Lemma 4.1 that In = (k2, . . . , kn) and Tr BIn = Tr Bk2 · · · Tr Bkn, +the average (4.2.11) simplifies to both the left- and right-hand sides of the following equation: +N2mk1+1,In = +�� +∂(XT)ab(JT)bc∂Xcd(Bk1)ad Tr BIn +�� +. +(4.2.15) +Proof. Expanding the argument of the average (4.2.11) shows that it is given by +�� +∂(XT)ab + NXba +� +(JT)bc +� +∂Xcd + N(XT)dc +� +(Bk1)ad Tr BIn +� += +� +∂(XT)ab(JT)bc∂Xcd(Bk1)ad Tr BIn +� ++ N +� +∂(XT)ab(JT)bc(XT)dc(Bk1)ad Tr BIn +� ++ N +� +Xba(JT)bc∂Xcd(Bk1)ad Tr BIn +� ++ N2 � +Xba(JT)bc(XT)dc(Bk1)ad Tr BIn +� +. +(4.2.16) +227 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +The first two terms on the right-hand side of this equation are manifestly zero by the +fundamental theorem of calculus (they are integrals of derivatives of the exponentially +decaying p.d.f. P(X) (4.2.1) multiplied by polynomials in entries of X, which must converge +to zero at the boundary of the domain of integration). Moreover, contracting indices in the +fourth term on the right-hand side of equation (4.2.16) while rewriting (JT)bc as Jcb shows +that this term is equal to the left-hand side of equation (4.2.15). Finally, to see that the third +term on the right-hand side of equation (4.2.16) vanishes, note that for f (X) a function of +the entries of X, integration by parts shows that +� +Xba(JT)bc∂Xcd f (X) +� += +� +∂XcdXba(JT)bc f (X) +� +− +� +{∂XcdXba} (JT)bc f (X) +� += +� +∂XcdXba(JT)bc f (X) +� +− +� +χa=d,b=c(JT)bc f (X) +� += +� +∂XcdXba(JT)bc f (X) +� +− +� +χa=d(JT)bb f (X) +� +, +which reduces to zero by application of the fundamental theorem of calculus to the first term +and by making the substitution (JT)bb = Tr JT = 0 in the second term. +Now, for f (X) again a function of the entries of X, observe that +� +∂(XT)ab + NXba +� +f (X) exp +� +− N +2 Tr(XTX) +� += +� +∂(XT)ab f (X) +� +exp +� +− N +2 Tr(XTX) +� ++ f (X) +� +∂(XT)ab exp +� +− N +2 Tr(XTX) +�� ++ NXba f (X) exp +� +− N +2 Tr(XTX) +� += +� +∂(XT)ab f (X) +� +exp +� +− N +2 Tr(XTX) +� +since ∂(XT)ab exp +�− N +2 Tr(XTX) +� +is equal to −NXba exp +�− N +2 Tr(XTX) +� +. We similarly have +� +∂Xcd + N(XT)dc +� +f (X) exp +� +− N +2 Tr(XTX) +� += {∂Xcd f (X)} exp +� +− N +2 Tr(XTX) +� +. +Thus, the average (4.2.11) simplifies as +�� +∂(XT)ab + NXba +� +(JT)bc +� +∂Xcd + N(XT)dc +� +(Bk1)ad Tr BIn +� += +�� +∂(XT)ab + NXba +� +(JT)bc +� +∂Xcd(Bk1)ad Tr BIn +�� +, +which can be seen to be precisely the right-hand side of equation (4.2.15) upon writing +f (X) = (JT)bc +� +∂Xcd(Bk1)ad Tr BIn +� +. +228 + +4.2. Loop Equations for the Antisymmetrised Laguerre Ensemble +Our claim is that using the Leibniz product rule to expand the derivative in the right-hand +side of equation (4.2.15) and then properly simplifying each of the averages in the resulting +sum will produce our sought loop equation. To demonstrate this claim, let us now write the +right-hand side of equation (4.2.15) as +�� +∂(XT)ab(JT)bc∂Xcd(Bk1)ad Tr BIn +�� += A + +n +∑ +i=2 +�Ai + Ai + Bi +� + ∑ +2⩽i,j⩽n, +i̸=j +Cij, +(4.2.17) +where +A = +�� +∂(XT)ab(JT)bc∂Xcd(Bk1)ad +� +Tr BIn +� +, +(4.2.18) +Ai = +�� +∂(XT)ab(Bk1)ad +� +(JT)bc +� +∂Xcd Tr Bki +� +Tr BIn\{ki}� +, +(4.2.19) +Ai = +�� +∂(XT)ab Tr Bki +� +(JT)bc +� +∂Xcd(Bk1)ad +� +Tr BIn\{ki}� +, +(4.2.20) +Bi = +�� +∂(XT)ab(JT)bc∂Xcd Tr Bki +� +(Bk1)ad Tr BIn\{ki}� +, +(4.2.21) +Cij = +�� +∂(XT)ab Tr Bki +� +(JT)bc +� +∂Xcd Tr Bkj +� +(Bk1)ad Tr BIn\{ki,kj}� +. +(4.2.22) +We show in the following lemma that each of these terms can be expressed in terms of the +mixed moments, so combining equations (4.2.15) and (4.2.17) will result in a loop equation +on said moments, as claimed. +Lemma 4.3. Let us retain the definitions In = (k1, . . . , kn) and Tr BIn = Tr Bk2 · · · Tr Bkn from +Lemma 4.1. Moreover, for k ∈ N, let [k]mod 2 equal zero when k is even and one when k is odd. Then, +the quantities (4.2.18)–(4.2.22) above are given by +A = +∑ +p1+p2=k1−1 +mp1,p2,In − +∑ +p1+p2+p3=k1−1 +mp1,p2,p3,In + 1 − k2 +1 +2 +mk1−1,In, +(4.2.23) +Ai = Ai = 2ki[ki + 1]mod 2 +� +mk1+ki−1,In\{ki} − +∑ +p1+p2=k1−1 +mki+p1,p2,In\{ki} +� +, +(4.2.24) +Bi = −2ki[ki + 1]mod 2 +� +mk1+ki−1,In\{ki} + +∑ +p1+p2=ki−1 +mk1+p1,p2,In\{ki} +� +, +(4.2.25) +Cij = Cji = −4kikj[ki + 1]mod 2[kj + 1]mod 2 mk1+ki+kj−1,In\{ki,kj}. +(4.2.26) +Proof. We present only the computations of A and Cij since their derivations contain all of +the ideas needed to prove equations (4.2.24) and (4.2.25). We first give the proof of equation +229 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +(4.2.26), as it is simpler than that of equation (4.2.23). Thus, we begin by using equation +(4.2.12) with (k1, f ) �→ (ki, e) to see that +∂(XT)ab Tr Bki = ∂(XT)ab(Bki)ee = +∑ +p1+p2=ki−1 +� +(JXBki−1)ba + (Bki−1XTJ)ab +� += (JXBki−1)ba +∑ +p1+p2=ki−1 +� +1 + (−1)ki +� += 2ki[ki + 1]mod 2(JXBki−1)ba. +(4.2.27) +Here, we have used the fact that (Bki−1XTJ)T = JTX(BT)ki−1 = (−1)ki JXBki−1 and that +(1 + (−1)ki) equals two if ki is even and vanishes, otherwise. Repeating this argument with +equation (4.2.13) likewise shows that +∂Xcd Tr Bkj = 2kj[kj + 1]mod 2(Bkj−1XTJ)dc. +Multiplying these two results together, in combination with (JT)bc, then reveals the identity +� +∂(XT)ab Tr Bki +� +(JT)bc +� +∂Xcd Tr Bkj +� += −4kikj[ki + 1]mod 2[kj + 1]mod 2(Bki+kj−1)da. +Further multiplying this by (Bk1)ad Tr BIn\{ki,kj}, summing over the repeated indices a, d, and +then taking the average yields the right-hand side of equation (4.2.26), as required. +Similar to above, we begin the proof of equation (4.2.23) by noting that equation (4.2.13) +with (e, f ) = (a, d) tells us that +(JT)bc∂Xcd(Bk1)ad = +∑ +p1+p2=k1−1 +� +(−1)p2(Bk1−1XT)ab − (Bp1XT)ab Tr Bp2 +� += [k1]mod 2(Bk1−1XT)ab − +∑ +p1+p2=k1−1 +(Bp1XT)ab Tr Bp2, +(4.2.28) +where we have used the fact that ∑p1+p2=k1−1(−1)p2 = ∑k1−1 +p2=0(−1)p2 equals one if k1 is odd +and zero, otherwise. Now, taking (k1, e, f ) �→ (k1 − 1, a, b) in equation (4.2.14) shows that +∂(XT)ab(Bk1−1XT)ab = +∑ +p1+p2=k1−2 +� +Tr Bp1 Tr Bp2+1 + (−1)p2 Tr Bk1−1� ++ N Tr Bk1−1 += +∑ +p1+p2=k1−1 +Tr Bp1 Tr Bp2 + [k1 + 1]mod 2 Tr Bk1−1 += +∑ +p1+p2=k1−1 +Tr Bp1 Tr Bp2; +(4.2.29) +the second line follows from making the replacement p2 �→ p2 − 1 in the sum and noting +that the new (p1, p2) = (k1 − 1, 0) term this introduces is equivalent to the term N Tr Bk1−1 +230 + +4.2. Loop Equations for the Antisymmetrised Laguerre Ensemble +that already appears at the end of the first line, while the final line follows from the fact that +[k1 + 1]mod 2 vanishes whenever k1 is odd, but Tr Bk1−1 vanishes whenever k1 is even. Moving +on to the second term in equation (4.2.28), using equation (4.2.27) with the replacement +ki �→ p2 and equation (4.2.29) with k1 �→ p1 shows that +∂(XT)ab(Bp1XT)ab Tr Bp2 = +� +∂(XT)ab(Bp1XT)ab +� +Tr Bp2 + (Bp1XT)ab +� +∂(XT)ab Tr Bp2 +� += +∑ +q1+q2=p1 +Tr Bq1 Tr Bq2 Tr Bp2 + [p1]mod 2 Tr Bp1 Tr Bp2 + 2p2[p2 + 1]mod 2 Tr Bp1+p2 += +∑ +q1+q2=p1 +Tr Bq1 Tr Bq2 Tr Bp2 + 2p2[p2 + 1]mod 2 Tr Bp1+p2, +(4.2.30) +where the last line follows from the fact that [p1]mod 2 vanishes if p1 is even, but Tr Bp1 +vanishes if p1 is odd. Applying the operator ∂(XT)ab to both sides of equation (4.2.28) and +then substituting in equations (4.2.29) and (4.2.30) shows that +∂(XT)ab(JT)bc∂Xcd(Bk1)ad = [k1]mod 2 +∑ +p1+p2=k1−1 +Tr Bp1 Tr Bp2 +− +∑ +p1+p2=k1−1 +� +2p2[p2 + 1]mod 2 Tr Bp1+p2 + +∑ +q1+q2=p1 +Tr Bq1 Tr Bq2 Tr Bp2 +� +. +(4.2.31) +The factor of [k1]mod 2 in the first term can be removed since it is superfluous: for Tr Bp1, Tr Bp2 +to be non-zero, each of p1, p2 must be even and, consequently, the sum over p1 + p2 = k1 − 1 +is non-zero only if k1 is odd. Likewise, the second term is non-zero for k1 odd and it +simplifies as +− +∑ +p1+p2=k1−1 +2p2[p2 + 1]mod 2 Tr Bp1+p2 = −2 Tr Bk1−1(2 + 4 + · · · + k1 − 1) = 1 − k2 +1 +2 +Tr Bk1−1, +while replacing p1 in the outer sum of the third term by q1 + q2 and then renaming the +summation indices shows that +− +∑ +p1+p2=k1−1 +∑ +q1+q2=p1 +Tr Bq1 Tr Bq2 Tr Bp2 = − +∑ +q1+q2+p2=k1−1 +Tr Bq1 Tr Bq2 Tr Bp2 += − +∑ +p1+p2+p3=k1−1 +Tr Bp1 Tr Bp2 Tr Bp3. +231 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +Combining these observations together, we thus see that equation (4.2.31) reduces to +∂(XT)ab(JT)bc∂Xcd(Bk1)ad = +∑ +p1+p2=k1−1 +Tr Bp1 Tr Bp2 + 1 − k2 +1 +2 +Tr Bk1−1 +− +∑ +p1+p2+p3=k1−1 +Tr Bp1 Tr Bp2 Tr Bp3. +Finally, multiplying the right-hand side of this equation by Tr BIn and taking the average +produces the right-hand side of equation (4.2.23), as required. +It is now a simple matter of substituting the expressions (4.2.23)–(4.2.26) into the right- +hand side of equation (4.2.17) while replacing the left-hand side by N2mk1+1,In, as prescribed +by Propositon 4.1, to obtain the desired loop equation on the mixed moments mk1,...,kn. This +loop equation is presented as equation (C.1.1) in Appendix C.1. +At this point, it is possible to proceed by substituting the moment-cumulants relation +(1.1.28) into the loop equation on the moments to obtain an analogous equation on the +cumulants ck1,...,kn. Simplifying this equation on the ck1,...,kn in an appropriate manner (using a +similar argument to that given in the proof of Proposition 4.3 in §4.2.2 upcoming), multiplying +the result by x−k1−1 +1 +· · · x−kn−1 +n +, and summing over k1, . . . , kn ⩾ 0 would then yield the loop +equation on the Wn(x1, . . . , xn) (cf. equation (4.2.4)). We do not go down this path, instead +opting to first derive the loop equation on the unconnected correlators Un(x1, . . . , xn) (4.2.3) +and then transforming it, through the identity (4.1.4), into the corresponding loop equation +on the Wn(x1, . . . , xn) — if necessary, the loop equation on the mixed cumulants ck1,...,kn +can be extracted from the loop equation on the Wn(x1, . . . , xn) by equating coefficients of +x−k1−1 +1 +· · · x−kn−1 +n +, possibly by taking suitable residues. (Note that we could have alternatively +skipped the derivation of the loop equation on the mixed moments altogether by starting +our analysis at a perturbation of the average (4.1.3), rather than (4.1.2); our choice of starting +point leads to the cleaner presentation.) +Proposition 4.2. Recall that Jn = (x2, . . . , xn), set U0 := 1, and define Un′ := 0 for n′ < 0. +Furthermore, define the auxiliary function +Ai(x1, Jn) = x2 +i (2Un(x1, x1, Jn \ {xi}) − Un(xi, Jn)) − x1xiUn(x1, Jn) ++ 1 +x1 +� +x1xiUn−1(Jn) − x2 +i Un−1(x1, Jn \ {xi}) +� +. +(4.2.32) +232 + +4.2. Loop Equations for the Antisymmetrised Laguerre Ensemble +Then, for n ⩾ 1, the unconnected correlators specified by equation (4.2.3) satisfy the loop equation +0 = x1Un+2(x1, x1, x1, Jn) − Un+1(x1, x1, Jn) + N2x1Un(x1, Jn) − N3Un−1(Jn) ++ 1 +2x1 +� +x2 +1 +∂2 +∂x2 +1 ++ x1 +∂ +∂x1 +− 1 +� +Un(x1, Jn) + 2 +n +∑ +i=2 +∂ +∂xi +� Ai(x1, Jn) +x2 +1 − x2 +i +� ++ 8 +∑ +2⩽i 1 extensions of equations +(4.2.11) and (4.3.9) require us to consider the action of differential operators of the form +∂(XTm)imim−1 · · · ∂(XT +1 )i1i0(JT)i0j0∂(X1)j0j1 · · · ∂(Xm)jm−1jm +with Xi, J defined analogously to the X, J of Section 4.2 and +∂(X†m)imim−1 · · · ∂(X† +1)i1i0∂( ˜H†)i0j0∂(X1)j0j1 · · · ∂(Xm)jm−1jm +with Xi, ˜H defined analogously to the X, ˜H of Section 4.3, respectively. Consequently, we +anticipate that the analogue of equation (4.2.45) pertaining to some fixed m > 1 should +contain a sum over partitions µ of the (2m + 1)-tuple (x1, . . . , x1) of the form seen in the first +257 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +line of equation (4.2.45) and sums of up to (2m)th order partial derivatives of the form seen +in the final two lines of equation (4.2.45); the m > 1 analogue of equation (4.3.31) should +likewise contain a sum over partitions µ of the (2m + 2)-tuple (x1, . . . , x1) in addition to +sums of up to (2m + 1)th order partial derivatives of the canonical extensions of Ai,j,h(x1, Jn) +(4.3.22) having up to (2m + 1) indices. +It was mentioned in §4.1.2 that applying residue calculus to the loop equation (4.1.28) for +the correlator expansion coefficients Wl +n of the one-cut 1-Hermitian matrix model to obtain +the simpler topological recursion formula (4.1.41) for the associated differential forms ωl +n +essentially removes the need for considering the terms Pl +n(x1, Jn) and +n +∑ +i=2 +∂ +∂xi +� +Wl +n−1(x1, Jn \ {xi}) − Wl +n−1(Jn) +x1 − xi +� +seen on the right-hand side of equation (4.1.28). Likewise, it stands to reason that one +option for alleviating the aforementioned complexities associated with the loop equations +for the general m antisymmetrised and Hermitised matrix product ensembles is to use the +methodology reviewed in §4.1.2 to instead derive topological recursion formulae for these +ensembles. This would very likely either replace the aforementioned general m analogues of +Ai,j,h(x1, Jn) with drastically simpler versions of themselves or possibly remove the need for +considering them altogether — it was commented in [74] that the loop equations therein for +the (N, N, N) complex Wishart product ensemble are also expected to reduce to relatively +simple topological recursion formulae upon applying the techniques of §4.1.2. It would +thus be very interesting to see if our loop equations (and those of [74]) can be simplified to +elegant topological recursion formulae in the proposed manner and, if so, to compare the +latter to the higher order topological recursion formulae of [45], [46]. Although this would +still be far off from obtaining loop equations or topological recursion formulae for general m, +it should at the very least be possible to take N → ∞ and observe a uniform structure across +all m ∈ N, thereby enabling us to recover the spectral curves known from equations (1.3.22), +(1.3.23), and (4.3.28) [29], [273], [127], [143], [144], +0 = x2m−1 +1 +� +W(Jm),0 +1 +(x1) +�2m+1 ++ x1W(Jm),0 +1 +(x1) − 1, +(4.4.1) +0 = x2m +1 +� +W(Hm),0 +1 +(x1) +�2m+2 +− x1W(Jm),0 +1 +(x1) + 1. +(4.4.2) +258 + +4.4. Concluding Remarks and Outlook +As an aside, a related endeavour that may be worth pursuing is to see if the theory of +[86], [63] concerning spectral curves corresponding to eigenvalue densities with hard edges +and that of [108] relating to topological recursion for general β ensembles can be used to +derive explicit topological recursion formulae from the loop equations [142] for the Laguerre +and Jacobi β ensembles discussed in §4.1.1. Such explicit formulae should be attainable as +these ensembles fall within the abstract setting considered in [41]. +4.4.2 +On 1-point recursions +As the loop equation formalism and topological recursion fully determine the resolvent +expansion coefficients Wl +1, which act as generating functions for the moment expansion +coefficients +˜Mk,l defined implicitly through equation (1.1.32), it is reasonable to say that +the data produced by 1-point recursions on the +˜Mk,l is a subset of that produced by the +loop equation formalism — the benefit in considering 1-point recursions is that they are +more efficient at generating this data. Thus, it should come as no surprise that many +ensembles that can be studied using 1-point recursions are also known to be susceptible to +loop equation analysis (see [61] and references therein for a brief review). Seeing as how the +1-point recursions given in §3.1.2 characterise random matrix ensembles that have also been +shown to be governed by the loop equations reviewed in §4.1.1, it is natural to ask if there +exist 1-point recursions on the ˜Mk,l = cl +k that are determined by the loop equation analysis +of §4.2.2 and §4.3.2. We suggest two approaches for obtaining such 1-point recursions: +1. It may be possible to derive 1-point recursions for the antisymmetrised Laguerre +ensemble by using the techniques of Chapters 2 and 3 in addition to the theory of [125] +relating the Laguerre Muttalib–Borodin ensemble to the Selberg integral. (Incidentally, +let us mention here that it remains to be seen how far the theory of Chapter 2 can be +pushed to uncover features of the classical β ensembles with general β ∈ 2N.) +2. It should be checked if Chekhov’s application [64] of an adaptation of the Br´ezin– +Hikami replica method [51] to derive the 1-point recursion (3.1.38) for the LUE with +a = 0 can be repurposed to treat the antisymmetrised and Hermitised Laguerre +ensembles. It would likewise be interesting to see if the 1-point recursions of §3.1.2 can +be obtained in this way, as well. +259 + +CHAPTER 4. Loop Equations for the Matrix Product Ensembles +Let us recall from the discussion at the end of §3.1.2 that while the moment expansion +coefficients Mk,l of the Gaussian and Laguerre ensembles have combinatorial interpretations +due to the theory reviewed in §3.3.1 and §3.3.2, these interpretations are not known to trans- +late to combinatorial interpretations for the 1-point recursions on said expansion coefficients. +However, preliminary experimentation by the present author suggests that dressing the +bijective proof of the Catalan recursion (3.1.25) sketched below Figure 3.2 (and demonstrated +in Example 3.1) with suitable additional rules may result in natural combinatorial proofs +of both the Harer–Zagier recursion (3.0.12) and the 1-point recursion (3.1.24) for the LUE. +Confirming this conjecture to be true and comparing the proposed bijections to those men- +tioned in [61, Sec. 1] would shed light on how the 1-point recursions of §3.1.2 relate to each +other. This is of particular interest to us since the 1-point recursions for the (shifted) JUE +presented in Corollary 3.4 are very similar to those for the GUE and LUE — it is hoped that +a combinatorial understanding of the GUE and LUE 1-point recursions in terms of ribbon +graphs could extend to a similar understanding of the 1-point recursions and, in turn, the +spectral moments of the (shifted) JUE. +4.4.3 +Combinatorics associated with the Jacobi unitary ensemble +In contrast to the approach described above for obtaining combinatorial interpretations for +the JUE moments through associated interpretations of its 1-point recursion, we now show +what happens if one tries to obtain such combinatorial interpretations by instead modifying +the arguments of Section 3.3. First, let G1, G2 be drawn from the M1 × N, respectively +M2 × N, complex Ginibre ensemble of Definition 1.3 so that for i = 1, 2, Wi = G† +i Gi is drawn +from the (Mi, N) LUE, in accordance with Definition 1.4. Then, according to this same +definition, Y = W1(W1 + W2)−1 represents the (M1, M2, N) JUE. Letting ⟨ · ⟩ denote averages +with respect to the product P(L)(W1)P(L)(W2), with P(L)(W) specified by equation (1.2.2), +equation (1.1.15) then tells us that the spectral moments of the JUE are given by +m(JUE) +k += +� +Tr Yk� +, +k ∈ N += +� +Tr +� +IN + W2W−1 +1 +�−k� += +∞ +∑ +l=0 +�l + k − 1 +l +� +(−1)l � +Tr (W2W−1 +1 )l� +, +(4.4.3) +260 + +4.4. Concluding Remarks and Outlook +where we have put aside issues of convergence and used the generalised binomial theo- +rem. This computation reduces the problem at hand to that of finding a combinatorial +interpretation for the spectral moments of W2W−1 +1 . Using the independence of W1, W2, it can +be shown [326], [25] that said spectral moments are given by particular sums of products +of mixed moments of W2 and W−1 +1 . Alternatively, one may use the cyclic property of the +trace to show that these same spectral moments are given by the averages ⟨Tr (G2W−1 +1 G† +2)l⟩, +which can be interpreted as the weighted count of LUE ribbon graphs, with the weights +being certain mixed moments of the inverse Wishart ensemble (cf. Remark 3.7 preceding +Proposition 3.17). The bottom line is that, since the mixed moments of the inverse Wishart +ensemble are currently not known to have any interpretations in terms of ribbon graphs, the +argument just given cannot produce such interpretations for the JUE spectral moments. +If one is content with combinatorial interpretations of JUE spectral moments that are +not related to ribbon graphs, then the situation is more fortuitous. Indeed, the above +construction can be made fully combinatorial by observing that the mixed moments of the +inverse Wishart ensemble have recently been characterised in terms of monotone double +Hurwitz numbers [73], [159]. More recently still, the authors of [159] have characterised the +mixed moments of the JUE itself in terms of monotone triple Hurwitz numbers [160]. These +works complement the earlier work [40], which characterised the mixed moments of the +GUE in terms of so-called 2-orbifold strictly monotone Hurwitz numbers. A curious feature +of these works is that the proofs therein of the relationships between mixed moments and +Hurwitz numbers do not have a combinatorial flavour of the sort discussed in Section 3.3, +even though Hurwitz numbers are very closely related to combinatorial (hyper)maps and +constellations (see [220], [16]). This gap has been addressed in the GUE case [44], but it +remains to be seen how the Hurwitz number combinatorics of the LUE and JUE could be +related to ribbon graphs. It would be interesting to see if the results of [73], [159], [160] +can be reverse engineered to determine classes of ribbon graphs that are enumerated by +the mixed moments of the JUE. 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A, 33(10): +2045–2057, 2000. doi: 10.1088/0305-4470/33/10/307. 14 +289 + +Appendices + +Appendix A +Quaternionic Matrices and Pfaffians +The quaternions H are a number system that contain the real and complex numbers R, C. +They are a finite-dimensional associative division algebra over R, of which there are exactly +three up to isomorphism (the other two being R, C) [146]. They are also referred to as a +skew-field since the only field axiom not satisfied by H is that of commutative multiplication. +The quaternions are best realised as the Clifford algebra Cl0,2(R), which is an R-algebra +generated by two elements that we label i and j, each squaring to negative one and together +satisfying the anti-commutation relation ij + ji = 0. Defining k := ij, one has the isomorphism +H ≃ {a + bi + cj + dk | a, b, c, d ∈ R} +(A.0.1) +with products induced by the following multiplication table: +× +1 +i +j +k +1 +1 +i +j +k +i +i +−1 +k +−j +j +j +−k +−1 +i +k +k +j +−i +−1 +Note A.1. Some authors refer to the right-hand side of (A.0.1) as the ‘real quaternions’ due to +the coefficients a, b, c, d being real, in contrast to the ‘complex quaternions’ defined by (A.0.1) +with a, b, c, d ∈ C. Throughout this thesis, H refers strictly to the real quaternions. +291 + +APPENDIX A. Quaternionic Matrices and Pfaffians +We can realise the well known embedding MM×N(H) �→ M2M×2N(C) by mapping each +matrix entry according to +a + bi + cj + dk �→ +� +� a + bi +c + di +−c + di +a − bi +� +� . +(A.0.2) +In fact, we identify MM×N(H) with the set of M × N matrices whose entries are themselves +2 × 2 matrices of the form given on the right-hand side of (A.0.2). +The quaternionic conjugate of q = a + bi + cj + dk is q = a − bi − cj − dk and the squared +norm is |q|2 = qq = qq = a2 + b2 + c2 + d2. The quaternionic dual of a 2N × 2M complex +matrix X is given by +XD := −J2MXTJ2N +where +J2N := IN ⊗ +� +�0 +−1 +1 +0 +� +� +is the 2N × 2N block-diagonal matrix with the diagonal blocks given by +J2 = +� +�0 +−1 +1 +0 +� +� . +If Q is the matrix representation of the quaternion q, then QD is the matrix representation of +q. More generally, if X ∈ MM×N(H), i.e., if X is a 2M × 2N complex matrix with 2 × 2 block +structure such that each block can be interpreted as a quaternion, then XD is the quaternionic +adjoint X† of X. Here, the quaternionic adjoint X† is equivalent to the transpose of the +quaternionic conjugate of X. With this in mind, the (unitary) symplectic group is defined by +an obvious generalisation of the orthogonal and unitary groups, +Sp(2N) := {U ∈ MN×N(H) | U†U = I2N}. +(A.0.3) +Due to the fact that H is not commutative, the determinant is not well-defined for +quaternionic matrices. When deciding on a definition, one needs to choose which properties +of the determinant must be retained and which can be jettisoned. For our purposes, it is +enough that the determinant correspond to the product of eigenvalues. Thus, we adopt a +definition of Dyson’s for self-adjoint quaternionic matrices [99]. +292 + +APPENDIX A. Quaternionic Matrices and Pfaffians +Definition A.1. Let X ∈ MN×N(H) be self-adjoint and for the sake of clarity, let ˜X be its +2N × 2N complex-matrix representation. Then, define the quaternionic determinant as +Det X := Pf J−1 +2N ˜X, +(A.0.4) +where Pf Y is the Pfaffian of Y. Likewise, to ensure that the trace of a quaternionic matrix is +the sum of its eigenvalues, we define the quaterionic trace as +Tr X := +N +∑ +i=1 +Re (Xii) = 1 +2 Tr ˜X. +The Pfaffian of an anti-symmetric matrix can be thought of as the square root of its +determinant. Strictly speaking, given an anti-symmetric matrix X ∈ M2N×2N(C), +Pf X = +1 +2NN! ∑ +σ∈S2N +sgn(σ)Xσ(1),σ(2)Xσ(3),σ(4) · · · Xσ(2N−1),σ(2N), +where S2N is the symmetric group of order (2N)! and sgn(σ) is the sign of the permutation σ. +Similar to the determinant, the Pfaffian can alternatively be computed through the following +cofactor expansion (expanding along the first row): +Pf X = +2N +∑ +j=2 +(−1)jX1j Pf X(1, j), +where the minor X(1, j) is obtained from X by deleting the first and jth rows and columns; if +one were instead computing the determinant, only the first row and jth column of X would +need to be deleted to produce the desired minor X(1, j). The Pfaffian has an interpretation +as a square root because of the fact that (Pf X)2 = Det X. Note that this means that the +square of the right-hand side of equation (A.0.4) is Det ˜X, so that eigenvalues are not +double-counted when computing the quaternionic determinant. +293 + +Appendix B +Particular Stieltjes Transforms +To take the Stieltjes transforms of equations (2.3.7) and (2.3.12), we need to compute terms +of the form +� 1 +0 +xp(1 − x)q +s − x +dn +dxn ρ(J)(x; N, β) dx +for β = 2 or 4, 0 ⩽ n ⩽ 5, n ⩽ q ⩽ n + 2, and q ⩽ p ⩽ q + 1 all integers. To this end, we +define +Iβ(s; p, q, n, k) := +� 1 +0 +xp(1 − x)q +(s − x)k +dn +dxn ρ(J)(x; N, β) dx +(B.0.1) +for integers 0 ⩽ n ⩽ q ⩽ p and k ⩾ 0. Then, integration by parts gives the identity +Iβ(s; p, q, n, k) = (p + q)Iβ(s; p, q − 1, n − 1, k) − pIβ(s; p − 1, q − 1, n − 1, k) +− kIβ(s; p, q, n − 1, k + 1). +(B.0.2) +Applying this identity n times allows us to reduce Iβ(s; p, q, n, k) to an expression involving +terms of the form Iβ(s; p, q, 0, k). Then, considering (s − x)−k−1 = (−1)k/k! ∂k +s(s − x)−1 for +k ⩾ 0 gives us +Iβ(s; p, q, 0, k + 1) = (−1)k +k! +dk +dsk Iβ(s; p, q, 0, 1), +k ⩾ 0, +(B.0.3) +which allows us to further reduce to an expression involving terms of the form Iβ(s; p, q, 0, 1). +Finally, factorisation of xp − sp for positive integer p yields +Iβ(s; p, q, 0, 1) = spIβ(s; 0, q, 0, 1) − +p−1 +∑ +l=0 +sp−l−1Iβ(s; l, q, 0, 0), +p ⩾ 1 +(B.0.4) +294 + +APPENDIX B. Particular Stieltjes Transforms +and likewise +Iβ(s; 0, q, 0, 1) = +q +∑ +m=0 +� q +m +� +(−1)m +� +smW(J) +1 (s; N, β) − +m−1 +∑ +l=0 +sm−l−1m(J) +l +� +, +q ⩾ 1, +(B.0.5) +where m(J) +l +is the lth spectral moment of ρ(J)(x; N, β). These last two equations thus reduce +our expression to one involving powers of s, derivatives of W(J) +1 (s; N, β), and spectral +moments of ρ(J)(x; N, β). +To begin, we list +Iβ(s; 1, 1, 1, 1) = s(1 − s) d +dsW(J) +1 (s; N, β) − N +Iβ(s; 2, 2, 2, 1) = s2(1 − s)2 d2 +ds2W(J) +1 (s; N, β) − 2N(s − 2) − 6m(J) +1 +Iβ(s; 3, 3, 3, 1) = s3(1 − s)3 d3 +ds3W(J) +1 (s; N, β) − 6N +� +s2 − 3s + 3 +� − 24m(J) +1 (s − 3) − 60m(J) +2 +Iβ(s; 4, 4, 4, 1) = s4(1 − s)4 d4 +ds4W(J) +1 (s; N, β) − 24N(s3 − 4s2 + 6s − 4) +− 120m(J) +1 +� +s2 − 4s + 6 +� − 360m(J) +2 (s − 4) − 840m(J) +3 +Iβ(s; 5, 5, 5, 1) = s5(1 − s)5 d5 +ds5W(J) +1 (s; N, β) − 120N(s4 − 5s3 + 10s2 − 10s + 5) +− 720m(J) +1 (s3 − 5s2 + 10s − 10) − 2520m(J) +2 (s2 − 5s + 10) +− 6720m(J) +3 (s − 5) − 15120m(J) +4 +All of the necessary Iβ(s; p, q, n, k) can be obtained from these through variants of identity +(B.0.4) and integration by parts. For example, +Iβ(s; n + 1, n, n, 1) = sIβ(s; n, n, n, 1) − Iβ(s; n, n, n, 0) += sIβ(s; n, n, n, 1) + (−1)n+1 +� 1 +0 +dn +dxn (xn(1 − x)n)ρ(J)(x; N, β) dx, +which only requires knowledge of the spectral moments m(J) +0 +to m(J) +n +of ρ(J)(x; N, β), in +addition to a term from the above list. It should be noted that applying the Stieltjes transform +to xp(1 − x)q dn +dxn ρ(J)(x) produces sp(1 − s)q dn +dsn W(J) +1 (s) plus terms that do not involve W(J) +1 (s). +It follows that the differential equations for the resolvents will be the same as those for the +eigenvalue densities, with additional inhomogeneous terms. +295 + +Appendix C +Loop Equations on Moments +C.1 +Loop Equations on the ˜m(J1) +k1,...,kn +Let mk1,...,kn = ˜m(J1) +k1,...,kn be as defined in equation (4.2.2). Substituting equations (4.2.23)– +(4.2.26) into equation (4.2.17) and then substituting the result into equation (4.2.15) yields +the following loop equation (recall that In = (k2, . . . , kn) and Tr BIn = Tr Bk2 · · · Tr Bkn): +0 = N2mk1+1,In + +∑ +p1+p2+p3=k1−1 +mp1,p2,p3,In − +∑ +p1+p2=k1−1 +mp1,p2,In + k2 +1 − 1 +2 +mk1−1,In ++ 8 +∑ +2⩽i0 of the form Dr X = exp(logr ·s)X +with s : g → g being a diagonalisable linear operator with 1 as its smallest eigenvalue. +Remark 2.2. The requirement that the smallest eigenvalue of s be 1 is purely cosmetic. Oth- +erwise, denoting the smallest eigenvalue by s1, one can work with the new operator ˜s = 1 +s1 s. +Remark 2.3. For a ∈ R, denote by Wa ⊂ g the eigenspace of s with eigenvalue a. Thus for +X ∈ Wa, Y ∈ Wb one has the condition +Dr [X ,Y ] = [Dr X ,Dr Y ] = r a+b[X ,Y ] +which in particular implies [Wa,Wb] ⊂ Wa+b and since Wa = {0} for a < 1 +g(j) ⊂ +� +a≥j +Wa . +In particular, if g admits a family of dilations, it is nilpotent. The converse is not necessarily +true. +Definition 2.2. A homogeneous Lie group G is a simply connected, connected Lie group where +its Lie algebra g is endowed with a family of dilations {Dr }r>0. For r > 0, we define the group +automorphism +x �→ r · x := exp◦Dr ◦exp−1 x . +A homogeneous norm on G is a continuous function |·| : G → [0,∞) satisfying the following +properties for all x ∈ G, r ∈ R +9 + +1. |x| = 0 if and only if x = e, the neutral element, +2. |x| = |x−1| , +3. |r · x| = r|x| . +The homogeneous norm naturally induces a topology generated by the open sets and in +turn a Borel σ-algebra. From now on, we will always assume that G is equipped with this +topology and σ-algebra. Furthermore, given a homogeneous group G we denote by {X j}d +j=1 ∈ +g a basis of eigenvectors of s with eigenvalues 1 = s1 ≤ s2 ≤ ... ≤ sd and such that +sX j = sj X j . +(2.2) +Given a measurable subset E ⊂ G we write |E| for its Haar measure which we assume to be +normalized such that the set B1 =: {x ∈ G : |x| ≤ 1} has measure 1, c.f. Proposition 2.1. In +integrals we use the standard notation dx. We define |s| := trace(s) as the homogeneous di- +mension of G, since for any measurable subset E ⊂ G and r > 0, one has +|r ·E| = r |s||E| . +We also define balls of radius r > 0, +Br(x) := {y ∈ G : |x−1y| < r} . +The topology induced by these balls agrees with the topology of G as a Lie group, see [FR16, +Sec. 3.1.6]. Note that due to the non-commutativity of G, in general |x−1y| ̸= |yx−1|. We +consistently work with the following choice a semi-metric on the group, +dG(x, y) := |x−1y| = |y−1x| . +For K ⊂ G we write ¯K := {z ∈ G : dG(z,K) := infy∈K|y−1z| ≤ 1} for the 1-fattening of K. +Remark 2.4. A function f : G → R is called homogeneous of degree λ ∈ R, if f (r · x) = r λf (x) +for all x ∈ G. One can show, c.f. [FS82, Prop. 1.5 & Prop. 1.6], that for any homogeneous norm +on G there exists γ > 0 such that +• |xy| ≤ γ(|x|+|y|) for any x, y ∈ G +• ||xy|−|x|| ≤ γ|y| for any x, y ∈ G such that |y| ≤ 1 +2|x| . +Furthermore all homogeneous norms are mutually equivalent and we may always choose a +homogeneous norm that is smooth away from e ∈ G, [FR16, Sec. 3.1.6]. +2.1 DERIVATIVES AND POLYNOMIALS +We identify g with the left invariant vector fields gL on G and write gR for the right invariant +vector fields. We write Xi for the the basis elements as in (2.2) seen as elements of gL and Yi +for the basis of gR satisfying Yi|e = Xi|e . Thus we can write +X j f (y) = ∂t f (y exp(t X j))|t=0 +and +Yj f (y) = ∂t f (exp(tYj)y)|t=0 +10 + +for any smooth function f ∈C ∞(G). +A map P : G → R is called a polynomial if P ◦ exp : g → R is a polynomial on g.1 Let ζi be +the basis dual to the basis Xi of g. We set η j = ζj ◦ exp−1, which maps G to R. Note that +η = (η1,...,ηd) forms a global coordinate system 2 and furthermore any polynomial map on G +can be written in terms of coefficients aI ∈ R as +P = +� +I +aIηI +with the sum running over a finite subset of Nd and where for a multi-index I = (i1,...,id) ∈ Nd +we write ηI = ηi1 +1 ·...·ηid +d . Define d(I) = � +j sji j and |I| = � +j i j, we call max{d(I) : aI ̸= 0} the +homogeneous degree and max{|I| : aI ̸= 0} the isotropic degree of P. For a > 0, we denote +by Pa the space of polynomials of homogeneous degree strictly less than a and define △ = +{d(I) ∈ R : I ∈ Nd}. 3 We can rewrite the group law on G explicitly in terms of η = (η1,...,ηd). +Proposition 2.5. For j ∈ {1,...,d} and multi-indices I, J s.t. d(I) + d(J) = sj, there exist con- +stants C I,J +j +> 0 such that the following formula holds, +η j(xy) = η j(x)+η j(y)+ +� +I,J̸=0, d(I)+d(J)=sj +C I,J +j ηI (x)ηJ(y) . +Proof. By the Baker–Campbell–Hausdorff formula (2.1) one has +η j(xy) = η j(x)+η j(y)+ +� +|I|+|J|≥2 +C I,J +j ηI (x)ηJ(y) . +By setting either x = e or y = e, we find that C I,J +j += 0 if I = 0 or J = 0. Since furthermore +η j((r x)(r y)) = r sj η j(xy) the claim follows. +Remark 2.6. Proposition 2.5 implies that for sj < 2 one has η j(xy) = η j(x) + η j(y) while for +sj = 2 one has η j(xy) = η j(x)+η j (y)+� +sk=sl=1C k,l +j ηk(x)ηl(y) . +Remark 2.7. It is also noteworthy to realise that Proposition 2.5 implies that Pa is invariant +under right and left-translations. (This is not true if one replaces homogeneous degree by +isotropic degree, except if G is abelian or a = 0). +Remark 2.8. If follows from Proposition 2.5 that one can write +ηK (xy) = ηK (x)+ηK (y)+ +� +I,J̸=0, d(I)+d(J)=d(K ) +C I,J +K ηI(x)ηJ(y) , +where the constants C I,J +K +can be written in terms of the constants C I,J +j . +1Recall that the space of polynomial functions on g is canonically isomorphic to � +n(g∗)⊗sn where ⊗s denotes +the symmetric tensor product. +2Occasionally we use the corresponding notation +∂ +∂ηi . +3We point out a possibly counter-intuitive quirk of our definition; for k ∈ △ the set Pk does not contain polyno- +mials of degree k but only those of degree less than k. +11 + +Lemma 2.9. For i, j ∈ {1,...,d}, +Xiη j = δi,j + +� +I̸=0, d(I)=sj −si +C I,ei +j +ηI , +(2.3) +where ei denotes the multi-index (0,...,0,1,0,...,0) with the 1 being in the i-th slot. +Proof. This follows directly from Proposition 2.5, applying Xi to the function +y �→ η j(xy) , +evaluating at y = 0 and using the fact that Xi|0 = +∂ +∂ηi |0. +Proposition 2.10 ([FS82, Prop. 1.26]). One has X j = �P j,k +� +∂ +∂ηk +� +where +P j,k = +� 1 +if k = j +0 +if sk ≤ sj,k ̸= j +and P j,k is a homogeneous polynomial of degree sk −sj if sk > sj . The analogous statement +holds for the vector fields Yj, +For a multi-index I = (i1,...,id) ∈ Nd we introduce the notation X I = X i1 +1 ...X id +d . Note that the +order of the composition matters since g is not in general Abelian. It is a well known fact that +any left invariant differential operator on G can (uniquely) be written as a linear combination +of {X I}I∈Nd . The next proposition follows as a direct consequence. +Proposition 2.11 ([FS82, Prop. 1.30]). The following maps from Pa → RdimPa are linear iso- +morphisms. +1. P �→ +�� +∂ +∂η +�I +P(e) +� +d(I) d(I) one has that +X I Pa +x[f ] = Pa−d(I) +x +[X I f ] +for all f ∈ C ∞(G). Indeed, this follows from the fact that for any d(J) < a − d(I) one has +X J(X I Pa +x[f ])(e) = X J(X I f )(x), since X J X I can be written as a linear combination of {X K }d(K )≤a. +12 + +Theorem 2.13 (Taylor’s Theorem). For each a ≥ 0 and every f ∈ C ∞(G) it holds that +f (xy)−Pa +x[f ](y) = +� +|I|≤[a]+1,d(I)≥a +� +G +X I f (xz)QI(y,dz) , +where for each multi-index I and y ∈ G the measure QI(y, ·) is supported on Bβ[a]+1|y|(e) for +some β > 0 depending only on G and satisfies +� +G |QI(y,dz)| ≲ |y|d(I). +Remark 2.14. Note that, while it is very useful to have a rather explicit form of the reminder in +order establish Schauder estimates in Section 3.5, there is a more common version of Taylor’s +Theorem on homogeneous groups, c.f. [FS82, Thm. 1.37], which states that the remainder +satisfies the estimate +|f (xy)−Pa +x[f ](y)| ≲a +� +|I|≤[a]+1,d(I)≥a +|y|d(I) +sup +|z|≤β[a]+1|y| +|X I f (xz)| +(2.4) +and which follows as a trivial consequence of Theorem 2.13. Furthermore the analogous +claim for the right Taylor polynomial, (PR)a +x[f ], holds. In this case the analogue of (2.4) reads +|f (yx)−(PR)a +x[f ](y)| ≲a +� +|I|≤[a]+1,d(I)≥a +|y|d(I) +sup +|z|≤β[a]+1|y| +|Y I f (zx)| . +(2.5) +Remark 2.15. If we define ˜Pa +x[f ](y) := Pa +x[f ](x−1y), then it follows that +f (y)− ˜Pa +x[f ](y) = +� +|I|≤[a]+1,d(I)≥a +� +G +X I f (xz)QI(x−1y,dz) +and in particular that +|f (y)− ˜Pa +x[f ](y)| ≲a +� +|I|≤[a]+1 +d(I)≥α +|x−1y|d(I) +sup +|z|≤C|x−1y| +|X I f (xz)| . +In the particular case when f is compactly supported, one can rewrite this for the rescaled +function f λ(x) = +1 +λ|s| f (λ· x) as +|f λ(y)− ˜Pa +x[f λ](y)| ≲a +� +|I|≤[a]+1 +d(I)≥a +λ−d(I)−|s||x−1y|d(I) sup +z∈G +|X I f (z)| +(2.6) +≲a,f +� +δ≥0 +λ−(a+δ)−|s||x−1y|a+δ , +where the sum over δ runs over some finite subset of (0,∞). +Remark 2.16. Given f ∈ C ∞(G), if we define F ∈ C ∞(G × G) by setting F(y,z) = f (z−1y), we +find that +˜Pa +x[F(·,z)](y) = ˜Pa +z−1x[f ](z−1y) . +Remark 2.17. For any p ∈ Pγ one has +˜Pγ +x[p](y) = p(y) . +13 + +Remark 2.18. In the proof of Theorem 2.13, given below, we use the following elementary +observations +• The map +Φ : Rn → G, +(t1,...,tn) �→ exp(t1X1)·...·exp(tnXn) +is a diffeomorphism. To see this note that it is clearly a local diffeomorphism since one +has +DΦ|0 : T Rd|0 ∼ Rd → T G|0 ∼ g +(t1,...,tn) �→ +� +i +ti Xi. +Then since, +Φ(r s1t1,...,r sn tn) = r ·Φ(t1,...,tn) +it is seen to be a global diffeomorphism. +• Setting for t ∈ Rd, +|t|s := +d� +i=1 +|ti|1/si , +one finds that there exists some β = β(G) > 0 such that +1 +β|t|s ≲ |Φ(t)| ≲ β|t|s +(2.7) +uniformly over t ∈ Rd. +• By repeated use of the commutator, for any I, J ∈ Nd and i ∈ N there exist coefficients +λi,J,I such that +Xi X J = +� +I +λi,J,I X I +(2.8) +and one has λi,J,I = 0 whenever d(J)+di ̸= d(I) . +• If a ∈ △, then ma := max{|I| : d(I) ≤ a} = [a] where [a] denotes the integer part of a. +This follows from the fact that d(I) ≤ a implies ma ≤ [a] and since ([a],0,...,0) is in the +set over which the maximum is taken. +Proof of Theorem 2.13. Define for i ∈ {1,...,d} the following measure on Rd with support on +Bs +|t|s(0) +˜Qei (t,ds) = +� +ji +δ0(ds j) +and define Qei (y, ·) = Φ∗ ˜Qei (Φ−1(y), ·) to be the push-forward measure. First we show that, +f (xy)− f (x) = +n� +i=1 +� +G +(Xi f )(xz)Qei (y,dz) . +(2.9) +14 + +Indeed one can write for y = y1y2... yd, where yi = exp(ti Xi) and t = (t1,...,td) = Φ−1(y) +f (xy)− f (x) = +d� +i=1 +f (xy1... yi−1yi)− f (xy1...yi−1) += +d� +i=1 +�ti +0 +∂s f (xy1... yi−1exp(sXi))|s=s′ds′ += +d� +i=1 +�ti +0 +(Xi f )(xy1 ... yi−1exp(sXi))ds += +d� +i=1 +� +Rn(Xi f )(xΦ(s)) ˜Qei (t,ds) += +d� +i=1 +� +G +(Xi f )(xz)Qei (y,dz) . +Using the fact that Φ is a diffeomorphism one easily checks that that Qei (y,dz) is supported +on Bβ|y|(0) where β is as in (2.7), and satisfies +� +G |Qei |(y,dz) ≲ |y|di , thus we have proved the +theorem in the special case a ∈ (0,1], also known as mean-value theorem. We turn to the +proof in the general case. +Claim 2.19. Set g(y) = f (xy)−Pa +x[f ](y), we note that X J g(0) = 0 for d(J) < a while X J g(y) = +X J f (xy) for d(J) ≥ a. We shall prove that for any multi-index J it holds that one can write +X J g(y) = +� +|I|≤[a]+1,d(I)≥a +� +G +X I f (xz)QI,J(y,dz) , +where the measures QI,J(x,·) satisfy the following properties +• QI,J(x,·) is supported on Bβm(J)|y|(0) ⊂ G where m(J) = min{m ∈ N : a −m < d(J)} , +• +� +G |QI,J|(x,d y) ≲ |x|d(I)−d(J) . +We shall prove by induction on m ∈ {1,...,⌈a⌉}, that for multi-indices J satisfying a − m < +d(J) ≤ a the claim holds. +• The case m = 1 follows from (2.8) and (2.9), +X J g(y) = X J g(y)− X J g(0) = +d� +i=1 +� +G +(Xi X J g)(z)Qei (y,dz) += +n� +i=1 +� +G +� +I : d(I)=d(J)+si +λi,J,I (X I g)(z)Qei (y,dz) += +n� +i=1 +� +G +� +I : d(I)=d(J)+si +λi,J,I (X I f )(xz)Qei (y,dz) += +� +|I|≤[a]+1,d(I)≥a +� +G +X I f (xz)QI,J(y,dz) +The properties of QI,J(y,dz) follow from the corresponding properties of Qei (y,dz), +completing the case m = 1. +15 + +• Suppose the claim holds for m −1. Then by the same argument as above +X J g(y) = +n� +i=1 +� +d(K )=d(J)+si +λi,J,K +� +G +(X K g)(xz)Qei (y,dz) += +� +i,K :d(K )=d(J)+si 0, there exists a continuous map G → Pα, x �→ ˜Px such that +∥f ∥Cα(K) = sup +x∈K +sup +φ∈B0 +sup +λ∈(0,1) +|〈f − ˜Px,φλ +x〉| +λα ++sup +x∈K +| ˜Px|Pα < +∞ , +(2.13) +where | · |Pα denotes any norm on the finite dimensional vector space Pα. +Furthermore,C(G) denotes the space of continuous functions (note that only the strict inclusion +C(G) ⊊ C0(G) holds). +Remark 2.21. We note that as in [Hai14], for α ∈ △ the Cα(G) spaces do not agree with the +usual notion of continuously α-differentiable functions. This definition, however, is more +canonical in the context of regularity structures, c.f. Theorem 3.12. +The following proposition gives some useful properties that also mirror those of Euclidean +Hölder spaces. +Proposition 2.22. Given any real number α ∈ R the following holds +1. Cα(G) ⊂ Cβ(G) for every β < α. +2. For any multiindex I the map X I : Cα(G) → Cα−d(I)(G), +f �→ X I f is well defined. +3. If α > 0, then any distribution f ∈ Cα(G) agrees4 with a continuous function. +Note that this proposition in particular implies that C ∞(G) = ∩α>0Cα(G). +4As usual, we say a distribution agrees with a continuous functions, if it lies in the image of the canonical em- +bedding C(G) → D′(G). +18 + +Proof. Points 1 and 2 follow directly. For Point 3, denote by ˜Px the polynomial in Defini- +tion 2.4, we set F = f − ˜P·(·), then for any ψ ∈ B0 such that +� +G ψ = c−1 > 0 and smooth com- +pactly supported function g we find +〈F,g〉 = c lim +λ→0〈F,g ∗ψλ〉 = c lim +λ→0 +� +F, +� +G +g(y)ψλ +yd y +� += c lim +λ→0 +� +G +〈F,ψλ +y〉g(y)d y = 0 , +where in the last step we used that since y → ˜Py is continuous we have +〈F,ψλ +y〉 = 〈f − ˜Py(·),ψλ +y〉+ +�� ˜Py(z)− ˜Pz(z) +� +ψλ +y (z)dz → 0 , +as λ → 0. Thus we find that the distribution f agrees with the continuous function x �→ ˜Px(x), +concluding the proof. +Remark 2.23. Note that by Proposition 2.22 for α > 0 and any f ∈ Cα(G) we know that X I f +agrees with a continuous function whenever d(I) < α. Thus, in particular Definition 2.3 of +Taylor polynomials Pα +x [f ] extends to Cα(G) ⊃C ∞(G). +2.3 DISCRETE SUBGROUPS +Recall that, given a discrete subgroup G ⊂ G acting on G (say) on the left, then the quotient +space S := G/G = {Gx : x ∈ G} is a smooth manifold. Furthermore, the quotient map π : G → S +is a smooth normal covering map , c.f. [Lee13, Thm. 21.29] and the canonical G right action +S ×G → S, +(Gx,x′) �→ Gxx′ +makes it a homogeneous space. We call G a lattice if S = G/G is a compact space. This is +equivalent to S carrying a finite G invariant measure, which we shall denote by d(Gx), c.f. +[Rag07, Thm. 2.1]. Throughout this article we assume that every homogeneous Lie group we +work with carries a lattice, the following theorem, [Rag07, Thm. 2.12], gives sufficient condi- +tions for this to be the case, which all our examples satisfy. +Theorem 2.24. Let G be a simply connected nilpotent Lie group and g its Lie algebra. Then, G +admits a lattice if and only if g admits a basis with respect to which the structure constants of +g are rational. +Let us point out that there do exist nilpotent Lie groups that do not admit a basis with re- +spect to which the structure constants are rational, see [Rag07, Rem. 2.14] for an example. +Denote by π∗ : C(S) →C(G) the pull-back under π. We define a convolution map +∗S : C ∞(S)×C ∞ +c (G) →C ∞(S), +(f ,φ) �→ f ∗S φ +as the unique map, such that the following diagram commutes +C ∞(S)×C ∞ +c (G) +C ∞(S) +C ∞(G)×C ∞ +c (G) +C ∞(G) , +∗S +π∗×id +π∗ +∗ +19 + +where convolution on the bottom row is convolution on the group as defined in 2.10. In order +to check that this map is well defined, we use the following terminology. A function f ∈C ∞(G) +is called left G periodic, if for every x ∈ G and n ∈ G it holds that +f (x) = f (nx) . +Thus, we only need to check that for any φ ∈ C ∞ +c (G) and any left G periodic function f ∈ +C ∞(G), the function f ∗φ is left G periodic. Indeed for any n ∈ G and x ∈ G, it holds that +f ∗φ(nx) = +� +G +f (y)φ(y−1nx)d y = +� +G +f (y)φ((n−1y)−1x)d y = +� +G +f (ny)φ(y−1x)d y = +� +G +f (y)φ(y−1x)d y . +By duality, one naturally extends the notion of convolution on S to pairs, (ξ,ζ) ∈ D′(S)×D′ +c(G), +where D′ +c(G) denotes distributions on G with compact support. +2.4 CONCRETE EXAMPLES +In order to cement ideas we present two concrete examples of non-abelian, homogeneous +Lie groups, both of which are identified with fixed global charts. These will be revisited in +Section 4.4 when we discuss the heat operator on the Heisenberg group and Kolmogorov +type operators. +2.4.1 THE HEISENBERG GROUP +Given n ≥ 1 we equip define the Heisenberg group Hn as the set R2n ×R with the group law, +(x, y,z)(x′, y′,z′) = +� +x + x′, y + y′,z + z′ + +n� +i=1 +� +x′ +i yi − xi y′ +i +� +� +. +Remark 2.25. Note that one may equally define the complex Heisenberg group on Cn × R +equipped with the group law, +(u,z)(u′,z′) = +� +v + v′,z + z′ +ℑ +n� +i=1 +ui ¯u′ +i +� +. +One sees that these definitions are equivalent after identifying Cn with R2n. +The origin e = (0,0,0) is clearly the identity and for (x, y,z) ∈ R2n × R one has (x, y,z)−1 = +(−x,−y,−z). The Lie algebra, h of Hn is identified with R2n+1 and spanned by the basis of +left-invariant vector fields, +Ai(x, y,z) = ∂xi + yi∂z, +Bi(x, y,z) = ∂yi − xi∂z, +C(x, y,z) = ∂z +and equipped with the Lie bracket [A,B] = AB −B A. We observe that for any (x, y,z) ∈ Hn and +i = 1,...,n, +[Ai,Bi](x, y,z) = −2C. +(2.14) +We say that a Lie algebra is graded if there exist vector spaces {Wk}k≥1 where only finitely +many Wk are non-zero, g = �∞ +k=1Wk and [Wi,Wj] ⊂ Wi+j. +20 + +Definition 2.5. A homogeneous Lie group G is called stratified (or a Carnot group) if its Lie +algebra is graded and generated by W1. +Due to (2.14), we see that if we equip Hn with the dilation map, +λ·(x, y,z) := (λx,λy,λ2z), +then Hn becomes a stratified group. Refer to [Bra14, Sec. 3.3.6] for a discussion of generalisa- +tions of this structure. +A simple example of a lattice on the Heisenberg group is the set of integer vectors (a,b,c) ∈ +Hn := Z2n × Z equipped with the group law as defined above. Note that it is not a normal +subgroup and thus the quotient space Hn/Hn not a group. However, since it is a lattice the +homogeneous space Hn/Hn is compact, c.f. Subsection 2.3. +2.4.2 MATRIX EXPONENTIAL GROUPS +For n ≥ 1, let B be a rational n ×n block matrix of the form, +B = + + +0 +B1 +0 +··· +0 +0 +0 +B2 +··· +0 +... +... +... +... +... +0 +0 +··· +0 +Bk +0 +0 +··· +0 +0 + + +with each Bi a pi−1×pi block matrix of rank pi, where n ≥ p0 ≥ p1 ≥ ··· ≥ pk and �k +i=0 pi = n. +Note that this implies the zero blocks on the diagonal are all pi × pi square matrices. We can +equip R×Rn with an associated Lie structure by defining the group law +(t,z)(s,z′) := +� +t + s,z′ +exp +� +sB⊤� +z +� +. +To define the dilation we begin by decomposing according to the structure of the block matrix +B, writing +Rn = Rp0 ×···×Rpn . +Thus the action of B⊤ on z = (z0,...,zk) ∈ Rp0×···×Rpk is written B⊤z = (B⊤ +1 z0, B⊤ +2 z1,...B⊤ +k zk−1) +etc. Then we set +λ·(t,z0,...,zk) := (λ2t,λz0,...,λ2k+1zk). +The origin e = (0,0) is again the identity element and (t,z)−1 = (−t,−exp(−tB⊤)z). The Lie +algebra is spanned by the translation invariant vector fields, +Xi(t,z) = ∂zi +for i = 1,...,p0 +and +Y (t,z) = ∂t −(Bz)·∇ = ∂t − +n� +i,j=p0+1 +bi j zi∂zi . +This defines the matrix exponential group associated to B and one sees that it is not stratified. +The simplest non-trivial example is to set n = 2 and +B = +�0 +1 +0 +0 +� +, +21 + +so that using the suggestive notation (t,v,x) ∈ R×R×R, the group law becomes, +(t,v,x)(s,w, y)= (t + s,v + w,x + y + sv). +The equivalent scaling as above is to set λ · (t,x,v) = (λ2t,λv,λ3x). We refer to [Man97] for +more details and Section 4.4 below for a discussion of natural, second order, hypoelliptic, +linear operators associated to these groups. +3 REGULARITY STRUCTURES AND MODELS +Definition 3.1. A regularity structure is a pair T = (T,G) consisting of the following elements: +1. A graded vector space T = � +α∈A Tα where +• the index set A ⊂ R is discrete, bounded from below and contains zero, +• each Tα is finite dimensional with a fixed norm |·|α. We write Qα : T → Tα for the +canonical projection, +• T0 is isomorphic to R with a distinguished element 1 ∈ T0, such that |1|0 = 1. +The space T is called the structure space. +2. A group G of linear operators acting on T, such that for every Γ ∈ G it holds that Γ|T0 is +the identity map and for all τ ∈ Tα: +Γτ−τ ∈ +� +β<α +Tβ . +The group G is called the structure group of T . +A sector is a G invariant subspace V ⊂ T and if V ̸= {0} the regularity of the sector V is defined +as min{α ∈ A : V ∩Tα ̸= {0}}. +We make use of the natural shorthands, T>α, T≥α, T<α, T≤α and the projections Q>α etc. +Definition 3.2. Given a regularity structureT = (T,G) and r ∈ N such that r > |min A|, a model +for T is a pair M = (Π,Γ), consisting of +• a realisation map Π : Rd → L(T,D′(G)), x �→ Πx, such that for any compact set K ⊂ G one +has ∥Π∥γ;K := supx∈K ∥Π∥γ,x < +∞ for all γ > 0, where +∥Π∥γ,x := +sup +ζ∈A∩(−∞,γ) +sup +τ∈Tζ +sup +λ<1 +sup +φ∈Br +|〈Πxτ,φλ +x〉| +|τ|ζλζ +, +• a re-expansion map Γ : Rd ×Rd → G, (x, y) �→ Γx,y, which satisfiesthe algebraic condition +ΠxΓx,y = Πy +and the analytic condition ∥Γ∥γ;K := supx,y∈K: |y−1x|<1∥Γ∥x,y,γ < +∞ for all γ > 0, where +∥Γ∥x,y,γ := +sup +ζ∈A∩(−∞,γ] +sup +A∋β<ζ +sup +τ∈Tζ +|Γx,yτ|β +|y−1x|ζ−β|τ|ζ +. +22 + +Lastly, we denote by MT the space of models for T equipped with the semi-norms ∥M∥γ;K := +∥Π∥γ;K +∥Γ∥γ;K. +Remark 3.1. In the rest of the article, often without any further comment, we will write +r := min{n ∈ N : n > |min A|} . +3.1 THE POLYNOMIAL REGULARITY STRUCTURE +As an important example we describe the polynomial regularity structure and canonical model +which are crucial in the analysis of singular SPDEs on homogeneous Lie groups. We de- +fine the structure space ¯T to be the symmetric tensor (Hopf) algebra generated by {ηηηi}d +i=1, +which we think of as abstract lifts of the monomials {ηi}d +i=1. For a multi-index I, we write +ηηηI :=ηηηi1 +1 ·...·ηηηid +d as well as 1 :=ηηη0. The group ¯G is given by a copy of G acting by +g �→ +� +Γg :ηηηj �→ηηηj + +� +I,J̸=0, d(I)+d(J)=sj +C I,J +j ηI(g)ηηηJ +η j (g)1 +� +and ΓgηηηI = +� +Γgηηη +�I . The canonical polynomial model is defined by setting +Πxηηηj(z) = η j(x−1z) +and +Γx,yηηηj =ηηηj + +� +I,J̸=0, d(I)+d(J)=sj +C I,J +j ηI (y−1x)ηηηJ +η j(y−1x)1 . +These maps are extended multiplicatively to all of ¯T. Using Prop. 2.5 in the third equality, +ΠxΓx,yηηηj(z) = Πx +� +ηηηj + +� +I,J̸=0, d(I)+d(J)=dj +C I,J +j ηI (y−1x)ηηηJ +η j (y−1x) +� +(z) += η j(x−1z)+ +� +I,J̸=0, d(I)+d(J)=dj +C I,J +j ηI(y−1x)ηJ(x−1z) +η j(y−1x) += η j(y−1xx−1z) += η j(y−1z) += Πyηηηj(z). +3.1.1 DERIVATIVES AND ABSTRACT POLYNOMIALS +Next we lift the vector fields X i to abstract differential operators X i on ¯T of degree sj, c.f. +Definition 3.6 given later. We set +X iηηηj = δi,j1+ +� +I̸=0, d(I)=sj −si +C I,ei +j +ηηηI +and extend this definition to the whole regularity structure by the Leibniz rule. The two con- +ditions to be an abstract differential operator are checked directly: +23 + +1. Indeed X i : Tα �→ Tα−si. +2. By the Leibinz rule the second property is checked by showing that +X iΓx,yηηηj = Γx,yX iηηηj . +(3.1) +Note that ΠxX iηηηj = XiΠxηηηj , since +ΠxX iηηηj(z) = Πx +� +δi,j1 + +� +I̸=0, d(I)=sj −si +C I,ei +j +ηηηI� += δi,j + +� +I̸=0, d(I)=sj −si +C I,ei +j +ηI(x−1z) +and using left invariance of the vector field Xi as well as (2.3) +XiΠxηηηj(z) = (Xiη j(x−1·))(z) = (Xiη j)(x−1z) = δi,j + +� +I̸=0, d(I)=sj −si +C I,ei +j +ηI (x−1z) . +Thus (3.1) follows from the injectivity of the maps Πx. +Remark 3.2. In addition to showing that Xi is an abstract differential operator, we have shown +that it also realizes Xi for the polynomial model, see Definition 3.6 in Section 3.4. +3.1.2 ABSTRACT TAYLOR EXPANSIONS +The following operator, which sends a smooth function to its abstract Taylor expansion, will +be used throughout the article. +Definition 3.3. For a > 0 and x ∈ G, we define the family of maps +PPPa +x : Ca(G) → ¯T +where PPPa +x[f ] is characterised by the fact that ΠePPPa +x[f ] = Pa +x[f ] ∈ Pa. +Remark 3.3. Note that the mapPPPa +x is well defined by Remark 2.23 . We shall almost exclusively +use it, when its argument is a smooth function f ∈C ∞(G) ⊂ Ca(G). +Lemma 3.4. Let f ∈C ∞(G) and a ≥ 0. Then for each multi-index I the coefficient of ηηηI in PPPa +x[f ] +is given by a linear combination (depending only on G) of {X K f (x)}d(K )≤d(I). Furthermore, for +b ≥ a one has +PPPa +x[f ] = Q≤aPPPb +x[f ] . +(3.2) +Proof. We first observe that (3.2) holds since the polynomial ΠeQ≤aPPPb +x[f ] satisfies the prop- +erty required in Definition 2.3. The first part of the lemma follows by combining Proposi- +tion 2.11 with (3.2). +Lemma 3.5. In the setting of the above lemma one has +|X IΠePPPa +x[f ](z)| = |X I Pa +x[f ](z)| ≲ +� +d(I)≤d(J) 0 and P ∈ ¯T<δ. If, for some ε ∈ [0,1], +��(X IΠeP)(e) +�� ≤ εδ−d(I) +for all I ∈ Nd such that δ > d(I), then it holds that |P|a ≲δ,G εδ−a for all a ≤ δ. +Proof. We observe that Pδ +x[ΠeP] = ΠeP. Thus, the claim follows directly from Lemma 3.4. +3.2 MODELLED DISTRIBUTIONS +Definition 3.4. Given a regularity structure T = (T,G), a model M = (Π,Γ) and γ ∈ R we define +Dγ +M as the space of all continuous maps f : G → T<γ, such that for all ζ ∈ A ∩ (−∞,γ) the +following bounds hold for every compact set K ⊂ G +sup +x∈K +|f (x)|ζ < +∞, +sup +x,y∈K, +0<|y−1x|≤1 +|f (y)−Γx,y f (x)|ζ +|y−1x|γ−ζ +< +∞ . +We define the corresponding semi-norm ∥ · ∥γ;K on Dγ +M by setting, +∥f ∥γ;K := sup +x∈K +sup +ζ<γ +|f (x)|ζ +sup +ζ<γ +sup +x,y∈K, +0<|y−1x|≤1 +|f (y)−Γx,y f (x)|ζ +|y−1x|γ−ζ +. +Given two models M = (Π,Γ), ¯M = ( ¯Π, ¯Γ), two modelled distributions f ∈ Dγ +M, ¯f ∈ Dγ +¯M and a +compact set K ⊂ G we define the quantity, +∥f ; ¯f ∥γ;K := sup +x∈K +sup +ζ<γ +|f (x)− ¯f (x)|ζ + +sup +(x,y)∈K +0<|y−1x|≤1 +sup +ζ<γ +|f (y)− ¯f (y)−Γx,y f (y)+ ¯Γy,x ¯f (x)|ζ +|y−1x|γ−ζ +. +(3.3) +For the set of modelled distributions taking values in a sector V ⊂ T we write Dγ +M(V ) and if the +regularity of the sector is α ∈ A we often use the shorthandDγ +α;M. We will freely drop the explicit +dependence on the model, image sector and its regularity when the context is clear. +3.2.1 RECONSTRUCTION +Theorem 3.7 (Reconstruction Theorem). Let T = (T,G) be a regularity structure with α = +min A. Then for every γ > 0 and M = (Π,Γ) ∈ MT , there exists a unique, continuous linear +map RM : Dγ +M → Cα, called the reconstruction operator associated to M, such that for any +compact K and λ ∈ (0,1] +sup +ψ∈Bm +|〈RM f −Πx f (x),ψλ +x〉| ≲K λγ∥f ∥γ;B2λ(x)∥Π∥γ;B2λ(x) , +(3.4) +25 + +uniformly over x ∈ K. Furthermore the map M → RM is locally Lipschitz continuous in the +sense that for a second model ¯M = ( ¯Π, ¯Γ) and ¯f ∈ Dγ +¯M, for every λ ∈ (0,1] +sup +ψ∈Bm +|〈RM f −R ¯M ¯f −Πx f (x)+ ¯Πx ¯f (x),ψλ +x〉| ≲K λγ� +∥f ; ¯f ∥γ;B2λ(x)∥ ¯Π∥γ;B2λ(x) ++∥f ∥γ;B2λ(x)∥Π− ¯Π∥γ;B2λ(x) +� +, +(3.5) +Remark 3.8. Existence of a reconstruction operator for γ ≤ 0 also holds, however, uniqueness +does not. In the case γ = 0 the analogous bound to (3.4) contains an additional logarithmic +correction on the right hand side, c.f. [CZ20]. +Remark 3.9. It follows from the definition of the reconstruction operator, that if f ∈ Dγ +α,M is a +modelled distributions with values in a sector V ⊂ T of regularity α ≥ α, then RM f ∈ C α. +In order to prove Theorem 3.7 we require two preliminary results. As in [FH20, Sec. 13.4] +we make the observation that for every N ≥ 0 there exists a ρ : G → R, smooth and compactly +supported in B1(0) and such that +� +ηI (x)ρ(x)dx = δI,0, +0 < d(I) ≤ N, +(3.6) +where the δ here denotes the Kronecker delta applied componentwise to the multi-index. For +r > 1 we define ρ(n)(x) := rn|s|ρ(rn · x) as well as, +ρ(n,m) = ρ(n) ∗ρ(n+1) ∗···∗ρ(m), +with the convention ρ(n,n) = ρ(n). We then have the following result. +Lemma 3.10. If r > ∥ρ∥L1 > 1, then there exists a smooth, compactly supported function ϕ(n) = +limm→∞ ρ(n,m), where the convergence takes place in D(G) and supp(ϕ(n)) ⊂ BCr−n for C = +r +r−1. +Proof. First, note that since ∥ρ(m)∥L1 = ∥ρ∥L1 and ρ(m) is supported on a ball of radius r, it +follows from the mean value theorem on G (c.f. (2.5) with a = 0) that +|f ∗ρ(m)(x)− f (x)| = +���� +� +G +(f (y)− f (x))ρ(m)(y−1x)dy +���� ≲ max +i=1,...,d ∥Yi f ∥L∞∥ρ∥L1r−m . +Secondly, since +Y Iρ(n,m) = Y I(ρ(n) ∗ρ(n+1,m)) = (Y Iρ(n))∗ρ(n+1,m) +(3.7) +one finds by applying Young’s convolution inequality m-times, c.f. Section 2.2, that +∥Yiρ(n,m)∥L∞ ≤ ∥Yiρ∥∞∥ρ∥m−n−1 +L1 +and therefore +∥ρ(n,m) −ρ(n,m−1)∥L∞ = ∥ρ(n,m−1) ∗ρ(m) −ρ(n,m−1)∥L∞ +≲ max +i=1,...,n∥Yiρ(n,m−1)∥L∞∥ρ∥L1r−m +≤ ∥Y ρ∥L∞∥ρ∥m−n−2 +L1 +r−m +≤ ∥Y ρ∥L∞ +∥ρ∥n+2 +L1 +�∥ρ∥L1 +r +�m +26 + +which is summable in m since we assumed that r > ∥ρ∥L1. Thus we may write, +ρ(n,m) = ρ(n) + +m−n−1 +� +k=0 +ρ(n,m−k) −ρ(n,m−1−k), +and it follows that ρ(n,m) converges uniformly as m → +∞. Using (3.7) we obtain convergence +in D(G). It remains to check the support of ϕ(n). For two functions f1, f2 such that supp(fi) ∈ +Bri one has supp(f1 ∗ f2) ∈ Br1+r2, hence it follows that ϕ(n) is supported in a ball of radius +�∞ +m=n r−m = +r−n +1−r−1 . +It follows from the definitions that ϕ(n) = ρ(n) ∗ϕ(n+1). We set +˜ρ(m,n) := ˜ρ(m) ∗ ˜ρ(m−1) ∗···∗ ˜ρ(n) +and using (2.12) we note that ˜ϕ(m+1) ∗ ˜ρ(m) = ˜ϕ(m) and ˜ρ(m,n) → ˜ϕ(n) in D(G) as m → ∞. +Lemma 3.11. Let r > 1 and ρ be as in Lemma 3.10 Let α > 0 and ξn : G → R be a sequence of +functions such that for every compact K ⊂ G there exists a CK such that supx∈K |ξn(x)| ≤ CKrαn +and such that ξn = ξn+1 ∗ ˜ρ(n). Then the sequence ξn is Cauchy in C−β(G) for every β > α and +the limit ξ satisfies ξn = ξ∗ ˜ϕ(n). If furthermore, for some x ∈ G and γ > −α one has the bound +|ξn(y)| ≤ rαn � +|x−1y|γ+α +r−(γ+α)n� +uniformly over n ≥ 0 and y ∈ G such that |x−1y| ≤ 1, then |〈ξ,ψλ +x〉| ≲ λγ for all λ ≤ 1 and φ ∈ Br , +where r = −[−α]. +Proof. The proof follows along the same steps as that of [FH20, Lem. 13.24], but one has to +be careful since convolution is non-commutative in our setting. Let λ ∈ (0,1] and ψ ∈ Br , we +first establish the bound +|〈ξn −ξn+1,ψλ +x〉| ≲ λ−βr(α−β)n +(3.8) +uniformly over ψ ∈ Br , λ ∈ (0,1] and locally uniformly over x ∈ G. First observe the trivial +bound, +|〈ξn −ξn+1,ψλ +x〉| ≤ sup +x∈K +(|ξn(x)|+|ξn+1(x)|)∥ψλ +x∥L1 ≤ (1+rα)CK ¯Crαn, +where ¯C := sup{ +� +|ψλ(x)|dx : ψ ∈ Br } < +∞ . Hence, when λ ≤ r−n the bound (3.8) holds +directly. +In the case r−n ≤ λ, using (2.11) we rewrite +|〈ξn −ξn+1,ψλ +x〉| = 〈ξn+1 ∗ ˜ρ(n) −ξn+1,ψλ +x〉| = |〈ξn+1,ψλ +x ∗ρ(n) −ψλ +x〉|. +By Taylor’s theorem and in particular Remark 2.15 +|ψλ +x(z)− ˜Pr +y[ψλ +x](z)| ≲r +� +δ>0 +λ−(r+δ)−|s||y−1z|r+δ , +(3.9) +where the sum runs over a finite set. It follows from (2.10), Remark 2.8 and (3.6) that we have +˜Pr +y[ψλ +x]∗ρ(n)(z) = +� +˜Pr +y[ψλ +x](zy−1)ρ(n)(y)d y = +� +˜P0 +y[ψλ +x](zy−1)ρ(n)(y)d y = ψλ +x(y) +27 + +and thus +ψλ +x ∗ρ(n)(y)−ψλ +x(y) = (ψλ +x − ˜Pr +y[ψλ +x])∗ρ(n)(y), +which by (3.9) is bounded uniformly by a multiple of � +δ>0 λ−(r+δ)−|s|r−n(r+δ) and supported +on a ball of radius λ+r−n ≤ 2λ. Using the bound |ξn+1| ≲α rαn we conclude that +|〈ξn+1,ψλ +x ∗ρ(n) −ψλ +x〉| ≲ +� +δ>0 +λ−(r+δ)r−n(r+δ)rαn ≲ λ−βr(α−β)n +where we used r−n ≤ λ and without loss of generality assumed that r ≥ β in the last line. +Hence {ξn}n≥1 is Cauchy in C−β(G) and for any test function, +〈ξn,ψ〉 = 〈ξn+1,ψ∗ρ(n)〉 = 〈ξm,ψ∗ρ(n,m)〉 = 〈ξ,ψ∗ϕ(n)〉, +showing that ξn = ξ∗ ˜ϕ(n). +To prove the second claim, for any test function ψ ∈ Br , λ > 0 and x ∈ G we write, +〈ξ,ψλ +x〉 = 〈ξn,ψλ +x〉+ +� +k≥n +〈ξk+1 −ξk,ψλ +x〉, +where n is chosen so that λ ∈ [r−(n+1),r−n] and as a consequence +|〈ξn,ψλ +x〉| ≤ λ−|s|rαn +� +Bλ(x) +� +|x−1y|γ+α +r−(γ+α)n� +dy,≲ λγ+αrαn +r−γn ≲ λγ. +To bound the summands 〈ξk+1 − ξk,ϕλ +x〉 we proceed as in the proof of the first claim to find +that +|〈ξk −ξk+1,ψλ +x〉| = |〈ξk+1,ψλ +x ∗ρ(k) −ψλ +x〉| += |〈ξk+1,(ψλ +x − ˜Pr +x[ψλ +x]))∗ρ(k)〉| +≲ +� +δ>0 +λ−(r+δ)−|s|r−k(r+δ) +� +B2λ(x) +|ξk+1(y)|dy +≲ +� +δ>0 +λ−(r+δ)−|s|rk(α−r−δ) +�� +B2λ(x) +|x−1y|γ+αdy +r−(γ+α)(k+1)λ|s| +� +≲ +� +δ>0 +� +λγ+α−r−δrk(α−r−δ) +λ−(r+δ)r−k(γ+r+δ)� +where the sum in δ is again over a finite set. Since r + δ > α the quantity on the left is +summable over k ≥ n and is of order λγ, concluding the proof. +We are now ready to proof Theorem 3.7. +Proof of Theorem 3.7. For m > 0 we first define the operators R(m,m) : Dγ →C(G) by setting, +�R(m,m) f +� +(y) := +� +Πy f (y)∗ ˜ϕ(m)� +(y) = 〈Πy f (y),ϕ(m) +y +〉. +We then set, for n < m, +R(m,n) f = R(m,m) f ∗ ˜ρ(m−1,n) +28 + +and recalling that ˜ϕ(m+1) ∗ ˜ρ(m) = ˜ϕ(m) we find +R(m,n) f −R(m+1,n) f = R(m,m) f ∗ ˜ρ(m−1,n) −R(m+1,m+1) f ∗ ˜ρ(m,n) += +�R(m,m) f −R(m+1,m+1) f ∗ ˜ρ(m)� +∗ ˜ρ(m−1,n) . +Using the identity +(F ∗ ˜φ)(x) = 〈F,φx〉 = +� +F(y)φx(y)d y , +it follows that +�R(m,n) f −R(m+1,n) f +� +(x) = +��R(m,m) f −R(m+1,m+1) f ∗ ˜ρ(m)� +(y)ρ(m−1,n) +x +(y)d y += +�� +Πy f (y)∗ ˜ϕ(m) −R(m+1,m+1)f ∗ ˜ρ(m)� +(y)ρ(m−1,n) +x +(y)d y += +��� +Πy f (y)∗ ˜ϕ(m+1) −R(m+1,m+1) f +� +∗ ˜ρ(m)� +(y)ρ(m−1,n) +x +(y)d y += +��� +Πy f (y)∗ ˜ϕ(m+1) −R(m+1,m+1) f +� +(z)ρ(m) +y +(z)dzρ(m−1,n) +x +(y)d y += +��� +Πy f (y)∗ ˜ϕ(m+1) −Πz f (z)∗ ˜ϕ(m+1)� +(z)ρ(m) +y +(z)dzρ(m−1,n) +x +(y)d y += +�� +〈Πy f (y)−Πz f (z),ϕ(m+1) +z +〉ρ(m) +y +(z)dzρ(m−1,n) +x +(y)d y . +Therefore, using the fact that Πz = ΠyΓyz, we have, +�R(m,n) f −R(m+1,n) f +� +(x) = +�� +〈Πy(f (y)−Γyz f (z)),ϕ(m+1) +z +〉ρ(m) +y +(z)dzρ(m−1,n) +x +(y)d y. +Then successively applying the facts that, +• supy,z∈Br−m (0) |〈Πyτ,ϕ(m+1) +z +| ≲ ∥Π∥γ;Br−m(0)r−αm|τ|ζ for τ ∈ Tζ, +• ∥f (y)−Γyz f (z)∥α ≲ ∥f ∥γ;Br−m (y)r(α−γ)m uniformly over |y−1z| ≲ r−m +• ∥ρ(n,m−1) +x +∥L1 ≲ 1 uniformly over m > n ≥ 0, +we establish the bound, +∥R(m,n) f −R(m+1,n) f ∥L∞(K) ≲ ∥Π∥γ, ¯K∥f ∥γ; ¯Kr−γm , +uniformly over m ≥ n ≥ 0, where ¯K denotes the two fattening of the set K. It follows directly +from the definition and properties of a model that we also have the bound, +∥R(n,n) f ∥L∞(K) ≲ ∥Π∥γ, ¯K∥f ∥γ; ¯Kr−αn, +(3.10) +where α = min A. It follows that R(m,n) f converges uniformly on compacts as m → ∞ to +R(n) f which also satisfies the bound(3.10). Since it also holds that for every m ≥ n +1, +R(m,n) f = R(m,m) f ∗ ˜ρ(m−1,n) = R(m,m) f ∗ ˜ρ(m−1,n+1) ∗ ˜ρ(n) = R(m,n+1) f ∗ ˜ρ(n) , +29 + +we find that +R(n) f = R(n+1) f ∗ ˜ρ(n) . +Therefore we may apply Lemma 3.11 to see that there exists a limit Rf := limn→∞ R(n) f . +With validity of the limit established we now turn to show the bounds (3.4) and (3.5); this +requires us to keep more careful track of the underlying sets in the proof. We begin with (3.4), +first noting that if we define fx(y) := Γy,x f (x) then one has R(n,n) fx = Πx f (x) ∗ ˜ϕ(n) so that +(3.4) can be written as the claim that for all λ ∈ (0,1], +sup +ψ∈Bm +|〈R(f − fx),ψλ +x〉| ≲K λγ∥f ∥γ;B2λ(x)∥Π∥γ;B2λ(x) . +(3.11) +Using that |(f − fx)(z)|α ≲ ∥f ∥γ;B|z−1x|(x)|z−1x|γ−α for x, z ∈ K, it follows that for all y ∈ Bλ(x) +one has +|(R(n,n)(f − fx)(y)| = |〈Πy(f (y)− fx(y)),ϕ(n) +y 〉| += |〈Πy(f (y)−Γx,y f (x)),ϕ(n) +y 〉| +≲ ∥Π∥γ;Bλ(y)∥f ∥γ;BCr−n (y) +� +α≤α≤γ +r−αn|y−1x|γ−α +≲ ∥Π∥γ;Bλ(y)∥f ∥γ;BCr−n (y)r−αn(|y−1x| +γ−α +r(α−γ)n), +(3.12) +where C(r) := +r +r−1 is as in Lemma 3.10. Given n > n0(λ) sufficiently larger, we have a uni- +form bound Cr−n ≤ λ. By the convergence of R(n,n) in the first exponent, it follows that R(n) +satisfies the same bound and so inspecting the proof of the second half of Lemma 3.11, in par- +ticular noticing that we integrate the above estimate over y ∈ Bλ(x), we conclude that (3.11) +holds for the limit R. +Using the obvious notation we can also rewrite (3.5) as +sup +ψ∈Bm +|〈R(f − fx)− ¯R( ¯f − ¯fx),ψλ +x〉| ≲ λγ � +∥f ; ¯f ∥γ;B2λ(x)∥ ¯Π∥γ;B2λ(x) +∥f ∥γ;B2λ(x)∥Π− ¯Π∥γ;B2λ(x) +� +, +uniformly over λ ∈ (0,1]. This is seen very similarly to (3.11) but using this time that for n +large enough, +|R(n,n)(f − fx)− ¯R(n,n)( ¯f − ¯fx)(y)| += |〈Πy(f (y)−Γx,y f (x))− ¯Πy( ¯f (y)− ¯Γy,x ¯f (x)),ϕ(n) +y 〉| += |〈Πy(f (y)−Γx,y f (x)− ¯f (y)+ ¯Γy,x ¯f (x))+(Πy − ¯Πy)( ¯f (y)− ¯Γy,x ¯f (x)),ϕ(n) +y 〉| +≲ +� +∥Π∥γ;Bλ(y)∥f ; ¯f ∥γ;Bλ(y) +∥Π− ¯Π∥γ;Bλ(y)∥ ¯f ∥γ;Bλ(y) +� +� +α≤α≤γ +r−αn|y−1x| +γ−α . +It remains to show that the reconstruction map is unique for γ > 0 and that Rf is indeed an +element of C α. This is done exactly as in [Hai14, Sec. 3]. +30 + +3.2.2 FUNCTIONS AS MODELLED DISTRIBUTIONS +Let ¯T = (¯T, ¯G) be the polynomial regularity structure equipped with the polynomial model. +We show that in this setting the reconstruction theorem and its inverse map Taylor polyno- +mials to Hölder functions and vice versa. +Theorem 3.12. For γ > 0 the reconstruction operator is an isomorphism between Dγ(G) and +Cγ(G). In particular the inverse of the reconstruction map is given by +Cγ(G) ∋ f (·) �→PPPγ +(·)[f ] ∈ Dγ(G), +where PPPa is defined in Definition 3.3 +Proof. Given a modelled distribution in Dγ and since we are working with the polynomial +regularity structure equipped with its canonical model, it follows directly from the bound +satisfied by the image of the reconstruction operator, i.e. Equation(3.4), and the definition of +Cγ in (2.13) that the reconstruction operator is a map Dγ(G) → Cγ(G). Continuity also follows +from (3.4) and by linearity. +To see the other direction recall that given a Hölder continuous distribution f ∈ Cγ(G) by +Proposition 2.22 the {X I f }d(I)<γ are actually functions and, in particular, that ˜Px = ΠxPPP[γ] +x [f ]. +Therefore, for λ = |x−1y|G +���Πx +� +PPP[γ] +x [f ]−Γx,yPPP[γ] +y [f ] +� +(ψλ +x) +��� = |〈 ˜Px − f ,ψλ +x〉|+|〈f − ˜Py,ψλ +x〉| ≲ λγ +(3.13) +uniformly in ψ ∈ Br . On the other hand, writing PPP[γ] +x [f ]−Γx,yPPP[γ] +y [f ] = � +d(I)<γ cI +x,yηηηI, we find +that +Πx +� +PPP[γ] +x [f ]−Γx,yPPP[γ] +y [f ] +� +(ψλ +x) = +� +d(I)<γ +cI +x,yΠxηηηI(ψλ +x) = +� +d(I)<γ +cI +x,yΠeηηηI(ψλ) = +� +d(I)<γ +cI +x,yλd(I)ΠeηηηI(ψ) . +Using that Pγ is a finite dimensional vector space and the surjectivity of the linear map +C ∞ +c (B1(e)) → RdimPγ, +ψ �→ {ΠeηηηI (ψ)}d(I)<γ +we find that +� +d(I)<γ +|cI +x,y|λd(I) ≲γ sup +ψ∈Br +����� +� +d(I)<γ +cI +x,yλd(I)ΠeηηηI(ψ) +����� . +(3.14) +Together, (3.13) and (3.14) imply that that |cI +x,y| ≲ λγ−d(I), we may then apply Lemma 3.6 to +see that PPP[γ] +(·)[f ] ∈ Dγ . +3.2.3 LOCAL RECONSTRUCTION +We will require the following further refinement of the reconstruction theorem which is an +analogue in our case of [Hai14, Prop. 7.2]. +31 + +Proposition 3.13. In the setting of Theorem 3.7 one has the improved bound, +sup +ψ∈Bm +|(Rf −Πx f (x))(ψλ +x)| ≲ λγ∥Π∥γ;B2λ(x) +sup +y,z∈supp(ψλ +x) +sup +ℓ<γ +|f (z)−Γzy f (y)|ℓ +|y−1z|γ−ℓ +, +(3.15) +as well as, given a second model ¯M( ¯Π, ¯Γ) and a modelled distribution ¯f ∈ Dγ +¯M, the analogous +bound, that for every λ ∈ (0,1], +sup +ψ∈Bm +|〈RM f −R ¯M ¯f −Πx f (x)+ ¯Πx ¯f (x),ψλ +x〉| ≲ λγ� +∥f ; ¯f ∥γ;supp(ψλ +x)∥ ¯Π∥γ;B2λ(x) ++∥f ∥γ;supp(ψλ +x)∥Π− ¯Π∥γ;B2λ(x) +� +, +(3.16) +for any x ∈ G and any λ ∈ (0,1]. +Proof. Since the right hand side of (3.15) is linear in f , as in [Hai14] we may assume it to be +equal to 1. We use the functions ρ(n) and ϕ(n) from Section 3.2.1 and recall that they satisfy +ϕ(n−1)(x) = +� +Br−n+1 +ρ(n−1)(y)ϕ(n) +y (x)d y, +ϕ(n−1) +y += +� +Br−n+1 +ρ(n−1)(w)ϕ(n) +yw(x)dw +in particular since +� +ρ(n)(x)dx = 1 we have +� +ϕ(n) +y (·)dy = 1 for all n ≥ 0. Define, for fixed ψλ +x, +the sets +Λn = +� +y ∈ G : supp(ϕ(n) +y )∩supp(ψλ +x) ̸= � +� +⊂ G +which by definition is contained Bλ+Cr−n(x) with C = +r +r−1 as in Lemma 3.10, as well as a (mea- +surable) function +πn : Λn → supp(ψλ +x) +such that πn(y) ∈ supp(ϕ(n) +y )∩supp(ψλ +x) for every y ∈ Λn. +Next, let +Rn = +� +Λn +�Rf −Ππn(y) f (πn(y)) +� +(ψλ +xϕn +y )d y += 〈Rf ,ψλ +x〉− +� +Λn +Ππn(y) f (πn(y))(ψλ +xϕn +y )d y . +It follows that for n0 = min{n ∈ N : r−n ≤ λ}, one has +���(Rf −Πx f (x))(ψλ +x)−Rn +��� = +���� +� +Λn +� +Πx f (x)−Ππn(y) f (πn(y)) +� +(ψλ +xϕn +y )d y +���� ≲ λγ +(3.17) +32 + +as well as for n > n0 +Rn−1 −Rn = +� +Λn−1 +Ππn−1(y) f (πn−1(y))(ψλ +xϕn−1 +y +)d y − +� +Λn +Ππn(z) f (πn(z))(ψλ +x ϕn +z )dz += +� +Λn−1 +Ππn−1(y) f (πn−1(y)) +� +ψλ +x +� +Br−n+1 +ρ(n−1)(w)ϕ(n) +ywdw +� +d y +− +� +Λn +Ππn(z) f (πn(z))(ψλ +xϕn +z )dz += +� +Br−n+1 +ρ(n−1)(w) +� +Λn−1 +Ππn−1(y) f (πn−1(y))(ψλ +xϕ(n) +yw)d ydw +− +� +Λn +Ππn(z) f (πn(z))(ψλ +xϕn +z )dz += +� +Br−n+1 +ρ(n−1)(w) +� +Λn−1 +Ππn−1(zw−1) f (πn−1(zw−1))(ψλ +xϕ(n) +z )dzdw +− +� +Λn +Ππn(z) f (πn(z))(ψλ +xϕn +z )dz += +� +Br−n+1 +ρ(n−1)(w) +� +Λn +� +Ππn−1(zw−1) f (πn−1(zw−1))−Ππn(z) f (πn(z)) +� +(ψλ +xϕn +z )dzdw . +Since |πn(z)−1πn−1(zw−1)| ≤ ¯Cr−n and for τ ∈ Tα such that |τ| ≤ 1 +|Ππn(z)τ(ψλ +xϕn +z )| ≲ λ|s|r−αn , +we find that +� +Λn +��� +� +Ππn−1(zw−1) f (πn−1(zw−1))−Ππn(z) f (πn(z)) +� +(ψλ +xϕn +z ) +���dz ≲ r−γn +and thus conclude +|Rn −Rn−1| ≲ r−γn. +A similar argument gives |Rn| → 0 as n → ∞ which combined with (3.17) concludes the proof +of (3.15). The proof of the analogous bound, (3.16), follows in a similar manner. +3.3 SINGULAR MODELLED DISTRIBUTIONS +As in [Hai14] we will eventually be concerned with solutions to SPDEs that take values in +spaces of modelled distributions with permissible singularities in some regions of the do- +main. Our main example will be modelled distributions on space-time domains that are al- +lowed to be discontinuous at {t = 0}, see Section 4. However, as in [Hai14] we build the notion +of singular modelled distributions allowing for singularities on more general sets, generalisa- +tions of which have been used in [GH19a, GH21] to study singular equations with boundary +conditions. +We fix a homogeneous sub-Lie group P ⊂ G with associated Lie algebra for which we write +p ⊂ g. The assumption that P be a homogeneous sub-Lie group means that the the scaling +map s restricts to a map ¯s := s|p : p → p which is diagonalisable. We fix a decomposition +33 + +g = p⊕pc such that pc is also invariant under s and define the homogeneous dimension of P +and its complement, Pc := exp(pc) as, +|¯s| := trace(s|p) +and +|m| = trace(s|pc ) . +(3.18) +Furthermore, we set +|x|P := 1∧dG(x,P) = 1∧inf +� +z ∈ P : |x−1z| +� +, +|x, y|P := |x|P ∧|y|P. +Note that since P is closed (being the image under exp of a linear subspace of g), one sees +easily that the infimum above is actually a minimum. Given K ⊂ G we define the set +KP := +� +(x, y) ∈ (K\P)2 : x ̸= y and |x−1y| ≤ |x, y|P +� +. +That is KP contains all the points in K that are closer to each other than they are to P. +Definition 3.5 (Singular Modelled Distributions). Given a regularity structure T and a sub- +group P as above, for any γ > 0, η ∈ R and maps f : G\P → T , we set +∥f ∥γ,η;K := sup +x∈K\P +sup +ζ<γ +|f (x)|ζ +|x|(η−ζ)∧0 +P +, +�f �γ,η;K := sup +x∈K\P +sup +ζ<γ +|f (x)|ζ +|x|η−ζ +P +. +Then given a model M = (Π,Γ) as well as a sector V , the space Dγ,η +P,M(V ) consists of all functions +f : G\P → V≤γ such that for every compact set K ⊂ G, +������f +������ +γ,η;K := ∥f ∥γ,η;K + +sup +(x,y)∈KP +sup +ζ<γ +|f (x)−Γx,y f (y)|ζ +|y−1x|γ−ζ|x, y|η−γ +P +< +∞. +For two models M = (Π,Γ), ¯M = ( ¯Π), ¯Γ) and two modelled distributions f ∈ Dγ,η +P,M, ¯f ∈ Dγ,η +P, ¯M we +also set, +������f ; ¯f +������ +γ,η;K := ∥f − ¯f ∥γ,η;K + +sup +(x,y)∈KP +sup +ζ<γ +|f (x)− ¯f (x)−Γx,y f (y)+ ¯Γxy ¯f (y)|ζ +|y−1x|γ−ζ|x, y|η−γ +P +. +If V is a sector of regularity α ∈ A, where appropriate we will use the shorthand Dγ,η +α;P,M = +Dγ,η +P,M(V ) and we will drop the dependence on the model when the context is clear. +Remark 3.14. We refer to [Hai14] for more intuition regarding the definition of these spaces +and their properties - all of which carry over to our setting. In particular [Hai14, Rem. 6.4] +discusses the relationship between the spaces Dγ,η +P +and Dγ. +Remark 3.15. The family of norms �f �γ,η;K and the two following lemmas play a role when we +consider fixed-point maps in Section 4 below. Their utility is in allowing us to extract small +scaling parameters in terms of the distance to the subgroup. In the semi-linear evolution +equation setting this allows us to obtain fixed points on sufficiently short time intervals. +34 + +Lemma3.16. Let K ⊂ G be a compact domain such that for every x ∈ K and ¯x := argminy∈P |x−1y| +one has that the points ¯x +� +λ·( ¯x−1x) +� +∈ K for every λ ∈ [0,1]. Also let f ∈ Dγ,η +P +for some γ > 0 and +assume that for every ζ < η the map x �→ Qζ f (x) extends continuously in such a way that for +x ∈ P one has Qζ f (x) = 0. Then one has the bound, +�f �γ,η;K ≲ +������f +������ +γ,η;K, +where the implied constant depends affinely on ∥Γ∥γ;K but is otherwise independent of K. Sim- +ilarly, if ¯f ∈ Dγ,η +P, ¯M with respect to a different model ¯M = ( ¯Π, ¯Γ) and is such that +lim +x→P Qζ(f (x)− ¯f (x)) = 0 +for every ζ < η. Then one has the bound, +�f − ¯f �γ,η;K ≲ +������f ; ¯f +������ +γ,η;K +∥Γ− ¯Γ∥γ;K +�������f +������ +γ,η;K + +������ ¯f +������ +γ,η;K +� +, +with proportionality constant also depending affinely on ∥Γ∥γ;K and ∥¯Γ∥γ;K. +Proof. The proof follows almost exactly as that of [Hai14, Lem. 6.5]. For completeness we +provide a sketched proof of the first inequality, the second follows analogously. +Firstly, note that for x ∈ G such that dG(x,P) ≥ 1 or for ζ ≥ η both bounds follow directly +from the definitions. Hence we restrict our attention to x ∈ K such that dG(x,P) < 1 and +ζ < η. We define a recursive sequence by setting x0 := x, x∞ := ¯x = argminz∈P |x−1z| and +xn := x∞ +� +2−n ·(x−1 +∞ x0) +� +. One has |x−1 +n x∞| = |2−n ·(x−1 +∞ x0)| = 2−n|x−1 +∞ x0| as well as +|x−1 +n xn+1| ≲G |2−n ·(x−1 +∞ x0)|+|2−(n+1) ·(x−1 +∞ x0)| = 2−n +�3 +2 +� +|x−1 +∞ x0| . +So using that we can write 2−n|x−1 +∞ x0| = |x−1 +n x∞| we find +|x−1 +n xn+1| ≲G |x−1 +n x∞| = 2−n|x−1 +∞ x0| = 2−ndG(x,P) = 2−n|x|P. +(3.19) +The main difference in the proof is to apply (3.19) in place of [Hai14, Equation (6.4)], then the +rest of the proof adapts closely. One uses (3.19) together with the definition of +������f +������ +γ,η;K, to +show that for any ζ ∈ A, +|f (xn+1)−Γxn+1xn f (xn)|ζ ≲G +������f +������ +γ,η;K2n(η−ζ)|x|η−ζ +P +. +To see this, note that for m ≥ η it trivially holds and so for |f (x)|m ≲K |x|η−m +P +, we proceed by +reverse induction, assuming that the required bound holds for all m > ζ and then show that +it also holds for ζ. Applying the triangle inequality, the inductive hypothesis, the properties +of Γ and (3.19) we find, as in the proof of [Hai14, Lem. 6.5], that +|f (xn+1)− f (xn)|ζ ≲ 2n(η−ζ)|x|η−ζ +P +, +where the constant depends affinely on ∥Γ∥γ;K. Then, using the assumption that Qζ f (x) = 0 +for all ζ < η and x ∈ P and applying the above inequality we find, +|f (x)|ζ = |f (x)− f (x∞)|ζ ≤ +� +n>0 +|f (xn+1)− f (xn)|ζ ≲K |x|η−ζ +P +� +n≥0 +δn(η−ζ). +Using that A is locally finite we may complete the induction which finishes the proof of the +first inequality. The proof of the second follows as in [Hai14, Lem. 6.5]. +35 + +Lemma 3.17. Let γ > 0, κ ∈ (0,1) and assume f , ¯f satisfy the assumptions of Lemma 3.16. +Then, for every compact K ⊂ G, one has +������f ; ¯f +������ +(1−κ)γ,η;K ≲ �f − ¯f �κ +γ,η;K +�������f +������ +γ,η;K + +������ ¯f +������ +γ,η;K +�1−κ +. +Proof. Follows by a direct adaptation of the proof of [Hai14, 6.6]. One need only replace Rd +there by G here. +3.3.1 RECONSTRUCTION THEOREM FOR SINGULAR MODELLED DISTRIBUTIONS +Since the reconstruction is purely local, it follows from our earlier proof that for any singular +modelled distribution, f ∈ Dγ,η +P , there exists a unique element ˜Rf ∈ S′(G\P), i.e. in the dual +of smooth functions compactly supported away from P, such that, +〈 ˜Rf −Πx f (x),ψλ +x〉 ≲ λγ, +for all x ∉ P and λ ≪ dG(x,P). However, we show below that under appropriate assumptions +there exists a natural extension of ˜Rf to an actual distribution on G with regularity Cα. +Proposition 3.18 (Singular Reconstruction). Let f ∈ Dγ,η +P (V ), γ > 0 and η ≤ γ. Then, provided +α ∧ η > −m, where m is the scaled dimension of the complement of the singular hyperplane +defined by (3.18), there exists a unique distribution Rf ∈ Cα∧η +s +such that (Rf )(ϕ) = ( ˜Rf )(ϕ) +for any smooth test function compactly supported away from P. If f and ¯f are given with +respect to two models M and ¯M then, for any compact K, it holds that +∥RM f −R ¯M ¯f ∥Cα∧η(K) ≲ +������f ; ¯f +������ +γ,η; ¯K + +������M; ¯M +������ +γ; ¯K, +where the constant depends on semi-norms of f , ¯f and Z, ¯Z on ¯K. +We provide the proof of Proposition 3.18 at the end of this section, let us first make some +preparatory observations. Recalling the decomposition g = p ⊕pc defined at the start of the +subsection, we define the projections πc : g → pc and πp : g → p. Then using the decomposi- +tion X = X p + X c ∈ p⊕pc, we define the map +˜Φ : g → G, +˜Φ(X ) = exp(X p)exp(X c). +(3.20) +Similarly to Remark 2.18 one sees that ˜Φ is a global diffeomorphism. We then define the map +NP : G → R+, +x �→ +��exp +� +πc ◦ ˜Φ−1(x) +��� +(3.21) +and observe the following properties. +• For x ∈ G one has that NP(x) = 0 if and only if x ∈ P . This follows from the fact that +P = exp(p). +• For x ∈ G and δ > 0 one has the identity +NP(δ· x) = δNP(x) . +(3.22) +Indeed, writing x = ˜Φ( ˜X ), we have +NP(δ· x) = |exp(πc(Dδ ˜X ))| = |exp(Dδ πc( ˜X ))| = |δ·exp(πc( ˜X ))| = δNP(x) . +36 + +• For any x ∈ G and y ∈ P one has +NP(yx) = NP(x) . +(3.23) +This follows from the observation that, writing x = ˜Φ( ˜X ) and y = ˜Φ(Y ) = exp(Y ) one +finds that yx = exp(H(Y , ˜X p))exp( ˜X c) where H was defined in (2.1). Identity (3.23) then +follows from the fact that H(Y , ˜X p) ∈ p. +• The map NP is Lipschitz continuous on G and it follows from Remark 2.4 that NP is +smooth on G\P. +• There exists a constant C > 0 such that for all x ∈ G +CNP(x) ≤ dG(x,P) . +(3.24) +In a neighbourhood of the origin this follows directly from the fact that NP is Lipschitz +continuous. Homogeneity of Np (given by (3.22) above) and of the distance function +x �→ dG(x,P) then shows that this constant is in fact uniform on all of G. +• For all x ∈ G +dG(x,P) ≤ NP(x) . +(3.25) +Indeed, write x = ˜Φ(X ) = exp(X p)exp(X c) and note that +d(x,P) ≤ |exp(X p)−1x| = |exp(X c)| = NP(x) . +Proof of Proposition 3.18. The proof is analogous to that of [Hai14, Lem.6.9], the only ele- +ment that does not adapt ad verbatim, is the construction of the partition of unity ϕx,n. We +therefore present an alternative construction of a partition of unity on G, which satisfies all +the required conditions. +First let ϕ : R+ → [0,1] be a smooth compactly, supported function such that supp(ϕ) = +[1/2,2] and with the property that for all r ∈ R+, +� +n∈Z +ϕ(2nr) = 1. +Secondly, let Z ⊂ P be a lattice (see Section 2.3) and let ˜ϕ be smooth, compactly supported, +such that +� +y∈Z +˜ϕy(x) = 1 +(3.26) +for all x in the 2 +C -fattening of P ⊂ G, where C is the constant in (3.24)and (3.25). For y ∈ Z we +then set +φy(x) := ϕ(CNP(x)) ˜ϕy(x) , +where the constant C is as in (3.24). To conclude, we then set for every n ≥ 0, y ∈ Z and x ∈ G, +φn,y(x) := φy(2n · x). +37 + +By using (3.22) and (3.23), it directly follows that φn,y(x) = (φ1,e) +� +y−1(2n · x) +� +. Since φ1,e has +compact support and is such that for all x ∈ supp(φ1,e), one has dG(x,P) ≥ CNP(x) ≥ 1/2, it +only remains to check that {φn,y}n∈Z,y∈Z is in fact a partition of unity. Indeed let x ∈ G, then +� +n∈Z,y∈Z +φn,y(x) = +� +n∈Z,y∈Z +ϕ(CNP(2n · x)) ˜ϕy(2n · x) = +� +n∈Z +ϕ(2nCNP(x)) +� +y∈Z +˜ϕy(2n · x) +� +�� +� +=1 given d(x,P)≤ 1 +C 2−n+1 += 1 , +where we used (3.26) in the last equality. The remainder of the proof of [Hai14, Lem. 6.9] then +adapts ad verbatim by also also making use of Proposition 3.13. +Remark 3.19. If the model M is smooth, i.e. Πxτ ∈ C ∞(G) for every τ ∈ T, one finds exactly as +in [Hai14, Rem. 3.15] that for any modelled distribution f ∈ Dγ with γ > 0 one has the identity +RM f (x) = +� +Πx f (x) +� +(x) (and in particular RM f is a continuous function). +3.4 LOCAL OPERATIONS +To handle SPDEs using regularity structures we require suitable extensions to modelled dis- +tributions of standard local operations such as differentiation, multiplication and composi- +tion with smooth functions. These extensions adapt easily from [Hai14], thus for brevity we +provide them directly for singular modelled distributions. +3.4.1 DIFFERENTIATION OF SINGULAR MODELLED DISTRIBUTIONS +Definition 3.6. Given a sector V of a regularity structure T = (T,G), a linear operator ∂ : V → T +defines an abstract differential operator of homogeneous degree β, if +• for any a ∈ Vα it holds ∂a ∈ Tα−β, +• for any a ∈ V and Γ ∈ G, one has Γ∂a = ∂Γa. +We say ∂ realizes L for the model M = (Π,Γ) if +• for any a ∈ V and x ∈ G, one has Πx∂a = LΠxa. +The following proposition is an immediate consequence of the definitions and the unique- +ness of the singular reconstruction operator. +Proposition 3.20. In the setting of Definition 3.6 let f ∈ Dγ,η +P (V ) for some γ > η, then ∂f ∈ +Dγ−β,η−β +P +. Furthermore, if the sector V has regularity α and it holds that γ > β as well as +α∧η > β−m, then one has R∂f = LRf . +Proof. For the first claim one may directly adapt the proofs of [Hai14, Prop. 5.28 & Prop. 6.15]. +The second claim follows by applying the Proposition 3.18 to Dγ−β,η−β +P +, +38 + +Remark 3.21. We note that any left invariant differential operator L of homogeneous degree +β on G is of the form +L = +� +d(I)=β +aI X I, +for some I ∈ Nd. Thus, in order to lift such an L, it suffices to lift each of the differential +operators {Xi }d +i=1 to abstract differential operators. +Remark 3.22. Recall that in Section 3.1.1 we lifted the differential operators Xi crucially using +the additional structure of the polynomial regularity structure. This is by no means the only +lift, though certainly the most canonical one. One observes that it is always possible to extend +a regularity structure and a model such that it carries some lift of a given differential operator. +3.4.2 MULTIPLICATION AND COMPOSITION WITH SMOOTH FUNCTIONS +We recall the notion of a product on a regularity structure. +Definition 3.7. Given a regularity structure T = (T,G) and two sectors V,W ⊂ T, a continuous +bilinear map +⋆ : (V,W ) → T, +(a,b) �→ a ⋆b +defines a product if +• one has a ⋆b ∈ Tα+β for every α, β ∈ A and a ∈ Tα, b ∈ Tβ, +• Γ(a ⋆b) = (Γa)⋆(Γb) for every Γ ∈ G and for every a ∈ V, b ∈ W . +We say that a sector V is stable under the product if V ⋆V ⊆ V . Given a product ⋆ and any +γ ∈ R, we introduce the truncated product ⋆γ, which is given by the composition of ⋆ with the +projections onto T<γ. +We show that the product of two singular modelled distributions is again a singular mod- +elled distribution with possibly new regularity and singularity parameters. +Proposition 3.23. Let P be a hyperplane as described above and f1 ∈ Dγ1,η1 +P +(V1) and f2 ∈ +Dγ2,η2 +P +(V2) for two sectors V1, V2 of respective regularities α1, α2 ∈ A and let ⋆ be a product +on (V1, V2). Then f = f1 ⋆γ f2 ∈ Dγ,η +α;P with +α = α1 +α2, +γ = (γ1 +α2)∧(γ2 +α1), +η = (η1 +α1)∧(η2 +α1)∧(η1 +η2). +Here ⋆γ is the projection of ⋆ onto T<γ. +Furthermore, for products between modelled distributions as above but on differing models, +writing f = f1 ⋆γ f2 and g = g1 ⋆γ g2 we have the bound, +������f ;g +������ +γ,η;K ≲ +������f1;g1 +������ +γ1,η1;K + +������f2;g2 +������ +γ2,η2;K +∥Γ− ¯Γ∥γ1+γ2;K, +uniformly over any compact set K ⊂ G. +Proof. One may directly adapt the proofs of [Hai14, Th. 4.7] and [Hai14, Prop. 6.12], recalling +that the homogeneous distance satisfies a triangle inequality with a constant, c.f. Remark2.4. +39 + +We define the composition of modelled distributions with smooth functions as in [Hai14]. +Given a ∈ V , a function-like sector, we decompose a = ¯a1 + ˜a with ˜a ∈ T>0 and ¯a = 〈1,a〉. +Further, let ζ > 0 be the smallest non-zero value such that Vζ ̸= 0 so that we actually have +˜a ∈ T≥ζ. +Given a smooth function F : Rn → R, for some n ≥ 1, and a function-like sector V which is +stable under some product ⋆γ, we lift F to a function ˆFγ : V n → V by the formula, +ˆFγ(a) := +� +k +DkF(Q0a) +k! +˜a⋆γk, +(3.27) +where we used the isomorphism T0 ∼ R for each component of a = (a1,...,an) ∈ V n, as well as +the notation ˜ai = ai −Q0ai. Here the sum runs over all possible multi-indices and we extend +the product ⋆ in a natural way for vectorial arguments and multi-index powers. +We make the same observation as in [Hai14, Sec. 4.2], that although the sum in (3.27) looks +infinite, since ζ > 0 we have ˜a⋆γk ∈ T≥|k|ζ and so only finitely many terms of the sum are +non-zero. In the following proposition we naturally extend the definition of a modelled dis- +tribution and associated norms componentwise. +Proposition 3.24. Let P be as above, γ > 0 and f = (f1,..., fn) ∈ Dγ,η +P (V n) be a collection of +modelled distributions for some function-like sector V which is stable under the product ⋆. Let +furthermore F ∈ C∞(Rn;R), then, provided η ∈ [0,γ], the modelled distribution +ˆFγ(f )(x) : G → V, +with ˆF(f ) defined as in (3.27), belongs to Dγ,η +P (V ). Furthermore, the map ˆFγ : Dγ,η +P (V ) → +Dγ,η +P (V ) is locally Lipschitz continuous in any of the semi-norms ∥ · ∥γ,η;K and |||·|||γ,η;K. +Furthermore, the analogous Lipschitz bound holds when working with two models. +Proof. One may follow ad verbatim the proof of [Hai14, Prop. 6.13], as well as [HP15, Prop. 3.11] +for the last sentence, since all Taylor expansions are carried out in Euclidean space. +3.5 CONVOLUTION WITH SINGULAR KERNELS +In this section we describe how to lift the action of singular kernels onto the regularity struc- +ture. While most arguments adapt from [Hai14] some care has to be taken due to the fact +that convolutions are not commutative and we do not have an explicit formula for Taylor ex- +pansions. This latter issue is circumvented using Lemma 3.6 which allows us to reduce our +analysis to similar expressions as appear in [Hai14]. The examples we have in mind are the +singular part of Greens functions of left invariant differential operators satisfying the follow- +ing assumption. +Assumption 3.25. The kernel K : G\{e} → R can be decomposed as +K (x) = +� +n∈N +Kn(x) +(3.28) +where the smooth functions Kn are supported on B2−n and +40 + +• for each I ∈ Nd there exists a constant C(I) > 0, uniform in n ∈ N such that +sup +x∈G +|X IKn(x)| ≤ C(I)2(|s|−β+d(I))n, +• for any multi-indices I, J ∈ Nd there exists a constant C(I, J) > 0 uniform in n ∈ N such +that, +���� +� +G +ηI(x)X JK (x)dx +���� ≤ C(I, J)2−βn, +• there exists an integer r, such that +� +G +ηI(x)K (x)dx = 0 +for all multi-indices I ∈ Nd with scaled degree d(I) ≤ r. +Remark 3.26. We note that all of the analysis in the remainder of this section also applies +to kernels of the form K : (G \{e})2 → R satisfying an analogue of [Hai14, Ass. 5.1 & Ass. 5.4] +adapted to our setting. Although Assumption 3.25 is somewhat less general we choose to +work with it for two reasons; firstly it is simpler to verify and secondly it highlights the role that +translation invariance plays in our applications. Lemma 4.4 which is an amalgam of [Hai14, +Lem. 5.5 & Lem. 7.7] in our setting, shows that fundamental solutions of left-translation in- +variant, homogeneous linear operators can always be decomposed into a compactly support +part satisfying Assumption 3.25 and a sufficiently well-behaved remainder. +Remark 3.27. Although we work explicitly with the one-parameter kernels of Assumption +3.25 it will sometimes convenient in the proofs below to define K (x, y) := K (y−1x) and use +the notation K (f )(x) := +� +K (x, y)f (y)dy = f ∗K (x). Note that under this convention, for any +left-translation invariant vector field X , one has (X K )(f ) = X (K f ). +From now on we shall exclusively work with regularity structures T models M satisfying +the following assumption. +Assumption 3.28. For each a ∈ △, the vector space Ta coincides with the linear span of abstract +monomials ηηηI with d(I) = a and the model M ∈ MT restricted to the polynomial sector ¯T = +� +a∈△Ta, is the canonical polynomial model. +We point out that the assumption K annihilates polynomials causes no real restriction on +the type of kernels since the result [Hai14, Lem. 5.5] adapts in a straightforward manner to +our setting, see also Lemma 4.4 below. We point out that this assumption is convenient but +not crucial for the theory, c.f. [HSa]. In the remainder of this subsection we show how the +action of kernels of this type are lifted onto the regularity structure and act on modelled dis- +tributions. Given a γ ∈ R∩△ we write Kγ for this lift; it corresponds to K in the sense that for +f ∈ Dγ +RKγ f = K (Rf ) +(3.29) +and in that it satisfies an appropriate version of the classical Schauder estimates. +41 + +Definition 3.8. Given a sector V , a map I : V �→ ¯T is called an abstract integration map of +order β > 0 if it satisfies the following properties: +1. For each α ∈ A, I : Vα → Tα+β, where Tα+β := {0} for α+β ∉ A. +2. It annihilates polynomials, that is I : ¯T∩V → {0}. +3. For each τ ∈ T, Γ ∈ G : (I Γ−ΓI)(τ) ∈ ¯T. +Assume that the kernel K satisfies Assumption 3.25 for some β > 0. We associate to K the +map J : Rd → L(T, ¯T) which for every τ ∈ Tα is given by +J (x)τ := +� +n +PPPα+β +x +(Πxτ∗Kn), +(3.30) +where the last sum is seen to converge absolutely by first observing that by Assumption 3.25 +for any τ ∈ Tα +|X I(Πxτ∗Kn)(x)| = |Πxτ∗ X IKn(x)| = |Πxτ(X I +1Kn(x,·))| ≲ 2−n(α+β−d(I)) , +and then using Lemma 3.4. +Definition 3.9. Given a regularity structureT equipped with an Integration map I, a kernel K +and a model M = (Π,Γ), we say the model M realises K for I if for each τ ∈ Tα and each x ∈ G, +ΠxIτ = K (Πxτ)−ΠxJ (x)τ. +Now we can define the lift of the kernel K, namely for f ∈ Dγ(V ) we set: +Kγ f (x) := I f (x)+J (x)f (x)+Nγ f (x), +where +(Nγf )(x) = +� +n +PPPγ+β +x +� +(Rf −Πx f (x))∗Kn +� +where the last sum converges by the same argument as for J (x)τ. +Remark 3.29. Given a kernel K satisfying Assumption 3.25 a regularity structure and model +satisfying Assumption 3.28, it turns out that one can always extend the regularity structure +and model to be equipped with an integration map I realizing the kernel K . This is the con- +tent of the extension theorem found as [Hai14, Thm. 5.14], which holds in our setting as well. +While we do not reproduce the whole proof since it is a straightforward adaptation of the +original one, we present the main steps below, see Lemmas 3.31, 3.32 and 3.33, so that the +interested reader will easily be able to fill in the remaining details. +The next theorem which is an analogue of [Hai14, Thm. 5.12], confirms that Kγ does indeed +correspond to K in the sense of (3.29) and satisfies the desired Schauder estimates. +42 + +Theorem 3.30. Let γ ∈ R \△ and β > 0 be such that γ + β ̸∈ △, let K : G \{e} → R be a kernel +satisfying Assumption 3.25 for r ≥ γ+β, let T = (T,G) be a regularity structure and M = (Π,Γ) +be a model satisfying Assumption 3.28. Furthermore assume that T is equipped with an ab- +stract integration operator and M realises K for I. Then for any sector V of regularity α ∈ A the +operator Kγ is a continuous linear map from Dγ +M(V ) to Dγ+β +(α+β)∧0 satisfying the identity +RKγ f = K (Rf ), +for all f ∈ Dγ +M(V ). Furthermore, if we denote by M = (Π,Γ), ¯M = ( ¯Π, ¯Γ) two admissible models +and by Kγ, respectively ¯Kγ the associated operators, one has the bound +∥Kγ f ; ¯Kγ ¯f ∥γ+β;K ≲C +������f , ¯f +������ +γ; ¯K +∥Π− ¯Π∥γ; ¯K +∥Γ− ¯Γ∥γ; ¯K , +where the implicit constant depends on the norms of M, ¯M and f ∈ Dγ +M(V ) and ¯f ∈ Dγ +¯M(V ). +The fact that the next lemma ([Hai14, Lem. 5.16]) still holds in our setting is the underlying +reason that all proofs extend in a rather straight forward manner from [Hai14] and one does +not require a more involved notion of abstract integration map, which is for example the case +on general Riemannian manifolds c.f.[HSa]. +Lemma 3.31. In the setting of Theorem 3.30 one has the identity +Γx,y(I +J (y)) = (I +J (x))Γx,y +for all x, y ∈ G. +Proof. The proof consists of unravelling the Definitions and using the fact that the map Πx is +injective when restricted to polynomials, exactly as in [Hai14]. +We introduce the following quantity; for I ∈ Nn,α > 0,n ∈ N and x, y,z ∈ G, set +K I,α +n,xy(z) := X I +1Kn(y,z)− ˜Pα+β−d(I) +x +[X I +1Kn(·,z)](y) = X I +1 +� +Kn(·,z)− ˜Pα+β +x +[Kn(·,z)] +� +(y) +(3.31) +where the second equality follows from Remark 2.12. Here we reiterate that we are using the +notation K (x, y) := K (y−1x) where K satisfies Assumption 3.25. Taylor’s theorem and Remark +2.15 then yield that +K I,α +n,xy(z) = +� +|J|≤[a]+1,d(J)≥α+β−d(I) +� +G +X J +1(X I +1Kn)(x ˜z,z)Q J(x−1y,d ˜z) . +(3.32) +43 + +As in [Hai14] the motivation for defining these quantities comes from the identity +Πx(Iτ)(φ) = K (Πxτ)(φ)−Πx J(x)τ(φ) += +� +n +�� +Πxτ(Kn(y,·))− ˜Pα+β +x +[Kn(Πxτ)](y) +� +φ(y)d y += +� +n +�� +Πxτ(Kn(y,·))− ˜Pα+β +x +� +(Πxτ)2(Kn(·,·)) +� +(y) +� +φ(y)d y += +� +n +�� +Πxτ(Kn(y,·))−(Πxτ)(˜Pα+β +x,1 [Kn(·,·)](y) +� +φ(y)d y += +� +n +� +Πxτ +� +Kn(y,·)− ˜Pα+β +x,1 [Kn(·,·)](y) +� +φ(y)d y += +� +n +� +Πxτ(K 0,α +n,xy)φ(y)d y +where we use the subscript in (Πxτ)2(Kn(·,·)) to clarify that this denotes the function w �→ +Πxτ(Kn(w,·)) and the analogous subscript for ˜Px,1[K (·,·)](y) to clarify that one expands in the +first coordinate. The next Lemma collects the results of [Hai14, Lem. 5.18, Lemma 5.19], the +proofs of which adapt ad verbatim. +Lemma 3.32. Let α ∈ A and τ ∈ Tα and assume α+β ∉ △, then the following bound holds +|(Πyτ)(K I,α +n,xy)| ≲ ∥Π∥α,B2(x)(1+∥Γ∥α,B2(x)) +� +δ>0 +2δn|y−1x|δ+α+β−d(I) , +(3.33) +where the sum over δ runs over a finite set of positive real numbers. The same bound holds for +|(Πxτ)(K I,α +n,xy)|, as well as the analogue bound on the difference of two models. +Furthermore one has the bound +� +n +���� +� +G +(Πxτ)(K I,α +n,xy)φλ +x(y)dy +���� ≲ λα+β∥Π∥α;B2(x)(1+∥Γ∥α;B2(x)) , +(3.34) +as well as the analogous bound for the difference of two models, where all these bounds hold +uniformly over x ∈ G, λ ∈ (0,1] and φ ∈ Br +As in [Hai14] we introduce for x, y ∈ G the following operator +Jx,y := J (x)Γx,y −Γx,yJ (y) +(3.35) +Lemma 3.33. Under the assumptions of Theorem 3.30 one has for each a ∈ ∆, τ ∈ Tα +|Jx,yτ|a ≲ ∥Π∥α,B2(x)(1+∥Γ∥α,B2(x))|y−1x|α+β−a|τ| +uniformly in x, y ∈ G satisfying x ∈ B1(y). Again, the analogous bound holds for the difference +of two models holds as well. +Proof. First we write Jx,y = � +n J n +x,y where each J n +x,y is defined by replacing K by only one +summand Kn in the definition of Jx,y. Observe that since J n +x,yτ ∈ ¯T<α+β it suffices by Lemma 3.6 +to show that +pn +τ (y) := Πe(J n +x,yτ)(y) ∈ Pα+β +44 + +satisfies +� +n +|(X I pn +τ )(e)| ≲ ∥Π∥α,B2(x)(1+∥Γ∥α,B2(x))|y−1x|α+β−d(I)|τ| . +(3.36) +We treat the cases |y−1x| ≤ 2−n and |y−1x| > 2−n separately. In the case |y−1x| ≤ 2−n, using +the definitions of the polynomial regularity structure, we find +pn +τ (x−1z) = Πe(J n +x,yτ)(x−1z) = Πx(J n +x,yτ)(z) +and thus +X I pn +τ (e) = X I(ΠxJ n +x,yτ)(x) . +Using the analogous notation J n(x) for the operator consisting only of one summand in +(3.30), we find +ΠxJ n(x)Γx,yτ(z) = +� +γ≤α +Πx +�J n(x)QγΓx,yτ +� +(z) += +� +γ≤α +ΠxPγ+β +x +[ +� +ΠxQγΓx,yτ +� +2Kn(·,·)](z) += +� +γ≤α +˜Pγ+β +x +[ +� +ΠxQγΓx,yτ +� +2Kn(·,·)](z) += +� +γ≤α +ΠxQγΓx,yτ +� +˜Pγ+β +x,1 [Kn(·,·)](z) +� += (Πyτ) +� +˜Pα+β +x,1 [Kn(·,·)](z) +� +− +� +γ≤α +ΠxQγΓx,yτ +�˜Pα+β +x,1 [Kn(·,·)]− ˜Pγ+β +x,1 [Kn(·,·)] +� +(z) += ˜Pα+β +x +[(Πyτ)2Kn(·,·)](z)− +� +γ<α +ΠxQγΓx,yτ +�˜Pα+β +x,1 [Kn(·,·)]− ˜Pγ+β +x,1 [Kn(·,·)] +� +(z) +and +ΠxΓx,yJ n(y)τ(z) = ΠyJ n(y)τ(z) = ˜Pα+β +y +[(Πyτ)2Kn(·,·)](z) += ˜Pα+β +x +� +˜Pα+β +y +[(Πyτ)2Kn(·,·)] +� +(z) += ˜Pα+β +x +�� +Πyτ +� +2 +�˜Pα+β +y,1 [Kn(·,·)] +�� +(z) , +where in the third equality we used the fact that ˜Pα+β +x +acts as the identity map on polynomial +functions of degree less then α+β. Therefore, +pn +τ (x−1z) = ˜Pα+β +x +� +(Πyτ)2 +� +Kn(·,·)− ˜Pα+β +y,1 [Kn(·,·)] +�� +(z) +� +�� +� +:=pn +τ,1(x−1z) +− +� +γ<α +ΠxQγΓx,yτ +�˜Pα+β +x,1 [Kn(·,·)]− ˜Pγ+β +x,1 [Kn(·,·)] +� +(z) +� +�� +� +:=pn,γ +τ,2 (x−1z) +Using the general formula X I(˜Pa +x[f ])(x) = X I(Pa +x[f ])(e) = X I f (x) for d(I) < a we find that +X I pn +τ,1(e) = X I� +(Πyτ)2 +� +Kn(·,·)− ˜Pα+β +y,1 [Kn(·,·)] +�� +(x) = (Πyτ)(K I,α+β +n,x,y ) +and so by Lemma 3.32 we find that +� +n :|y−1x|≤2−n +|X I pn +τ,1(e)| ≲ |y−1x|α+β−d(I)|τ| . +45 + +Concerning pn,γ +τ,2 , since one has the identity +pn,γ +τ,2 (x−1z) = −˜Pγ+β +x +[(ΠxQγΓx,yτ)2Kn(·,·)](z)+ ˜Pα+β +x +[(ΠxQγΓx,yτ)2Kn(·,·)](z) , +we find that X I pn,γ +τ,2 (e) = 0 unless d(I) ∈ [γ+β,α+β), in which case +X I pn,γ +τ,2 (e) = ΠxQγΓx,yτ +� +X I +1Kn(x,·) +� +. +Thus, +� +n :|y−1x|≤2−n +|X I pn,γ +τ,2 (e)| ≲ +� +n :|y−1x|≤2−n +|y−1x|α−γ2−n(γ+β−d(I))|τ|a ≲ |y−1x|α+β−d(I)|τ| , +where we used that the case d(I) = γ+β does not arise as by Assumption 3.28 +we have γ+β ∉ △. This concludes the case |y−1x| > 2−n. +To treat the case |y−1x| > 2−n, one writes +pn +τ (z) = +� +γ≤α +ΠxJ n(x)(QγΓx,yτ)(xz) +� +�� +� +=:qn,γ +τ,1 (z) +−ΠxΓx,yJ n(y)τ(xz) +� +�� +� +=:qn +τ,2(z) +and using the fact that γ+β ∉ △ one finds +|X I qn,γ +τ,1 (e)| = +���ΠxQγΓx,yτ +� +X I +1Kn(x,·) +���, +if d(I) < γ+β, +0, +otherwise, +which is bounded uniformly over |y−1x| < 1 by a constant multiple of |y−1x|α−γ2−n(γ+β−d(I)). +Concerning qn +τ,2, we find that +qn +τ,2(z) = ΠxΓx,yJ n(y)τ(xz) = ΠyJ n(y)τ(xz) = Pγ+β +y,1 [(Πyτ)2Kn(·,·)](y−1xz)) +and therefore by Lemma 3.5 +|X I qn +2 (e)| ≲ +� +d(I) 2−n� +. +Proof of Theorem 3.30. First, we need to check that for a ∈ A it holds that +|Kγ f (x)−Γx,yKγ f (y)|a ≲ |y−1x|γ+β−a . +(3.37) +For a ∉ △ this bound follows from Lemma 3.33 and properties of modelled distributions by +an ad verbatim adaptation of the same argument in the proof of [Hai14, Thm. 5.12]. The only +46 + +place, where the proof [Hai14, Thm. 5.12] does not adapt directly is when showing the bound +for a ∈ △. As in the proof of Lemma 3.33 we use Lemma 3.6 to circumvent the difficulty of +not having explicit Taylor expansions in our setting. The rest of the proof adapts almost ad +verbatim. +We first note that the polynomial part of Kγ f (x)−Γx,yKγ f (y) is given by +P := Γx,yNγf (y) +� +�� +� +=:P1 +−Nγf (x) +� �� � +=:P2 ++J (x)(Γx,y f (y)− f (x)) +� +�� +� +=:P3 +(3.38) +and thus, in order to prove (3.37) for the polynomial part it suffices by Lemma 3.6 to show +that for any d(I) < γ+β, p(e) := ΠeP satisfies +|X I p(e)| ≲ |y−1x|γ+β−d(I) . +(3.39) +Using (3.28) we define the analogous decompositions J = � +n J n and Nγ = � +n N n +γ and simi- +larly write P = � +n=0 Pn, etc. +As usual we use different strategies for small and large scales, starting with the case 2−n ≤ +|y−1x|. In this case we separately estimate pi(e) := ΠePi for i ∈ {1,2,3} . +• Noting that +pn +1 (z) = ΠxΓx,yNγf (y)(xz) = ΠyNγ f (y)(xz) = Pγ+β +y,1 +� +(Rf −Πy f (y))2 +� +Kn(·,·) +�� +(y−1xz), +we find by Lemma 3.5 +|X I pn +1 (e)| ≲ +� +d(I)≤d(J)<γ+β +sup +d(J′)≤d(J) +��� +�Rf −Πy f (y) +�� +X J′ +1 Kn(y,·) +����|y−1x|d(J)−d(I) +≲ +� +d(I)≤d(J)<γ+β +sup +d(J′)≤d(J) +2−n(γ+β−d(J′))|y−1x|d(J)−d(I) +≲ +� +d(I)≤d(J)<γ+β +2−n(γ+β−d(J))|y−1x|d(J)−d(I) . +• Regarding pn +2 we find that X I pn +2 (e) = +�Rf −Πx f (x) +�� +X I +1Kn(x,·) +� +and thus by the Re- +construction, see Theorem 3.7, +|X I pn +2 (e)| ≲ 2−n(γ+β−d(I)) . +• Lastly, for pn, using the definition of J (x) and properties of models and modelled dis- +tributions we find +|X I pn +3 (e)| ≤ +� +δ∈A: d(I)−β<δ<γ +��� +ΠxQδ(Γx,y f (y)− f (x) +� +(X I +1K (x,·)) +�� +≲ +� +δ∈A: d(I)−β<δ<γ +|y−1x|γ−δ2−n(β+δ−d(I)) . +47 + +Therefore summing over 2−n ≤ |y−1x|, since β+δ ∉ △ for any δ ∈ A and β+γ ∉ △, we find +� +n :2−n≤|y−1x| +|X I pn(e)| ≤ +� +n : 2−n≤|y−1x| +3� +i=1 +|X I pn +i (e)| ≲ |y−1x|γ+β−d(I) . +Next we turn to the case 2−n > |y−1x|; here a computation analogous to the one preceding +[Hai14, Eq. (5.48)] gives +X I pn(e) = (Πy f (y)−Rf )(K I,γ +n,x,y)− +� +ζ≤d(I)−β +� +ΠxQζ(Γx,y f (y)− f (x) +�� +X I +1Kn(x,·) +� +(3.40) +which can be bounded exactly the way it is done there. +Finally, one may follow the concluding steps of the proof of [Hai14, Thm. 5.12], finding that +for any φ ∈ B, +� +Πx(Kγ f (x))−K (Rf ) +� +(φλ +x) = +� +n +�� +Πx f (x)−Rf +� +(K γ +n,xy)φλ +x(y)d y +and thus one obtains the identity RKγ f = K (Rf ). +The proof of the difference bound follows by similar steps as to those presented here and +applied in the proof of [Hai14]. +3.6 SCHAUDER ESTIMATES FOR SINGULAR MODELLED DISTRIBUTIONS +Since our main application will be to semi-linear evolution equations we will often require +a Schauder estimate for modelled distributions with permissible singularities near a given +subgroup P ⊂ G, as in Section 3.3 on singular modelled distributions. +Proposition 3.34. In the setting of Section 3.3 and Theorem 3.30, given a sector V of regularity +α ∈ A, the operator Kγ is well defined on Dγ,η +P (V ) for η < γ provided that γ > 0 and η∧α > −|m|. +Furthermore, if γ+β ∉ △ and η+β ∉ △, one has Kγ f ∈ Dγ+β,(η∧α)+β +P +continuity bound +������Kγ f ; ¯Kγ ¯f +������ +γ+β,(η∧α)+β;K ≲ +������f ; ¯f +������ +γ,η; ¯K +∥Π− ¯Π∥γ; ¯K +∥Γ− ¯Γ∥γ; ¯K +(3.41) +over any compact set K ⊂ G and admissible models, where the implicit constant is uniform in +the semi-norms of M, ¯M and f ∈ Dγ,η +P,M(V ), ¯f ∈ Dγ,η +P, ¯M(V ) on ¯K. +Proof. The proof follows exactly along the same lines as the one of [Hai14, Prop. 6.16], where +as in the proof of Theorem 3.30 the only modification needed is due to the fact that our Taylor +expansions are non explicit. Using the same notation as in the proof of Theorem 3.30, we +recall (3.38) +P := Γx,yNγf (y) +� +�� +� +=:P1 +−Nγ f (x) +� �� � +=:P2 ++J (x)(Γx,y f (y)− f (x)) +� +�� +� +=:P3 +. +(3.42) +This time, in order to show that Kγ f ∈ Dγ+β,(η∧α)+β +P +, by Lemma 3.6, we are required to show +that for p(e) := ΠeP = �3 +i=1ΠePi, +|X I p(e)| ≲ |y−1x| +γ+β−d(I)|x, y|(η∧α)+β−γ +P +, +(3.43) +There are three scales, which need separate arguments. +48 + +• In the case 2−n ≤ |y−1x| one proceeds exactly as in the proof of Theorem 3.30 using the +decomposition pn = �3 +i=1 pn +i . +• In the cases 2−n ∈ +� +|y−1x|, 1 +2|x, y|P +� +and 2−n ≥ 1 +2|x, y|P one uses Equation (3.40) and +proceeds exactly as in [Hai14] from there onwards. +Again, the proof of the difference bound follows analogously. +3.7 SYMMETRIES +As in [Hai14, Sec. 3.6], we shall consider modelled distributions which respect certain sym- +metries of G. In this work we will only consider the symmetries of G under the canonical left +action of a discrete subgroup G ⊂ G as in Section 2.3 acting by G×G ∋ (n,x) �→ (nx) ∈ G. We +extend this to an action on function ψ : G → R by pull-back, i.e. +(n∗ψ)(x) := ψ(n−1x) = ψn(x). +For a regularity structure T = (T,G) we give the following definition, by analogy with [Hai14, +Def. 3.33] but restricted to this more straightforward setting. +Definition 3.10. Given a discrete sub-group G ⊂ G as above, we say that a model M = (Π,Γ) is +adapted to the action of G if, for every test function, φ ∈C ∞ +c (G), x ∈ G, τ ∈ T and n ∈ G one has +(Πnxτ)(n∗ψ) = (Πxτ)(ψ) +and +Γnx,ny = Γx,y. +A modelled distribution f : G → T is said to be symmetric if f (nx) = f (x) for every x ∈ G and +n ∈ G. +The following proposition is an amalgam of [Hai14, Prop. 3.38 & Prop. 5.23]. +Proposition 3.35. Let T be a regularity structure, G ⊂ G be a discrete subgroup as above and +M = (Π,Γ) be adapted to the action of G (according to Definition 3.10). Then, for every mod- +elled distribution f ∈ Dγ, for some γ > 0, symmetric with respect to the action of G, the follow- +ing hold; +1. For every φ ∈C ∞ +c (G) and n ∈ G one has (Rf )(n∗φ) = (Rf )(φ). +2. For any x ∈ G and n ∈ G one has (Kγ f )(nx) = (Kγ f )(x). +Proof. The proof of the first point follows exactly the steps of the proof of [Hai14, Prop. 3.38] +in our simplified setting. +The proof of the second follows similarly along the lines of the proof of [Hai14, Prop. 5.23], +in particular using the assumption that the model is adapted to the action of G and the vector +fields {Xi}d +i=1 are left-translation invariant. +49 + +3.8 BOUNDS ON MODELS +We state two results which, as in the Euclidean setting, allow one to reduce the number of +stochastic estimates needed in order to obtain convergence of models to only Π|T<0. The +proof of the next proposition, [Hai14, Prop 3.31], adapts ad verbatim to our setting +Proposition 3.36. Let T = (T,G) be a regularitystructure and (Π,Γ) a model. For α ∈ (0,∞)∩A, +the action of Π on Tα is fully determined by the action of Π on T<α as well as the action of Γ on +Tα. Furthermore, one has the bound +sup +x∈K +sup +λ<1 +sup +φ∈Br +sup +τ∈Tα \{0} +|Πxτ(φλ +x)| +λα|τ|α +≤ ∥Π∥α; ¯K∥Γ∥α; ¯K, +as well as the analogous difference bound. +Furthermore, one has the following simple consequence of Lemma 3.31 and Lemma 3.33. +Proposition 3.37. Under the assumptions of Theorem 3.30 one has for each α ∈ A \△, τ ∈ Tα +and δ ∈ A one has +sup +x,y∈K:|y−1x|<1 +|Γx,yIτ|δ+β +|τ|α−β|y−1x|α−δ ≲ (1+∥Γ∥α; ¯K)∥Π∥α; ¯K + +sup +x,y∈K:|y−1x|<1 +|Γx,yτ|δ +|τ|α−β|y−1x|α−δ . +Again, the analogous bound for the difference of two models holds as well. +4 APPLICATIONS TO SEMILINEAR EVOLUTION EQUATIONS +In this section we specialize to the setting, when the homogeneous Lie group G has distin- +guished time direction, see Section 4.4 below for illustrative examples. Specifically, assume +we are given decomposition of the Lie algebra g = pc⊕p where both summands are s-invariant +subspaces and furthermore pc is one dimensional. In line with this decomposition, in this +section we deviate from the convention stated above Equation (2.2) and reorder the basis of g +so that we always have the time component, i.e. X1 ∈ pc, in the first slot. Under this assump- +tion we also fix the basis ¯X = {Xi}d +i=2 of p where sXi = si . We note that P := exp(p) ⊂ G is a +homogeneous subgroup as in 3.3 and we recall the notation defined there, +¯s := s|p +and +|¯s| := trace(s|p). +In particular we have |s| = s1 +|¯s|. From now on we identify the Lie group G with R×P under +the diffeomorphism +R×P → G, +(t,x) �→ x exp(t X1) . +and we shall often use the notation z = (t,x) ∈ R×P = G. Let us collect some useful facts; +• Under this identification we see that R×{e} ⊂ G is a homogeneous subgroup. +• The scaling map behaves as expected, in that for any z = (t,x) ∈ G and δ > 0 +δ·(t,x) = (δs1t,δ· x) , +where δ· x is understood to be the restriction of the dilation to P. +50 + +• The map ˜Φ from (3.20) is related to our decomposition here, in that for any X p ∈ p and +t ∈ R, we have ˜Φ(X p + t X1) = (t,exp(X p)). +• Recall the map NP : G → R+ defined by (3.21). There exists a constant c′ > 0 such that +NP(t,x) = c′|t| +1 +s1 . In particular, by (3.24) and (3.25) there exist a constant c > 0 such that +for any (t,x) ∈ G +1 +c |t| +1 +s1 ≤ d ((t,x),P) ≤ c|t| +1 +s1 . +(4.1) +A core assumption for the remainder of this section will be that of non-anticipativity. +Assumption 4.1. We say that G : G \{e} → R is non-anticipative if for any (t,x), (s, y) ∈ G one +has G((s, y)−1(t,x)) = 0 whenever s ≥ t. +We will also require an assumption of prescribed homogeneity on the kernel, with respect +to the dilation map. +Assumption 4.2. For σ ∈ R, we say that that G : G\{e} → R is smoothly σ-homogeneous if it is +smooth and for any z ∈ G\{e} and λ > 0, +G(λ· z) = λσG(z). +We now have the following analogue of [Hai14, Lem. 7.4]. +Lemma 4.3. Given G : G\{e} → R satisfying Assumption 4.1 and Assumption 4.2 with σ = −|¯s|, +then there exists a smooth function ˆG : P → R such that for all (t,x) ∈ G with t > 0, +G(t,x) = t− ¯s +s1 ˆG +� +t− 1 +s1 · x +� +. +(4.2) +For every multi-index I ∈ Nd−1 and every n > 0 there exists a constantC > 0 such that uniformly +over x ∈ G, +| ¯X I ˆG(x)| ≤ C(1+|x|2)−n. +(4.3) +Proof. The proof follows exactly the same steps of [Hai14, Lem. 7.4] after replacing the usual +derivatives there with the vector fields ¯X = {Xi}d +i=2. +Lemma 4.4. Let G : G\{e} → R satisfy Assumption 4.1 and Assumption 4.2 with σ = −|¯s|. Then, +for any r > 0, there exist smooth functions K : G \ {e} → R and K−1 : G → R, both satisfying +Assumption 4.1 and such that G = K +K−1 and +• K is compactly and satisfies Assumption 3.25 with β = s1. +• K−1 : G → R is globally smooth and such that X IK−1 ∈ L∞ +loc(R;L1(P)) for any I ∈ Nd. +Proof. The proof readily adapts from [Hai14, Lem. 5.5 & Lem. 7.7] with the only real change +being to skip the last step in the proof of [Hai14, Lem. 7.7] and instead using Faa di Bruno’s +formula along with (4.3) to obtain the claimed integrabillity of K−1. +Remark 4.5. Note that more careful analysis yields improved bounds on K−1, however, K−1 ∈ +L∞ +loc(R;L1(P)) suffices for our purposes. +51 + +4.1 SHORT TIME BEHAVIOUR OF KERNEL CONVOLUTIONS +Together, the following two results show, in our setting, that the lift of the kernel applied to a +modelled distribution, f , can be controlled on sets of the form OT := {z ∈ G : d(z,P) ≤ T } only +using information about f on the same set. Note that using (4.1) there exists a constant c > 0 +such that [0,cT ]×P ⊂ OT . Hence, the corresponding notion of local solutions, described in +Section 4.3, corresponds to the usual one. +First, we introduce some final pieces of notation, let R+ : R×P → R be a map such that for +all x ∈ P one has R+(t,x) = 1 for t > 0 and R+(t,x) = 0 for t ≤ 0. From now on we shall also +use subspaces of the previously introduced Hölder spaces (Definition 2.4), modelled distri- +butions (Definition 3.4) and singular modelled distributions (Definition 3.5) writing, for ex- +ample +¯Cα(G) ⊂ Cα(G), +¯Dγ +α ⊂ Dγ +α, +¯Dγ,η +α,P ⊂ Dγ,η +α,P, +where we allow the set K in the relevant definitions to be any closed subset K ⊆ OT for some +T > 0 (in particular K is not necessarily compact). We shall often write the corresponding +semi-norms for example as +∥ · ∥α;OT ∩K, +|||·|||γ;OT ∩K, +|||·|||γ;η;OT∩K , +where in particular we allow K = OT . +With these notions at hand the proof of the next theorem adapts directly from the proof of +[Hai14, Thm. 7.1]. +Theorem 4.6. Let γ > 0, T be a regularity structure and M := (Π,Γ) and ¯M = ( ¯Π, ¯Γ) models +and K : G\{e} → R a non-anticipative kernel (see Assumption 4.1) such that the assumptions of +Theorem 3.30 are satisfied for some β > 0 and a sector V of regularity α > −s1. Then, for every +T ∈ (0,1], and η > −s1 and for κ > 0 small enough +������KγR+ f +������ +γ+β,(η∧α)+β−κ;OT ≲ T κ/s1������f +������ +γ,η;OT +(4.4) +������KγR+ f ; ¯KγR+ f +������ +γ+β,(η∧α)+β−κ;OT ≲ T κ/s1 +�������f ; ¯f +������ +γ,η;OT + +������M; ¯M +������ +γ;O2 +� +. +(4.5) +The constant in the first bound depends only on |||M|||γ;O2, while in the second it may also de- +pend on +������f +������ +γ,η;OT ∨ +������ ¯f +������ +γ,η;OT and |||M|||γ;O2 ∨ +������ ¯M +������ +γ;O2. +Proof. The proof follows almost ad verbatim the proof of [Hai14, Thm. 7.1], only using our +modified definition of the sets OT := {z ∈ G : dG(z,P) ≤ T }. +Remark 4.7. In fact, given any closed set K ⊂ G the bounds (4.4) and (4.5) both hold if OT is +replaced on the left hand side by OT ∩K and on the right hand side by OT ∩ ¯K. However, we +will not make use of this fact in this article. +While Theorem 4.6 treats the lift of the singular part of the kernel the following lemma +shows that the application of the smooth remainder can be lifted into the polynomial regu- +larity structure in a similar manner. +52 + +Lemma 4.8. Let K−1 : G → R be a smooth, non-anticipative kernel on G, such that for any +I ∈ Nd the functions X IK−1(t, ·) are bounded in L1(P), locally uniformly in t ∈ R. Then, under +the assumptions of Theorem 4.6 one has the bound +��� +��� +���PPPγ +(·)[K−1RMR+ f ] +��� +��� +��� +γ+β,¯η;OT ≲ T +������f +������ +γ,η;OT +as well as the analogous difference bound +��� +��� +���PPPγ +(·)[K−1RMR+ f ];PPPγ +(·)[K−1R ¯MR+ ¯f ] +��� +��� +��� +γ+β,¯η;OT ≲ T +�������f ; ¯f +������ +γ,η;OT + +������M; ¯M +������ +γ,O1 +� +. +Proof. The argument adapts mutatis mutandis form the proof of [Hai14, Lem. 7.3], replacing +the compact support assumption therein by the integrability property of K−1. +4.2 INITIAL CONDITIONS +We will only consider evolution equations on domains without boundary so that our only +boundary data is the initial condition and proceed as in [Hai14, Sec. 7.2]. +Lemma 4.9. Let u0 ∈ Cα(P) such that ∥u0∥Cα(P) < ∞ and T = (T,G) be a regularity structure +containing the polynomial regularity structure over G and let G be a kernel satisfying Assump- +tions 4.1 and 4.2. We set for t ̸= 0 +G(u0)(t,x) := +� +T×R +G((0, y)−1(t,x))u0(y)dy, +where we used the suggestive but only formal notation if α ≤ 0. Then, for every γ > 0 the map +PPPγ +(·)[G(u0)] : G\P → T belongs to Dγ,α +0;P for every γ > (α∧0) and furthermore, for any T > 0, one +has +��� +��� +���PPPγ +(·)[G(u0)] +��� +��� +��� +γ,η;OT < ∞. +Proof. We first decompose the kernel as in Lemma 4.4 and write +G(u0)(t,x) = K (u0)(t,x)+K−1(u0)(t,x) . +For the summands of K (u0)(t,x), one can argue as in the proof of [Hai14, Lem. 7.5] or as in +Section 3.5. The desired bound for K−1(u0)(t,x) follows exactly as in Lemma 4.8. +4.3 AN EXAMPLE FIXED POINT THEOREM +At this point, we have all ingredients at hand to straightforwardly see that the very general +fixed point Theorem [Hai14, Theorem 7.8] as well as the other parts of [Hai14, Section 7.3] +adapt to the setting of homogeneous Lie Groups. For the sake of conciseness and with the +primary example of Anderson-type models in mind, we refrain from presenting this material +in all generality and instead show an example fixed point theorem. This result covers Ander- +son type equations, such as the one treated in Section 6. Recall here the notation introduced +at the start of this section, in particular the identity |s| = s1 +|¯s|. +53 + +Theorem 4.10. Let α ∈ (−s1,0], γ > −α, η + α > −s1 and G : G \ {e} → R be a kernel satisfy- +ing Assumptions 4.1 and 4.2 with σ = −|¯s| and the decomposition G = K + K−1 as in Lemma +4.4. Furthermore, let T = (T,G) be a regularity structure containing the polynomial regularity +structure, equipped with an abstract integration operator I of order s1 and containing an el- +ement Ξ ∈ Tα, where α = min A. Then, given a model M = (Π,Γ) which realises K for I and +satisfies the assumptions of Theorem 3.30 along with any v ∈ Dγ,η +0;P,, the map +MT ;v : Dγ,η +0 +→ Dγ,η +0 +U �→ KγR+(UΞ)+Pγ +(·)[K−1R+(UΞ)]+ v, +(4.6) +is well-defined and there exists a unique fixed point for some T > 0. +Furthermore, if ΠeΞ and v are G-periodic and the model is adapted to the action of G on the +group (as defined in Section 3.7), then the unique fixed point is as well. +Crucially, the solution map depends continuously on the model. +Remark 4.11. Note that using that the abstract mild equation given by (4.6) is linear in U, the +local time of existence T > 0 can be chosen independ of the initial condition. Thus, given +suitable bounds on the model over a suitably large set of the form {z ∈ G : d(z,P) ≤ T +1}, by +restarting the equation one obtains existence of a fixed point for arbitrary T > 0. We refer to +[HL18, Theorem 5.2] for details. +Remark 4.12. In practice we will usually take v = Pγ[G(u0)] so that by Lemma 4.9 the con- +dition v ∈ Dγ,η +0;P is satisfied provided u0 ∈ Cη(P), where η + α > −s1. The notation K and K−1 +is carried over from Section 3 and Section 4, in particular K is the lift of K , the compactly +supported, singular part of the fundamental solution, see Theorem 4.6 and K−1 the smooth +remainder, see Lemma 4.8. +Proof sketch of Theorem 4.10. The proof is a straightforward adaptation of the proof of [Hai14, +Theorem 7.8] and follows by applying Proposition 3.23, Theorem 4.6 and Lemma 4.8 to show +that the map M ¯T ;v has a unique fixed point in Dγ,η +0 +for some T > 0. Since α = min A, the +modelled distribution x �→ Ξ is an element of Dγ,γ +α;P for any γ > 0. Assuming U ∈ Dγ,η +0;P by +Proposition 3.23 one therefore has, +UΞ ∈ Dγ+α,η+α +α;P +. +So by using the singular Schauder estimate, Proposition 3.34, along with the assumption (η+ +α)∧α > −s1, one has +K(UΞ) ∈ Dγ+α+β,((η+α)∧α)+β +(α+β)∧0;P +. +Then, we see that +γ+α+β > γ, +η < η+α+β +and +(α+β)∧0 > 0 . +Hence, one finds Dγ+α+β,((η+α)∧α)+β +(α+β)∧0;P +�→ Dγ,η +0;P which concludes the proof that MT ;v is a self- +mapping on Dγ,η +α;P. By comparing MT ;vU and MT ;v ¯U for U, ¯U ∈ Dγ,η +0 +, one then shows that +MT ;v is a contraction for sufficiently small T > 0. +From the steps outlined above one furthermore obtains the the claims of G-periodicity and +the continuous dependence of the solution on the model. +54 + +4.4 CONCRETE EXAMPLES OF DIFFERENTIAL OPERATORS AND KERNELS +The theory developed in the proceeding sections provides the analytic framework to treat +singular SPDEs using the tools of regularity structures where the linear part of the equation is +given by an operator L satisfying the assumptions of Folland’s theorem, Theorem 1.1. In this +section we present a non-exhaustive list of homogeneous Lie groups and linear differential +operators, L, to which our results apply. Generically, these equations are of the form, +Lu = ∂tu − ¯Lu = F(u, ¯Xu,...,ξ), +u|t=0 = u0, +where ξ is a suitable noise, the non-linearity is linear in the noise and only depends on lower +order derivatives of u than are contained in L which satisfies the assumptions of Theorem 1.1, +c.f. [Fol75]. +Setting 1 P is a stratified Lie Group (see Definition 2.5) with a basis {Xi }m +i=1 of W1, generating the +Lie Algebra. We equip G := R×P with the trivial homogeneous Lie Group structure +G×G ∋ +� +(t,x),(t′,x′) +� +�→ (t + t′,xx′) +with the extended scaling λ·(t,x) = (λ2t,λ·x). The heat type operator associated to the +sub-Laplacian +L = ∂t − +m +� +i=1 +X 2 +i . +satisfies the assumptions of Theorem 1.1. and it follows from [Fol75, Thm. 3.1 & Prop. 3.3] +that L is non-anticipative. We shall discuss a notable example of the setting, that of the +heat operator on the Heisenberg group, below. +Setting 2 A generalisation of the above setting is to take P a homogeneous Lie Group and equip +G := R×P with the same structure as above, but this time define the scaling λ·(t,x) = +(λ2q t,λ· x) for q ≥ 2 and an integer. A natural class of operators is given by +LQ = ∂t −Q(X1,..., Xm) , +where Q is a polynomial of homogeneous degree 2q and such that LQ and L∗ +Q are hy- +poelliptic and LQ is non-anticipative, see [Hai14, Eq. 7.8]. A notable example is the heat +type operator associated to the bi-sub-Laplacian on a stratified Group. +Setting 3 The homogeneous Lie Group G = R×P is equipped with a non-trivial Lie group struc- +ture. In this case we look at differential operators of the form, +L = X0 −Q(X1,..., Xm) . +where X0 = ∂t + ¯X0 and the operators L, L∗ satisfy the criteria of Folland’s theorem +and L is non-anticipative. Examples are parabolic/restricted Hörmander operators, +c.f. [Hai16, Def. 1.1] or [Bel95, Ch. 3] and the kinetic Fokker–Planck operator fits into +this setting. +55 + +4.4.1 HEAT OPERATOR ON THE HEISENBERG GROUP +In the context of Setting 1 and Setting 2 we recall that the Heisenberg group, Hn, defined in +Section 2.4, is a stratified Lie group. Identifying Hn = R2n ×R we recall that +Ai(x, y,z) = ∂xi + yi∂z, +Bi(x, y,z) = ∂yi − xi∂z, +C(x, y,z) = ∂z +are left-translation invariant vector fields. It is readily checked that the operator +L = ∂t − +n� +i=1 +(A2 +i +B2 +i ), +is homogeneous with respect to the scaling, +λ·(t,x, y,z) := (λ2t,λx,λy,λ2z), +Applying [Fol75, Thm. 2.1] there exists a unique, fundamental solution K : Hn → R associated +to L which is smooth away from e and satisfies the assumptions of Lemma 4.4. As a result our +framework allows for the study of semi-linear evolution equations of the form, +∂tu −Lu = F(u, A1u,..., Anu,B1u,...,Bnu,ξ), +u|t=0 = u0. +4.4.2 MATRIX EXPONENTIAL GROUPS AND KOLMOGOROV TYPE OPERATORS +A wide class of operators fitting into Setting 3 above are the Kolmogorov (or K-type) operators +on R×Rn for n ≥ 1. The group structure is a matrix exponential group, as defined in Section +2.4. Given two, rational, n ×n block matrices, +A = + + +A0 +··· +0 +... +... +... +0 +··· +0 + +, +B = + + +0 +B1 +0 +··· +0 +0 +0 +B2 +··· +0 +... +... +... +... +... +0 +0 +··· +0 +Bk +0 +0 +··· +0 +0 + + +. +with A0 constant, positive definite and of rank q ≤ n and each Bi a pi−1 × pi block matrix of +rank pi, where q = p0 ≥ p1 ≥ ··· ≥ pk and �k +i=1 pi = n. Then the linear operator, defined for +u : R×Rn → R by +Lu(t,z) = ∂tu(t,z)−∇·(A∇u(t,z))+Bz ·∇u(t,z), +satisfies Hörmander’s condition. Equipping R×Rn with the matrix exponential group struc- +ture associated to B (and defined in Section 2.4) it is readily checked that L satisfies the as- +sumptions of Theorem 1.1. In fact, there is an explicit formula for the fundamental solution. +We first let +C(t) := 1 +t +�t +0 +exp(−sB⊤)A exp(−sB)ds +and then using the same notation for the effective spatial dimension, |¯s| := �k +i=0(2i +1)pi, +K (t,z) = +� +1 +(4π)nt|¯s| detC(t) +�1/2 +exp +� +−C(t)−1z · z +4t +� +. +56 + +The kinetic Fokker–Plank operator falls into this class. Consider the domain R × R2d, with +variables (t,x,v) and set, as block matrices, +A = +�Id +0 +0 +0 +� +, +B = +�0 +Id +0 +0 +� +. +Then the associated operator is +Lu(t,v,x)= ∂tu(t,v,x)−∆vu(t,v,x)− v ·∇xu(t,v,x). +and the fundamental solution in fact has the explicit form, +K (t,v,x) = +2 +� +3 +d(4π)d t2d+1 exp +� +−|v|2 +4t − +3 +��x + v +2 t +��2 +t3 +� +. +See [IK64, Sec. 7] for a derivation in the case d = 1 and [Man97] in the general case. +5 A REGULARITY STRUCTURE FOR ANDERSON EQUATIONS +In this section we present a brief construction of a sufficiently rich regularity structure T = +(T,G) in order to solve abstract fixed point equations of the form +U = I(UΞ)+U0 , +(5.1) +as treated by Theorem 4.10, where I denotes the abstract part of the lift of a 2 regularising +kernel as in Section 3.5, Ξ is the lift of a noise of regularity α and U0 is the polynomial part of +U. In order for the equation to be subcritical one needs to impose α > −2 and thus it follows +from the previous discussions that we can look for a fixed point in a space Dγ +P for γ < 2. Thus +we only need to incorporate abstract polynomials ηηηi with si < 2 into our regularity structure +and since one has the following identities in this case +ηi(xy) = ηi(x)+ηi (y), +Pγ +x[f ](·) = f (x)+ +� +i :si<γ +ηi(·)Xi f (x) +(5.2) +one can work with essentially a truncated version algebraic framework as on the abelian +group Rd developed in [BHZ19]. Therefore, in the remainder of this short section, we freely +use notations and definitions from [BHZ19], often without further explanation. +We define two edge types L = {t,Ξ} and declare |t| = 2 and |Ξ| = α and as well as a scaling +in the sense of [BHZ19] s = (s1,...,sn), where n = max{i ≤ d : si < 2}. 5 Motivated by (5.1) we +define the naive (normal) rule, c.f. [BHZ19, Def. 5.7] +˚R(t) = {(t,Ξ),(t),(Ξ),()}, +˚R(Ξ) = {()} +(5.3) +and consider its completion R constructed in [BHZ19, Prop. 5.21] (note that this second +step is only necessary if α ≤ −3/2). We denote by T ex and T ex ++ ,T− ⊂ T ex +− +the corresponding +spaces constructed in [BHZ19, Def. 5.26, Def. 5.29, Def. 6.22] and summarise some important +properties of these spaces. +5Recall form Section 2 that s1, s2,..,sn are the first n eigenvalues of the scaling s in increasing order and that Xi +are the corresponding eigenvectors +57 + +Proposition 5.1. The triple {T ex,T ex ++ ,T−} have the following algebraic structure. +• The spaces T ex ++ ,T−,T ex +− +are graded Hopf algebras with coproduct given by △+ and △− +respectively. +• The space T ex is a right comodule over T ex ++ . +• The spaces T ex and T ex ++ +are left comodules over T− . +• These spaces satisfy the co-interaction property, i.e. the following diagram commutes +T ex +T ex ⊗T ex ++ +T ex +− ⊗T ex +T ex +− ⊗T ex ⊗T ex ++ +△+ +△− +M(1,3)(2)(4)◦(△−⊗△−) +id⊗△+ +Note that the fact that we can leverage the framework developed in [BHZ19] relies crucially +on the fact that it suffices to include polynomials of degree < 2, which translates into the +algebraic relation △+(ηηηj) =ηηηj ⊗1+1⊗ηηηj whenever sj < 2. In general one would need a mod- +ification ˜△+ of the map △+ such that ˜△+(ηηηj) = � +d(I)+d(J)=sj C I,J +j ηηηI ⊗ηηηJ. Furthermore, one +would need a modification ˜△− of triangle △− and both modifications, ˜△+ and ˜△−, would +depend on the group structure of G, since the form of higher order Taylor polynomials de- +pends crucially on it. To circumvent these considerations we restrict ourselves to working +with the following subspaces. +Let T consists of the span of those basis vectors (trees) T ∈ T ex where T and its grading |T |+, +c.f. [BHZ19, Def. 5.3], satisfies the following properties +• it is planted and satisfies |T |+ < γ or T = T ′Ξ where T ′ is planted and satisfies |T ′|+ < γ, +• the only polynomial decorations appearing are at the second highest node and of de- +gree < γ. +Similarly, we set T+ ⊂ T ex ++ +to consist of the subalgebra generated by those trees T ∈ T ex ++ satis- +fying +• |T |+ < γ +• the only polynomial decorations appearing are at the second highest node and of de- +gree < γ. +Observe that Proposition 5.1 still holds with T ex ++ +replaced by T+ and T ex replaced by T. We +define G+ to be the character group of T+. From now on our regularity structure shall be given +by +T = (T,G) , +(5.4) +where G consists of the elements Γ ∈ L(T,T ) of the form Γ = (id⊗ f )△+ for some f ∈ G+. +58 + +5.1 SMOOTH MODELS +In order to define the models, we have to deviate slightly from [BHZ19]. For i ∈ {1,...,n} we +write ηηηi instead of Xi for abstract polynomials and we declare a map ΠΠΠ : T → C ∞(G) (c.f. +[BHZ19, Def. 6.8]) to be admissible for a smooth function ξ ∈ C ∞(G) and a kernel K as in +Assumption 3.25, if for every x ∈ G +ΠΠΠΞ(x) = ξ(x), +Πηηηi(x) = ηi(x) +and +ΠΠΠIτ = KΠΠΠτ, +ΠΠΠIiτ = (XiK )ΠΠΠτ , +where we write I resp. Ii for the operation of attaching an edge of type t, resp. (t,ei) to the +root. For ΠΠΠ an admissible map as above and z ∈ G we recursively define fz ∈ G+ and Πz by +fz(Iτ) = − +� +d(I)<|It(τ)|+ +ηI (z−1)X IK (Πzτ)(z) +ΠzIτ(¯z) = K (Πzτ)(¯z) − +� +d(I)<|I(τ)|+ +ηI(z−1 ¯z)X IK (Πzτ)(z) , +and by the analogous expression for Iiτ, where i ∈ {1,...,n}, c.f. [BHZ19, Lem. 6.9]. We say +that ΠΠΠ is canonical, if it satisfies ΠΠΠΞτ = ΠΠΠΞΠΠΠτ for every τ such that Ξτ ∈ T , and it does not +see the extended decoration, i.e is reduced in the sense of [BHZ19, Def. 6.2.1]. One can check +that for such canonical lifts ΠΠΠ the maps Πz = (ΠΠΠ⊗ fz)△+ and Γγx,y where γx,y = (f −1 +x +⊗ fx)△+ +form a model +M = (Π,Γγ) . +(5.5) +The renormalisation group G− is defined to be the character group of the Hopf algebra T ex +− . +We observe that in our case T ex +− +coincides (as an algebra) with the free algebra spanned by +a family of linear trees. (In the case α > −3/2, one furthermore has T ex +− = T− and we list the +trees explicitly in the next section). The group G− acts on models as follows. For g ∈ G− let +Rg = (g ⊗id)△−, we set +ˆM g = ( ˆΠg, ˆΓg) = (ΠRg,ΓγRg ) . +(5.6) +One can check that every element the orbit of a smooth canonical model as in (5.5) is indeed +a model. +Remark 5.2. Note that in order to show convergence of a family of models ˆM(ε) = ( ˆΠ(ε), ˆΓ(ε)) +of the form (5.6) for the regularity structure T = (T,G) for some smooth underlying noises +ξǫ and a sequence of elements gǫ ∈ G−, it is sufficient to show convergence of ˆΠε|T≤0 due to +Propositions 3.36 and 3.37. +We emphasise again that this section relied on the the assumption γ < 2. +6 THE ANDERSON EQUATION ON STRATIFIED LIE GROUPS +In this section we apply the machinery developed in this article in order to solve a class of +parabolic Anderson type models on a compact quotients of an arbitrary d-dimensional, strat- +ified, Lie groups. Let G be a stratified Lie group (recall Definition 2.5) with Lie algebra g and +59 + +G a lattice subgroup with its canonical left action on G. Recall from Section 2.3 the definition +of the quotient map π : G → S = G/G, we shall, without loss of generality assume that there +exists a fundamental domain of this action, K ⊂ G, such that e ∈ BN(e) ⊂ K for some N ∈ N, +large enough.6 For m ≤ d, let X1,..., Xm be a basis of W1, the family of left invariant vector +fields generating g and we write ˜Xi = π∗Xi for the push-forward vector fields on S. Finally we +recall the notion of convolution ∗S on S induced by the right action of G on S, as defined in +Section (2.3). +We shall solve equations where the driving noise satisfies the following assumption for +some ¯α > 0. +Assumption 6.1. For ¯α ∈ (−|s|/2,0), assume that ξα is of the form ξ = ˜ξ∗S c ¯α, where ˜ξ denotes a +Gaussian white noise7 on S and c ¯α : G\{e} → R is a function with bounded support, satisfying +|c ¯α(x)| ≲ |x|−|s|+(|s|/2+¯α) as well as c ¯α(x−1) = c ¯α(x) for all x ∈ G. +Remark 6.2. For ξ satisfying Assumption 6.1, a simple calculation shows that for any φ,ψ ∈ +L2(S), +E[〈ξ,φ〉〈ξ,ψ〉] = +� +S +(φ∗S c ¯α)(x)(ψ∗S c ¯α)(x)dx . +In particular, coloured noise where the regularisation is given by a negative fractional power +of the sub-Laplacian fits into our setting, c.f. [BOTW22]. +Remark 6.3. The assumption that c ¯α(z) = c ¯α(z−1) is not crucial, but allow us to reuse com- +putations previously carried out for equations on Euclidean domains in [HP15]. A robust +treatment of similar equations in the Euclidean setting, driven by more general driving noise, +even beyond the Gaussian setting, is given in [CS17]. A similar remark holds for the choice of +mollifier ρ in Theorem 6.4 below. +In the above setting, we have the following theorem. +Theorem 6.4. Fix T > 0 arbitrary and let ξ be a noise satisfying Assumption 6.1 with ¯α ∈ +(−3/2,−1) and η ∈ (−(2 + ¯α),0). For ε ∈ (0,1), let uε : [0,T ] × S → R be the unique solution +to the random PDE +∂tuε = +m +� +i=1 +˜X 2 +i uε +uε(ξε −cε), +u|t=0 = u0 ∈ Cη(S) , +(6.1) +where ξε := ξα ∗S ρε, for ρ ∈ C∞ +c (B1(e)) satisfying ρ(z) = ρ(z−1) and {cε(ρ)}ε∈(0,1) is a family of +(diverging) constants, depending on ρ and such that |cε| ≲ ε2+2 ¯α. Then, there exists a random +function u : [0,T ]×S → R, independent of the chosen mollifier, ρ ∈ C∞ +c (B1(e)), such that +sup +(t,x)∈[0,T ]×S +t−η|uε(t,x)−u(t,x)| → 0 +in probability as ε → 0. +6This is for example achieved by replacing the homogeneous norm on G with a multiple of itself. We only require +this condition in order to make the integrands in Section 6.1 supported on the fundamental domain K . +7Recalling that S is a measure space, ˜ξ can be defined as a centred Gaussian field, over L2(S) and with covariance +given by the L2(S) inner product. +60 + +Remark 6.5. We point out that the proof of Theorem 6.4 given below modifies mutatis mu- +tandis to the case where the noise is allowed to depend on time. In this case the natural mod- +ification of Assumption 6.1 is to assume that the noise is of the form ξ = ˜ξ∗c ¯α, where ˜ξ is a +space-time white noise on R×S and c ¯α satisfies |c ¯α(t,x)| ≤ (|x|+|t|1/2)−|s|/2−1+ ¯α. The only dif- +ference in the proof is to replace all convolutions and integrals over S with convolutions and +integrals over R×S. We point out that this form of noise is not covered by [BOTW22], where +the white in time assumption is required in order to make use of martingale arguments, see +also Remark 6.7 below. +Remark 6.6. The range of ¯α considered in Theorem 6.4 does not cover the full subcritical +regime of the Anderson equation on a stratified Lie group. Indeed one expects an analogue of +Theorem 6.4 to hold for any ¯α ∈ (−2,0). The main obstacle is the absence of a BPHZ type the- +orem in the setting of homogeneous Lie groups, see [CH16, HSb] for the corresponding result +on Rd. Let us further point out that in order for the Carnot group G to be non-trivial it must +have scaled dimension |s| ≥ 4, hence the sub-critical regime for ¯α is necessarily contained in +Assumption 6.1. +Remark 6.7. Let us point to the work [BOTW22], where, using Itô calculus methods, the au- +thors construct a solution-theory in the case, when the noise is white in time and coloured in +space, corresponding to the full subcritical regime on the Heisenberg group. This probabilis- +tic notion of solution circumvents the need for explicit renormalisation as in (6.1). Further- +more, their results are obtained on the infinite volume group Hn, rather than the compact +quotient space that we consider here. We expect that by using weighted modelled distribu- +tions, c.f. [HL18], together with the techniques developed in [HP15], one can recover the +solution constructed in [BOTW22] together with a Wong–Zakai type theorem in the analogue +of the regime ¯α ∈ (−3/2,−1), treated by Theorem 6.4. If one is interested in the full subcriti- +cal regime however, this would not just require a suitable BPHZ-type theorem in the setting +of homogeneous Lie groups, but furthermore a systematic way to single out the Itô model +among an (arbitrarily) high dimensional family of models obtained. +Proof of Theorem 6.4. We follow a usual strategy in the theory of regularity structures, show- +ing the equivalent theorem for the pulled back equations on G. For this, we work with the +regularity structure T = (T,G) constructed in (5.4) with |Ξ| = (−3/2, ¯α). Note that since we are +in Setting 1 we can directly apply Theorem 1.1 to see that there exists a unique, fundamental +solution G : (R×G)\(0,e) → R and satisfying the assumptions of Lemma 4.4, so that we have +G = K +K−1 enjoying the properties described therein. +Denote by M(ε) = (Π(ε),Γε) the canonical model for T , satisfying Π(ε) +z Ξ = ξε and realising +K for I. From now on we shall use the common tree notation, with the obvious changes in +interpretation, as defined for example in [HP15, Sec. 4.1], and write +:= Ξ, += IΞ, += ΞIΞ, +etc. +Since ¯α ∈ (−3/2,−1) the rule defined in (5.3) is complete and T− coincides (as an algebra) +with the free commutative algebra generated by +� +, +, +� +. Since the noise is Gaussian and +centred, it is sufficient to work with the subgroup ˜G− ⊂ G− consisting of g ∈ G− satisfying +g(τ) = 0 +⇒ +τ ∈ span +� +, +� +. +61 + +For each ε ∈ (0,1) we fix gε ∈ ˜G− to be the character specified by +gε( +) = E[ξε(e)K (ξε)(e)] . +Thus, applying Theorem 4.10, we obtain a solution Uε : [0,T ]×G → T<γ to the abstract lifted +equation, (4.6), for the model ˆM(ε) := M gε +ε . Next we show that uR +ε := R ˆM(ε)Uε actually solves +Equation (6.1) in several steps. +• First, note that for any z = (t,x) ∈ [0,T ]×G, the abstract solution has the explicit form +Uε(z) = u0 +ε(z) +� +1+ ++ +� ++ +� +si<γ +ui +ε(z)ηηηi , +since we are working with the expansion up to order γ < 3 +2. In particular ˆΠ(ε) +z Uε(z)(z) = +u0 +ε(z). +• We observe that therefore +(Uε )(z) = u0 +ε(z) +� ++ ++ +� ++ +� +si<γ +ui +ε(z)ηηηi +. +• We also note that ˆΠ(ε) +z +�� +si<γui +ε(z)ηηηi +� +(z) = 0 and +ˆΠ(ε) +z +� +u0 +ε(z) +� ++ ++ +�� +(z) = u0 +ε(z) +� +ˆΠ(ε) +z +(z)+ ˆΠ(ε) +z +� ++ +� +(z) +� += u0 +ε(z)(ξε(z)−cε) . +• Thus, we can conclude using Remark 3.19 that +R ˆM(ε)(U g ) = u0 +ε(z)(ξε(z)−cε) = +�R ˆM(ε)U g � +(z)(ξε(z)−cε) . +Thus it only remains to show is that the family of models ˆM(ε) converges to a limiting model +M, which is the content of Proposition 6.8, stated below. +Proposition 6.8. The familly of models ˆM(ε) = ( ˆΠ(ε), ˆΓε) constructed in the proof of Theorem 6.4 +converges as ε → 0. +Proof. Note that by Remark 5.2 it is sufficient to show convergence of ˆΠ(ε)|T<0 in the model +topology, which follows from a stright-forward adaptation of Kolmogorov’s criterion to mod- +els using the molifiers ϕ(n) constructed in Lemma 3.10 together with the estimates obtained +in Proposition 6.10. +Remark 6.9. For a detailed proof of a Kolmogorov type criterion for general models on Eu- +clidean domains, i.e. G = Rd, in the spirit above, we refer to [HSb, Appendix B]. +62 + +6.1 STOCHASTIC ESTIMATES FOR THE RENORMALISED MODEL +In this subsection we obtain convergence the renormalised models. We shall freely use tech- +niques of [Hai14, Sec 10] and as well as graphical notiation similar to that in [HP15, Sec. 5], +with some slight changes in interpretation. For example we write +� +Π(ε) +(0,e) +� +(ϕλ) = ++ +, +with the following interpretations; +• The node +represents the origin, (0,e) ∈ R×G while the edge +represents integra- +tion against the rescaled test function ϕλ. +• The node +represents an instance of the G left periodic white noise on G while the +edge +represents the kernel c ¯α ∗ρε. Note therefore, that when ε = 0 our diagrams +will still contain dotted lines, as opposed to [HP15]. +• The nodes +represent dummy variables z = (t,x) ∈ R × G which are to be integrated +out and the edges +represent integration against the kernel K . A barred arrow +represents a factor of K (t − s, y−1x) − K (−s, y−1), where (s, y) and (t,x) are the +coordinates of the start and end points of the arrow respectively. +As in [Hai14, Sec. 10.2] and [HP15, Sec. 5.1.1] we will also use the notation W(ε;k)τ to denote +the kernel associated to the kth-homogeneous Wiener chaos component of Π(0,e)τ and sim- +ilarly ˆ +W(ε;k)τ for the renormalised model. For example, we find that ˆW(ε;2) +((s, y);x1,x2) = +W(ε;2) +((s, y);x1,x2) is given by +W(ε;2) +((s, y);x1,x2) := +� +R×G +(c ¯α∗ρε)(y−1 +1 x1)(c ¯α∗ρε)(y−1x2) +� +K (s − s1, y−1 +1 y)−K (−s1, y−1 +1 ) +� +ds1dy1. +Proposition 6.10. Fix δ > 0 small enough. Then, for every λ ∈ (0,1), ϕ ∈ Br and z = (t,x) ∈ +R×G, there exist random variables +( ˆΠz )(ϕλ), +( ˆΠz +)(ϕλ), +( ˆΠz +)(ϕλ) +(6.2) +such that for any p ≥ 1 the following estimates hold uniformly over λ,ε ∈ (0,1) +1a) E +� +|( ˆΠ(ε) +z +)(ϕλ)|p� +≲ λp( ¯α−δ), +2a) E +� +|( ˆΠ(ε) +z +)(ϕλ)|p� +≲ λp(2+2 ¯α−δ), +3a) E +� +|( ˆΠ(ε) +z +)(ϕλ)|p� +≲ λp(4+3 ¯α−δ), +1b) E +� +|( ˆΠ(ε) +z +)(ϕλ)−( ˆΠz )(ϕλ)|p� +≲ εpδλp( ¯α−δ) , +2b) E +� +|( ˆΠ(ε) +z +)(ϕλ)−( ˆΠz +)(ϕλ)|p� +≲ εpδλp(2+2 ¯α−δ) , +3b) E +� +|( ˆΠ(ε) +z +)(ϕλ)−( ˆΠz +)(ϕλ)|p� +≲ εpδλp(4+3 ¯α−δ) . +Proof. Firstly, we note that by translation invariance of the noise it is enough to consider +z = (0,e). Furthermore, since all random variables belong to a finite Wiener chaos, it suffices +63 + +to show these estimates for p = 2, [Hai14, Lem. 10.5]. The first two estimates, 1a) and 1b), +follow readily by Ito’s isometry. +For the next items we recall [Hai14, Lem. 10.14 & 10.18], the natural analogues of which +hold in our setting as well. We can write (with the adaptation of the meaning of the diagrams +described above) the Wiener chaos decomposition of the second symbol as +� ˆΠ(ε) +(0,e) +� +(ϕλ) = +− +. +(6.3) +The first summand is the graphical representation of the iterated integral, I2(( ˆ +W(ε;2) +)(z)) +integrated against a test function centred at (0,e) ∈ R×G; see for example the analogous sec- +ond term in [Hai14, Eq. (10.23)]. The second summand is the graphical analogue of the first +term in [Hai14, Eq. (10.23)] given by I0(( ˆ +W(ε;0) +)(z)). Next note that by virtually an identical +calculation as in the beginning of the proof of [Hai14, Thm. 10.19] one finds, +|〈(( ˆ +W(ε;1) )(z),( ˆ +W(ε;1) )(¯z)〉| ≲ |z|2 ¯α+4 +|¯z|2 ¯α+4 , +(6.4) +which implies +|〈( ˆW(ε;2) +)(z),( ˆ +W(ε;2) +)(¯z)〉| ≲ (|z|2 ¯α+4 +|¯z|2 ¯α+4)|¯z−1z|2 ¯α . +Together with the simple estimate I0(( ˆ +W(ε;0) +)(z))| ≲ |z|2 ¯α+2 this gives 2a). +Next we turn to show 3a). First, note that by a similar calculation as for (6.4) one finds +|〈( ˆW(ε;2) +)(w),( ˆW(ε;2) +)( ¯w)〉| ≲ |w|2(4+2 ¯α) +| ¯w|2(4+2 ¯α) . +Therefore, together with the bound I0(( ˆ +W(ε;0) +)(w))| ≲ |w|2 ¯α+4 we find +E +��� +� +( ˆΠ(ε) +(0,e) +)⋄( ˆΠ(ε) +(0,e) ) +� +(ϕλ) +��� +2 ≲ λ2(4+3 ¯α) . +As in [HP15, Sec. 5.3.2] we find +� +ˆΠ(ε) +(0,e) +� +(ϕλ) = ++ + + +− + + ++ +− +and introduce the notationQε(s, y) = K (s, y)(cα∗ρε)∗2(y). We see that cε = +� +R×GQε(z)dz, and +define the renormalised kernel RQε(s, y) := Qε(s, y)−cεδ(0,e). Then, as in [HP15, Eq. (5.14)], +using the notation +for the renormalised kernel, one finds +� +ˆΠ(ε) +(0,e) +� +(ϕλ) = ++ +− ++ +− +. +64 + +Similarly, +� +( ˆΠ(ε) +(0,e) +)⋄( ˆΠ(ε) +(0,e) ) +� +(ϕλ) = +− +and thus it only remains the bound the covariance of the diagrams +, +, +, +(6.5) +The latter two can be bounded by repeatedly using the analogue of [Hai14, Lem. 10.14], while +for the first diagram we use additionally the natural analogue of [Hai14, Lem. 10.16]. This +concludes the proofs of 3a) and therefore the first column. The bounds on the random vari- +ables in (6.2) are obtained analogously by replacing c ¯α ∗ρε by just c ¯α throughout (and appro- +priately interpreting the renormalised kernel, Qε, for ε = 0). 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Springer, Berlin, 1986. +69 + diff --git a/INE4T4oBgHgl3EQfhA00/content/tmp_files/load_file.txt b/INE4T4oBgHgl3EQfhA00/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e282d88f04ad43c7a0b6498adb64a9b6bd219a3c --- /dev/null +++ b/INE4T4oBgHgl3EQfhA00/content/tmp_files/load_file.txt @@ -0,0 +1,3337 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf,len=3336 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='05121v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='PR] 12 Jan 2023 Singular SPDEs on Homogeneous Lie Groups Avi Mayorcas1 and Harprit Singh2 1Institute of Mathematics, Technische Universität Berlin, Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' des 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Juni 136, 10587 Berlin, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Email: avimayorcas@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' ORCID iD: 0000-0003-4133-9740 2Department of Mathematics, Imperial College London, South Kensington Campus, SW7 2AZ, UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Email: h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='singh19@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' ORCID iD: 0000-0002-9991-8393 January 13, 2023 Abstract The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular SPDEs of the form ∂tu = Lu +F(u,ξ) , where the differential operator L fails to be elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' This is achieved by interpreting the base space Rd as a non-trivial homogeneous Lie group G such that the differential oper- ator ∂t −L becomes a translation invariant hypoelliptic operator on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Prime examples are the kinetic Fokker-Planck operator ∂t − ∆v − v · ∇x and heat-type operators associ- ated to sub-Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' As an application of the developed framework, we solve a class of parabolic Anderson type equations ∂tu = � i X 2 i u +u(ξ−c) on the compact quotient of an arbitrary Carnot group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Keywords: Regularity structures, homogeneous Lie groups, hypoelliptic operators, stochastic partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 2020 MSC: 60L30 (Primary);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 60H17, 35H10, 35K70 (Secondary) CONTENTS 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 Related Literature .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': 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Behaviour of Kernel Convolutions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 59 6 The Anderson Equation on Stratified Lie Groups 59 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 Stochastic Estimates for the Renormalised Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 63 1 INTRODUCTION The theory of regularity structures, [Hai14], provides a framework for the study of subcritical stochastic partial differential equations (SPDEs) of the form ∂tu −Lu = F(u,ξ) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1) when the operator L is uniformly elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' This article extends the theory of regularity struc- tures in order to solve equations where the differential operator stems from a large class of hy- poelliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' This is achieved by building on the fundamental idea of Folland [Fol75] to reinterpret the differential operator in question as a differential operator on a homoge- neous Lie group and extending the theory of regularity structures to these non-commutative spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The treatment of singular SPDE via the theory of regularity structures can be divided into two parts, an analytic step and an algebraic and stochastic step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The first, analytic step, is to introduce the notions of a regularity structure, models and modelled distributions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' generalised Taylor expansions of both functions and distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Given this set-up, sufficient analytic tools are then developed to allow for the study of abstract, fixed-point equations in the spaces of modelled distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Cru- cially, a reconstruction operator maps modelled distributions to genuine distributions (or functions) on the underlying domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For the case of constant coefficient, hypoel- liptic equations on compact quotients of Euclidean domains, this aspect of the theory was already worked out in full generality in [Hai14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The second step, which was carried out in a case by case basis in [Hai14], was then fully automated in the subsequent works [CH16, BHZ19, BCCH21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In [BHZ19] the au- thors construct concrete (equation-dependent) regularity structures and models to- gether with a large enough renormalisation group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Then, in [BCCH21], the question of how this renormalisation group acts on the SPDE in question was answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Con- vergence of renormalised models for an extremely large class of noises, known as the BPHZ theorem, was proved in [CH16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In much of the above work translation invariance on Rd, of both the equation and the driving noise, plays a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In this paper we will instead work with equations that are translation invariant with respect to a homogeneous Lie group G (Euclidean spaces being special case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The bulk of this this paper is dedicated to implementing the first analytic part of the theory in the case when the underlying space is a general (non-Abelian) homogeneous Lie group, the second step is then carried out for a specific class of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' While many of the key ideas from [Hai14] carry over to our setting, we encounter a number of significant devia- tions from the arguments presented therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The main cause of these deviations are the non- commutative structure of the base space and the fact that the notion of Taylor expansion, a crucial element of the theory, heavily depends on the underlying group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' While this article aims at being relatively self-contained, we focus mainly on these required deviations from [Hai14] and do not reproduce arguments which carry over directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The central motivation for this extension to homogeneous Lie groups is to allow for the study of singular SPDE with linear part given by a hypoelliptic operator, which may fail to be uniformly parabolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Two motivating examples are the heat operator associated to the sub-Laplacian on the Heisenberg group and the kinetic Fokker-Planck operator on R×R2n, u(t,x, y,z) �→ ∂tu −(∆x +∆y +(x2 + y2)∆z)u and u(t,x,v) �→ ∂tu −∆vu − v ·∇xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2) We will carry these two operators and their associated homogeneous Lie groups, throughout the paper as working examples of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In the final sections we develop a full solution theory for Anderson type equations associated to sub-Laplacians on general stratified Lie groups (Carnot groups), of which the Heisenberg group is a well-studied example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' More generally, however, the analytic results of this paper apply to any differential operator, ¯L on Rd−1, such that the following combination of results by Folland applies to L := ∂t − ¯L, with respect to some homogeneous Lie group structure on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 ([Fol75, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 & Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Let G be a homogeneous Lie group of homoge- neous dimension |s| and L be a left-translation invariant (with respect to G), homogeneous 3 differential operator of degree β ∈ (0,|s|) on G, such that L and its adjoint L∗ are both hypoel- liptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Then there exists a unique homogeneous distribution K , of order β−|s| such that for any distribution, ϕ ∈ D′(G) L(ϕ∗K ) = (Lϕ)∗K = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' where the convolution is with respect to the given Lie structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Referring to Section 2 for a more thorough discussion of homogeneous Lie groups and their properties, we recall here that a differential operator D on Rd is called hypoelliptic, if for every open subset Ω ⊂ Rd one has that, Du ∈C ∞(Ω) ⇒ u ∈C ∞(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The following celebrated result of Hörmander (almost) entirely classifies the family of second order hypoelliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2 ([Hö67, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Let r ≤ d and {Xi }r i=0 be a collection of first order differential operators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' vector fields) on Rd and recursively define V1 = span{Xi : i = 0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',r}, Vn+1 = Vn ∪{[V,W ] : V ∈ V1, W ∈ Vn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' If a differential operator can be written in the form, D = r� i=1 X 2 i + X0 +c, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='3) for some c ∈ R, and there exists an N ≥ 1 such that dim(VN) = d at every point in Ω ⊆ Rd then D is hypoelliptic on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' By Froebenious’s theorem, [Fro77], if the condition of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2 fails in some open set then D is not hypoelliptic on that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' However, the statement is not truly sharp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' for example the Grushin operator ∂2 vv − v∂x is hypoelliptic, while the conditions of Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2 are violated on sets intersecting {v = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' On the other hand this type of exception cannot occur if the sections Vi are continuous, since in this case it can be shown that the map x �→ 1{dimVi(x)=d} is upper semi-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' While Hörmander’s theorem gives an almost sharp characterisation of second order, hy- poelliptic, operators it does not say much about fine properties of the fundamental solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For example, it gives no direct route to a refined regularity theory for hypoelliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' This observation highlights the contribution of Folland’s theorem (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' since trans- lation invariance allows access to many additional tools in the Euclidean setting, such as harmonic analysis and singular integral methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Viewing the class of translation invariant operators satisfying Folland’s theorem as analogous to constant coefficient, elliptic operators on Rd a programme was successfully carried out in the works [FS74a, FS74b, Fol75, RS76], es- tablishing a full Lp regularity theory for general, smooth coefficient, hypoelliptic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We refer to [Bra14, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 3] for a concise introduction to this programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 RELATED LITERATURE Non-translation invariant and non-uniformly elliptic SPDEs have been well-studied in the classical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' non-singular, regime and we do not attempt to present this large literature here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 4 We refer to standard texts such as [DPZ14, LR17, DKM+09, PR07] for a general overview and references to more specific works contained therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' However, in the more specialised set- ting of semi-linear SPDEs on homogeneous Lie groups, we mention the works [TV99, TV02, PT10] which treat both hypoelliptic and parabolic SPDEs on some classes of Lie groups and sub-Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A solution theory for conservative SPDEs based on the kinetic Fokker–Planck equation (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='4) in the Itô case was developed in [FFPV17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Using the theory of paracontrolled calculus this was extended to the singular regime in [HZZZ21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The kinetic Fokker–Plank operator and its associated homogeneous Lie group fall into the analytic framework developed in this paper, however, we postpone a concrete application of this theory towards a kinetic Anderson type equation to a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Recently, in [BOTW22], an Anderson type equation on the Heisenberg group with white in time and coloured in space noise was studied using Itô calculus techniques, up the to the full sub-critical regime of the noise regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We treat a closely related problem in the final sections of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A fuller discussion of the similarities and differences between the results of [BOTW22] and those of our approach is postponed to Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Singular SPDEs with non-translation invariant, but uniformly parabolic or elliptic linear part, have also been considered, especially using the recently developed pathwise techniques of [Hai14, GIP15, OSSW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Some of these works are discussed in more detail in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Quasilinear SPDEs have been studied using regularity structures, rough path based methods and paracontrolled distribution theory, see [GH19b, OW19, BM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Recently, an ap- proach inspired by the theory of regularity structures, but technically distinct, has been devel- oped in [OSSW21, LOT21, LO22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' SPDEs on domains with boundaries have been treated us- ing both regularity structures and paracontrolled distribution based methods, [Lab19, GH19a, CvZ21, GH21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Finally, a number of works have considered parabolic, singular SPDEs on Rie- mannian manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' a paracontrolled approach using the spectral decomposition associated to the Laplace–Beltrami operator has been developed in [BB16, BB19, Ant22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' An approach via regularity structures has been applied to the 2d parabolic Anderson equation on a Rie- mannian manifold in [DDD19], while [BB21] develops some aspects of the general algebraic structure required to treat non-translation invariant, uniformly parabolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Finally, the upcoming work [HSa] gives a comprehensive extension of regularity structures to sin- gular SPDEs on Riemannian manifolds, with only the renormalisation of suitable stochastic objects left to be done by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2 A MOTIVATING CLASS OF EXAMPLES In this paper we restrict ourselves to linear operators satisfying the criteria of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 and use the Anderson equation as a main motivating example, although we stress that our main analytic results apply in the full generality treated by [Hai14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The parabolic Anderson model ∂tu = ∆u −uξ, u|t=0 = u0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='4) on R+×Rd describes the conditional, expected density of particles, where each particle moves according to an independent Brownian motion and branches at a rate proportional to the random environment ξ, see [Kön16, CM94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Rigorously, this description is derived by discrete approximations and it is well known that in the case ξ is a spatial white noise, when d = 2 and 5 d = 3 one needs to recentre the potential in order to obtain a non-trivial limit as the discreti- sation is removed, [Hai14, HL15, HL18, GIP15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We point out that if the environment is also allowed to depend on time and is for example, white in time, then martingale methods can be used instead to develop a probabilistic solution theory, see [Wal86, Dal99, Dal01, Che15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' If one replaces the Brownian motions with a general diffusion, dxt = � 2 r� i=1 Xi(xt)dW i t , where the vector fields satisfy Hörmander’s rank condition (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2), then, formally the conditional expected density of particles is described by the hypoelliptic Anderson equation, ∂tu − ¯Lu := ∂tu − r� i=1 X 2 i u = uξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5) When realisations of the environment are sufficiently singular then a pathwise solution the- ory for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5) is expected to require renormalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Since the vector fields {Xi }r i=1 satisfy Hör- mander’s condition, the operator ¯L is smoothing and therefore in principle, an extension of the theory of regularity structures should be applicable to find a renormalised, pathwise, so- lution theory for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In Section 6 we apply the analytic tools developed in the paper to demonstrate such an extension to the case where ¯L is the sub-Laplacian on a compact quo- tient of stratified Lie group (Carnot group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We show a result analogous to those of [Hai14, HL18], finding a notion of renormalised so- lution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5), when ξ is a coloured periodic noise on a stratified Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' More precisely, we show that when ξ is replaced with a mollified, recentred noise ξε − cε, for (specific) di- verging constants {cε}ε∈(0,1), solutions uε to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5) converge in probability to a unique limit independent of the specific choice of mollification-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We stress that this result does not cover the full subcritical regime of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The treatment of this full regime would require an analogue of the BPHZ theorem on homogeneous Lie groups, see [CH16, HSb].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A notable example of a stratified Lie group is the Heisenberg group, Hn ∼= R2n ×R, see Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 for a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In this case the collection of left-translation invariant vector fields, which generate the associated Lie algebra are, Ai(x, y,z) = ∂xi + yi∂z, Bi(x, y,z) = ∂yi − xi∂z, for i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' It is readily checked that the collections {Ai ,Bi}n i=1 are left-translation invariant with respect to the group action described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 and that one has C(x, y,z) := [Ai,Bi] = −2∂z for all i = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The associated sub-Laplacian is the linear differential operator, ¯Lu = n� i=1 (A2 i +B2 i )u, naturally extended to a heat type operator as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A phenomenon of interest for Anderson equations is that of localisation, the con- centration of the solution at large times, taking arbitrarily large values on islands of arbitrarily small size, [CM94, Kön16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Since one expects the geometry of the underlying domain to have 6 effect on the emergence of this phenomenon, as also noted in [BOTW22] and since pathwise approaches to the parabolic Anderson model have proved fruitful in studying finer proper- ties of solutions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' [AC15, HL15, HL18, GUZ19, Lab19, BDM22], it is our hope that the tools developed herein may prove useful in analysing similar equations on more complex domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The solution theory exposited in Section 6 applies to precisely this example on a compact quotient of the Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='3 OPEN PROBLEMS AND WIDER CONTEXT As discussed above, translation invariant (and for example) parabolic operators on Euclidean domains serve as a starting point from which non-translation invariant parabolic problems on Euclidean domains as well as parabolic problems on Riemannian manifolds can be stud- ied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A somewhat parallel progression holds starting from translation invariant operators on homogeneous Lie groups, moving to general Hörmander operators as well as heat type equa- tions on sub-Riemannian manifolds, see [Bra14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' This parallel progression in PDE analysis leads one to ask, how far the theory of regularity structures can be extended in these direc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The table below gives a schematic presentation of the progress so far, presents some open problems and places this work within the context of the study of subcritical parabolic- type SPDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The two rows describe the two parallel progressions outlined above in the con- text of regularity structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Translation invariant operators on Rd Non-translation operators on Rd On Riemannian manifolds The works [Hai14, BHZ19, BCCH21, CH16] present a general and complete picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Mostly understood: analytic step in [Hai14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' aspects of renormalisation addressed in [BB21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In [DDD19] 2d-PAM is treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A general account in forthcoming work [HSa].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Translation invariant operators on homogeneous Lie groups General Hörmander operators On sub-Riemannian manifolds Analytic theory covered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Renormalisation and convergence by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Open problem A Open problem B Each problem in the second row is closely related to its equivalent problem in the first row, we discuss the posed open problems in a little more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Open Problem A is expected to be more involved than its counterpart in the Euclidean setting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' going from translation invariant operators to non-translation invariant oper- ators by local approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In the case of general Hörmander operators, the un- derlying Lie group structure would in general vary from point to point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' An interesting application would be the study of general hypoelliptic Anderson models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5) where the underling particles perform a general diffusion with generator of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='3), c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Open Problem B is motivated by the fact that analogous to the tangent space being an appropriate local approximations of a Riemannian manifold, the non-holonomic tan- gent space is an appropriate local approximations of a sub-Riemannian manifold and 7 each fibre of the non-holonomic tangent space is a stratified Lie group, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' [ABB20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Note that in view of Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='3 the difficulties in Problem B are expected to be some- what complementary to those of Problem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' NOTATION: Given α ∈ R we let [α] := max{r ∈ Z : r ≤ α} denote the integer part of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We often write ≲ to mean that an inequality holds up to multiplication by a constant which may change from line to line but is uniform over any stated quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In general we reserve the notation D for the usual derivative on Euclidean space, and X , Y for elements of a Lie algebra thought of as vector fields on the associated Lie group, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We will use | · | to denote the size of various quantities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' elements of a Lie group, the Haar measure of subsets of the group, the homogeneity of members of the structure space in a regularity structure, traces of linear maps and the absolute value function on R etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The meaning will usually be clear from the context but in cases where it is not we will make sure to specify in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For integrals we will use the standard notation, dx, for integration against the Haar measure on a given homogeneous, Lie group, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Throughout most of the article we take an intrinsic point of view and do not equip the homogeneous Lie group with an explicit chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' STRUCTURE OF THE PAPER: We begin with a discussion of analysis on homogeneous Lie groups sufficient for our purpose, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The most important result in this section is The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='13 which provides us with an intrinsic version of Taylor’s theorem on homogeneous Lie groups and will be used frequently throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Section 3 contains the bulk of the paper and establishes an extension to the analytic aspects of regularity structures to the setting of homogeneous Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In Section 4 we demonstrate the application of this general theory to semi-linear evolution equations of the type discussed in the introduction and provide an example fixed point theorem for such generalised multiplicative stochastic heat equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We note that there is no difficulty in extending the general fixed point theorem of [Hai14] to our setting, we only specialise to aid the clarity of presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Finally, in Sections 5 and 6 we the construct suitable regularity structures for our specialised setting and demonstrate a solution theory for Anderson-type equations associated to sub-Laplacians on quotients of stratified Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The very last sections draw heavily on the notation and arguments of [Hai14, HP15, BHZ19], which carry over to homogeneous Lie Groups to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors wish to thank Martin Hairer, Rhys Steele, Ilya Chevyrev and Ajay Chandra for helpful and insightful conversations during the preparation of this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' AM wishes to thank the INI and DPMMS for their support and hospitality which was sup- ported by Simons Foundation (award ID 316017) and by Engineering and Physical Sciences Research Council (EPSRC) grant number EP/R014604/1 as well as DFG Research Unit FOR2402 for ongoing support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' HS acknowledges funding by the Imperial College London President’s PhD Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' He wishes to thank Josef Teichmann and Máté Gerencsér for their hospitality during his visit to ETH Zürich and TU Wien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 8 2 ANALYSIS ON HOMOGENEOUS LIE GROUPS We collect some basic facts on homogeneous Lie groups and smooth function on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Let g be a Lie algebra and G be the unique, up to Lie algebra isomorphism, corresponding sim- ply connected Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We write [·,·] for its Lie bracket and inductively set, g(1) = g and g(n) = [g(n−1),g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Recalling the exponential map exp : g → G, we have the Baker–Campbell– Hausdorff formula exp(X )exp(Y ) = exp(H(X ,Y )) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1) where H is given by H(X ,Y ) = X + Y + 1 2[X ,Y ] + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' with the remaining terms consisting of higher order iterated commutators of X and Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' crucially H is universal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' it does not depend on the underlying Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 ([FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Assume that g is nilpotent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' g(n) = 0 for some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Then, the exponential map is a diffeomorphism and through this identification of g with G, the map G×G ∋ (x, y) �→ xy ∈ G becomes a polynomial map (between vector spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' the pull-back to G of the Lebesgue measure on g is a bi-invariant Haar measure on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A dilation on g is a group of algebra automorphisms {Dr}r>0 of the form Dr X = exp(logr ·s)X with s : g → g being a diagonalisable linear operator with 1 as its smallest eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The requirement that the smallest eigenvalue of s be 1 is purely cosmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Oth- erwise, denoting the smallest eigenvalue by s1, one can work with the new operator ˜s = 1 s1 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For a ∈ R, denote by Wa ⊂ g the eigenspace of s with eigenvalue a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Thus for X ∈ Wa, Y ∈ Wb one has the condition Dr [X ,Y ] = [Dr X ,Dr Y ] = r a+b[X ,Y ] which in particular implies [Wa,Wb] ⊂ Wa+b and since Wa = {0} for a < 1 g(j) ⊂ � a≥j Wa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In particular, if g admits a family of dilations, it is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The converse is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A homogeneous Lie group G is a simply connected, connected Lie group where its Lie algebra g is endowed with a family of dilations {Dr }r>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For r > 0, we define the group automorphism x �→ r · x := exp◦Dr ◦exp−1 x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A homogeneous norm on G is a continuous function |·| : G → [0,∞) satisfying the following properties for all x ∈ G, r ∈ R 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' |x| = 0 if and only if x = e, the neutral element, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' |x| = |x−1| , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' |r · x| = r|x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The homogeneous norm naturally induces a topology generated by the open sets and in turn a Borel σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' From now on, we will always assume that G is equipped with this topology and σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Furthermore, given a homogeneous group G we denote by {X j}d j=1 ∈ g a basis of eigenvectors of s with eigenvalues 1 = s1 ≤ s2 ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' ≤ sd and such that sX j = sj X j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2) Given a measurable subset E ⊂ G we write |E| for its Haar measure which we assume to be normalized such that the set B1 =: {x ∈ G : |x| ≤ 1} has measure 1, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' In integrals we use the standard notation dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We define |s| := trace(s) as the homogeneous di- mension of G, since for any measurable subset E ⊂ G and r > 0, one has |r ·E| = r |s||E| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We also define balls of radius r > 0, Br(x) := {y ∈ G : |x−1y| < r} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The topology induced by these balls agrees with the topology of G as a Lie group, see [FR16, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Note that due to the non-commutativity of G, in general |x−1y| ̸= |yx−1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We consistently work with the following choice a semi-metric on the group, dG(x, y) := |x−1y| = |y−1x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For K ⊂ G we write ¯K := {z ∈ G : dG(z,K) := infy∈K|y−1z| ≤ 1} for the 1-fattening of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A function f : G → R is called homogeneous of degree λ ∈ R, if f (r · x) = r λf (x) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' One can show, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' [FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5 & Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='6], that for any homogeneous norm on G there exists γ > 0 such that |xy| ≤ γ(|x|+|y|) for any x, y ∈ G ||xy|−|x|| ≤ γ|y| for any x, y ∈ G such that |y| ≤ 1 2|x| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Furthermore all homogeneous norms are mutually equivalent and we may always choose a homogeneous norm that is smooth away from e ∈ G, [FR16, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 DERIVATIVES AND POLYNOMIALS We identify g with the left invariant vector fields gL on G and write gR for the right invariant vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We write Xi for the the basis elements as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='2) seen as elements of gL and Yi for the basis of gR satisfying Yi|e = Xi|e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Thus we can write X j f (y) = ∂t f (y exp(t X j))|t=0 and Yj f (y) = ∂t f (exp(tYj)y)|t=0 10 for any smooth function f ∈C ∞(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' A map P : G → R is called a polynomial if P ◦ exp : g → R is a polynomial on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1 Let ζi be the basis dual to the basis Xi of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' We set η j = ζj ◦ exp−1, which maps G to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Note that η = (η1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',ηd) forms a global coordinate system 2 and furthermore any polynomial map on G can be written in terms of coefficients aI ∈ R as P = � I aIηI with the sum running over a finite subset of Nd and where for a multi-index I = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',id) ∈ Nd we write ηI = ηi1 1 ·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='·ηid d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Define d(I) = � j sji j and |I| = � j i j, we call max{d(I) : aI ̸= 0} the homogeneous degree and max{|I| : aI ̸= 0} the isotropic degree of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For a > 0, we denote by Pa the space of polynomials of homogeneous degree strictly less than a and define △ = {d(I) ∈ R : I ∈ Nd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 3 We can rewrite the group law on G explicitly in terms of η = (η1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',ηd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',d} and multi-indices I, J s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' d(I) + d(J) = sj, there exist con- stants C I,J j > 0 such that the following formula holds, η j(xy) = η j(x)+η j(y)+ � I,J̸=0, d(I)+d(J)=sj C I,J j ηI (x)ηJ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' By the Baker–Campbell–Hausdorff formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='1) one has η j(xy) = η j(x)+η j(y)+ � |I|+|J|≥2 C I,J j ηI (x)ηJ(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' By setting either x = e or y = e, we find that C I,J j = 0 if I = 0 or J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Since furthermore η j((r x)(r y)) = r sj η j(xy) the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5 implies that for sj < 2 one has η j(xy) = η j(x) + η j(y) while for sj = 2 one has η j(xy) = η j(x)+η j (y)+� sk=sl=1C k,l j ηk(x)ηl(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' It is also noteworthy to realise that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5 implies that Pa is invariant under right and left-translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' (This is not true if one replaces homogeneous degree by isotropic degree, except if G is abelian or a = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' If follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5 that one can write ηK (xy) = ηK (x)+ηK (y)+ � I,J̸=0, d(I)+d(J)=d(K ) C I,J K ηI(x)ηJ(y) , where the constants C I,J K can be written in terms of the constants C I,J j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1Recall that the space of polynomial functions on g is canonically isomorphic to � n(g∗)⊗sn where ⊗s denotes the symmetric tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 2Occasionally we use the corresponding notation ∂ ∂ηi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 3We point out a possibly counter-intuitive quirk of our definition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' for k ∈ △ the set Pk does not contain polyno- mials of degree k but only those of degree less than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' For i, j ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',d}, Xiη j = δi,j + � I̸=0, d(I)=sj −si C I,ei j ηI , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='3) where ei denotes the multi-index (0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',0,1,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',0) with the 1 being in the i-th slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' This follows directly from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='5, applying Xi to the function y �→ η j(xy) , evaluating at y = 0 and using the fact that Xi|0 = ∂ ∂ηi |0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='10 ([FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' One has X j = �P j,k � ∂ ∂ηk � where P j,k = � 1 if k = j 0 if sk ≤ sj,k ̸= j and P j,k is a homogeneous polynomial of degree sk −sj if sk > sj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The analogous statement holds for the vector fields Yj, For a multi-index I = (i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=',id) ∈ Nd we introduce the notation X I = X i1 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='X id d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Note that the order of the composition matters since g is not in general Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' It is a well known fact that any left invariant differential operator on G can (uniquely) be written as a linear combination of {X I}I∈Nd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The next proposition follows as a direct consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='11 ([FS82, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content='30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' The following maps from Pa → RdimPa are linear iso- morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/INE4T4oBgHgl3EQfhA00/content/2301.05121v1.pdf'} +page_content=' P �→ �� ∂ ∂η �I P(e) � d(I) d where d is the spa- +tial dimension, the dispersion is gapless and of Goldstone- +type. In particular, for α > d + 2 (denoted as standard +Goldstone regime), the dispersion of the LR model re- +duces to the short-ranged case with ω ∼ |q|2, and for +d < α < d + 2 (denoted as anomalous Goldstone regime) +the dispersion is ω ∼ |q|α−d. +For α ≤ d (denoted as +Higgs regime) the generalized Higgs mechanism [85] in- +duces a discrete spectrum and thus the system is gapped, +corresponding to a power of zero in the dispersion. As +will be shown below, our QMC results are roughly con- +sistent with this picture as we also find a Higgs regime +with α ≤ 2 (see Fig. 1 (d)). +As for the antiferromagnetic case, it is worth not- +ing that Ref. [85] predicts the Higgs regime occurs at +α ≤ d−2 for the system with H = J � +i̸=j +1 +|ri−rj|α Si ·Sj, +therefore for d = 2 there will be no finite α values with +gapped spectra, and the anomalous and standard Gold- +stone regimes are d − 2 < α < d and α > d respectively. +However, we consider a sign-problem-free Hamiltonian +of Eq. (2) which does not host frustrations and conse- +quently we get different boundaries of the three regimes. +Our QMC and SWT analysis find a Higgs regime with +α ≤ 2.2 (see Fig. 1 (e)). +In order to obtain the low-energy spectra of HF M +and HAF M, we compute the imaginary time correla- +tion function Gq(τ) ≡ ⟨Sz +q(τ)Sz +−q(0)⟩ − ⟨Sz +q⟩2, where +Sz +q ≡ +1 +√ +N +� +r eiq·rSz +r, via the SSE QMC method [78, 92– +94]. Here we consider the periodic boundary conditions +in the simulation so that (qx, qy) = (± 2πm +L , ± 2πn +L ) with +m and n being integers are physical momenta on a L×L +finite size lattice. The loop update scheme of SSE QMC +is purposely adapted to cope with the long range inter- +actions by assigning each bond with a separate bond +weight and bond type (ferromagnetic or antiferromag- + +3 +(a) +(d) +(b) +(c) +(d) +(e) +(f) +FIG. 2. +Dynamical properties of 2D LR ferromagnetic Heisenberg model. +Dispersion relations along the path +(Γ → X → M → Γ) with panels (a)-(f) for different decay exponents α. Results of various sizes L are plotted together in each +panel and share the same legend on the top. Inset of panel (b) indicates that at α = 2 the first excitation gaps near Γ for +various sizes converge to a finite value and the system has a gaped spectrum, i.e., inside the Higgs-regime. Insets of (c), (d) +and (e) show the fitting of power-law dispersions ωq ∼ |q|s(α) near Γ (with |∆q| denotes the relative momentum away from +Γ) in range 2 < α ≤ 4. Red dashed line in (f) is the SWT dispersion for 2D nearest neighbor FM Heisenberg model with +ωq = |J| zS(1 − γq) where S = 1/2, the coordination number z = 4, and γq = 1 +z +� +δ eiqδ. +netic) [106]. +To obtain the spectrum in QMC, notice +that +⟨Sz +q(τ)Sz +−q(0)⟩ = +� +eHτSz +q(0)e−HτSz +−q(0) +� += +�� +l=0 +e−βEl +�−1 +× +� +n,m=0 +|⟨n|Sz +q|m⟩|2e−(Em−En)τe−βEn +(3) +where H|n⟩ = En|n⟩ and E0 is the ground state energy +of the system. When β∆E1 ≫ 1 where ∆En = En − E0, +we can estimate +Gq(τ) ≈ +� +n=1 +|⟨0|Sz +q|n⟩|2 � +e−∆En(q)τ + e−∆En(q)(β−τ)� +. +(4) +When the imaginary time is sufficiently large, we assume +that the system will gradually evolve to the ground state, +so that the correlation function can be further approxi- +mated by +Gq(τ) ≈ |⟨0|Sz +q|1⟩|2e−∆E1(q)τ. +(5) +If |⟨0|Sz +q|1⟩|2 is finite (which is usually the case), we can +then extract the energy gap for each q point by fitting +Gq(τ) with an exponentially decaying function. Exam- +ples of such fitting and their finite size extrapolations to +the thermodynamic limit are shown in Sec.II in SM [102]. +Results.— Fig. 2 shows the obtained QMC spectra along +the high symmetry path Γ(0, 0) → X(π, 0) → M(π, π) → +Γ(0, 0) for the ferromagnetic case. At α = 100 (panel +(f)), the system reduces to the short-ranged case with +only nearest-neighbor couplings [10–12], and our QMC- +obtained spectra matches well with the spectra obtained +from SWT analysis. +Both of them show a ωq ∼ |q|2 +dispersion close to Γ, and surprisingly, the QMC and +SWT spectra match well along the whole path. As α gets +smaller, as shown in panels (c), (d) and (e), we find the +dispersion enters the anomalous Goldstone region [85], +i.e., the dispersion close to Γ deviates from a quadratic +one. We use ωq ∼ |q|s(α) to fit the dispersion close to +Γ and find the power s(α) gradually decreases as α gets +smaller. +Insets of these three panels demonstrate the +power law fitting of s(α) using L = 64 QMC results. We +find, at α = 3 (panel (d)), s = 1.076 which agrees well +with the relation of s(α) = α − 2 suggested in Ref. [85]. +However, for α = 4 (panel (e)) and α = 2.5 (panel c)) +our results show apparent derivations from s(α) = α − 2. +Fig. 1(d) collects the fitted power s(α) by QMC (red +dots) at various α and we observe a satisfactory match +with our SWT results. At α = 2, we find ωq near Γ, +i.e., q = ( 2π +L , 0) for different system sizes converge to a + +4 +(a) +(c) +(d) +(e) +(f) +(b) +FIG. 3. Dynamical properties of 2D LR (staggered) antiferromagnetic Heisenberg model. Dispersion relations +along the path (Γ → X → M → Γ) with panels (a)-(f) for different decay exponents α. Results of various sizes L are plotted +together in each panel and share the same legend on the top. Inset of panel (b) indicates that the first excitation gaps near +M = (π, π) for various sizes converge to a finite value and thus the system is inside the Higgs regime at α ≤ 2.2. Insets of (c), +(d) and (e) show the fitting of power-law dispersion as ω ∼ |q|s(α) near M (with |∆q| denotes the relative momentum away +from M) in range 2.2 < α < 4.5. Dashed red line in (f) is the nearest-neighbor SWT dispersion ωq = |J| zS +� +(1 − γq)2 with +an additional coefficient ∼1.158 to approximate the second order spin wave effects [79, 99, 104, 105]. +large and finite value of ω ≈ 3.41 as indicated in the +inset of Fig. 2 (b). This phenomenon is fundamentally +different from a gapless excitation in which the finite size +gap ω(2π/L,0) converges to zero as L → ∞ and results in +a continuous spectra. Our result reveals that at α = 2 +the system enters the Higgs regime where the Goldstone +mode acquires mass due to the LR interation and the +excitation spectrum becomes gapped. For α < 2 (α = 1.5 +in panel (a)) we find the gaps begin to diverge with the +system size L. Therefore, we conclude that α = 2 is the +separation power between the Higgs-type and Goldstone- +type spectra in HF M from our QMC results. +Fig. 3 illustrates the QMC dispersion relation for +HAF M along the high symmetry path. Similarly, in panel +(f), we benchmark the spectrum at α = 100 with SWT +result for the short-range antiferromagnetic Hamiltonian +(with an extra coefficient ∼1.158 multiplied to approxi- +mate the second order spin wave effects [79, 99, 104, 105]) +and find QMC results agree well with SWT dispersion +close to M and the dispersion relation is ωq ∼ |q|. As +α decreases, the system also enters the anomalous Gold- +stone region with ωq ∼ |q|s(α) and 0 < s(α) < 1 close to +M. Fitted powers via QMC at various α are displayed in +Fig. 1(e) and agree well with the SWT results. In Fig. 3 +(b) at α = 2.2, ωq close to M converges to a large and +finite value of ω = 4.57. This means HAF M is in the +Higgs-regime with gapped spectra when α ≤ 2.2 (panel +(a) with α = 2 shows the divergent gap close to M). +In contrast to the vanishing of the Higgs spectra in the +purely AFM case in Ref. [85] for d = 2, our staggered +AFM case (Eq. (2)) offers finite boundaries between the +Higgs-type and Goldstone-type spectra, and it is of in- +terest to perform similar theoretical analysis as done in +Ref. [85] for Eq. (2) to further reveal the subtle working +of the generalized Higgs mechanism, with different types +of LR interactions. +Discussions.— With the unbiased large-scale QMC sim- +ulations and SWT analysis, we systematically investi- +gate the dynamical properties of 2D spin-1/2 Heisenberg +model with LR interactions. We find in contrast to the +well accepted low-energy customs such as Hohenberg- +Mermin-Wagner theorem and gapless Goldstone mode, +the LR quantum many-body systems offer richer tun- +ability and exhibit new phenomena. As the interaction +exponent α varies, the Goldstone modes can be strongly +modified, in that they can be either distorted (in the +anomalous Goldstone regime), or even be gapped via a +generalized Higgs mechanism. + +5 +Most remarkably, these dynamical properties have im- +mediate relevance to the on-going experiments with ul- +tracold atom arrays and quantum moiré materials. For +example, the long-range Coulomb interaction in quan- +tum moiré systems can be easily tuned by varying dielec- +tric environment, electrostatic gating and twisting angles, +and in this way, different observed thermodynamical and +dynamical properties (such as switching between gapped +and gapless spectra) [30, 61, 62, 67, 68, 70] can be iden- +tified with different LR interaction types and regimes, +when compared unbiased results such as ours. Similar +tunability can also be realised in dressed Rydberg atom +arrays whose interaction can be modified [107], one can +then compare different responses from experiments with +our results to identify the LR interaction and the novel +phases. +Acknowledgment.- We thank Zheng Yan, Tianyu Wu, +Ting-Tung Wang, Meng Cheng, Fakher Assaad and Qi +Yang for valuable discussions on related topics. We ac- +knowledge the support from the Research Grants Coun- +cil of Hong Kong SAR of China (Grant Nos. 17303019, +17301420, 17301721, AoE/P-701/20, 17309822 and A- +HKU703/22), the K. C. Wong Education Foundation +(Grant No. GJTD-2020-01), +and the Seed Funding +“Quantum-Inspired explainable-AI” at the HKU-TCL +Joint Research Centre for Artificial Intelligence. +We +thank the HPC2021 system under the Information Tech- +nology Services and the Blackbody HPC system at the +Department of Physics, the University of Hong Kong for +their technical support and generous allocation of CPU +time. The authors also acknowledge Beijng PARATERA +Tech Co.,Ltd. +for providing HPC resources that have +contributed to the research results reported within this +paper. +∗ jrzhao@connect.hku.hk +† zhouchk@connect.hku.hk +‡ zymeng@hku.hk +[1] P. C. Hohenberg, Existence of long-range order in one +and two dimensions, Phys. Rev. 158, 383 (1967). +[2] N. D. Mermin and H. 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Phys. 12, 71 +(2016). + +9 +Supplementary Materials +In this supplementary material, we present the linear +spin wave analysis for the LR FM and staggered AFM +Heisenberg model, in which the dispersion relation of +the low-lying magnetic excitations at different decaying +power α are extracted. From here, we make comparison +with the dispersion obtained from the QMC simulations +in the main text. Moreover, we also provide detailed data +on the fitting of the excitation gaps from the dynamical +correlation functions in QMC, such as representative data +points where the extrapolation of the converged gaps at +thermodynamic limit are shown. +Linear spin wave analysis +We applied the linear spin wave theory (SWT) to an- +alyze the dispersion of the low energy excitation in the +LR spin-1/2 Heisenberg model with power law decaying +couplings in both ferromagnetic and staggered antiferro- +magnetic cases [12, 95–99]. Taking the staggered antifer- +romagnetic cases as an example, it calls for the definition +of two sublattices, A and B. The spin on each sublattice +is pointing in the same direction. Then, we rewrite the +spin operators by S+ = Sx +iSy and S− = Sx −iSy and +apply the Holstein-Primakoff transformation up to order +S that for sublattice A +Sz +i = S − a† +iai, +S+ +i = +√ +2Sai, +S− +i = +√ +2Sa† +i, +(6) +and for sublattice B +Sz +i = b† +ibi − S, +S+ +i = +√ +2Sb† +i, +S− +i = +√ +2Sbi. +(7) +Here we take S = 1/2 and the Hamiltonian in the mo- +mentum space is given by +Hsw = +� +q +γ†(q)Hqγ(q), +Hq = +�Jd +0 + Js +0 − Js +q +Jd +q +Jd +q +Jd +0 + Js +0 − Js +q +� +, +(8) +in which γ†(q) = (a† +q, bq) and a† +q is the Fourier trans- +formed that a† +q += +N 1/2 � +q a† +ie−iqr. +And, Js +q += +� +rs∈same e−iqrsJs +r refers to the coupling between the +spins +belong +to +the +same +sublattice, +and +Jd +q += +� +rd∈diff e−iqrdJd +r to that of the different sublattices. +Jd(s) +r += 1/|∆r|α is the coupling strength. Finally, the +single magnon dispersion relation of the LR Heisenberg +model with staggered antiferromagnetic power-law de- +caying couplings is given by +ωAFM +q += +� +(Jd +0 + Js +0 − Jsq + Jdq)(Jd +0 + Js +0 − Jsq − Jdq). +(9) +Similarity, in the ferromagnetic case, the dispersion re- +lation of single magnon can be read as ωFM +q += |J0 − Jq| +with Jq = � +r e−iqrJr. +To capture the dependence of the dispersion relation +to α in Eqs. (1) and (2) in the main text, we numerically +calculate the linear SWT results by applying a cut-off +of longest range coupling as rmax, meaning that we only +consider the coupling between the sites (rx, ry) and (rx + +∆rx, ry +∆ry) with ∆rx and ∆ry ranging from −rmax/2 +to rmax/2, and we have computed rmax up to 1000. +Fig. 4 (a) describes the dispersion relation of the linear +SWT along the momentum path Γ → X → M → Γ with +α changing from α = 2.0 to 4.0. For large α, the LR +coupling rapidly decays which makes its disperion rela- +tion of single magnon similar to that of the typical anti- +ferromagnetic square lattice only with nearest-neighbor +coupling. +Here, the single magnon dispersion relation +is gapless only at Γ and M. +As α decreases, the LR +couplings strongly distort the dispersion relation, which +makes the magnon excitation cost more energy and the +dispersion relation goes higher as α decreasing. But this +dispersion relation obtained from the linear SWT theory +still remains gapless at Γ and M. Actually, the single +magnon dispersion relation would become discrete at Γ +and M in the limit rmax → ∞. +With ∆q the relative momentum away from the M +point, we plot the dispersion relation along the M → Γ +direction in Fig. 4 (b) with a double logarithmic scale at +α = 3.5 with varying rmax, which shows the power-law +dependence between ∆q and ω. We fit the linear SWT +results with ωq = A|∆q|s near the M point. Note that +for small rmax (rmax ≤ 100), the single magnon dispersion +around the M point still depends on rmax, which is shown +in Fig. 4 (b) with rmax changing from 16 to 1000. Such a +dependence would disappear and s would finally converge +in the limit rmax → ∞. In order to preform this changing +as rmax → ∞, we plot two black dashed lines that ωq ∝ +|∆q|0.48 in (b), where s = 0.48 comes from the fitting +of the linear SWT result with rmax = 1000. In Fig. 4 +(b), as rmax increasing, s converges to 0.48. Finally, with +rmax = 1000, Fig. 4 (c) presents our fitting about the +relation between the power s(α) and α, which preserve +in the limit rmax → ∞ and is also plotted as the blue +dots in Fig. 1 (d) +Similarly, we also plot our linear SWT results of the +ferromagnetic case in Fig. 4 (d), (e), and (f). Fig. 4(f) +also shows the relation between the power s(α) and α +taking rmax = 1000, which is also given as the blue dots +in Fig. 1 (e). + +10 +0.0 +5.0 +10.0 +15.0 +20.0 +25.0 +q +(a) += 2.0 += 3.0 += 4.0 +X +M +q +0.0 +5.0 +10.0 +15.0 +20.0 +25.0 +q +(d) += 2.0 += 3.0 += 4.0 +10 +1 +100 +101 +q +(b) +rmax = 1000 +rmax = 100 +rmax = 16 +10 +1 +100 +| q| +10 +1 +100 +101 +q +(e) +rmax = 1000 +rmax = 100 +rmax = 16 +0.00 +0.25 +0.50 +0.75 +1.00 +s( ) +(c) +staggered, AFM +0.0 +2.0 +4.0 +6.0 +8.0 10.0 +0.00 +0.50 +1.00 +1.50 +2.00 +s( ) +(f) +FM +FIG. 4. The linear SWT results. Panels (a), (b), and (c) are the linear spin wave result of the staggered antiferromagnetic case +while (d), (e), and (f) are the ferromagnetic case. Panel (a) and (d) are the dispersion relation plotted along the momentum +path Γ → X → M → Γ with rmax = 1000. +(b) is the spin wave dispersion relation near the M point for the staggered +antiferromagnetic lattice with α = 3.5 and rmax ranges from 16 to 1000 while (e) for Γ point in the ferromagnetic case. And +two black dashed lines here refer to the relation ωq ∝ |∆q|0.48 in (b) and ωq ∝ |∆q|0.96 in (e). (c) and (f) describe the relation +between the power of the gap changing s and α. +Fitting with QMC data +We fit the data of Gq(τ) by the relation Gq(τ) ∝ e−∆qτ +and the fitting process is shown in Fig. 5. We first choose +the data points for fitting according to their relative er- +rors. If the relative error of one data point is less than +0.2, then the data point is chosen to be used for fitting. In +the fitting process, we gradually omit the first Nτ data +points and then do the curve fitting to find the most +probable gap. As shown in the inset of Fig. 5, the fitting +error becomes intolerant when Nτ = 10 and the fitted +gap converges around ∆ = 2.35 when Nτ gradually de- +creases to 0. +In this case, we choose ∆ = 2.35 to be +the fitted gap for the data. Note that we find that for +all the q points at different α, the fitted gap does not +change evidently with Nτ, which means that higher ex- +cited states have much bigger energy gaps then the first +excited states(∆E2 ≫ ∆E1) so that e−∆E1τ term in G(τ) +contributes much more then other terms for the range of +τ we consider. +Fig. 6(a) shows the dispersion of HF M near Γ and +Fig. 6(b) shows the dispersion of HAF M near M for var- +ious system sizes L at α = 3. |∆q| denotes the relative +momentum away from the Γ in (a) (M in (b)). Plotting +under double logarithm scale, it is demonstrated that the +power of low-momentum dispersion s(α) depends on the +system size L. However, as the system size L increases, +the dispersion gradually converge and s(α) will finally re- +main unchanged as L → ∞. Noticing that the vast ma- +jority of L = 56 and L = 64 data collapses in both FM +and AFM cases, we thus obtain s(α) by fitting L = 64 +data and y ∝ xs(α) is plotted as red lines in Fig. 6. +FIG. 5. The fitting of energy gap with the data of the cor- +relation function Gq(τ) versus τ for L = 64 and α = 2.5 at +q = (3×2π/L, 0) for the ferromagnetic case. The inset shows +the obtained gap when the first Nτ data points are omitted +before fitting. + +11 +(a) FM +(b) AFM +FIG. 6. Dispersion relation at α = 3 for various system sizes +L. (a) Dispersion of HF M near Γ with ∆q denotes the rel- +ative momentum away from Γ. +Red line y ∝ x1.076 shows +the fitted power s(α = 3) = 1.076 using L = 64 QMC data. +(b) Dispersion of HAF M near M with ∆q denotes the rela- +tive momentum away from M. Red line y ∝ x0.469 shows the +fitted power s(α = 3) = 0.469 using L = 64 QMC data. + diff --git a/M9AyT4oBgHgl3EQf6_oJ/content/tmp_files/load_file.txt b/M9AyT4oBgHgl3EQf6_oJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae0cd90e656a761e0fe05260664695f5301f2db4 --- /dev/null +++ b/M9AyT4oBgHgl3EQf6_oJ/content/tmp_files/load_file.txt @@ -0,0 +1,1405 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf,len=1404 +page_content='Dynamical properties of quantum many-body systems with long range interactions Menghan Song,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='1 Jiarui Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' ∗ Chengkang Zhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' † and Zi Yang Meng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' ‡ 1Department of Physics and HKU-UCAS Joint Institute of Theoretical and Computational Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The University of Hong Kong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Pokfulam Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Hong Kong SAR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' China (Dated: January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 2023) Employing large-scale quantum Monte Carlo simulations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' we systematically compute the energy spectra of the 2D spin-1/2 Heisenberg model with long-range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' With the 1/rα fer- romagnetic and staggered antiferromagnetic interactions, we find the explicit range in α for the Goldstone-type (gapless) and Higgs-type (gapped) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Accompanied with the spin wave anal- ysis, our results vividly reveal how the long-range interactions induce a mass to the Goldstone mode via the generalized Higgs mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' This work provides the first set of unbiased dynamical properties of long-range quantum many-body systems and suggest that many universally accepted low-energy customs for short-range systems need to be substantially modified for long-range ones which are of immediate relevance to the on-going experimental efforts from quantum simulators to 2D quantum moiré materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='— Quantum many-body systems with long range (LR) interactions exhibit different and exotic prop- erties compared with their short-range counterparts, as the LR nature of the interaction differentites them from many universally accepted long-wavelength and low-energy customs governing the short-range ones over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' For example, the well-known Hohenberg- Mermin-Wagner theorem [1, 2] that forbids spontaneous symmetry-breaking of continuous symmetry at finite temperature in low dimensions can be easily circum- vented and generate interesting finite temperature tran- sitions [3–9] and new critical phenonema [10–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The bedrock in the research of highly entangled quantum matter – the area law scaling of the entanglement entropy – can also be bypassed in LR systems, and the consequent new scaling behavior points towards new guiding princi- ple of quantum entanglement that awaits to be worked out [8, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' recently the field of LR quantum-many sys- tems becomes even more active due to their fast exper- imental realizations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' such as the Rydberg atom arrays with long-range van der Waals or dipole-dipole interac- tion where topological ordered state of matter and quan- tum criticality have been realized [17–20],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' magic angle twisted bilayer graphene (TBG) and other 2D quantum moiré materials in which flat-band topology and long- range Coulomb interaction give rise to a plethora of cor- related phases beyond semi-classical or band-theory de- scription [21–71],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' as well as the quantum gases coupled to optical cavities [72] and many other programmable quantum simulators [73–78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Despite of such fast developments, theoretical and nu- merical investigations on the dynamical properties of the LR quantum many-body systems are however still lack- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' This is mainly due to the fact that dynamical proper- ties, such as spectral and response functions [65, 79–84], are usually difficult to compute without approximation in analytic theory and numerical simulations, even for the systems with short-range interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' And therefore by now there only exist few perturbative works such as Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' [85–89], which are mainly valid either in higher di- mensions or at various mean-field limits where the fluctu- ations are suppressed, and previous algorithmic develop- ments in non-perturbative numerical approaches for LR system are mainly focused on classical systems [90, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' However, in the aforementioned experiments of quantum LR systems, it is actually the dynamical and spectral information that can be easily detected by means of neu- tron scattering, nuclear magnetic resonance, scanning tunneling spectroscopy, nonlinear and non-equilibrium transport and optical probes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' To overcome the dilemma between the fast experimen- tal developments and the slow progresses in theoretical reality in LR quantum many-body systems, the need to develop and carry out unbiased approaches such as large- scale quantum Monte Carlo (QMC) simulations, to sys- tematical investigate the dynamical properties therein is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' And only in this way, can one fully reveal the interplay between the LR interaction and quantum topol- ogy and fluctuations to explain the aforementioned fas- cinating experimental outcomes and predict new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' This is the focus of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Here we develop and em- ploy the stochastic series expansion (SSE) QMC [78, 92– 94] simulation for the LR quantum many-body systems, to compute the energy spectra of the 2D spin-1/2 Heisen- berg model with 1/rα interaction where α is the decaying power, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' With the interaction types of ferromagnetic (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1 (a)) and antiferromagnetic (staggered without introducing frustration, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1 (b)), we find the explicit range in α for the Goldstone-type (where the spectra are gapless) and Higgs-type (where the spectra are gapped) spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1 (d) and (e), accompanied with spin-wave theory (SWT) anal- ysis [12, 95–99], our results reveal how the quantum fluc- tuations and long-range interactions conspire the Gold- stone mode to accuqire mass via the generalized Higgs mechanism [85] and therefore provide the first set of un- biased dynamic properties of LR quantum many-body arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='00829v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='str-el] 2 Jan 2023 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 (c) (a) (b) AFM FM FM (c) (d) FM (e) AFM Anomalous Goldstone Higgs Standard Goldstone 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 (c) (a) (b) AFM FM FM (c) (d) FM (e) AFM Anomalous Goldstone Higgs Goldstone FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 2D LR Heisenberg model with Higgs and Goldstone spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The schematic plots of 2D Heisenberg model with LR ferromagnetic interaction (a) and staggered antiferromagnetic interaction (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The power-law decay of J(r) ∼ 1/rα for three different α is shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (d) and (e) show the power s(α) of low-energy spectra ω ∼ |q|s(α) obtained from QMC and SWT versus α for both the ferro- magnetic and antiferromagnetic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The green-shaded area represents the Higgs regime where the spectra are gapped, the yellow shaded area represents the anomalous Goldstone regime where the dispersion powers change with α, and the white area is the standard Goldstone regime as those of the short-range systems where s = 1 for antiferrmagnetic and s = 2 for ferromagnetic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The red dots are fitting re- sults from QMC (L = 64) and the red stars denote QMC boundaries of α = 2 (for ferromagnetic) and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='2 (for antiferromagnetic) which speparate the Higgs and Goldstone regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The blue dashed lines are fitting results from SWT, with a cut-off of longest coupling distance rmax = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' systems where universally accepted low-energy physics for short-range ones are substantially modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Implica- tions of on-going experiments in quantum simulators and 2D quantum moiré materials are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Model and Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='— We consider the 2D spin-1/2 LR Heisenberg model with power-law decaying couplings on the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The Hamiltonians for the ferromag- netic case and the antiferromagnetic case (with staggered interaction to avoid the sign problem [100, 101]) are given by HF M = −J � i̸=j 1 |ri − rj|α Si · Sj, (1) HAF M = J � i̸=j (−1)|xi+yi−xj−yj+1| |ri − rj|α Si · Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (2) The schematic spin configurations and the decaying LR interactions of J(r) = 1/rα for the ferromagnetic case and the antiferromagnetic case are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1(a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Here we set J = 1 and simulate the linear system sizes upto L = 64 and inverse timperature β = L/2, and the decay exponent α from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5 to 100, with the focus on α ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Detailed implementation and finite size analysis of the obtained dispersions in QMC are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' II of SM [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' [85, 103], for HF M, the SWT analysis accompanied with a continuum approximation leads to the conclusion that for α > d where d is the spa- tial dimension, the dispersion is gapless and of Goldstone- type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' In particular, for α > d + 2 (denoted as standard Goldstone regime), the dispersion of the LR model re- duces to the short-ranged case with ω ∼ |q|2, and for d < α < d + 2 (denoted as anomalous Goldstone regime) the dispersion is ω ∼ |q|α−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' For α ≤ d (denoted as Higgs regime) the generalized Higgs mechanism [85] in- duces a discrete spectrum and thus the system is gapped, corresponding to a power of zero in the dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As will be shown below, our QMC results are roughly con- sistent with this picture as we also find a Higgs regime with α ≤ 2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As for the antiferromagnetic case, it is worth not- ing that Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' [85] predicts the Higgs regime occurs at α ≤ d−2 for the system with H = J � i̸=j 1 |ri−rj|α Si ·Sj, therefore for d = 2 there will be no finite α values with gapped spectra, and the anomalous and standard Gold- stone regimes are d − 2 < α < d and α > d respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' However, we consider a sign-problem-free Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (2) which does not host frustrations and conse- quently we get different boundaries of the three regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Our QMC and SWT analysis find a Higgs regime with α ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1 (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' In order to obtain the low-energy spectra of HF M and HAF M, we compute the imaginary time correla- tion function Gq(τ) ≡ ⟨Sz q(τ)Sz −q(0)⟩ − ⟨Sz q⟩2, where Sz q ≡ 1 √ N � r eiq·rSz r, via the SSE QMC method [78, 92– 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Here we consider the periodic boundary conditions in the simulation so that (qx, qy) = (± 2πm L , ± 2πn L ) with m and n being integers are physical momenta on a L×L finite size lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The loop update scheme of SSE QMC is purposely adapted to cope with the long range inter- actions by assigning each bond with a separate bond weight and bond type (ferromagnetic or antiferromag- 3 (a) (d) (b) (c) (d) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Dynamical properties of 2D LR ferromagnetic Heisenberg model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Dispersion relations along the path (Γ → X → M → Γ) with panels (a)-(f) for different decay exponents α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Results of various sizes L are plotted together in each panel and share the same legend on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Inset of panel (b) indicates that at α = 2 the first excitation gaps near Γ for various sizes converge to a finite value and the system has a gaped spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=', inside the Higgs-regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Insets of (c), (d) and (e) show the fitting of power-law dispersions ωq ∼ |q|s(α) near Γ (with |∆q| denotes the relative momentum away from Γ) in range 2 < α ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Red dashed line in (f) is the SWT dispersion for 2D nearest neighbor FM Heisenberg model with ωq = |J| zS(1 − γq) where S = 1/2, the coordination number z = 4, and γq = 1 z � δ eiqδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' netic) [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' To obtain the spectrum in QMC, notice that ⟨Sz q(τ)Sz −q(0)⟩ = � eHτSz q(0)e−HτSz −q(0) � = �� l=0 e−βEl �−1 × � n,m=0 |⟨n|Sz q|m⟩|2e−(Em−En)τe−βEn (3) where H|n⟩ = En|n⟩ and E0 is the ground state energy of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' When β∆E1 ≫ 1 where ∆En = En − E0, we can estimate Gq(τ) ≈ � n=1 |⟨0|Sz q|n⟩|2 � e−∆En(q)τ + e−∆En(q)(β−τ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (4) When the imaginary time is sufficiently large, we assume that the system will gradually evolve to the ground state, so that the correlation function can be further approxi- mated by Gq(τ) ≈ |⟨0|Sz q|1⟩|2e−∆E1(q)τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (5) If |⟨0|Sz q|1⟩|2 is finite (which is usually the case), we can then extract the energy gap for each q point by fitting Gq(τ) with an exponentially decaying function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Exam- ples of such fitting and their finite size extrapolations to the thermodynamic limit are shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='II in SM [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='— Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 2 shows the obtained QMC spectra along the high symmetry path Γ(0, 0) → X(π, 0) → M(π, π) → Γ(0, 0) for the ferromagnetic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' At α = 100 (panel (f)), the system reduces to the short-ranged case with only nearest-neighbor couplings [10–12], and our QMC- obtained spectra matches well with the spectra obtained from SWT analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Both of them show a ωq ∼ |q|2 dispersion close to Γ, and surprisingly, the QMC and SWT spectra match well along the whole path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As α gets smaller, as shown in panels (c), (d) and (e), we find the dispersion enters the anomalous Goldstone region [85], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=', the dispersion close to Γ deviates from a quadratic one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' We use ωq ∼ |q|s(α) to fit the dispersion close to Γ and find the power s(α) gradually decreases as α gets smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Insets of these three panels demonstrate the power law fitting of s(α) using L = 64 QMC results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' We find, at α = 3 (panel (d)), s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='076 which agrees well with the relation of s(α) = α − 2 suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' However, for α = 4 (panel (e)) and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5 (panel c)) our results show apparent derivations from s(α) = α − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1(d) collects the fitted power s(α) by QMC (red dots) at various α and we observe a satisfactory match with our SWT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' At α = 2, we find ωq near Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=', q = ( 2π L , 0) for different system sizes converge to a 4 (a) (c) (d) (e) (f) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Dynamical properties of 2D LR (staggered) antiferromagnetic Heisenberg model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Dispersion relations along the path (Γ → X → M → Γ) with panels (a)-(f) for different decay exponents α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Results of various sizes L are plotted together in each panel and share the same legend on the top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Inset of panel (b) indicates that the first excitation gaps near M = (π, π) for various sizes converge to a finite value and thus the system is inside the Higgs regime at α ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Insets of (c), (d) and (e) show the fitting of power-law dispersion as ω ∼ |q|s(α) near M (with |∆q| denotes the relative momentum away from M) in range 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='2 < α < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Dashed red line in (f) is the nearest-neighbor SWT dispersion ωq = |J| zS � (1 − γq)2 with an additional coefficient ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='158 to approximate the second order spin wave effects [79, 99, 104, 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' large and finite value of ω ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='41 as indicated in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' This phenomenon is fundamentally different from a gapless excitation in which the finite size gap ω(2π/L,0) converges to zero as L → ∞ and results in a continuous spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Our result reveals that at α = 2 the system enters the Higgs regime where the Goldstone mode acquires mass due to the LR interation and the excitation spectrum becomes gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' For α < 2 (α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5 in panel (a)) we find the gaps begin to diverge with the system size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Therefore, we conclude that α = 2 is the separation power between the Higgs-type and Goldstone- type spectra in HF M from our QMC results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 3 illustrates the QMC dispersion relation for HAF M along the high symmetry path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Similarly, in panel (f), we benchmark the spectrum at α = 100 with SWT result for the short-range antiferromagnetic Hamiltonian (with an extra coefficient ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='158 multiplied to approxi- mate the second order spin wave effects [79, 99, 104, 105]) and find QMC results agree well with SWT dispersion close to M and the dispersion relation is ωq ∼ |q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As α decreases, the system also enters the anomalous Gold- stone region with ωq ∼ |q|s(α) and 0 < s(α) < 1 close to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Fitted powers via QMC at various α are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1(e) and agree well with the SWT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 3 (b) at α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='2, ωq close to M converges to a large and finite value of ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' This means HAF M is in the Higgs-regime with gapped spectra when α ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='2 (panel (a) with α = 2 shows the divergent gap close to M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' In contrast to the vanishing of the Higgs spectra in the purely AFM case in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' [85] for d = 2, our staggered AFM case (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (2)) offers finite boundaries between the Higgs-type and Goldstone-type spectra, and it is of in- terest to perform similar theoretical analysis as done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' [85] for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (2) to further reveal the subtle working of the generalized Higgs mechanism, with different types of LR interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='— With the unbiased large-scale QMC sim- ulations and SWT analysis, we systematically investi- gate the dynamical properties of 2D spin-1/2 Heisenberg model with LR interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' We find in contrast to the well accepted low-energy customs such as Hohenberg- Mermin-Wagner theorem and gapless Goldstone mode, the LR quantum many-body systems offer richer tun- ability and exhibit new phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As the interaction exponent α varies, the Goldstone modes can be strongly modified, in that they can be either distorted (in the anomalous Goldstone regime), or even be gapped via a generalized Higgs mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 5 Most remarkably, these dynamical properties have im- mediate relevance to the on-going experiments with ul- tracold atom arrays and quantum moiré materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' For example, the long-range Coulomb interaction in quan- tum moiré systems can be easily tuned by varying dielec- tric environment, electrostatic gating and twisting angles, and in this way, different observed thermodynamical and dynamical properties (such as switching between gapped and gapless spectra) [30, 61, 62, 67, 68, 70] can be iden- tified with different LR interaction types and regimes, when compared unbiased results such as ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Similar tunability can also be realised in dressed Rydberg atom arrays whose interaction can be modified [107], one can then compare different responses from experiments with our results to identify the LR interaction and the novel phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='- We thank Zheng Yan, Tianyu Wu, Ting-Tung Wang, Meng Cheng, Fakher Assaad and Qi Yang for valuable discussions on related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' We ac- knowledge the support from the Research Grants Coun- cil of Hong Kong SAR of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 17303019, 17301420, 17301721, AoE/P-701/20, 17309822 and A- HKU703/22), the K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Wong Education Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' GJTD-2020-01), and the Seed Funding “Quantum-Inspired explainable-AI” at the HKU-TCL Joint Research Centre for Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' We thank the HPC2021 system under the Information Tech- nology Services and the Blackbody HPC system at the Department of Physics, the University of Hong Kong for their technical support and generous allocation of CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The authors also acknowledge Beijng PARATERA Tech Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=',Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' for providing HPC resources that have contributed to the research results reported within this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' ∗ jrzhao@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='hk † zhouchk@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='hk ‡ zymeng@hku.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' [106] The diagonal and off-diagonal operators sit on the same bond are still equal weighted even now, different weights and type bonds are assigned to different bonds, and this ensures that the loop update scheme does not need to be amended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' [107] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Jau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' M.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 12, 71 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 9 Supplementary Materials In this supplementary material, we present the linear spin wave analysis for the LR FM and staggered AFM Heisenberg model, in which the dispersion relation of the low-lying magnetic excitations at different decaying power α are extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' From here, we make comparison with the dispersion obtained from the QMC simulations in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Moreover, we also provide detailed data on the fitting of the excitation gaps from the dynamical correlation functions in QMC, such as representative data points where the extrapolation of the converged gaps at thermodynamic limit are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Linear spin wave analysis We applied the linear spin wave theory (SWT) to an- alyze the dispersion of the low energy excitation in the LR spin-1/2 Heisenberg model with power law decaying couplings in both ferromagnetic and staggered antiferro- magnetic cases [12, 95–99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Taking the staggered antifer- romagnetic cases as an example, it calls for the definition of two sublattices, A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The spin on each sublattice is pointing in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Then, we rewrite the spin operators by S+ = Sx +iSy and S− = Sx −iSy and apply the Holstein-Primakoff transformation up to order S that for sublattice A Sz i = S − a† iai, S+ i = √ 2Sai, S− i = √ 2Sa† i, (6) and for sublattice B Sz i = b† ibi − S, S+ i = √ 2Sb† i, S− i = √ 2Sbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (7) Here we take S = 1/2 and the Hamiltonian in the mo- mentum space is given by Hsw = � q γ†(q)Hqγ(q), Hq = �Jd 0 + Js 0 − Js q Jd q Jd q Jd 0 + Js 0 − Js q � , (8) in which γ†(q) = (a† q, bq) and a† q is the Fourier trans- formed that a† q = N 1/2 � q a† ie−iqr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' And, Js q = � rs∈same e−iqrsJs r refers to the coupling between the spins belong to the same sublattice, and Jd q = � rd∈diff e−iqrdJd r to that of the different sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Jd(s) r = 1/|∆r|α is the coupling strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Finally, the single magnon dispersion relation of the LR Heisenberg model with staggered antiferromagnetic power-law de- caying couplings is given by ωAFM q = � (Jd 0 + Js 0 − Jsq + Jdq)(Jd 0 + Js 0 − Jsq − Jdq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (9) Similarity, in the ferromagnetic case, the dispersion re- lation of single magnon can be read as ωFM q = |J0 − Jq| with Jq = � r e−iqrJr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' To capture the dependence of the dispersion relation to α in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (1) and (2) in the main text, we numerically calculate the linear SWT results by applying a cut-off of longest range coupling as rmax, meaning that we only consider the coupling between the sites (rx, ry) and (rx + ∆rx, ry +∆ry) with ∆rx and ∆ry ranging from −rmax/2 to rmax/2, and we have computed rmax up to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 4 (a) describes the dispersion relation of the linear SWT along the momentum path Γ → X → M → Γ with α changing from α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' For large α, the LR coupling rapidly decays which makes its disperion rela- tion of single magnon similar to that of the typical anti- ferromagnetic square lattice only with nearest-neighbor coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Here, the single magnon dispersion relation is gapless only at Γ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As α decreases, the LR couplings strongly distort the dispersion relation, which makes the magnon excitation cost more energy and the dispersion relation goes higher as α decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' But this dispersion relation obtained from the linear SWT theory still remains gapless at Γ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Actually, the single magnon dispersion relation would become discrete at Γ and M in the limit rmax → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' With ∆q the relative momentum away from the M point, we plot the dispersion relation along the M → Γ direction in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 4 (b) with a double logarithmic scale at α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5 with varying rmax, which shows the power-law dependence between ∆q and ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' We fit the linear SWT results with ωq = A|∆q|s near the M point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Note that for small rmax (rmax ≤ 100), the single magnon dispersion around the M point still depends on rmax, which is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 4 (b) with rmax changing from 16 to 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Such a dependence would disappear and s would finally converge in the limit rmax → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' In order to preform this changing as rmax → ∞, we plot two black dashed lines that ωq ∝ |∆q|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='48 in (b), where s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='48 comes from the fitting of the linear SWT result with rmax = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 4 (b), as rmax increasing, s converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Finally, with rmax = 1000, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 4 (c) presents our fitting about the relation between the power s(α) and α, which preserve in the limit rmax → ∞ and is also plotted as the blue dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1 (d) Similarly, we also plot our linear SWT results of the ferromagnetic case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 4 (d), (e), and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 4(f) also shows the relation between the power s(α) and α taking rmax = 1000, which is also given as the blue dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 1 (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 q (a) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 X M q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 q (d) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 10 1 100 101 q (b) rmax = 1000 rmax = 100 rmax = 16 10 1 100 | q| 10 1 100 101 q (e) rmax = 1000 rmax = 100 rmax = 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='00 s( ) (c) staggered, AFM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='00 s( ) (f) FM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The linear SWT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Panels (a), (b), and (c) are the linear spin wave result of the staggered antiferromagnetic case while (d), (e), and (f) are the ferromagnetic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Panel (a) and (d) are the dispersion relation plotted along the momentum path Γ → X → M → Γ with rmax = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (b) is the spin wave dispersion relation near the M point for the staggered antiferromagnetic lattice with α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5 and rmax ranges from 16 to 1000 while (e) for Γ point in the ferromagnetic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' And two black dashed lines here refer to the relation ωq ∝ |∆q|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='48 in (b) and ωq ∝ |∆q|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='96 in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (c) and (f) describe the relation between the power of the gap changing s and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Fitting with QMC data We fit the data of Gq(τ) by the relation Gq(τ) ∝ e−∆qτ and the fitting process is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' We first choose the data points for fitting according to their relative er- rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' If the relative error of one data point is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='2, then the data point is chosen to be used for fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' In the fitting process, we gradually omit the first Nτ data points and then do the curve fitting to find the most probable gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' As shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 5, the fitting error becomes intolerant when Nτ = 10 and the fitted gap converges around ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='35 when Nτ gradually de- creases to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' In this case, we choose ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='35 to be the fitted gap for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Note that we find that for all the q points at different α, the fitted gap does not change evidently with Nτ, which means that higher ex- cited states have much bigger energy gaps then the first excited states(∆E2 ≫ ∆E1) so that e−∆E1τ term in G(τ) contributes much more then other terms for the range of τ we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 6(a) shows the dispersion of HF M near Γ and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 6(b) shows the dispersion of HAF M near M for var- ious system sizes L at α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' |∆q| denotes the relative momentum away from the Γ in (a) (M in (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Plotting under double logarithm scale, it is demonstrated that the power of low-momentum dispersion s(α) depends on the system size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' However, as the system size L increases, the dispersion gradually converge and s(α) will finally re- main unchanged as L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Noticing that the vast ma- jority of L = 56 and L = 64 data collapses in both FM and AFM cases, we thus obtain s(α) by fitting L = 64 data and y ∝ xs(α) is plotted as red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The fitting of energy gap with the data of the cor- relation function Gq(τ) versus τ for L = 64 and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='5 at q = (3×2π/L, 0) for the ferromagnetic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' The inset shows the obtained gap when the first Nτ data points are omitted before fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 11 (a) FM (b) AFM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Dispersion relation at α = 3 for various system sizes L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (a) Dispersion of HF M near Γ with ∆q denotes the rel- ative momentum away from Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Red line y ∝ x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='076 shows the fitted power s(α = 3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='076 using L = 64 QMC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' (b) Dispersion of HAF M near M with ∆q denotes the rela- tive momentum away from M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content=' Red line y ∝ x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='469 shows the fitted power s(α = 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} +page_content='469 using L = 64 QMC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/M9AyT4oBgHgl3EQf6_oJ/content/2301.00829v1.pdf'} diff --git a/NtAzT4oBgHgl3EQfWPy8/vector_store/index.pkl b/NtAzT4oBgHgl3EQfWPy8/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..229e32d726c00cc7cfd6fe03848d00e9822bf26c --- /dev/null +++ b/NtAzT4oBgHgl3EQfWPy8/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ed4ad8d1a2471ea683a922c2286a520776ff2befb1b24a7862689b6f0f781b4a 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b/Q9A0T4oBgHgl3EQfDf--/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..679cb73d28b59c060bf865aac5daeb6a0e025de9 --- /dev/null +++ b/Q9A0T4oBgHgl3EQfDf--/content/tmp_files/load_file.txt @@ -0,0 +1,458 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf,len=457 +page_content='1 Rainbows in a bottle: Realizing microoptic effects by polymerizable multiple emulsion particle design Naresh Yandrapalliab*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Baris Kumrua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Tom Robinsonb*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Markus Antoniettia a Max Planck Institute of Colloids and Interfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Department of Colloid Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Am Mühlenberg 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 14424 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Germany b Max Planck Institute of Colloids and Interfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Department of Theory & Bio-Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Am Mühlenberg 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 14424 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Germany Introduction In nature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' structural colour generation is based on discriminative light propagation associ- ated with physical structures in the range of the wavelengths of light1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' These iridescent struc- tural colours are of immense significance2 but not easy to control experimentally and there- fore difficult to exploit for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' In this work, we employ microfluidics to produce polymerizable double emulsions that can not only induce the already known lensing effect3 but also result in the spectral separation of white light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Here, liquids of varying refractive index that constitute the emulsions resulted in patterns of iridescent colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' After polymer- ization, the inner emulsion cores collapse and this results in curved concave surfaces on these polymeric microspheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Interestingly, the light propagation along the curved surfaces undergo total internal reflection, followed by near-field interference along exit structures on the polymerized microspheres4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' These structured polymeric particles that are able to gener- ate colour dispersions can be exploited for optical devices, displays and even sensing tech- nologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Main From the formation of glories in the sky to the spectacular vibrant colours observable on various living organisms, humans have learned that light can be controlled by materials 2 structures/structured materials that exist and evolved in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' We are continuously discov- ering that the majority of the everyday iridescent spectra, either colourful butterflies or the plumage of birds, is the result of micrometre scale material structures5,6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Simple physical phenomena such as absorption, reflection, and refraction, as well as diffraction and interfer- ence that result from light-matter interactions can combine into complex structural colour generations 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' An attempt to classify the physical interaction of light with microstructures resulted in identifying processes such as thin-film interference7, multi-film interference8, dif- fraction grating9, scattering (coherent & incoherent)10 and photonic crystal diffraction11 that leads to structural colouration observed in nature12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Recently, Goodling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' have shown that light interaction with concave interfaces formed by materials with two different refractive indices (η) will result in structural colouration due to the interference of total internally re- flected rays4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This proves that two materials (with varying refractive indices) forming a simple interface with a degree of curvature can produce iridescent colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Our results show that water-in-oil-in-water (W/O/W) double emulsions produced with a highly refractive oil layer (styrene (η-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='516) interfaced with water (η-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='333) on either side can generate iridescent colours with spectral separation similar to that observed in glories13 but with more complexity and can still be exploited for their light focusing effect14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The double emulsions produced here contain an aqueous inner core surrounded by a layer of styrene in a continuous aqueous phase (shown in figure 1) stabilized with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='5 wt% F108 surfactant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Because the two liquids have different refractive indices, light undergos both refraction and reflection based on the layering of the two liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The spectral dispersions or rainbows observed in nature are an effect of light interacting with water drops in the air, however, in the case of double emulsions (here, W/O/W), having layers of aqueous-oil-aqueous-oil- aqueous phase result in a similar dispersion but are followed by merging of the spectral wavelengths to produce structural colouring (figure 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' A z-stack of the diffused structural 3 Figure1: Microfluidic generation of liquid emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (a) Double and multi-core emulsion production with vary- ing inner cores (scale bar – 200 µm) and the iridescent nature of the dispersed light at the equatorial plane, off-focus plane (30 µm below the equatorial plane) of the emulsions and the intense light spots observed along the z-axis of the inner cores (scale bar – 10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (b) bright-field z-stacks of the light scattering from a 1-in-1 emulsion with each slice separated by 10 µm (top left to bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (c) 2D ray-tracing of a 1-in-1 emulsion with varying inner droplet sizes (diameters 5, 10, 30, 40 and 50 µm) showing the effect on the focal length and TIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (d) schematic representation of a 1-in-1 emulsion showing refraction, TIR and the effect of inner to outer droplet density on the TIR (black arrows represent the direction of light with a single wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' S – corre- sponds to styrene phase and W – corresponds to aqueous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' colouring produce from a 1-in-1 W/O/W double emulsion is presented in supplementary video 1,and a clear spectral separation is modelled by ray tracing (see supplementary video 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' As depicted schematically in figure 1d(4), the light rays from the source passing through the double emulsion initially undergo refraction at the first convex interface formed by the (a) W/o/W emulsion Equilateral off-focus Microfluidicproduction (b) 1-in-1 Z-stack array (4x4) Z-project plane plane 1-in-1 2-in-1 3-in-1 4-in-1 5-in-1 (c) (p) Double Totalinternalreflection Decreasing outer droplet to inner droplet diameter ratio (TIR) TIRangle (α)= 145° emulsion 1) W styrene-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='51 non-TiR regime TIRregime Increasingtotal internalreflectionwithdecreasingsecondaryfocusing Refraction RefractionandTIR4 outer aqueous solution and the oil layer as well as through the second convex interface across the aqueous inner core present inside the oil drop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Thanks to the spherical configu- ration of the emulsion, a single light ray can undergo a minimum of two or a maximum of four refraction events as it passes out of the emulsion (1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This results in more complex colour formations than observed from single emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Our observations suggest that the focusing effect from these double emulsions is also prom- inent (see figure 1c), which was not observed in previous works on diverging light rays pass- ing through low refractive index inner core surrounded by a high refractive index liquid3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Moreover, a point source of light passing through the emulsions such as the present case resulted in dual focusing – one right below the double emulsion (resulting from the rays passing through the inner aqueous core) and the other farther away (resulting from the rays passing through the rest of the emulsion) (see supplementary figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' In agreement with that, z-stack images of the double emulsion indeed show a bright spot right below the emul- sion due to the said focusing of the dispersed light beyond the emulsion (see supplementary video 1 as well as z-project from figure 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The second focus vanishes for the double emulsions whose inner aqueous core diameter is equal to or above the radius of the diam- eter of the W/O/W emulsion (see figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This is because the rays that are otherwise contributing to the second focus are then refracted through the bigger inner aqueous core converge into the primary focusing area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This is observed by the increase in the intensity of the primary focusing area as the inner aqueous droplet size increases (figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Previous studies have shown that refraction through equidistant alternating layers of liquid or transparent material is associated with multi-layer interference8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This phenomenon is not observable in the case of the double emulsions described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The inner aqueous droplet being heavier (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='9982 g/mL at 20 ºC) than the oil phase (ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='909 g/mL at 20 ºC) sinks to the bottom of the oil drop (figure 1d(1)), resulting in non-equidistant layers and a decrease in peak reflectivity8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' As the light is passing through a high refractive index medium to a lower 5 refractive index medium, followed by the curved nature of the interface4, a total internal re- flection (TIR) of light is possible at particular angular incidences, observable from the 2D ray-tracing result (figure 1c) and presented in the scheme (figure 1d(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Only refracted rays whose angle of incidence (α) is ~145° can undergo TIR when they pass through the inner aqueous core and encounter the concave interface across styrene (high refractive index) and the aqueous outer solution (low refractive index) (figure 1d(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The schematic produced with the ray tracing data figure 1d(6) shows the density-dependent TIR with ρstyrene/ρwater less than 1, while values greater than 1 result in no TIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This suggests that a double emulsion with a high refractive index outer layer and a low refractive index inner layer should require inversed density values to take advantage of TIR as shown previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='4 Similarl to double emulsions, the emulsions with multiple inner droplets inside one W/O/W double emulsion generated by the same single inlet microfluidic device (see Methods) in- deed showed light scattering and iridescent colours (figure 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Producing such multi-core emulsions using a single inlet microfluidic design was made possible in this work by taking advantage of the swelling property of PDMS in the presence of specific solvents like sty- rene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='15,16 The swelling of PDMS upon the uptake of styrene resulted in narrowed channels width, especially at first cross junction where water-in-oil (W/O) droplets are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Through carefull manipulation of flow rates, when can achieve W/O/W emulsions wih mul- tiple inner cores, all with the same device design (see figure 1a and Supplementary Video 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' In the absence of such a reliable system, multiple device designs with varied channel widths would have to be fabricated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The light scattering observed in all these multi-core emulsions, 2-in-1, 3-in-1, 4-in-1, and 5-in-1, are presented in the supplementary figure S1 as arrays of z-stack slices and in supplementary video 4, 5, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' At any given z-slice, the light scattering observed is similar for each different type of multi-core emulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' There is no observable influence of the light scattering produced by one inner droplet on the other inner droplet within such multi-core emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Moreover, the iridescence within and around each 6 inner droplet is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This is only possible if the inner droplets are in the same plane (at the bottom) and are not stacked one above the other (see ray tracing data in supplemen- tary figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Although the formation of emulsions with multiple inner droplets has been presented previously, their light scattering and high iridescence properties have never been explored before17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Figure 2: Light scattering from polymerized emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (a) Electron micrographs of polymerized multi-core emulsion, inset showing the close-up of the curved surfaces on the microsphere (scale bar – 10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (b) Bright- field images of the polymerized emulsions with a varying number of inner cores (scale bar – 50 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (c) Con- focal 3D reconstruction of the reflected light from the curved surfaces of two polymerized 1-in-1 emulsions facing the light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The curved dashed lines represent the surface boundary of the concave surface on the microparticle .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (d) Color images of the polymerized multi-core emulsions showing spectral dispersion of the reflected light from the curved surfaces (scale bar – 10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (e) Spectral separation of dispersed reflected light ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='Depth ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='Distance(μm)Distance(μm)Distance(μm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='Distance(μm)7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='from the curved surfaces of polymerized 1-in-1 emulsion aligned 0° (parallel) and 90° (perpendicular) angle to the light source (scale bar – 10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' We then further explored the possibility to polymerize the styrene within the double and multi-core emulsions, with the aim of preserving some of these effects into a more stable, polymer structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' We have demonstrated that microfluidics can be used to generate uni- form sized hollow microspheres 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' However, it is not possible to produce structural colour- ation/iridescent colours with individual microspheres or hollow microspheres unless they are transparent or with surface patterning20 or regularily packed to form so-called photonic balls21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' In this work, the above produced double emulsions and multi-core emulsions with styrene as the oil-phase are polymerized to generate microspheres with lensing curved surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Unlike for the formation of hollow polymer microspheres where the inner aqueous cores have to be stabilized, microspheres with concaved dimples as shown in figure 2 are made from inner aqueous cores devoid of any surfactant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The surfactant F108 used in this study is a tri-block copolymer with two polypropylene hydrophobic blocks separated by a central hydrophilic polyethene oxide block, which is known to form rather stable block copolymer bilayers along two aqueous droplets, similar to that of lipids in a cell membrane22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' However, in the absence of surfactant in the inner aqueous core, there will only be a monolayer of the surfactant stabilizing the emulsion interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' We hypothesize that during polymerization, de- stabilization of the monlayer could result in the formation of a concaved dimple on the spher- ical polystyrene microsphere as the inner aqueous core fuses with the outer aqueous solu- tion (as shown schematically in supplementary figure S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The scanning electron micro- graphs of these polymeric spheres show the resulting concave surfaces (see figure 2(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Similarly, polymerization of 2-in-1, 3-in-1, and multi-in-1 emulsions resulted in an equivalent 8 number of surface lenses on the respective polystyrene microspheres formed (see bright- field images in figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Unlike the averaged light scattering observed in liquid phase emulsions, single polystyrene microspheres can be described via the rules of the reflection of light23 and thereby show optical near field effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Like concave mirrors, the dimples on the polymeric microsphere reflect and focus the light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' True to this assumption, the 3D reconstruction of the confocal z- stack images suggests both reflection and focused light from the structured surface of the polymerized 1-in-1 W/O/W emulsion (figure 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The z-stack was acquired with the curved surface facing towards the source of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Similar to the liquid emulsions, our observations of their polymerized versions using a colour camera have revealed the iridescence nature of these focused surface emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Initial observations of these particles have shown their surface dimples/curves and iridescent col- ouring around the particles as well as bright spots from the lensing regions (figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' A closer look reveals more spectacular details of iridescence from the lensing regions of the particles (figure 2(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Concaved structures that are facing the light source reflect the light and appear as intense bright white spots – a combination of all the spectral colours, as seen in figure 2(b) and 1-in-1 polymerized emulsions of figure 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Interestingly, particles with concaved structures which are not facing but at an angle to the light source show a dramatic colour separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This is exhibited in figure 2(d) for 2-in-1, 3-in-1 and multi-in-1 configura- tions of the polymerized emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' In the case of multi-in-1 particles, with multiple curved surfaces facing the light source at different angles (figure 2(d)), one can observe that ap- proximately 20 micron “rainbows” show no apparent angular dependency, within the ob- served upper hemisphere of the particle .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Light reflected from the concave interface of pol- ymeric microspheres creates optical interference and the dispersion of light4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' In our experi- ments with the results shown in figure 2, this is clearly visible that the longer wavelengths of 9 the reflected light are closer to the surface of the microsphere while rays of shorter wave- lengths are farther away from the surface (see figure 2(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The line profile data taken from the centre of the images plotted to suggest the mixing and separation of the three main wavelengths - red, green and blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' A similar patterning of light reflection and dispersion is observed in polymeric microspheres, 2-in-1, 3-in-1, and multi-in-1 (figure 2(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This sug- gests that not only the liquid emulsions but also their polymerized versions can be exploited for light scattering properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This study highlights some interesting optical phenoma that are enabled by manufacturing controlled double and multi-core emulsions with a high refractive index differences using microfluidics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' We have shown that simple emulsions like W/O/W can induce spectral sepa- ration of white light under very local light focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The observed light enhancements are mainly due to refraction and to a lesser extent from total internal reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' More dramatic colour patterns can be achieved by simple alteration of the double emulsions to complex multi-core emulsions, also involving morphological transitions occurring under polymeriza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' While structural colouring in ordered 2D-arrays is a clear use case as a multilens array, because of the focusing nature, the applications and use of the local, close-to particle light fields with their colour gradients are yet to be discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The possibility to embed catalysts within the polymerizable liquid emulsions was demonstrated earlier19 and opens applications towards focused photocatalysis and photocatalytic gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Well ahead of applicability and exploitation in multiple fields however, we want to underline that the particles presented above are simple and rather effective to make, and that the observed effects are also just stunningly beautiful and unexpected: little rainbows in a bottle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 10 References 1 Sun J, Bhushan B, Tong J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Structural coloration in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' RSC Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 2013;' metadata={'source': 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Solvent Compatibility of Poly(dimethylsiloxane)- Based Microfluidic Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Anal Chem 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='1021/ac0346712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 16 Dangla R, Gallaire F, Baroud CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Microchannel deformations due to solvent-induced PDMS swelling.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 25 Tu R, Johnson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Ray Optics Simulation - Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' https://ricktu288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='io/ray-optics/ (accessed 11 Feb2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Supplementary information Materials, supplementary Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 1-4 and caption for supplementary Video 1-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Acknowledgments The authors thank the Max Planck Society for funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' acknowledge support from the MaxSynBio consortium, which is jointly funded by the Federal Ministry of Education and Research of Germany and the Max Planck Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Author Contributions N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' conceptualized the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' performed the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=', T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=', and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' analyzed the data and wrote the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Author Information Naresh Yandrapalli Department of Colloid Chemistry, Max Planck Institute of Colloids and InterfacesAm Müh- lenberg 1, Potsdam 14424, Germany Email: naresh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='yandrapalli@mpikg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='de Baris Kumru Department of Colloid Chemistry, Max Planck Institute of Colloids and InterfacesAm Müh- lenberg 1, Potsdam 14424, Germany Email: baris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='kumru@mpikg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='de Tom Robinson Department of Theory & Bio-Systems, Max Planck Institute of Colloids and Interfaces, Am Mühlenberg 1, Potsdam 14424, Germany E-mail: tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='robinson@mpikg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='de 12 Markus Antonietti Department of Colloid Chemistry, Max Planck Institute of Colloids and InterfacesAm Müh- lenberg 1, Potsdam 14424, Germany E-mail: markus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='antonietti@mpikg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='de Data Availability All data generated or analysed during this study are included in the published article and supplementary information, and are available from the corresponding authors upon rea- sonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Competing Interest Declaration No competing interests to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Corresponding Author Correspondence to Naresh Yandrapalli and Tom Robinson Figure Legends Figure1: Microfluidic generation of liquid emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (a) Double and multi-core emulsion production with vary- ing inner cores (scale bar – 200 µm) and the iridescent nature of the dispersed light at the equatorial plane, off-focus plane (30 µm below the equatorial plane) of the emulsions and the intense light spots observed along the z-axis of the inner cores (scale bar – 10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (b) bright-field z-stacks of the light scattering from a 1-in-1 emulsion with each slice separated by 10 µm (top left to bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (c) 2D ray-tracing of a 1-in-1 emulsion with varying inner droplet sizes (diameters 5, 10, 30, 40 and 50 µm) showing the effect on the focal length and TIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (d) schematic representation of a 1-in-1 emulsion showing refraction, TIR and the effect of inner to outer droplet density on the TIR (black arrows represent the direction of light with a single wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' S – corre- sponds to styrene phase and W – corresponds to aqueous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Figure 2: Light scattering from polymerized emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (a) Electron micrographs of polymerized multi-core emulsion, inset showing the close-up of the curved surfaces on the microsphere (scale bar – 10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (b) Bright- field images of the polymerized emulsions with a varying number of inner cores (scale bar – 50 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (c) Con- focal 3D reconstruction of the reflected light from the curved surfaces of two polymerized 1-in-1 emulsions 13 facing the light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The curved dashed lines represent the surface boundary of the concave surface on the microparticle .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (d) Color images of the polymerized multi-core emulsions showing spectral dispersion of the reflected light from the curved surfaces (scale bar – 10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (e) Spectral separation of dispersed reflected light from the curved surfaces of polymerized 1-in-1 emulsion aligned 0° (parallel) and 90° (perpendicular) angle to the light source (scale bar – 10 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Methods Device Fabrication Master mould for the microfluidic device was created using UV-based photolithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Ini- tially, a 4 inch silicon wafer was pre-heated for 30 min at 200 °C and 80 µm thick layer of photoresist (SU8 2025) was spin-coated on top (model no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' WS-650MZ-23NPPB, Laurell Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Corp) as per the specifications provided by the manufacturer at 23 °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' After the coating step, the wafer is heated at 65 °C for 3 min and 95 °C for 9 min before UV exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' After 8 sec of UV exposure through a specific design (see the supplementary figure S4) using kloe photolithographic instrument (model no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' UV-KUB 2), the wafer was post-baked at 65 °C for 2 min and 95 °C for 7 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The device design was revealed on the wafer after the washing steps with a developer solution and isopropanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Finally, the wafer was baked at 200 °C for 2 h before performing overnight surface passivation with 50 µL of 1H,1H,2H,2H-per- fluorodecyltrichlorosilane in a dessicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' To produce the microfluidic chips, a PDMS:curing agent (10:1) mixture was thoroughly mixed and degassed for 30 min in a desiccator connected to low pressure (150 millibars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The degassed mixture was poured on top of the surface passivated wafer and cured at 90 °C for 3 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Crosslinked PDMS was pealed from the master mould and diced into small pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' The inlets and outlets were created using 1 mm biopsy puncher (Kai Europe GmbH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Finally, plasma cleaned (at 600 mbar for 1 min) (Plasma Cleaner PDC-002-CE, Harrick Plasma) glass coverslips and diced PDMS chips with the desired design were bonded to form the microfluidic chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' These chips were further heated at 60 °C for 2h to complete the bonding process and retention of hydrophobic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Device Surface Passivation Surface passivation of the double emulsion microfluidic design is necessary for the formation of stable double emulsions24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' A series of solutions are flown through the outlet to the outer aqueous (OA) solution inlet to render the hydrophobic PDMS surface hydrophilic (see the supplementary figure S4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' To achieve this, initially, a 2:1 mixture of H2O2-HCl solution was flushed for 30 sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This was followed by flushing of 10 wt% of PDADMAC solution for 2 min and later by 5 wt% of PSS solution for another 2 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' After every step, MilliQ® water was flushed for 30 sec to remove excess material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Thus, flushed solutions form a hydrophilic 14 polyelectrolyte layer on top of the hydrophobic PDMS surface along the OA solution inlet to the outlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Production of Double and Multi-core emulsions Briefly, the inner aqueous solution (IA) is flushed through the first cross-junction to form a water-in-oil (W/O) emulsion, followed by a second shearing step at the second cross-junc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' This results in the formation of water-in-oil-in-water (W/O/W) double emulsion with a single inner aqueous core surrounded by styrene (S) which is stabilized through F108 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='5 wt%) containing outer aqueous solution (OA) (see the supplementary figure S4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Further- more, to form multi-core emulsions, the swelling property of the PDMS in the presence of styrene is exploited to reproducibly narrow the channels at the first junction (at its lowest dimension) where water-in-oil (W/O) droplets are created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' A careful alteration of the flow pressures resulted in controlling the number of aqueous droplets that get encased inside the final double emulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Using this property, double emulsions with double, triple, quadru- ple, and quintuple cores are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Since styrene is used as the oil phase, thus produced droplets can be polymerized to yield polymeric styrene microspheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' For fluid flow control, four-channel pressure devices are used (MFCS-EZ, Fluigent Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Emulsion polymerization Double and multi-core emulsions prepared with styrene:octanol (95:5 %) oil phase dissolved with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='5 wt% BAPO, were placed under UV illumination (395-400 nm, custom made device, 50W LED chips (Foxpic High Power 50 W LED Chip Bulb Light DIY White 3800LM 6500 K) and 30 W UV chip (Fdit, 395-400 nm UV LED chip) were connected to a self-made circuit and cooling system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=') for 4 hours for complete polymerization of styrene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Produced micro- particles were washed via centirifugation before imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Freeze drying was performed be- fore scanning electron microscopy analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Microscopy Microfluidic production of multi-core emulsions is recorded with a MicroLab 310 camera at full-frame and ~3000 fps (Vision Research Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=') that is connected to wide-field Olympus IX73 microscope using a x5 objective in bright-field transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Monochrome confocal images are acquired with Leica TCS SP8 (Leica Microsystems Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=') confocal microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Colour images with Nikon DS-Fi3 high definition camera fitted to a wide-field Olympus IX73 microscope using a x40 objective in bright-field transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Additionally, red (580/30 nm), green (510/30 nm) and blue (420/30 nm) filters are also used to image the spectral separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' In all the cases, the 50 µL of emulsion suspensions were pipetted on to a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='17 mm glass coverslip fitted with imaging spacers (SecurteSealTM) for imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' General image processing is performed using ImageJ/Fiji, z-stack arrays with Huygens Professional (Sci- entific Volume Imaging Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='), and 3D rendering of confocal z-stacks with LAS X Core (Leica) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Ray tracing 15 Ray tracing simulations are performed using Ray-Optics Simulation software25 and COM- SOL Multiphysics® (COMSOL AB, Stockholm, Sweden).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Dimensions of the droplets and emulsions simulated were obtained from the microscopic images of the emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Further- more, the COMSOL ray tracing box setup with dimensions is visualized in the supplementary figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Supplementary Information for Rainbows in a bottle: Realizing microoptic effects by polymerizable multiple emulsion particle design Naresh Yandrapalliab*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Baris Kumrua,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Tom Robinsonb*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Markus Antoniettia a Max Planck Institute of Colloids and Interfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Department of Colloid Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Am Mühlenberg 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 14424 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Germany b Max Planck Institute of Colloids and Interfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Department of Theory & Bio-Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Am Mühlenberg 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 14424 Potsdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Germany Materials All materials were used as purchased unless noted otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 1-octanol (99 %, Sigma Aldrich), phenylbis(2,4,6-trimethylbenzoyl)phosphine oxide (BAPO initiator, 97%, Sigma Aldrich), Synperonic® F 108 surfactant (Sigma Aldrich).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Polydi-methylsiloxane (PDMS) and curing agent were obtained as SYLGARD® 184 silicone elastomer kit from Dow Corning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 1H,1H,2H,2H-Perfluorodecyltrichlorosilane was purchased from abcr GmbH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Poly(diallyldimethylammonium chloride (PDADMAC) and poly(sodium 4-styrenesulfonate (PSS) were obtained from Sigma Aldrich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' SU8 2025 (Microchem Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='), Silicon wafer (Siegert Wafers), SU8 developer solution (Microchem Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=') Styrene (99 %, Sigma Aldrich) was passed through alumina column to remove inhibitor before use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Figure S1: Bright-field z-stack of the spectral iridescence from 2-in-1, 3-in-1, 4-in-1, and 5-in-1 multi-core emulsions with each slice separated by 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Figure S2: Comparative 2D ray-tracing of multiple emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Ray tracing data reveals the formation of multiple focal points from single light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 1 represents primary focusing point and 2- secondary focusing point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' S – Corresponds to Styrene pphase and W – corresponds to aqueous phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 1 in 1 2 in 1 3 in 1 4 in 1 5 in 1 s s S S S A W W W W Direction of light X axis Figure S3: Mechanism for the formation of curved microspheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' 3 4 5 2 2 interface collapse andfusion to form curved surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' UV induced polymerization Styrene Aquoeus solution F108surfactant Polystyrene Figure S4: Microfluidic chip design with flow directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (a) Inlet and outlets for successful surface passivation of the microfluidic chip with solutions flowing from outlet to the OA solution inlet and (b) showing multiple inlets and outlet with respective solutions and their flow direction to produce emulsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' (a) Coating Pressure solutions connection goes in here goes here (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content='05 bars) (q) 2nd Junction Outlet Outer Aqueous (OA) Inner Aqueous (IA) solution solution inlet Middle inlet styrene inlet Figure S5: 3D COMSOL raytracing setup with example double emulsion with two inner cores (2-in-1) surrounded by water box topped with a glass coverslip toward the direction of the point light source positioned at (-165,0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' μm 50 100 0 50 50 μm 100 0 150 100 50 wn 50 0 50 100 100Supplementary Video 1 Z-stack video of 1-in-1 W/O/W double emulsion produced using a microfluidic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Scale bar corresponds to 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Supplementary Video 2 Comsol Multiphysics® ray-tracing simulation of ray-path (1000 rays) and dispersion of light along 1-in-1 W/O/W double emulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Colour legend depicts the wavelength of dispersed light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Supplementary Video 3 Microfluidic production of 2-in-1 (IA - 66 mbar, Styrene – 102 mbar, OA – 112 mbar), 3-in- 1 (IA - 66 mbar, Styrene – 106 mbar, OA – 105 mbar) , 4-in-1 ( IA - 66 mbar, Styrene – 104 mbar, OA – 94 mbar ) and 5-in-1 (IA - 66 mbar, Styrene – 105 mbar, OA – 86 mbar) multi-emulsions containing aqueous inner and outer solution and middle styrene solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Image sequence was acquired using high speed camera at ~3000 fps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Supplementary Video 4 Z-stack video of 2-in-1 W/O/W double emulsion produced using a microfluidic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Scale bar corresponds to 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Supplementary Video 5 Z-stack video of 3-in-1 & 4-in-1 W/O/W double emulsion produced using a microfluidic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Scale bar corresponds to 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Supplementary Video 6 Z-stack video of 5-in-1 W/O/W double emulsion produced using a microfluidic device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} +page_content=' Scale bar corresponds to 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9A0T4oBgHgl3EQfDf--/content/2301.02005v1.pdf'} diff --git a/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf b/QdFJT4oBgHgl3EQfJizI/content/2301.11461v1.pdf new file mode 100644 index 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b/R9E2T4oBgHgl3EQfBwbf/content/tmp_files/2301.03607v1.pdf.txt @@ -0,0 +1,2748 @@ +Prepared for submission to JHEP +MPP-2022-142 +Fits of αs using power corrections in the three-jet region +Paolo Nason,a,b Giulia Zanderighib,c +aUniversit`a di Milano-Bicocca and INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3,20126 +Milano, Italy +bMax-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany +cPhysik-Department, Technische Universit¨at M¨unchen, James-Franck-Strasse 1, 85748 Garching, +Germany +E-mail: paolo.nason@mib.infn.it, zanderi@mpp.mpg.de +Abstract: In this work we study the impact of recent findings regarding non-perturbative +corrections in the three-jet region to e+e− hadronic observables, by performing a simul- +taneous fit of the strong coupling constant αs and the non-perturbative parameter α0. +We extend the calculation of these power corrections, already known for thrust and C- +parameter, to other e+e− hadronic observables. We find that for some observables the +non-perturbative corrections are reasonably well behaved in the two-jet limit, while for +others they have a more problematic behaviour. If one limits the fit to the three-jet region +and to the well-behaved observables, one finds in general very good results, with the ex- +tracted value of αs agreeing well with the world average. This is the case in particular for +the thrust and C-parameter for which notably small values of αs have been reported when +non-perturbative corrections have been computed using analytic methods. Furthermore, +the more problematic variables are also well described provided one stays far enough from +the two-jet limit, while in this same region they cannot be described using the traditional +implementation of power-corrections based on two-jet kinematics. +Keywords: Perturbative QCD, QCD Phenomenology, electron-positron scattering +arXiv:2301.03607v1 [hep-ph] 9 Jan 2023 + +Contents +1 +Introduction +1 +2 +Observable definitions +4 +3 +Hadron mass ambiguities +6 +4 +Power correction calculation +6 +4.1 +Thrust +8 +4.2 +Other observables +10 +4.3 +The shift in the cumulative cross section +12 +4.4 +Numerical checks +14 +5 +Calculation of the observable distributions +15 +6 +Fit to ALEPH data +18 +6.1 +Treatment of uncertainties +18 +6.1.1 +Statistical and systematic errors, and correlations +18 +6.1.2 +Perturbative theory uncertainties +19 +6.1.3 +Non-perturbative theory uncertainty +20 +6.2 +Correction for heavy-quark mass effects +20 +6.3 +Hadron mass-effects corrections +20 +7 +Fit results +21 +7.1 +Including higher energy data +23 +7.2 +Discussion of the results +23 +7.3 +Comparison to results obtained by setting ζ(v) = ζ2J(v) +25 +7.4 +On the structure of αsλ/Q corrections +25 +8 +Conclusions +29 +A Impact of resummation +32 +1 +Introduction +The study of shape variables in e+e− annihilation is one of the simplest contexts in which +to test perturbative QCD, and it is potentially among the cleanest frameworks where +one can measure the strong coupling constant αs at high energy by probing directly the +quark-antiquark-gluon vertex. Shape variables have been computed up to order α3 +s [1– +4], and resummations near the two-jet region have been performed at different levels of +– 1 – + +accuracy, either using traditional resummation methods [5–12], or using Soft Collinear +Effective Theory (SCET) [13–16], leading to very precise predictions at high energies. +It is well known, however, that shape variables are affected by linearly suppressed +power corrections, i.e. of the order of Λ/Q, where Λ is a typical hadronic scale and Q is +the annihilation energy. Since in the 3-jet region the shape variables are of order αs, this +implies a relative error of order (Λ/Q)/αs, that affects at the same level the measured +value of αs. If we assume that Λ is of the order of 0.5 GeV (i.e. the typical additional +transverse energy per unit of rapidity due to hadronization), on the Z peak we estimate an +error of the order of 5%. In practice, power corrections can reach the 10% level for some +observables. +A commonly adopted approach for dealing with power corrections in the determina- +tions of the strong coupling constant from shape variables is to use Monte Carlo mod- +els [17–23]. +A shower Monte Carlo is used to construct a migration matrix for shape +variables computed from final-state hadrons, and from partons before hadronization. The +migration matrix is then applied to the measured differential distribution of hadrons to ob- +tain the shape distribution in terms of partons. This is in turn compared to perturbative +QCD, and a value of αs is extracted. This method is often criticized, because the Monte +Carlo hadronization model does not bear a clean relation to field-theoretical calculations. +An alternative strategy for the inclusion of power corrections makes use of analytic +approaches. In this case, the theoretical calculation including power corrections is compared +directly to the shape variable measurement using hadrons. These methods can be classified +into two broad classes. +One approach makes use of an effective coupling for the emission of very soft gluons +(called “gluers”) [24–27]. The average value of the effective coupling in a given low-energy +range plays the role of a parameter to be fitted to data together with the value of αs. The +Particle Data Group [57] (PDG) currently includes two fits of the strong coupling based on +NNLO+NLL [28] or NNLO+NNLL [29] accurate perturbative results combined with this +approach to the non-perturbative corrections. +This approach is also motivated by the large-nf limit of QCD (see [30] and references +therein), where the effective coupling can be actually computed. +It is argued that the +non-perturbative parameter in this contest is universal, i.e. it is the same for a large class +of shape variables. The coefficient of the power correction is computed by simply adding a +gluer to an initial q¯q state. For shape variables that are additive in soft radiation near the +two jet limit, the emission of the gluer acts as a shift in the value of the shape variable. This +behaviour is then extrapolated to the three-jet region, i.e. the non perturbative correction +is included as a shift in the argument of the shape variable computed in perturbation +theory. +The other approach relies upon factorization in QCD [31–34]. This begins with the +computation of the shape variables including resummation of the soft-collinear singular- +ities arising from gluon emission from the primary quark and antiquark. The region of +very soft emissions is parameterized by a shape function that is factorized out of the dis- +tribution. +In the three-jet region, a single moment of the shape function controls the +linear non-perturbative corrections. This approach arises naturally in SCET [35, 36]. Two +– 2 – + +determinations of αs(MZ) included in the PDG [37, 38] currently rely on such analytic +SCET-based approaches and notably lead to low values of the strong coupling accompa- +nied by small uncertainties. +A common feature of these two approaches is that they rely upon the extrapolation +of the non-perturbative correction from the two-jet to the three-jet limit. This extrap- +olation has been shown not to agree with the direct calculation of the non-perturbative +correction for the C parameter near the three-jet symmetric limit [39], where it leads to +an overestimate by approximately a factor of two. +In refs. [40, 41] it was shown that linear power corrections in the bulk of the three +parton final state region can be computed in large-nf QCD in the process e+e− → q¯qγ, +and, under some further assumptions, also in the e+e− → q¯qg process. In ref. [41] the +result for the C-parameter and thrust was given, but the method is quite general and can +be extended to a wide class of shape variables. In the case of the C-parameter it leads +to a result consistent with ref. [39] in the three-jet symmetric limit. +In general as for +the C-parameter case, one finds considerable violations of the assumption that the non- +perturbative correction can be implemented as a constant shift of the perturbative result +well into the three-jet region. +The purpose of this work is to investigate whether there are some indications that the +newly computed power corrections are preferred by available data. In order to do this, we +considered Z-peak data from the ALEPH experiment [42] that are publicly available on +HEPDATA and quite precise, and consider a set of shape variables such that the compu- +tation along the lines of ref. [41] can be carried out. Besides thrust and the C-parameter, +ref. [42] provides data for other shape variables for which we are in a position to compute +non-perturbative corrections in the three-jet region, namely the square mass of the heavy +hemisphere M2 +H, the difference of the squares masses of the heavy and light hemisphere M2 +D, +the broadening of the wide jet BW , and the 3-jet resolution parameter y3 in the Durham +scheme. In this work we have then computed the non-perturbative coefficients for M2 +H, M2 +D, +BW , and, with some caveats to be detailed in the following, also for y3. We thus supple- +ment the α3 +s calculation of these shape variables with the inclusion of the non-perturbative +corrections that we have computed as a shift in the argument of the cumulative cross sec- +tion Σ(v). More precisely, calling V a generic shape variable, defined in such a way that it +vanishes in the two jet limit, Σ(v) is defined as the cross section for producing events such +that V < v. In our approach, the shift in the argument is given by v → v −ζ(v)HNP, where +HNP is a coefficient suppressed by a power of Q, equal for all shape variables, and ζ(v) is a +shape-variable specific, dimensionless function. In contrast, in the traditional form of the +power corrections the variable-specific function ζ(v) is evaluated in the two-jet limit, where +it is in most cases replaced by a constant.1 +We stress that, somewhat unconventionally, we do not include resummation effects in +our result, while it is common practice to include them also very far away from the two-jet +limit. They generally lead to an increase of the shape variable distributions, and thus to +a smaller value of αs. We take here the point of view that if we consider ranges of the +1In the case of the broadening the shift is not a constant, see ref. [43]. +– 3 – + +shape variables that are far enough from the 2-jet limit, resummation can be neglected. +The reader may keep in mind that if resummation effects were included we would generally +obtain smaller values of αs. +A further reason for not including resummation in our result is that it is not clear +whether including the constant non-perturbative shifts in the singular contributions is an +acceptable procedure. In fact, such corrections would propagate into the three-jet region, +where (as we will see later) they sharply differ from their two-jet limit. Furthermore, in +this work we will not try to give a preferred value of αs with an error. Rather, our aim +is only to see whether and where the newly computed non-perturbative corrections are in +some way preferred by data, and to assess their impact. +The rest of the paper is organized as follows. In Sec. 2 we define the observables that +we consider in this work. In Sec. 3 we discuss ambiguities in the event-shape definitions +that arise when dealing with massive hadrons, as opposed to massless QCD partons, and +recall three alternative definitions that differ for massive hadrons but agree for massless +partons. +In Sec. 4 we present the calculation of the power-corrections in the three-jet +region for all observables considered in this work and show that they give rise to a non- +constant shift of the perturbative distribution. We also discuss numerical checks of the +analytic calculations. In Sec. 5 we discuss how to combine perturbative O(α3 +s) results with +non-perturbative corrections. In particular, we define various schemes that differ by higher +order terms. In Sec. 6 we discuss our treatment of uncertainties and correlations, as well as +the corrections that we apply to account for the heavy-quark masses. Finally, in Sec. 7 we +present the results of our fits of αs. We discuss various ambiguities and uncertainties, as +well as their difference from fits relying on the calculation of non-perturbative corrections +in the two-jet region. We conclude in Sec. 8. In App. A we discuss the impact of all-order +resummation effects for the observables used in our fit. +2 +Observable definitions +The choice of event shapes considered in this work is based on whether ALEPH data are +available for them, and whether their associated non-perturbative corrections in the three +jet region can be calculated along the lines of ref. [41], as discussed in detail in Sec. 4. +Unless otherwise specified, all sums in the definitions below run over all particles in the +event. +• The thrust T , or τ = 1 − T, is defined as +T = max +⃗nT +�� +i |⃗pi · ⃗nT | +� +i |⃗pi| +� +, +(2.1) +where the axis ⃗nT , that maximises the sum, is the thrust axis of the event. +• The heavy-jet mass: the plane through the origin of the event, orthogonal to the +thrust axis ⃗nT , divides each event into two hemispheres Hj (j = 1, 2), the invariant +– 4 – + +mass of each is defined as +M2 +j = +1 +E2 +vis +� +� � +pi∈Hj +pi +� +� +2 +, +j = 1, 2 , +(2.2) +where Evis = � +i Ei. The heavy-jet mass is the larger of the two +M2 +H = max +� +M2 +1 , M2 +2 +� +. +(2.3) +• The jet mass difference is defined as the difference between the larger and smaller +of the two masses +M2 +D = |M2 +1 − M2 +2 | . +(2.4) +• The C-parameter is computed from the three eigenvalues λi of the momentum +tensor Θαβ +Θαβ = +1 +� +i |pi| +� +i +pα +i pβ +i +|⃗pi| , +α, β = 1, 2, 3 , +(2.5) +as +C = 3 · (λ1λ2 + λ1λ3 + λ2λ3) . +(2.6) +• The wide broadening: given the thrust axis nT , the hemisphere broadenings Bj +(j = 1, 2) measure the amount of transverse momentum in each hemisphere +Bj = +� +pi∈Hj |⃗pi × ⃗nT | +2 � +i |⃗pi| +, +j = 1, 2 . +(2.7) +The wide broadening BW is the larger of the two hemisphere broadenings +BW = max (B1, B2) . +(2.8) +• The three-jet resolution y3: we take the Durham jet clustering, whose distance +measure reads +yij = +2 min(E2 +i , E2 +j )(1 − cos θij) +E2 +vis +. +(2.9) +(Pseudo)-jets are recombined sequencially summing the four-momenta of the pair of +particles with the smallest yij. The three-jet resolution y3 is defined as the value of +ycut for which an event changes from being classified as 2- to 3-jet. +The published data are already corrected using Monte Carlo generators in such a way that +all particles produced by the e+e− annihilation are included, comprising also the neutrinos +from meson decays. +– 5 – + +3 +Hadron mass ambiguities +When computing shape variables in perturbative QCD, one always deals with massless +partons. However, the measurements use the four-momenta of massive hadrons. It turns +out that shape variable definitions may differ for massive hadrons and be identical for +massless partons, and this introduces an ambiguity in the experimental definition of the +event shapes. This problem has been studied in detail in ref. [44] (see also [45]), where +three alternative schemes where suggested: the p-scheme, the E-scheme and the D-scheme. +In the p-scheme one uses only the three-momenta of the particles ⃗pi, and the energies Ei +are replaced by |⃗pi|. Instead, in the E-scheme the energies of the particles are preserved, +but the three-momenta are rescaled so as to have massless four-momenta, ⃗pi → ⃗pi · Ei +|⃗pi|. +It is clear that in the p-scheme energy conservation is violated, while in the E-scheme +the three momentum is not conserved, the violation being in both cases of the order of the +hadron masses. +In the so called D-scheme, final state hadrons are decayed isotropically in their rest +frame into two fictitious massless particles. The event shape is then computed using only +massless particles. This scheme has the advantage that the full four-momentum of the +event is conserved, and that no reference to a particular frame needs to be invoked in +its implementation. Notice also that it can happen that long-lived, unstable hadrons are +produced that decay to lighter particles. Therefore the event shape depends on the level +at which the measurement is performed, i.e. it becomes relevant whether the measurement +is performed before or after these decays. Unlike all other schemes, the D-scheme has the +advantage that it is rather insensitive to the particular hadron level chosen to perform the +measurement [44]. +In ref. [44], the advantages and disadvantages of each of these schemes are discussed. +In particular it is argued that in the E-scheme non-universal mass effects are absent. +The arguments used there are based upon an analysis near the two-jet limit, and their +applicability to the case of three widely separated jets is unclear. One may also argue that +the D-scheme should be preferred, since it mimics to some extent the models of hadron +formation. In the present work we will adopt the E-scheme as our default choice and use +the additional three schemes to gauge the hadron-mass sensitivity of our results. +4 +Power correction calculation +According to ref. [41], provided an event shape satisfies specific conditions, as explained in +detail later, its power correction in the three-jet region can be computed according to the +formula +[Σ(v)]NP = +� +� +� +� +dσB(ΦB)δ(v(ΦB) − v) +� +dip +� +−M × 4αsCdip +2π +1 +Q +� +dη dφ +2π hv(η, φ) +�� +� +� × INP, +(4.1) +where Q is the total center of mass (CM) energy and the sum runs over all radiating dipoles +associated with the given Born configuration. Thus, for the two jet case there is just a +– 6 – + +single q¯q dipole, while for the three-jet case we have a q¯q, qg and ¯qg dipole.2 We stress +that the function hv depends also upon ΦB, and that for ease of notation we do not show +explicitly this dependence. The colour coefficients Cdip for the three-jet case are given by +Cq¯q = CF − CA +2 , +Cqg = C¯qg = CA +2 . +(4.2) +The Milan factor M is given in analytic form in ref. [46] +M = 3 +64 +(128π(1 + log 2) − 35π2)CA − 10π2TRnF +11CA − 4TRnF +, +(4.3) +that agrees with the numerical result given earlier in ref. [27] +M = 1 + (1.575CA − 0.104nf)/β0, +(4.4) +where β0 = (11CA − 4nfTR)/3. The coefficient INP depends upon the model used to im- +plement power corrections. In the large-nF theory, it has the expression (see e.g. Ref. [47]) +INP = +1 +b0,nf αs(µ) +� µC +0 +dλ +π arctan +πb0,nf αs(µ) +1 + b0,nf αs(µ) log λ2e−5/3 +µ2 += +1 +αs(µ) +� µC +0 +dλarctan(πb0,nf αs(λe−5/6)) +πb0,nf +, +(4.5) +where b0,nf = −nf/(6π). The upper limit of integration in eq. (4.5) is quite arbitrary. It +should be large enough to cover the region where the argument of the arctangent diverges, +corresponding to the Landau pole. In phenomenological models INP is replaced by the +integral over a non-perturbative effective coupling, given as function of a scale λ. +The function hv(η, φ) depends upon the shape variable. It is defined as +hv(η, φ) = lim +|l⊥|→0 +1 +|l⊥| (v({P} , l) − v({p})) , +(4.6) +where {P} denote the momenta of the hard final state partons after the radiation of a soft +massless parton of momentum l, and {p} denote the momenta of the final state partons +in the absence of radiation. The arguments η and φ are the rapidity and azimuth of the +soft parton, and l⊥ denotes its transverse momentum, all evaluated in the rest frame of +the radiating dipole. The mapping from {P} and l to {p} must have certain smoothness +properties, namely the momenta {P} must be functions of {p} and l that are linear in l +for small l. +There are two further requirements for formula (4.1) to hold. The first one is that it +applies to variables that are additive in the emission of more than one soft parton in the +three-jet region. This property is violated by y3, as discussed in the following. The second +one is that the function hv(η, φ), after azimuthal integration, should yield a convergent +integral in η. This property is violated, for example, by the total broadening, and that is +the reason why we do not consider it in this work. +2The same formula is also applicable to higher multiplicity Born processes, that we do not consider here. +– 7 – + +Notice that in the large nf limit the Milan factor becomes equal to 15π2/128, and +the expression in the curly bracket of eq. (4.1) becomes equal to eq. (4.7) of Ref. [41], up +to the λ factor. In fact, according to eq. (A.1) of ref. [41], the linear non-perturbative +correction to an observable in the large nf limit is proportional to the first order coefficient +of its expansion in λ, where λ is a (fictitious) gluon mass introduced in the calculation, +multiplied by the factor given in eq. (4.5). +In ref. [41] the integration in η and φ was performed analytically for the C-parameter +and for thrust. Here we have set up a numerical code to perform the η and φ integration +numerically, since a sufficient precision can be easily reached, and this allows us to add +more observables with relatively minor effort. +4.1 +Thrust +We illustrate now how the hv(η, φ) function is computed in our code, using thrust as an +example. We generate the Born momenta {p} according to the three-body phase space. +Let us assume for definiteness that p1, p2 is the radiating dipole. We generate η and φ, +and construct the four-vector +l = l+ + l− + l⊥, +l+ = +p1 +√2p1 · p2 +exp(η), +l− = +p2 +√2p1 · p2 +exp(−η), +(4.7) +where +l⊥ · l+ = 0, +l⊥ · l− = 0, +l2 +⊥ = −1, +(4.8) +and l⊥ has an azimuthal angle φ relative to the p1/2 axis in the dipole rest frame. Notice +that by construction l2 = 0. +Let us call ⃗t0 the trust axis in the CM frame, defined to have the direction of the largest +⃗pi, i = 1 . . . 3. The thrust variation due to the emission of a parton with momentum λl is +given by +δτ = −δT = − 1 +Q +� +�max +⃗t +� +� � +i=1,3 +|⃗Pi · ⃗t| + λ|⃗l · ⃗t| +� +� − +� +i=1,3 +|⃗pi · ⃗t0| +� +� . +(4.9) +We need to expand this expression for small λ, keeping only the linear terms. We have +three terms +δT = +� +i=1,3 |(⃗pi + δ ⃗Pi) · ⃗t0| − � +i=1,3 |⃗pi · ⃗t0| +Q ++ +� +i=1,3 |⃗pi · (⃗t0 + δ⃗t)| − � +i=1,3 |⃗pi · ⃗t0| +Q ++ λ|⃗l · ⃗t0| +Q +. +(4.10) +– 8 – + +The second line of eq. (4.10) can be worked out as follows. We must have δ⃗t · ⃗t0 = 0, +since ⃗t has fixed length. Thus we have δ⃗t · ⃗pk = 0, where k is the hardest parton. For the +remaining two partons, with i, j ̸= k we have +|⃗pi · (⃗t0 + δ⃗t)| = |⃗pi · ⃗t0| × +� +1 + δ⃗t · ⃗pi +⃗pi · ⃗t0 +� += |⃗pi · ⃗t0| − δ⃗t · ⃗pi, +(4.11) +where we have used the fact that ⃗pi · ⃗t0 < 0 for i ̸= k. Thus +� +i=1,3 |⃗pi · (⃗t0 + δ⃗t)| − � +i=1,3 |⃗pi · ⃗t0| +Q += − 1 +Qδ⃗t · ( +� +i̸=k +⃗pi) = 1 +Qδ⃗t · ⃗pk = 0, +(4.12) +since δ⃗t is orthogonal to ⃗t0 and thus to ⃗pk. Thus only the terms in the first and last line +of eq. (4.10) contribute. The first line is linear in δPi, and thus (in an appropriate recoil +scheme) also in l. It must have the form +λ +Q(A exp(η) + B exp(−η) + C sin φ + D cos φ) +(4.13) +with A, B, C and D depending only upon the Born kinematics. The full result is +δT = λ +Q(|⃗l · ⃗t0| + A exp(η) + B exp(−η) + C sin φ + D cos φ). +(4.14) +The above expression must however not lead to a divergent integral for large rapidity. +Looking, for example, at the large η limit of the above expression (see eqs. (4.7)), we have +|⃗l · ⃗t0| = +���� +⃗p1 · ⃗t0 +√2p1 · p2 +exp(η) + +⃗p2 · ⃗t0 +√2p1 · p2 +exp(−η) +⃗l⊥ · ⃗t0 +���� +(4.15) += +���� +⃗p1 · ⃗t0 +√2p1 · p2 +���� +� +exp(η) + ⃗p2 · ⃗t0 +⃗p1 · ⃗t0 +exp(−η) + +⃗l⊥ · ⃗t0 +⃗p1 · ⃗t0 +� +(4.16) +We thus see that by choosing +A = − +���� +⃗p1 · ⃗t0 +√2p1 · p2 +���� +(4.17) +we cancel that exponential growth in η. With an analogous choice for B we can cancel the +exponential divergence for η → −∞. Terms with constant behaviour for large η do remain, +but they cancel after azimuthal integration. Thus, our final expression for the hv function +for τ is obtained by changing sign to the previous expression, +hτ(η, φ) = −hT (η, φ) = −|⃗l · ⃗t0| + |⃗l+ · ⃗t0| + |⃗l− · ⃗t0|. +(4.18) +In order to explicitly get rid of the constant φ-dependent term, in the numerical integration +process we sum the two contributions obtained with the replacement φ → φ + π. +– 9 – + +4.2 +Other observables +With a similar procedures we find the expression of hv for all shape variables of our interest, +for which we report here only the final results. For the C parameter, starting from the +expression +C = 3 − 3 +2 +� +i,j +(pi · pj)2 +(pi · q)(pj · q), +(4.19) +valid for massless partons, we obtain +hC(η, φ) = −3 +3 +� +i=1 +� (l · pi)2 +l · q pi · q − (l+ · pi)2 +l+ · q pi · q − (l− · pi)2 +l− · q pi · q +� +, +(4.20) +where q = � +i pi. +The negative terms in the square bracket of eq. (4.20) are there to +cancel the divergent rapidity behaviour of the positive term, and are clearly linear in the +momentum components of l. +For the heavy-jet mass we find +hM2 +H(η, φ) =θ(t0 · l) +�q · l +Q (2 − T0) − T0 t0 · l +� +− θ(t0 · l+) +�q · l+ +Q +(2 − T0) − T0 t0 · l+ +� +− θ(t0 · l−) +�q · l− +Q +(2 − T0) − T0 t0 · l− +� +, +(4.21) +where T0 stands for the value of the thrust at Born level, and, as before, the vector t0 is +obtained by adding a zero time-component to the thrust three-vector. Also in this case the +subtraction terms are clearly identified. Notice that the theta functions involving l+ and +l− are actually independent upon l, since +θ(±t0 · l+/−) = θ(±t0 · p1/2) . +(4.22) +The light jet mass is given by +hM2 +l (η, φ) =θ(−t0 · l) T0 +� +t0 · l + q · l +Q +� +− θ(−t0 · l+) T0 +� +t0 · l+ + q · l+ +Q +� +− θ(t0 · l−) T0 +� +t0 · l− + q · l− +Q +� +. +(4.23) +From the heavy- and light-jet mass we also obtain the mass difference +hM2 +D(η, φ) = hM2 +H(η, φ) − hM2 +l (η, φ). +(4.24) +The wide jet broadening is given by +hBW (η, φ) =θ(t0 · l)1 +2 +��l · q +Q +�2 +− (t0 · l)2 − θ(t0 · l+)1 +2 +��l+ · q +Q +�2 +− (t0 · l+)2 +− θ(t0 · l−)1 +2 +��l− · q +Q +�2 +− (t · l−)2 ++ +3 +� +i=1 +θ(t · pi)θ(−t · l) (⃗l · ⃗pi)t · pi − l · t(t · pi)2 +T0 +� +(pi · q)2 − Q2(pi · t)2 . +(4.25) +– 10 – + +The first two lines in eq. (4.25) represent the variation in BW at fixed thrust axis, and the +last line is the contribution due to the fact that if the emission is in the hemisphere of the +hardest parton, the thrust axis is tilted, and this affects BW . Notice also that while the +first term requires a subtraction (the two following terms), the last term does not. In fact, +l cannot be collinear with the partons opposite to the hardest one. Thus, assuming for +example that parton p1 is in the hemisphere opposite to the hardest parton, there will be a +cut-off for large values of η. Therefore, large values of η will only be allowed if l is collinear +to the hardest parton, i.e. the one aligned with the thrust axis. In this case however, it is +easy to check that the numerator in the last line of eq. (4.25) vanishes. +For the calculation of y3, we assume first that the two soft partons arising from gluon +splitting are not the first pair to be clustered together. Under this assumption, also y3 +becomes additive in the soft partons, and the calculation can be done in analogy with the +other variables. Assume for definiteness that, at the Born level, the hard partons pair +yielding the smallest y3 is given by the parton labels j, k, that the remaining parton is +labeled i, and that p0 +k < p0 +j. Then the change in y3 is given by +hy3(η, φ) = +� +θ(dk,l < min(dj,l, di,l))2 +� +2p0 +kl0(1 − cos ψkj) − (p0 +k)2 +� ⃗l · ⃗pj +|⃗pj||⃗pk| − ⃗pj · ⃗pk ⃗pk ·⃗l +|⃗pj||⃗pk|3 +� � ++ θ(dj,l < min(dk,l, di,l))2 +� +− (p0 +k)2 +� ⃗l · ⃗pk +|⃗pk||⃗pj| − ⃗pj · ⃗pk ⃗pj ·⃗l +|⃗pk||⃗pj|3 +� � +− θ(dk,l+ < min(dj,l+, di,l+))2(2p0 +k(l+)0)(1 − cos ψkj) +− θ(dk,l− < min(dj,l−, di,l−))2(2p0 +k(l−)0)(1 − cos ψkj) +� +(4.26) +where +dh,l = 1 − ⃗ph ·⃗l +|⃗ph||⃗l| +, +and +cos ψkj = ⃗pk · ⃗pj +|⃗pk||⃗pj|. +(4.27) +The term proportional to p0 +kl0 is due the change in the energy of parton k when it combines +with l, while the terms proportional to (p0 +k)2 are due to the change in the angle between +parton k combined with l and parton j (the first instance), and between parton j combined +with l and parton k (second instance). Notice that there are no subtractions associated with +the change in angle. In fact, because of the theta functions, the only collinear singularity +that can arise in this case is when l is collinear to j or k, but then the angle does not +change. +As stated earlier, the y3 variable is really not additive, i.e. the y3 modification due to +several soft emissions is not the sum of the modifications due to each emission since partons +can be clustered together.3 In order to estimate the magnitude of the error associated with +this assumption, it is interesting to compute the non-perturbative correction to y3 also in +the case when the two partons are always clustered together. In this case formula (4.26) +still holds, with l equal to the total momentum of the pair of partons, l2 = λ2, and, in the +3This is also discussed in ref. [48]. We thank Andrea Banfi for pointing this out to us. +– 11 – + +left hand side, hy3(η, φ) is replaced by hy3(l). The non-perturbative correction can then be +written as +[Σ(v)]NP = +� +� +� +� +dσB(ΦB)δ(v(ΦB) − v) +� +dip +� +4αsCdip +2π +1 +Q +� +dy dφ +4π +dl2 +⊥ +l2 +⊥ + λ2 hy3(l) +�� +� +� +λ +× INP, +(4.28) +where the suffix λ in the closing curly bracket indicates that we should extract the coefficient +of the term proportional to λ in the enclosed expression. We will use formula (4.28) in the +following to assess the error due to our approximation in eq. (4.26). +4.3 +The shift in the cumulative cross section +In eq. (4.1) we have given the formula for the non-perturbative correction to the leading +order 3-jet cross section. It is customary to express the non-perturbative correction as a +shift in Σ(v), i.e. to write +ΣB+NP(v) − ΣB(v) = ΣB (v − δv) − ΣB(v) = −dσB +dv δv, +δv = HNPζ(v), +(4.29) +where ΣB(v) is the Born level value +ΣB(v) = +� v +0 +dσB +dv dv, +(4.30) +and using eq. (4.1) for the left-hand side of eq. (4.29), we thus define +ζ(v) = +�dσB +dv +�−1 +� +� +� +� +dσB(ΦB)δ(v(ΦB) − v) +� +�� +dip +Cdip +CF +� +dη dφ +2π hv(η, φ) +� +� +� +� +� , +(4.31) +HNP = M × 4αsCF +2π +× INP +Q . +(4.32) +With the above normalization, the shift function ζ in the two-jet case assumes the values +3π for the C parameter, 2 for τ = 1 − T, 1 for M2 +H and 0 for M2 +D and y3. For the wide +jet broadening in the two-jet limit the linear λ term is actually accompanied by a log λ/Q, +and thus the linear term does not have a finite coefficient. +We computed the functions ζ(v) for the variables listed above. The results are displayed +in Fig. 1. With an angular-ordering argument, one can show that the limit ζ(v) for v → 0 +should tend to the corresponding two-jet limit values. In fact, in this limit, the emitted +hard gluon becomes collinear to either the quark or the antiquark, let us say to the quark +for sake of discussion, as shown in Fig. 2. Because of coherence, the soft gluon associated +with the power corrections sees the collinear quark-gluon pair as a single colour source, with +the same colour of q. Thus the emission pattern is the same as that of a quark-antiquark +pair. Alternatively, one may consider the emissions of the three dipoles q¯q, qg, and ¯qg, +that carry the colour factors CF − CA/2 for q¯q and CA/2 for qg and ¯qg. The qg dipole does +not emit in the small angle limit (the eikonal formula vanishes there), and the ¯qg dipole +becomes equal to the q¯q dipole, giving CF −CA/2+CA/2 = CF, i.e. the same soft radiation +– 12 – + + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 0 + 0.1 0.2 0.3 0.4 0.5 0.6 0.7 +ζ(C) +C + 0 + 0.5 + 1 + 1.5 + 2 + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +ζ(τ) +τ +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 + 0 + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +ζ(y3) +y3 +-2 +-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +ζ(M2H) +M2H +-4 +-3.5 +-3 +-2.5 +-2 +-1.5 +-1 +-0.5 + 0 + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +ζ(M2D) +M2D +-1 +-0.8 +-0.6 +-0.4 +-0.2 + 0 + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +ζ(BW) +BW +Figure 1: The function ζ plotted for C, 1 − T, y3, M2 +H, M2 +D and BW . +of a q¯q dipole. This must happen, however, when the logarithm of the shape variable is +so large that it clearly prevails over single logs and constant terms. In the case of C and +1 − T, one finds that for values of the shape variable v ≈ 10−3 the ζ function differs from +the two-jet limit value by roughly 10%, i.e. of the order of 1/ log(v), that is the natural +size of single-log corrections. +The case of M2 +H and M2 +D, however, are much more extreme. In this case, in order to +– 13 – + +Figure 2: Dominant double logarithmic region near the two jet limit. The qg dipole does +not radiate, while the q¯q and ¯qg dipoles differ only by their colour factor. +-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 1.5 + 0 + 0.05 0.1 0.15 0.2 0.25 0.3 +ζ(M2H) +M2H +-1.5 +-1 +-0.5 + 0 + 0.5 + 1 + 1.5 +10-12 +10-10 +10-8 +10-6 +10-4 +10-2 +M2H +Figure 3: The ζ(M2 +H) function at very small value of its argument. The dots are obtained +by performing a quadruple precision calculation and binning the results uniformly in a +logarithmic scale. The left/right plot use a linear/logarithmic scale for the x axis. +check that the two jet limit of 1 and 0 respectively are actually reached, we had to perform +a dedicated calculation in quadruple precision in the small v region. +As an example, +we show in Fig. 3 the result of this calculation for M2 +H. It is evident that M2 +H changes +sign and reaches the value 1 very near zero, varying by about 2 units in a very narrow +neighbourhood around zero. M2 +D undergoes an even stronger variation, changing by three +units, and reaching zero from negative values. +Such an abrupt change in the three-jet +distribution as we approach the two-jet limit suggests that subleading soft terms in the +two-jet limit remain more important than double logarithms all the way down to very small +values of the shape variable, questioning on one side the possibility to associate the two-jet +limit non-perturbative correction to the resummation of soft radiation, and, on the other +side, the application of our newly computed non perturbative correction as we approach +the two-jet limit. +4.4 +Numerical checks +As a numerical check of the above calculations we also computed the ζ functions by directly +generating the phase space comprising the three hard partons and the soft one, fixing its +transverse momentum to a value λ0 = Q0/100. +More explicitly, we first generate the +underlying Born momenta pi, i = 1 . . . 3, choose λ0 = 1 GeV and Q0 = 100 GeV, and +construct the momentum of the radiated parton as in eqs. (4.7) to (4.8). Assuming for +– 14 – + +sake of argument that p1 and p2 are the momenta of the radiating dipole, we construct the +recoil-corrected momenta as +P1 = p1 − l+ − 1 +2l⊥ , +P2 = p2 − l− − 1 +2l⊥ . +(4.33) +In this way the total momentum is conserved, and the on-shell property of P1/2 are main- +tained up to terms of order λ2/Q2 = 1/104. The event comprising P1, P2, p3 and l is then +used to compute directly the values of the shape variables, and its difference with respect +to the value obtained for momenta p1, p2 and p3 is computed. Using this method, we find +good agreement with the λ → 0 calculations described in the previous section, except near +the zero value of the shape variable and, in the case of the C parameter, near the upper +end-point of 3/4, i.e. the 3-jet symmetric limit. We will make use of this method to give +an estimate of corrections suppressed by higher powers of λ, as illustrated later. +5 +Calculation of the observable distributions +We are interested in fitting αs from event shapes in the three-jet region, where the novel +results for the non-perturbative corrections can be used. Furthermore, in the three-jet +region the relation between the observables and the value of αs is more direct. For this +reason, at the perturbative level we consider here only fixed-order predictions and, when +determining the fit range, we will make sure that all-order resummed predictions, not +included here, have a small effect. +Perturbative predictions for e+e− → 3 jets are available up to next-to-next-to-leading +order (NNLO) accuracy and are implemented in the public code EERAD3 [1–3], which +is based on the antenna subtraction formalism [49] and in a private code [4], which is +based on the CoLoRFulNNLO subtraction method [50]. We have used here predictions +from EERAD3 up to NNLO and have checked that they agree with predictions using the +CoLoRFulNNLO subtraction method up to NLO accuracy.4 +Denoting by v a generic event shape, the normalized integrated distribution at center- +of-mass energy Q and at the renormalization scale µR can be written as +ΣNNLO(v) = +� v +0 +dv′ +1 +σNNLO +dσNNLO(v′, Q) +dv′ += αs(µR) +2π +dA(v) +dv ++ +�αs(µR) +2π +�2 dB(v, xµR) +dv ++ +�αs(µR) +2π +�3 dC(v, xµR) +dv +, +(5.1) +where xR = µR/Q and +B(v, xµR) = B(v, 1) + A(v) (β0 ln xR − σ1) , +(5.2) +C(v, xµR) = C(v, 1) + B(v, 1) (2β0 ln xR − σ1) ++ A(v) +�1 +2β1 ln xR + β2 +0 ln2 xR + σ2 +1 − σ2 +� +, +4We thank Adam Kardos for providing results up to NLO using the CoLoRFulNNLO subtraction method. +– 15 – + +with β0 = (11CA − 4nfTR)/3, β1 = (34C2 +A − 20CAnfTR − 12CFTRnf)/3, and where the +expansion of the total cross section reads +σNNLO = σ0 +� +1 + αs(µR) +2π +σ1 + +�αs(µR) +2π +�2 +σ2 +� +, +(5.3) +with σ1 = 3CF/2 and σ2 = CF ((123/8 − 11ζ3)CA − 3/8CF + (4ζ3 − 11/2) nf TR). For our +central predictions we choose xR = 1/2, and we estimate the error due to missing higher- +order terms by varying this scale up and down by a factor of two. The choice of xR = 1/2 +for the central value is motivated by the fact that the scale entering in the production of +the third jet is somewhat lower than Q. +The non-perturbative corrections discussed in Sec. 4 can be included as a shift in the +argument of the cumulative cross section, i.e. according to eq. (4.29), but using instead +the full NNLO cross section. We now depart from the large-nf parameterization of the +shift, and switch instead to the dispersive model of ref. [27], where the role of the effective +coupling of eq. (4.5) is played by a parameter α0(µ2 +I). So, rather than using the definition +of eqs. (4.31) and (4.32), the shift (see eq. (4.29)) can be written as +δv = ζ(v)MµI +Q +4CF +π2 +� +α0(µ2 +I) − αs(µ2 +R) − α2 +s(µ2 +R)β0 +π +� +2 ln µR +µI ++ K(1) +2β0 ++ 2 +� +(5.4) +− α3 +s(µ2 +R)β2 +0 +π2 +� +4 ln2 µR +µI ++ 4 +� +ln µR +µI ++ 1 +�� +× +� +2 + β1 +2β2 +0 ++ K(1) +2β0 +� ++ K(2) +4β2 +0 +� +, (5.5) +where the observable dependent part ζ(v) has been discussed in detail in Sec. 4, the Milan +factor is given in eq. (4.4) and α0(µ2 +I) is defined as a mean value of the strong coupling in +the CMW [51] scheme below an infrared scale µI which is conventionally taken equal to 2 +GeV: +α0(µ2 +I) = 1 +µI +� µI +0 +dµ ˜αs(µ2) , +(5.6) +where in the perturbative region the MS and CMW couplings are related as +˜αs(µ2) = αs(µ2) +� +1 + αs(µ2) +2π +K(1) + +�αs(µ2) +2π +�2 +K(2) + O(α3 +s) +� +, +(5.7) +with [52, 53] +K(1) = CA +�67 +18 − π2 +6 +� +− 5 +9nf , +(5.8) +K(2) = C2 +A +�245 +24 − 67 +9 ζ2 + 11 +6 ζ3 + 11 +5 ζ2 +2 +� ++ CFnf +� +−55 +24 + 2ζ3 +� +(5.9) ++ CAnf +� +−209 +108 + 10 +9 ζ2 − 7 +3ζ3 +� +− 1 +27n2 +f + β0 +2 +� +CA +�808 +27 − 28ζ3 +� +− 224 +54 nf +� +. +The last terms in Eq. (5.5) are subtraction terms of contributions already accounted for in +the perturbative calculation. This assumes that non-inclusive corrections are described by +– 16 – + +the same multiplicative Milan factor M, that applies to all observables we consider with +the exception of y3, as discussed at the end of section 4.2. +Notice that in the large nf limit we found (see eq. (4.32)) +δv = HNPζ(v) = M × 4αsCF +2π ζ(v) × INP +Q , +(5.10) +where INP can be interpreted as the integral of the large-nf, CMW effective coupling. In +fact, expanding the second line of eq. (4.5) for small αs we find +INP = +1 +αs(µ) +� µC +0 +dλ αs(λe−5/6), +(5.11) +and +αs(µe−5/6) ≈ αs(µ) + 5 +3b0,nf αs(µ)2 = αs(µ) +� +1 − 5 +9nf +αs(µ) +2π +� +, +(5.12) +consistently with eqs. (5.7) and (5.8). +However, formula (5.10) differs by a factor π/2 with respect to eq. (5.5), i.e. +the +factor CF/(2π) is replaced by CF/π2 in eq. (5.5). This replacement (for more details see +ref. [27] near formula (6.3)) is irrelevant for the purposes of this work, but we follow this +prescription in order to fit values of α0 that can be compared to those found in previous +publications. +It is possible to implement the non-perturbative corrections in different ways, leading +to results that differ by terms of order O(αs/Q). +We use this ambiguity to assign an +uncertainty related to our treatment of non-perturbative corrections. For this purpose we +define four schemes. Our default predictions, scheme “(a)”, are obtained by shifting the +perturbative distribution ΣNNLO(v) by the non-perturbative correction computed in Sec. 4 +Σ(a)(v) = ΣNNLO(v − δv) . +(5.13) +Furthermore, in scheme (a), we also add to δv an approximate estimate of quadratic cor- +rections. These are obtained from the difference between the numerical evaluation at finite +transverse momentum described in Sec. 4.4 with respect to our standard calculation. More +specifically, calling ˜ζ(v) the evaluation of Sec. 4.4, we correct δ(v) as follows +δ(v) = ζ(v)HNP + +� +˜ζ(v) × Q0 +λ0 +− ζ(v) +� +× Q0 +λ0 +× H2 +NP . +(5.14) +Alternatively, instead of shifting the full NNLO distribution, one can shift only in the +leading order term ΣB of the integrated distribution (scheme (b)): +Σ(b) +FULL(v) = ΣB(v − δv) + Σ(v) − ΣB(v) . +(5.15) +Yet another option is to expand the integrated distribution around the perturbative value +(scheme (c)): +Σ(c) +FULL(v) = Σ(v) − δvΣB(v) +dv +. +(5.16) +Scheme (d) is defined as scheme (a) but without the quadratic correction of eq. (5.14) +included in the other schemes. +– 17 – + +6 +Fit to ALEPH data +We now compare the theoretical predictions including power corrections to the ALEPH +data of ref. [42], where several shape variables were analyzed in the centre-of-mass energy +range from 91.2 to 206 GeV. Here we focus upon the 91.2 GeV data. Including higher +energy data does not lead to noticeable differences in the results, as we will discuss briefly +in Sec. 7.1. +Our goal is to fit several observables at once. We need to select observables such that +the power corrections in the three jet region can be computed with our methods, and +that are at the same time available in ALEPH. These are the C-parameter, τ = 1 − T, +y3 in the Durham scheme, the heavy-jet mass M2 +H, the mass difference M2 +D, and the wide +jet broadening BW . Since the y3 variable is not really additive, we need to provide an +estimate of the error associated with this. We will do so along the lines discussed at the +end of Sec. 4.2. +The non-perturbative corrections to M2 +H, M2 +D and BW have a common feature: in the +3-jet regime they differ drastically from their value in the two-jet limit. Such an abrupt +change is quite worrisome, and may be taken as an indication that higher-order emissions +may be associated with large corrections to the non-perturbative coefficient. +For this +reason, initially we leave these variables out of the fit, and only fit the C-parameter, τ and +y3. We fit the value of αs(MZ) and the non-perturbative parameter α0, defined in Sec. 5. +6.1 +Treatment of uncertainties +6.1.1 +Statistical and systematic errors, and correlations +The ALEPH data (available at the site https://www.hepdata.net/record/ins636645) +includes statistical and systematic errors. Our method of choice for computing the error +is the following. Calling Ri the statistical error, Si the systematic error, Ti the theoretical +error relative to bin i, Cij the statistical correlation matrix, and Cov(Sys) +ij +the covariance +matrix for the systematic errors, we compute the full covariance matrix as +Vij = δij(R2 +i + T 2 +i ) + (1 − δij)CijRiRj + Cov(Sys) +ij +, +(6.1) +where the indices i and j run over all the bins of all observables that have been included +in the fit. The ALEPH data quotes two kinds of systematic errors for the data taken at +the Z pole. We add these two errors in quadrature to obtain the global systematic error +that we use in our analysis. +We computed the statistical correlation coefficients Cij using Pythia8. Calling Ni and +Nj the number of events that fall into bin i and bin j, and Nij the number of events that +contribute to both bins, we have +Cij = +Nij +N − NiNj +N2 +� +Ni +N − N2 +i +N2 +� +Nj +N − +N2 +j +N2 +, +(6.2) +where N is the total number of events. Note that Nij is zero for different bins of the same +observable, so that a negative correlation is expected for all pairs of bins in this case. +– 18 – + +Statistical, systematic and theoretical errors are assumed to be uncorrelated among +each other. For the covariance of the systematic errors we adopt the so called “minimum +overlap” assumption (denoted in the following as MO), and set them equal to the minimum +of the square of the systematic errors for the bins in question, i.e. +Cov(Sys) +ij += δijS2 +i + (1 − δij) min(S2 +i , S2 +j ) . +(6.3) +As an alternative, we computed the covariance matrix for the case of C, T and y3, by using +the 24 systematic variations of the resulting distributions that were obtained by ALEPH +in order to determine the systematic errors.5 We compute the covariance matrix and the +central value as follows. We call v(r) +i +the value of a shape variable for the bin i, where again +i denotes both the bin and the observable, and where r labels the 25 replicas (a central +value plus 24 variations.). We then define +¯vi = 1 +Nr +� +r +v(r) +i , +(6.4) +¯vij = 1 +Nr +� +r +v(r) +i v(r) +j , +(6.5) +Cov(Sys) +ij += +� +r +� +v(r) +i +− ¯vi +� � +v(r) +j +− ¯vj +� += Nr (¯vij − ¯vi¯vj) . +(6.6) +We use Cov(Sys) +ij +as covariance matrix, and for the central value we use either the replica +corresponding to the ALEPH default setup, or the average over all replicas ¯vi. +Some +of the variations provided are double sided (i.e. they are associated with a positive and +negative variation of a parameter). For these variations we have included a factor 1/2 in +the computation of ¯vij. In the following we call this the “replica method”, and denote it +with R. +The covariance matrix is used to compute the χ2 value according to the standard +formula +χ2 = +� +ij +� +vi − v(th) +i +� +Vij +� +vj − v(th) +j +� +. +(6.7) +6.1.2 +Perturbative theory uncertainties +As a consequence of the high precision of the LEPI data, in order to obtain reasonable χ2 +values when performing the fits we add the theoretical uncertainty in quadrature to the +experimental one. We do not assume any correlations for the theoretical errors. +We define the perturbative theoretical error by considering three values for the renor- +malization scale: Q, Q/2 and Q/4. Calling O(µr) the value of a shape variable in a bin, +we define the perturbative central value Ocv and the associated perturbative error Oerr of +the theoretical prediction as follows, +Ocv = max(O(Q), O(Q/2), O(Q/4)) + min(O(Q), O(Q/2), O(Q/4)) +2 +, +(6.8) +Oerr = max(O(Q), O(Q/2), O(Q/4)) − min(O(Q), O(Q/2), O(Q/4)) +2 +. +(6.9) +5We thank Hasko Stenzel for providing these data to us. +– 19 – + +The perturbative theoretical error is quite small at the NNLO level we are working with. +6.1.3 +Non-perturbative theory uncertainty +Non-perturbative corrections can be sizeable, up to the order of 10%, and thus we must +also include an error associated with them. As seen in Sec. 4, there is evidence that power +suppressed corrections of second order are not negligible, especially near the two jet region. +We have estimated them and used them to correct the central value, see Eq. (5.14). We +thus associated an uncertainty equal to twice the quadratic correction. As a further point, +we expect that the ζ functions may receive perturbative corrections of order αs ∼ 0.1. We +thus define the following associated error to δ(v) +δerr(v) = 2 · +����˜ζ(v) × Q0 +λ0 +− ζ(v) +���� × Q0 +λ0 +× H2 +NP + 0.1 · δ(v) . +(6.10) +6.2 +Correction for heavy-quark mass effects +Our NNLO calculation deals with massless quarks, while the data includes primary charm +and bottom pairs. We correct the data by multiplying, for each bin of each observable +denoted globally as i, the ratio of the Monte Carlo predictions for the corresponding ob- +servables vi evaluated without and with the c and b primary production processes +v(corr) +i += vi × vMC,uds +i +vMC,udscb +i +. +(6.11) +The correction factors obtained using Pythia8 are shown in Fig. 4. Notice that corrections +are quite modest, although not totally negligible in some cases. +6.3 +Hadron mass-effects corrections +As already discussed in Sec. 3, the theoretical calculation of shape-variable distributions +deals with massless particles and the massless definition can be extended to deal with +massive particles using different schemes. Since full particle identification is not available +in an experimental context, this lack of information is filled by the Monte Carlo simulation +when correcting from the detector level to the generator level. +We also use a Monte +Carlo generator to compute shape variables in the different schemes, and then construct +migration matrices to correct from the scheme adopted by the experiment to any another +scheme. More specifically, for each Monte Carlo event, we compute the shape variable in +the standard scheme (the one adopted by the experiment, as defined in Sec. 2) and another +scheme S. Assuming that the shape variable in the standard scheme falls into bin i, and +the same shape variable in scheme S falls into bin j, we increase by one unit a migration +matrix Tij. This matrix is used to correct the real data according to the formula +n +(S) +j += +� +i +n +(data) +i +Tij +� +k Tik +, +(6.12) +designed in such a way that if one replaces the n +(data) +i +with its Monte Carlo prediction, one +obtains by construction the Monte Carlo prediction for n +(S) +j . In the following, we use this +method to assess the hadron-mass sensitivity of our results. +– 20 – + +0.96 +1 +1.04 + 0 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 + 0.35 + 0.4 +C/2 +0.96 +1 +1.04 +T +Shape variable +0.96 +1 +1.04 +MH2 +(1/σ dσ(u,d,s)/dv) / (1/σdσ(u,d,s,c,b)/dv) +0.96 +1 +1.04 +y3 +0.96 +1 +1.04 +MD2 +0.96 +1 +1.04 +BW +Figure 4: Heavy-flavours correction factors. The coloured band marks the range where +fits are usually performed. Notice that we plot C/2 rather than C. +7 +Fit results +Our default fit is based on the ALEPH data of ref. [42] at 91.2 GeV, and includes the thrust +variable τ = 1 − T, the C-parameter and the Durham 3-jet resolution variable y3. In our +perturbative predictions we fix the renormalization scale to µR = Q/2. Non-perturbative +effects are included as a shift of the total integrated distribution, corresponding to scheme +(a) in Eq. (5.13). Our default mass scheme is the E scheme discussed in Sec. 3, since +it yields intermediate results with respect to the other schemes, and is also closer to the +result obtained in the standard scheme (i.e. the scheme used by ALEPH, as defined in +Sec. 2). The treatment of correlations is described in Sec. 6.1.1. In particular, we chose +the minimum-overlap method as our default choice, see Eq. (6.3). We apply the heavy- +to-light correction factors described in Sec. 6.2 and illustrated in Fig. 4. We use Pythia8 +as our standard Monte Carlo to compute the heavy-to-light correction factor and, when +using a different mass scheme, to compute the migration matrix to be used to correct from +the scheme adopted by the experiment to any another scheme. +To perform our fit we +use the default fit ranges listed in the second column of Table 1. The lower edges of the +ranges are determined in such a way that the impact of the resummation remains small +(see Appendix A), while the upper edge is close to the three-particle kinematic bound of +the observable. The result of the simultaneous fit of αs and α0, together with the total χ2 +and χ2 per degree of freedom is shown in the first line of Table 2. +In the same table we illustrate how the fit results change if we vary any of the default +choice made. In particular, we show the fit results when fixing the central value of the +– 21 – + +observable +default +Fit ranges (2) +Fit ranges (3) +C +[ 0.25 : 0.6 ] +[ 0.17 : 0.6 ] +[ 0.375 : 0.6 ] +τ +[ 0.1 : 0.3 ] +[ 0.067 : 0.3 ] +[ 0.15 : 0.3 ] +y3 +[ 0.05 : 0.3 ] +[ 0.033 : 0.3 ] +[ 0.075 : 0.3 ] +Table 1: Default fit range used (second column), and alternative choices (obtained by +multiplying the default lower bound by 2/3 and 3/2) used to estimate the impact of the +choice of the fit range (third and forth column). +Variation +αs(MZ) +α0 +χ2 +χ2/Ndeg +Default setup +0.1182 +0.64 +7.3 +0.17 +Renormalization scale Q/4 +0.1202 +0.60 +9.1 +0.21 +Renormalization scale Q +0.1184 +0.68 +8.7 +0.20 +NP scheme (b) +0.1198 +0.77 +7.0 +0.16 +NP scheme (c) +0.1206 +0.80 +5.4 +0.12 +NP scheme (d) +0.1194 +0.66 +5.8 +0.13 +P-scheme +0.1158 +0.62 +10.7 +0.24 +D-scheme +0.1198 +0.79 +5.7 +0.13 +Standard scheme +0.1176 +0.58 +9.2 +0.21 +No heavy-to-light correction +0.1186 +0.67 +6.8 +0.16 +Herwig6 +0.1180 +0.59 +15.9 +0.36 +Herwig7 +0.1180 +0.60 +12.0 +0.27 +Ranges (2) +0.1174 +0.62 +12.7 +0.23 +Ranges (3) +0.1188 +0.69 +2.7 +0.08 +Replica method (around average) +0.1192 +0.61 +7.0 +0.16 +Replica method (around default) +0.1192 +0.61 +7.0 +0.16 +y3 clustered +0.1174 +0.66 +8.2 +0.19 +C +0.1256 +0.48 +1.3 +0.07 +τ +0.1194 +0.64 +0.8 +0.04 +y3 +0.1214 +1.81 +0.2 +0.02 +C, τ +0.1238 +0.51 +2.6 +0.07 +Table 2: Default fit result for αs(MZ) and α0 (first line) and other fit results obtained by +varying the setup. See text for more details. +renormalization scale to µR = Q/4 or Q. We investigate the impact of the way in which non- +perturbative corrections are implemented, using the alternative schemes (b, c, d) presented +in Sec. 5 (near Eq. (5.15)). We also present the result obtained using the P- and D- scheme +to define the observables, as discussed in Sec. 6.3, and the result obtained in the standard +scheme. To assess the impact of the heavy-to-light correction factor we switch it completely +off. We vary the Monte Carlo used to compute the migration matrix for the scheme and +– 22 – + +the heavy-to-light correction factor, and consider Herwig6 [54] and Herwig7 [55]. We vary +the fit ranges adopted, as detailed in columns three and four of Table 1. Since correlations +play an important role, we also use the replica method, see Eq. (6.6), using variations either +around the average values of the replicas, or around the values of the default replica. +In the case of y3 there is one further uncertainty, associated with the fact that we +computed the non-perturbative correction assuming that the two soft partons from the +splitting of the soft gluon are not clustered together. In order to estimate an associated +uncertainty, we also computed the non-perturbative correction assuming that the two soft +partons are always clustered together, see Eq. (4.28). The ratio of the latter to the former +results ranges from 0.7 up to 0.85 in the fit window adopted for y3. We have therefore +performed the fit using alternatively the approximation where the soft partons are always +clustered together. The corresponding result is reported in the table labeled as y3-clustered. +The central value for αs in the simultaneous fit of y3, C and T is reduced by 0.7%. Given +the fact that we have chosen the lowest extreme of the variation, and that the correct result +must lie between the always-clustered and the never-clustered cases, our estimate of this +uncertainty is very conservative.6 +Finally, we examine how the fit results change if we consider one observable at the +time, or if we exclude y3 from the fits. +7.1 +Including higher energy data +In the ALEPH publication [42], data are also available for centre-of-mass energies of 133, +161, 172, 183, 189, 200 and 206 GeV. Including these data does not appreciably change the +result of the fit. For our default setup we get αs(MZ) = 0.1184 and α0 = 0.64, compared +to αs(MZ) = 0.1182 and α0 = 0.64 of the fit on the Z peak. We get a χ2/Ndeg = 0.70, +larger than the 0.17 of the table. This is easily understood, since higher energy data have +dominant statistical errors, and thus the χ2/Ndeg is more in line with the expectation from +statistical dominated data. +7.2 +Discussion of the results +Our findings can be summarized as follows. +For all results presented in the table, we +observe an excellent χ2 of the fit. In particular the χ2 over number of degrees of freedom +is always well below one. This is a consequence of our treatment of the theoretical error, +that has been added bin-by-bin to the experimental one without correlations. Because of +this, the theoretical prediction has considerable flexibility to adapt to data. +The choice of renormalization scale changes the fit by about 1.5%, the largest change +driven by the variation to lower scales. A similar change can be observed when examining +alternative schemes to implement non-perturbative corrections. The mass-scheme defini- +tions bring in an effect of about 2%. The heavy-to-light correction factor changes αs by just +about one permille, hence the uncertainty associated to this correction seems negligible. A +few permille differences are found when using a different Monte Carlo to change from the +standard definition to the E-scheme and to perform the heavy-to-light correction. These +6Notice that the anti-kt algorithms [56] are such that the softest particles are never clustered together. +– 23 – + +small differences are not surprising since all the Monte Carlos we use are tuned to these +data. The choice of the fit range has an impact on the result of about one percent. This +confirms that the range chosen is such that the impact of the resummation is modest. The +choice of how to treat statistical correlations has also a similar impact, and confirms that +our minimal overlap approach provides a sensible description of the correlations. For y3, +the difference between the two limiting cases (where soft emissions are always-clustered or +never-clustered) amounts also to about a one percent effect on the full fit. +Finally, we note that if one fits αs and α0 from the three observables considered +separately, one tends to get a larger value of the strong coupling, but with very different +values of α0. Indeed, there is a tension in the fitted value of α0, where both thrust and C- +parameter prefer a lower value, while y3 prefers a higher one. When fitting all observables +at the same time, the overall effect is that one finds an intermediate value for α0 and a +lower value of αs. The χ2 of the fits remain excellent, which justifies a simultaneous fit. +The role of each variable in the common fit is illustrated in Fig. 5. As one can see, for C + 0 + 0.5 + 1 + 1.5 + 2 + 0.09 + 0.1 + 0.11 + 0.12 + 0.13 + 0.14 + 0.15 +α0 +αs(MZ) +C +τ +y3 +C+τ +C+τ+y3 +Figure 5: Contours at ∆χ2 = 1 for fitting, C, τ and y3 individually, and then in the +combinations C + τ and C + τ + y3. +and τ, α0 and αS are strongly anti-correlated, and with a similar anti-correlation. On the +other hand, y3 has a ζ function that is small and of opposite sign, and thus α0 and αS +are only weakly correlated. The combined fit is then strongly constrained leading to an +intermediate value of α0 and a smaller value of αs. +Altogether, we conclude by remarking that our fit results agree very well with the +world average. In particular, we do not find low values of αs for the thrust or C-parameter +which are included in the current PDG average [57]. However, our results also clearly show +that a fit of αs from event shapes with an overall uncertainty below the percent level seems +today not feasible. In particular, by changing certain choices that we have made, like the +central renormalization scale or the mass scheme, one can easily obtain higher values of +αS. +– 24 – + +7.3 +Comparison to results obtained by setting ζ(v) = ζ2J(v) +It is natural now to ask what the results of the fits would have been if we had used the +non-perturbative correction as estimated in the two-jet limit. For C, τ, y3, M2 +H and M2 +D +this amounts to setting the ζ(v) functions plotted in Fig. 1 to a constant value, according +to the table 3.7 For BW the function ζ2J(v) can be found in Appendix F of ref. [43]. The +v +C +τ +y3 +M2 +H +M2 +D +BW +ζ2J(v) +3π +2 +0 +1 +0 +App. F of [43] +Table 3: The non-perturbative coefficients in the two jet limit. +complete results are reported in table 4. As shown there, the values of αs found in this +way are consistently lower than those of table 2. For example, for our default setup we +have αs(MZ) = 0.1182 and α0 = 0.64, while using ζ2J we get 0.1132 and 0.55 respectively. +On the other hand, the χ2 values are also quite acceptable.8 +A more detailed comparison of our default fit with the newly calculated ζ functions, +and with the ζ2J functions corresponding to what has been available until now is shown +in Fig. 6. As mentioned earlier, both fits look plausible, was it not for the fact that the +ζ2J result favours values of αs lower than the world average. The quality of the fits is +displayed in Fig. 7. As one can see, the fit with the full ζ(v) dependence seems slightly +better, while the one with the ζ2J functions exhibits some tensions among the different +observables. However, on the basis of the χ2/Ndeg values, both fits are quite acceptable. +It is now interesting to see what happens to the remaining shape variables, M2 +H, M2 +D and +BW evaluated with the same parameters used for our default fits. The result is displayed +in Fig. 8. There we see distinctly that the full ζ(v) fit works very well towards the three jet +region for all the observables. The ζ2J fit, on the other hand, does not work in the three- +jet limit, while its description of data improves in the two-jet region, with the noticeable +exception of M2 +D. +7.4 +On the structure of αsλ/Q corrections +Higher-order corrections to the linear λ term are certainly present. The important issue is +whether these corrections are of order αs(Q) or rather αs(λ). In this work we are implicitly +assuming that they are suppressed by a power of αs(Q). We do not have a solid argument to +prove this assumption. However, by examining the structure of the linear power corrections +near the two-jet limit we gain some insight into how this may actually work. In fact one +can write schematically the correction of order λ to a shape variable v in the two-jet limit +7 For the case of y3, the coefficient is known to be zero [25], since y3 is quadratic in the transverse +momentum for soft emissions. +As for the case of M 2 +D, a colour coherence argument would lead to the +conclusion that in the dominant collinear limit the corrections to the two hemispheres are identical, leading +again to a null value. For the remaining variables, see for example table 1 of ref. [44]. +8We do not ascribe any significance to the larger χ2 values in the two-jet limit, because in this case in +eq. (6.10) we have assumed rather arbitrarily ζ(2)/ζ = 0.1. +– 25 – + +Variation +αs(MZ) +α0 +χ2 +χ2/Ndeg +Default setup +0.1132 +0.55 +15.8 +0.36 +Renormalization scale Q/4 +0.1174 +0.53 +8.5 +0.19 +Renormalization scale Q +0.1126 +0.57 +22.0 +0.50 +NP scheme (b) +0.1126 +0.63 +25.7 +0.58 +NP scheme (c) +0.1134 +0.72 +16.4 +0.37 +NP scheme (d) +0.1132 +0.55 +15.8 +0.36 +P-scheme +0.1108 +0.53 +21.8 +0.50 +D-scheme +0.1126 +0.66 +16.1 +0.37 +Standard scheme +0.1134 +0.51 +15.9 +0.36 +No heavy-to-light correction +0.1130 +0.58 +15.9 +0.36 +Herwig6 +0.1136 +0.51 +31.1 +0.71 +Herwig7 +0.1136 +0.52 +21.8 +0.49 +Ranges (2) +0.1122 +0.54 +30.0 +0.55 +Ranges (3) +0.1134 +0.58 +10.5 +0.33 +Replica method (around average) +0.1158 +0.53 +13.4 +0.31 +Replica method (around default) +0.1160 +0.53 +13.5 +0.31 +y3 clustered +0.1132 +0.55 +15.8 +0.36 +C +0.1238 +0.45 +1.3 +0.08 +τ +0.1202 +0.51 +1.2 +0.06 +y3 +0.1160 +– +1.4 +0.18 +C, τ +0.1222 +0.46 +2.7 +0.08 +Table 4: Default fit result for αs(MZ) and α0 (first line) and other fit results obtained by +varying the setup, using the ζ2J values of table 3. See text for more details. +as9 +dσ +dv +���� +λ += Nλ +� +δ′(v)ζ2j + (δ′(v)V1 + δ(v)V2)αs + d +dv +�dσq¯qg +dv ζ(v) +�� +, +(7.1) +where the first term is the correction to the leading (two-parton) configuration, the second +term is the virtual correction of order αs, and the third term is the correction we have +computed, and where we implicitly assume some regularization of the v → 0 region. The +derivative of the delta function in the first term is necessary to guarantee that upon in- +tegration in v there are no linear corrections left at order zero in αs, since we know that +they are absent in the total cross section. The terms V1 and V2 incorporate corrections +where the hard gluon is virtual and the gluer is real or virtual. In this case, besides the +derivative of the δ-function, we also include an explicit δ-function to indicate that terms +that do not vanish upon integration in v must exist and are in fact divergent. We do not +include virtual corrections to the q¯qg process for the exchange of a virtual gluon of mass +λ, since it was shown in ref. [40] that these do not lead to linear terms in λ. The absence +9This holds for all the observables that we are considering with the exception of BW . +– 26 – + + 0.4 + 0.45 + 0.5 + 0.55 + 0.6 + 0.65 + 0.7 + 0.75 + 0.8 + 0.112 + 0.114 + 0.116 + 0.118 + 0.12 + 0.122 +ζ(v) +ζ2J(v) +α0 +αs(MZ) +Figure 6: Central values and δχ2 = 4 (dashes) and 1 (solid) contours for our default fit +of table 2 (blue) and the fit obtained with the ζ2J functions, corresponding to the default +fit of table 4 (magenta). +of linear corrections to the total cross section leads us to conclude that the integral of the +above formula from v = 0 up to any finite value of v must be finite. In fact, if that was +not the case, such divergence could not be canceled when performing the integral in the +whole range of the shape variable. Thus the argument of αs must be taken equal to the +hard scale (that in this case is not quite Q, but is related to the typical transverse mo- +mentum of the perturbative gluon that sets the value of v). We have thus shown that the +singular contributions of the hard gluon (hard relative to the scale λ) in the real emission +and virtual exchanges cancel each other also in the coefficient of the linear term. +The argument given above also suggests a possible way to match the linear corrections +in the three-jet limit to those in the two-jet limit, that are entangled with resummation +effects. +If we recall that the two-jet limit of the functions ζ(v) for C, τ, M2 +H and M2 +D +approach the value ζ2j, we could conclude that the part of the last term in the square +bracket of eq. (7.1) that is singular in the two jet limit must combine with the virtual +correction to yield a finite result. This combined result is precisely what one gets when +expanding in powers of αs the Sudakov form factor, including the shift for the two-jet non- +perturbative correction. Thus, it is tempting to conclude that the singular part of the last +term function should be combined with the resummation component of the cross section, +while only the regular part should be applied to the 3-jet region. It is unlikely, however, +that this approach will work for observables like M2 +H and M2 +D, since in their case the limiting +value is approached extremely slowly, and in the first case it has even opposite sign with +respect to the average value of the ζ function in the fit range. It is however reassuring to +see that if we restrict ourselves to regions far away the two-jet region, all shape variables +– 27 – + + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 +1/σ dσ/dC +ζ(v) +no NP corr. +data +0.95 +1 +1.1 + 0.2 + 0.25 + 0.3 + 0.35 + 0.4 + 0.45 + 0.5 + 0.55 + 0.6 +data/theory +C + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 +ζ2J(v) +no NP corr. +data +0.95 +1 +1.1 + 0.2 + 0.25 + 0.3 + 0.35 + 0.4 + 0.45 + 0.5 + 0.55 + 0.6 +C + 0 + 2 + 4 + 6 + 8 + 10 + 12 +1/σ dσ/dτ +ζ(v) +no NP corr. +data +0.95 +1 +1.1 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +data/theory +τ + 0 + 2 + 4 + 6 + 8 + 10 + 12 +ζ2J(v) +no NP corr. +data +0.951 +1.1 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +τ +-0.5 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 + 3 + 3.5 +1/σ dσ/dy3 +ζ(v) +no NP corr. +data +0.95 +1 +1.05 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +data/theory +y3 + 0 + 0.5 + 1 + 1.5 + 2 + 2.5 +ζ2J(v) +no NP corr. +data +0.95 +1 +1.05 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +y3 +Figure 7: Theoretical predictions compared to data for our default setup on the left side, +and the default setup with the ζ2J functions on the right side. The gray band represents the +theoretical errors, while the red bars indicate the experimental ones, with the smaller one +representing the statistical error, and the green lines show the pure perturbative results. +The highlighted region represents the fit range. +are well described with the ζ functions computed here, while this is not the case with the +values of table 3. +– 28 – + + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 +1/σ dσ/dM2h +ζ(v) +no NP corr. +data +0.9 +1 +1.1 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +data/theory +M2h + 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 +ζ2J(v) +no NP corr. +data +0.7 +0.8 +0.9 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +M2h + 0 + 1 + 2 + 3 + 4 + 5 + 6 +1/σ dσ/dM2d +ζ(v) +no NP corr. +data +0.8 +0.9 +1 +1.2 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +data/theory +M2d + 0 + 1 + 2 + 3 + 4 + 5 + 6 +ζ2J(v) +no NP corr. +data +0.8 +0.9 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +M2d + 0 + 2 + 4 + 6 + 8 + 10 +1/σ dσ/dBw +ζ(v) +no NP corr. +data +0.9 +1 +1.1 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 +data/theory +Bw + 0 + 2 + 4 + 6 + 8 + 10 +ζ2J(v) +no NP corr. +data +0.8 +0.9 +1 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 +Bw +Figure 8: Theoretical predictions compared to data for our default setup on the left side, +and the default setup with the ζ2J functions on the right side, for the M2 +H, M2 +D and BW +shape variables. The gray band represents the theoretical errors, while the red bars indicate +the experimental ones, with the smaller one representing the statistical error. The green +lines show the pure perturbative results. +8 +Conclusions +In this work, we study the effect of power corrections in e+e− observables in comparison +to data, under the light of the new findings of refs. [40, 41], where it was shown that +power corrections can be computed directly in the three-jet configuration, rather than +– 29 – + +extrapolating them from the two-jet region. In refs. [40, 41] these power corrections were +computed for the C-parameter and for thrust. Here we also computed them for the three- +jet resolution parameter in the Durham scheme y3, for the squared mass of the heavy +hemisphere M2 +H, for the squared-mass difference of heavy-light hemispheres M2 +D, and for +the wide jet broadening BW . The observables we considered are those that can be computed +in the approach of refs. [40, 41], and that are included in the ALEPH data of ref. [42]. +For simplicity we stick to a single data set, and we perform our calculation using the +NNLO results for e+e− hadronic observables, plus the newly computed power corrections. +We do not attempt to include resummation effects. +Rather, we stick to ranges of the +observables that are far enough from the two jet region so that no visible depletion of the +resummed result with respect to the fixed-order one is present. +We stress that in this work we are assuming that the non-perturbative corrections +as estimated according to the results of ref. [40, 41] are not drastically modified by the +inclusion of soft radiation. Our argument concerning the two-jet limit region near eq. (7.1) +seems to indicate that this is not the case. However, we are unable to provide a solid +argument for the three-jet region. +Our main results can be summarized as follows. First of all, for all the shape variables +that we considered, with the exclusion of the wide-jet broadening, the function that pa- +rameterised the non-perturbative correction, called ζ(v), approaches its two jet-limit value +when its argument approaches the two-jet limit value (set conventionally to v = 0), as one +expects according to simple physics arguments. However, with the exception of y3, the +limit, is approached only for exponentially small values of the shape variable, so that, in +practice, one sees an effective jump of the function near v = 0. This jump is not very +important for C and for the thrust τ = 1 − T, where it is around 10-20% of the two-jet +limit value. It is instead quite large for M2 +H and M2 +D, where it is such that the two-jet limit +value cannot be considered representative of the value of the function even very close to the +two-jet limit. In view of these observations, we exclude these observables from our fit, and +also exclude BW that is positive and divergent in the two-jet limit, and is instead negative +in the three-jet region. +We thus fitted C, τ and y3, extracting a value for the strong coupling constant on the +Z peak, and for the non-perturbative parameter α0. The result of the fits yield a value +of αs in acceptable agreement with the world average, although we find that a number of +variations of our procedure can lead easily to differences of the order of a percent. Using +the same value of αs and α0, we see that we can describe quite well also the remaining +observables M2 +H, M2 +D and BW , as long as we remain far enough from the two-jet limit. +Conversely, with the traditional implementation of power corrections, good fits to C, τ and +y3 can also be obtained, however the description of M2 +H, M2 +D and BW in the three-jet region +is totally unacceptable. +We stress again that the inclusion of resummation effects in the bulk of the three jet +region leads smaller values of αs.10 +10In particular, for fits to the C-parameter one finds values of αs smaller by about ten percent (private +communication by P. Monni). +– 30 – + +We are aware that the present work should only be considered as a preliminary explo- +ration of the implications of the results of refs. [40, 41]. In fact, there are few directions +that need further exploration in order to fully exploit these new results. +First of all, it would be interesting and important to also include resummation effects +in our analysis. Some ideas regarding this are discussed in the text, suggesting that perhaps +the two-jet limit shift should be applied to the resummed component of the cross section, +while the full ζ(v) dependent part should be applied to the finite part. Yet, whether this +approach is sensible also when including resummation effects far from the two-jet region is +a question that needs to be examined more closely, since for most observables ζ(0) differs +considerably from ζ(v) in the three-jet region. +A second direction of improvement regards the choice of the hadron mass-scheme. +Lacking a theoretically sound treatment of this problem, a possible development would be +to see if there is a scheme that is preferred by data. This in turn would require considering +enough observables that display different behaviour regarding the mass-scheme choice. +This brings us to consider a third extension of this work, which is to examine more +variables, and find a sufficiently large set such that the requirements for the applicability +of the results of refs. [40, 41] are met, and such that their behaviour near the two-jet limit +are closer to that of the thrust and the C-parameter. These new variables, could also +be analyzed at present using preserved LEP data [58], while waiting for the beginning of +operation of new e+e− colliders. +Acknowledgments +P. N. would like to thank the Max Planck Institute for hospitality while part of this work +was carried out. We thank Andrea Banfi, Adam Kardos, Stephan Kluth, Pier Francesco +Monni, Silvia Ferrario Ravasio, Gavin Salam, Hasko Stenzel, Roberto Tenchini, and Andrii +Verbytskyi for useful discussions. +– 31 – + +A +Impact of resummation +The fits of αs carried out in this work rely on fixed order NNLO predictions, rather than +on all-order (NNLL) predictions matched to fixed order, as computed in Ref. [9, 10, 13– +15, 59–61] for event-shapes and in Ref. [12] for the Durham three-jet resolution parameter +y3. Although it is customary to include resummation effects also far away from the two- +jet region, in this work we made the assumption that resummation effects should not be +included when the logarithm of the shape variable is not large. +In order to determine +a range for the fit, we thus compare in Fig. 9 NNLO and NNLO+NNLL predictions for +the thrust variable τ = 1 − T, the C-parameter, and the Durham three jet resolution +variable y3 and exclude in our fits the regions where matched predictions clearly depart +from the fixed order. Each plot shows the ratio to the NLO prediction obtained with central +renormalization scale µR,0 = Q/2. The green band shows the uncertainty of the NLO and +the blue band of the NNLO, and are obtained by varying µR up and down by a factor two +around the central value. For the NNLO+NNLL matched predictions we fix our default +setup as follows: we set the central renormalization scale to µR,0 = Q/2, the resummation +scale to µQ,0 = Q/2, we use the modified logarithm L = 1/p ln +� +1/vp − 1/vp +lim + 1 +� +, where +vlim denotes the kinematic limit of the event shapes, with p=3, and we use the log-R +matching scheme (see e.g. ref. [10]). The uncertainty band is then obtained as follows. +Around the above described default setup, we vary, one at the time, µR,0/2 ≤ µR ≤ 2µR,0, +µQ,0/2 ≤ µQ ≤ 2µQ,0, we vary p to p = 2 and p = 5, and, finally, we use the R-matching +scheme. This gives a total of eight matched predictions. The red uncertainty band shown +in Fig. 9 is obtained by taking the envelope of all these predictions. +The onset of resummation effects is signalled both by a drop of the distribution of the +resummed result and by an increase of the NLO result with respect to the NNLO one. We +choose the lower bound of our fit ranges to be to the right of this region. Furthermore, for +the three observables used in the fit we observe the following features: for the thrust, the +uncertainty bands of the NNLO and matched predictions overlap, with the resummation +band being a few percent higher, which would lead to slightly smaller values of αs. For the +C-parameter one observes a somewhat similar behaviour. However, the difference between +the center of the resummed and NNLO bands now reach up to 10% and the resummed +band has a slightly different shape compared to the NNLO one. For y3 one observes small +effects, at the level of a 2%, however in this case the uncertainty bands do not overlap +since the NNLO band is extremely small. From all three plots it is also clear that the +difference between NNLO and matched predictions does not vanish even for large values +of the observables. This is due to the fact that, even with the modified logarithms, the +resummation is not switched off fast enough even close to the end-point of the distributions. +From the figures it can be seen that in the case of the thrust, the resummed predic- +tion seems to follow the trend of the NLO and NNLO corrections, possibly approximating +higher-order results if they follow the same trend. However, in the case of the C-parameter +the resummed result has a slope that is not present in the NLO and NNLO results. Fur- +thermore, in the case of y3, the trend is to have the NNLO distribution smaller than the +NLO one, while the resummed result is larger. +In conclusion, although it has become +– 32 – + +common practice, we see no reason in principle to include resummation effects also in the +three-jet region. + 0.8 + 0.9 + 1 + 1.1 + 1.2 + 1.3 + 1.4 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 + 0.35 + 0.4 +Ratio to dσNLO/dτ at µR=Q/2 +τ + NLO + NNLO + NNLL+NNLO + 0.6 + 0.7 + 0.8 + 0.9 + 1 + 1.1 + 1.2 + 1.3 + 1.4 + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 +Ratio to dσNLO/dC at µR=Q/2 +C + NLO + NNLO + NNLL+NNLO + 0.9 + 0.95 + 1 + 1.05 + 1.1 + 0.05 + 0.1 + 0.15 + 0.2 + 0.25 + 0.3 +Ratio to dσNLO/dy3 at µR=Q/2 +y3 + NLO + NNLO + NNLL+NNLO +Figure +9: +Comparison between NLO (green bands), +NNLO (blue bands) and +NNLO+NNLL (red bands) predictions for the thrust (left), C-parameter (central), y3 +(right). See text for more details. +– 33 – + +References +[1] A. Gehrmann-De Ridder, T. Gehrmann, E. W. N. 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Zuberi, Combining Higher-Order Resummation with Multiple NLO +Calculations and Parton Showers in GENEVA, JHEP 09 (2013) 120, [1211.7049]. +– 37 – + diff --git a/R9E2T4oBgHgl3EQfBwbf/content/tmp_files/load_file.txt b/R9E2T4oBgHgl3EQfBwbf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..83feb3f3d7b2ae798ab6e6916f34eeec99d6dcde --- /dev/null +++ b/R9E2T4oBgHgl3EQfBwbf/content/tmp_files/load_file.txt @@ -0,0 +1,1759 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf,len=1758 +page_content='Prepared for submission to JHEP MPP-2022-142 Fits of αs using power corrections in the three-jet region Paolo Nason,a,b Giulia Zanderighib,c aUniversit`a di Milano-Bicocca and INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3,20126 Milano, Italy bMax-Planck-Institut f¨ur Physik, F¨ohringer Ring 6, 80805 M¨unchen, Germany cPhysik-Department, Technische Universit¨at M¨unchen, James-Franck-Strasse 1, 85748 Garching, Germany E-mail: paolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='nason@mib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='it, zanderi@mpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='mpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='de Abstract: In this work we study the impact of recent findings regarding non-perturbative corrections in the three-jet region to e+e− hadronic observables, by performing a simul- taneous fit of the strong coupling constant αs and the non-perturbative parameter α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We extend the calculation of these power corrections, already known for thrust and C- parameter, to other e+e− hadronic observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We find that for some observables the non-perturbative corrections are reasonably well behaved in the two-jet limit, while for others they have a more problematic behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' If one limits the fit to the three-jet region and to the well-behaved observables, one finds in general very good results, with the ex- tracted value of αs agreeing well with the world average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This is the case in particular for the thrust and C-parameter for which notably small values of αs have been reported when non-perturbative corrections have been computed using analytic methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Furthermore, the more problematic variables are also well described provided one stays far enough from the two-jet limit, while in this same region they cannot be described using the traditional implementation of power-corrections based on two-jet kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Keywords: Perturbative QCD, QCD Phenomenology, electron-positron scattering arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='03607v1 [hep-ph] 9 Jan 2023 Contents 1 Introduction 1 2 Observable definitions 4 3 Hadron mass ambiguities 6 4 Power correction calculation 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 Thrust 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 Other observables 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 The shift in the cumulative cross section 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 Numerical checks 14 5 Calculation of the observable distributions 15 6 Fit to ALEPH data 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 Treatment of uncertainties 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 Statistical and systematic errors, and correlations 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 Perturbative theory uncertainties 19 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 Non-perturbative theory uncertainty 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 Correction for heavy-quark mass effects 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 Hadron mass-effects corrections 20 7 Fit results 21 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 Including higher energy data 23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 Discussion of the results 23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 Comparison to results obtained by setting ζ(v) = ζ2J(v) 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 On the structure of αsλ/Q corrections 25 8 Conclusions 29 A Impact of resummation 32 1 Introduction The study of shape variables in e+e− annihilation is one of the simplest contexts in which to test perturbative QCD, and it is potentially among the cleanest frameworks where one can measure the strong coupling constant αs at high energy by probing directly the quark-antiquark-gluon vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Shape variables have been computed up to order α3 s [1– 4], and resummations near the two-jet region have been performed at different levels of – 1 – accuracy, either using traditional resummation methods [5–12], or using Soft Collinear Effective Theory (SCET) [13–16], leading to very precise predictions at high energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is well known, however, that shape variables are affected by linearly suppressed power corrections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' of the order of Λ/Q, where Λ is a typical hadronic scale and Q is the annihilation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Since in the 3-jet region the shape variables are of order αs, this implies a relative error of order (Λ/Q)/αs, that affects at the same level the measured value of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' If we assume that Λ is of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 GeV (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the typical additional transverse energy per unit of rapidity due to hadronization), on the Z peak we estimate an error of the order of 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In practice, power corrections can reach the 10% level for some observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' A commonly adopted approach for dealing with power corrections in the determina- tions of the strong coupling constant from shape variables is to use Monte Carlo mod- els [17–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' A shower Monte Carlo is used to construct a migration matrix for shape variables computed from final-state hadrons, and from partons before hadronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The migration matrix is then applied to the measured differential distribution of hadrons to ob- tain the shape distribution in terms of partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This is in turn compared to perturbative QCD, and a value of αs is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This method is often criticized, because the Monte Carlo hadronization model does not bear a clean relation to field-theoretical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' An alternative strategy for the inclusion of power corrections makes use of analytic approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In this case, the theoretical calculation including power corrections is compared directly to the shape variable measurement using hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' These methods can be classified into two broad classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' One approach makes use of an effective coupling for the emission of very soft gluons (called “gluers”) [24–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The average value of the effective coupling in a given low-energy range plays the role of a parameter to be fitted to data together with the value of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The Particle Data Group [57] (PDG) currently includes two fits of the strong coupling based on NNLO+NLL [28] or NNLO+NNLL [29] accurate perturbative results combined with this approach to the non-perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This approach is also motivated by the large-nf limit of QCD (see [30] and references therein), where the effective coupling can be actually computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is argued that the non-perturbative parameter in this contest is universal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' it is the same for a large class of shape variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The coefficient of the power correction is computed by simply adding a gluer to an initial q¯q state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For shape variables that are additive in soft radiation near the two jet limit, the emission of the gluer acts as a shift in the value of the shape variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This behaviour is then extrapolated to the three-jet region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the non perturbative correction is included as a shift in the argument of the shape variable computed in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The other approach relies upon factorization in QCD [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This begins with the computation of the shape variables including resummation of the soft-collinear singular- ities arising from gluon emission from the primary quark and antiquark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The region of very soft emissions is parameterized by a shape function that is factorized out of the dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the three-jet region, a single moment of the shape function controls the linear non-perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This approach arises naturally in SCET [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Two – 2 – determinations of αs(MZ) included in the PDG [37, 38] currently rely on such analytic SCET-based approaches and notably lead to low values of the strong coupling accompa- nied by small uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' A common feature of these two approaches is that they rely upon the extrapolation of the non-perturbative correction from the two-jet to the three-jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This extrap- olation has been shown not to agree with the direct calculation of the non-perturbative correction for the C parameter near the three-jet symmetric limit [39], where it leads to an overestimate by approximately a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [40, 41] it was shown that linear power corrections in the bulk of the three parton final state region can be computed in large-nf QCD in the process e+e− → q¯qγ, and, under some further assumptions, also in the e+e− → q¯qg process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [41] the result for the C-parameter and thrust was given, but the method is quite general and can be extended to a wide class of shape variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the case of the C-parameter it leads to a result consistent with ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [39] in the three-jet symmetric limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In general as for the C-parameter case, one finds considerable violations of the assumption that the non- perturbative correction can be implemented as a constant shift of the perturbative result well into the three-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The purpose of this work is to investigate whether there are some indications that the newly computed power corrections are preferred by available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In order to do this, we considered Z-peak data from the ALEPH experiment [42] that are publicly available on HEPDATA and quite precise, and consider a set of shape variables such that the compu- tation along the lines of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [41] can be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Besides thrust and the C-parameter, ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [42] provides data for other shape variables for which we are in a position to compute non-perturbative corrections in the three-jet region, namely the square mass of the heavy hemisphere M2 H, the difference of the squares masses of the heavy and light hemisphere M2 D, the broadening of the wide jet BW , and the 3-jet resolution parameter y3 in the Durham scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In this work we have then computed the non-perturbative coefficients for M2 H, M2 D, BW , and, with some caveats to be detailed in the following, also for y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We thus supple- ment the α3 s calculation of these shape variables with the inclusion of the non-perturbative corrections that we have computed as a shift in the argument of the cumulative cross sec- tion Σ(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' More precisely, calling V a generic shape variable, defined in such a way that it vanishes in the two jet limit, Σ(v) is defined as the cross section for producing events such that V < v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In our approach, the shift in the argument is given by v → v −ζ(v)HNP, where HNP is a coefficient suppressed by a power of Q, equal for all shape variables, and ζ(v) is a shape-variable specific, dimensionless function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In contrast, in the traditional form of the power corrections the variable-specific function ζ(v) is evaluated in the two-jet limit, where it is in most cases replaced by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 We stress that, somewhat unconventionally, we do not include resummation effects in our result, while it is common practice to include them also very far away from the two-jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' They generally lead to an increase of the shape variable distributions, and thus to a smaller value of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We take here the point of view that if we consider ranges of the 1In the case of the broadening the shift is not a constant, see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 3 – shape variables that are far enough from the 2-jet limit, resummation can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The reader may keep in mind that if resummation effects were included we would generally obtain smaller values of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' A further reason for not including resummation in our result is that it is not clear whether including the constant non-perturbative shifts in the singular contributions is an acceptable procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact, such corrections would propagate into the three-jet region, where (as we will see later) they sharply differ from their two-jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Furthermore, in this work we will not try to give a preferred value of αs with an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Rather, our aim is only to see whether and where the newly computed non-perturbative corrections are in some way preferred by data, and to assess their impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 2 we define the observables that we consider in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 3 we discuss ambiguities in the event-shape definitions that arise when dealing with massive hadrons, as opposed to massless QCD partons, and recall three alternative definitions that differ for massive hadrons but agree for massless partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4 we present the calculation of the power-corrections in the three-jet region for all observables considered in this work and show that they give rise to a non- constant shift of the perturbative distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We also discuss numerical checks of the analytic calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 5 we discuss how to combine perturbative O(α3 s) results with non-perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In particular, we define various schemes that differ by higher order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 6 we discuss our treatment of uncertainties and correlations, as well as the corrections that we apply to account for the heavy-quark masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 7 we present the results of our fits of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We discuss various ambiguities and uncertainties, as well as their difference from fits relying on the calculation of non-perturbative corrections in the two-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We conclude in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' A we discuss the impact of all-order resummation effects for the observables used in our fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 2 Observable definitions The choice of event shapes considered in this work is based on whether ALEPH data are available for them, and whether their associated non-perturbative corrections in the three jet region can be calculated along the lines of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [41], as discussed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Unless otherwise specified, all sums in the definitions below run over all particles in the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The thrust T , or τ = 1 − T, is defined as T = max ⃗nT �� i |⃗pi · ⃗nT | � i |⃗pi| � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) where the axis ⃗nT , that maximises the sum, is the thrust axis of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The heavy-jet mass: the plane through the origin of the event, orthogonal to the thrust axis ⃗nT , divides each event into two hemispheres Hj (j = 1, 2), the invariant – 4 – mass of each is defined as M2 j = 1 E2 vis � � � pi∈Hj pi � � 2 , j = 1, 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2) where Evis = � i Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The heavy-jet mass is the larger of the two M2 H = max � M2 1 , M2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3) The jet mass difference is defined as the difference between the larger and smaller of the two masses M2 D = |M2 1 − M2 2 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4) The C-parameter is computed from the three eigenvalues λi of the momentum tensor Θαβ Θαβ = 1 � i |pi| � i pα i pβ i |⃗pi| , α, β = 1, 2, 3 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5) as C = 3 · (λ1λ2 + λ1λ3 + λ2λ3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6) The wide broadening: given the thrust axis nT , the hemisphere broadenings Bj (j = 1, 2) measure the amount of transverse momentum in each hemisphere Bj = � pi∈Hj |⃗pi × ⃗nT | 2 � i |⃗pi| , j = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7) The wide broadening BW is the larger of the two hemisphere broadenings BW = max (B1, B2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8) The three-jet resolution y3: we take the Durham jet clustering, whose distance measure reads yij = 2 min(E2 i , E2 j )(1 − cos θij) E2 vis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9) (Pseudo)-jets are recombined sequencially summing the four-momenta of the pair of particles with the smallest yij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The three-jet resolution y3 is defined as the value of ycut for which an event changes from being classified as 2- to 3-jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The published data are already corrected using Monte Carlo generators in such a way that all particles produced by the e+e− annihilation are included, comprising also the neutrinos from meson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 5 – 3 Hadron mass ambiguities When computing shape variables in perturbative QCD, one always deals with massless partons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, the measurements use the four-momenta of massive hadrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It turns out that shape variable definitions may differ for massive hadrons and be identical for massless partons, and this introduces an ambiguity in the experimental definition of the event shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This problem has been studied in detail in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [44] (see also [45]), where three alternative schemes where suggested: the p-scheme, the E-scheme and the D-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the p-scheme one uses only the three-momenta of the particles ⃗pi, and the energies Ei are replaced by |⃗pi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Instead, in the E-scheme the energies of the particles are preserved, but the three-momenta are rescaled so as to have massless four-momenta, ⃗pi → ⃗pi · Ei |⃗pi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is clear that in the p-scheme energy conservation is violated, while in the E-scheme the three momentum is not conserved, the violation being in both cases of the order of the hadron masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the so called D-scheme, final state hadrons are decayed isotropically in their rest frame into two fictitious massless particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The event shape is then computed using only massless particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This scheme has the advantage that the full four-momentum of the event is conserved, and that no reference to a particular frame needs to be invoked in its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Notice also that it can happen that long-lived, unstable hadrons are produced that decay to lighter particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Therefore the event shape depends on the level at which the measurement is performed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' it becomes relevant whether the measurement is performed before or after these decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Unlike all other schemes, the D-scheme has the advantage that it is rather insensitive to the particular hadron level chosen to perform the measurement [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [44], the advantages and disadvantages of each of these schemes are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In particular it is argued that in the E-scheme non-universal mass effects are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The arguments used there are based upon an analysis near the two-jet limit, and their applicability to the case of three widely separated jets is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' One may also argue that the D-scheme should be preferred, since it mimics to some extent the models of hadron formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the present work we will adopt the E-scheme as our default choice and use the additional three schemes to gauge the hadron-mass sensitivity of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4 Power correction calculation According to ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [41], provided an event shape satisfies specific conditions, as explained in detail later, its power correction in the three-jet region can be computed according to the formula [Σ(v)]NP = � � � � dσB(ΦB)δ(v(ΦB) − v) � dip � −M × 4αsCdip 2π 1 Q � dη dφ 2π hv(η, φ) �� � � × INP, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) where Q is the total center of mass (CM) energy and the sum runs over all radiating dipoles associated with the given Born configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus, for the two jet case there is just a – 6 – single q¯q dipole, while for the three-jet case we have a q¯q, qg and ¯qg dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 We stress that the function hv depends also upon ΦB, and that for ease of notation we do not show explicitly this dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The colour coefficients Cdip for the three-jet case are given by Cq¯q = CF − CA 2 , Cqg = C¯qg = CA 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2) The Milan factor M is given in analytic form in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [46] M = 3 64 (128π(1 + log 2) − 35π2)CA − 10π2TRnF 11CA − 4TRnF , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3) that agrees with the numerical result given earlier in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [27] M = 1 + (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='575CA − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='104nf)/β0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4) where β0 = (11CA − 4nfTR)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The coefficient INP depends upon the model used to im- plement power corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the large-nF theory, it has the expression (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [47]) INP = 1 b0,nf αs(µ) � µC 0 dλ π arctan πb0,nf αs(µ) 1 + b0,nf αs(µ) log λ2e−5/3 µ2 = 1 αs(µ) � µC 0 dλarctan(πb0,nf αs(λe−5/6)) πb0,nf , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5) where b0,nf = −nf/(6π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The upper limit of integration in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5) is quite arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It should be large enough to cover the region where the argument of the arctangent diverges, corresponding to the Landau pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In phenomenological models INP is replaced by the integral over a non-perturbative effective coupling, given as function of a scale λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The function hv(η, φ) depends upon the shape variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is defined as hv(η, φ) = lim |l⊥|→0 1 |l⊥| (v({P} , l) − v({p})) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6) where {P} denote the momenta of the hard final state partons after the radiation of a soft massless parton of momentum l, and {p} denote the momenta of the final state partons in the absence of radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The arguments η and φ are the rapidity and azimuth of the soft parton, and l⊥ denotes its transverse momentum, all evaluated in the rest frame of the radiating dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The mapping from {P} and l to {p} must have certain smoothness properties, namely the momenta {P} must be functions of {p} and l that are linear in l for small l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' There are two further requirements for formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The first one is that it applies to variables that are additive in the emission of more than one soft parton in the three-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This property is violated by y3, as discussed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The second one is that the function hv(η, φ), after azimuthal integration, should yield a convergent integral in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This property is violated, for example, by the total broadening, and that is the reason why we do not consider it in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 2The same formula is also applicable to higher multiplicity Born processes, that we do not consider here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 7 – Notice that in the large nf limit the Milan factor becomes equal to 15π2/128, and the expression in the curly bracket of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) becomes equal to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [41], up to the λ factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact, according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [41], the linear non-perturbative correction to an observable in the large nf limit is proportional to the first order coefficient of its expansion in λ, where λ is a (fictitious) gluon mass introduced in the calculation, multiplied by the factor given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [41] the integration in η and φ was performed analytically for the C-parameter and for thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Here we have set up a numerical code to perform the η and φ integration numerically, since a sufficient precision can be easily reached, and this allows us to add more observables with relatively minor effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 Thrust We illustrate now how the hv(η, φ) function is computed in our code, using thrust as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We generate the Born momenta {p} according to the three-body phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Let us assume for definiteness that p1, p2 is the radiating dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We generate η and φ, and construct the four-vector l = l+ + l− + l⊥, l+ = p1 √2p1 · p2 exp(η), l− = p2 √2p1 · p2 exp(−η), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7) where l⊥ · l+ = 0, l⊥ · l− = 0, l2 ⊥ = −1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8) and l⊥ has an azimuthal angle φ relative to the p1/2 axis in the dipole rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Notice that by construction l2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Let us call ⃗t0 the trust axis in the CM frame, defined to have the direction of the largest ⃗pi, i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The thrust variation due to the emission of a parton with momentum λl is given by δτ = −δT = − 1 Q � �max ⃗t � � � i=1,3 |⃗Pi · ⃗t| + λ|⃗l · ⃗t| � � − � i=1,3 |⃗pi · ⃗t0| � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9) We need to expand this expression for small λ, keeping only the linear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We have three terms δT = � i=1,3 |(⃗pi + δ ⃗Pi) · ⃗t0| − � i=1,3 |⃗pi · ⃗t0| Q + � i=1,3 |⃗pi · (⃗t0 + δ⃗t)| − � i=1,3 |⃗pi · ⃗t0| Q + λ|⃗l · ⃗t0| Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='10) – 8 – The second line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='10) can be worked out as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We must have δ⃗t · ⃗t0 = 0, since ⃗t has fixed length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus we have δ⃗t · ⃗pk = 0, where k is the hardest parton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the remaining two partons, with i, j ̸= k we have |⃗pi · (⃗t0 + δ⃗t)| = |⃗pi · ⃗t0| × � 1 + δ⃗t · ⃗pi ⃗pi · ⃗t0 � = |⃗pi · ⃗t0| − δ⃗t · ⃗pi, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='11) where we have used the fact that ⃗pi · ⃗t0 < 0 for i ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus � i=1,3 |⃗pi · (⃗t0 + δ⃗t)| − � i=1,3 |⃗pi · ⃗t0| Q = − 1 Qδ⃗t · ( � i̸=k ⃗pi) = 1 Qδ⃗t · ⃗pk = 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='12) since δ⃗t is orthogonal to ⃗t0 and thus to ⃗pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus only the terms in the first and last line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='10) contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The first line is linear in δPi, and thus (in an appropriate recoil scheme) also in l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It must have the form λ Q(A exp(η) + B exp(−η) + C sin φ + D cos φ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='13) with A, B, C and D depending only upon the Born kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The full result is δT = λ Q(|⃗l · ⃗t0| + A exp(η) + B exp(−η) + C sin φ + D cos φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='14) The above expression must however not lead to a divergent integral for large rapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Looking, for example, at the large η limit of the above expression (see eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7)), we have |⃗l · ⃗t0| = ���� ⃗p1 · ⃗t0 √2p1 · p2 exp(η) + ⃗p2 · ⃗t0 √2p1 · p2 exp(−η) +⃗l⊥ · ⃗t0 ���� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15) = ���� ⃗p1 · ⃗t0 √2p1 · p2 ���� � exp(η) + ⃗p2 · ⃗t0 ⃗p1 · ⃗t0 exp(−η) + ⃗l⊥ · ⃗t0 ⃗p1 · ⃗t0 � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='16) We thus see that by choosing A = − ���� ⃗p1 · ⃗t0 √2p1 · p2 ���� (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='17) we cancel that exponential growth in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' With an analogous choice for B we can cancel the exponential divergence for η → −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Terms with constant behaviour for large η do remain, but they cancel after azimuthal integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus, our final expression for the hv function for τ is obtained by changing sign to the previous expression, hτ(η, φ) = −hT (η, φ) = −|⃗l · ⃗t0| + |⃗l+ · ⃗t0| + |⃗l− · ⃗t0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='18) In order to explicitly get rid of the constant φ-dependent term, in the numerical integration process we sum the two contributions obtained with the replacement φ → φ + π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 9 – 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 Other observables With a similar procedures we find the expression of hv for all shape variables of our interest, for which we report here only the final results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the C parameter, starting from the expression C = 3 − 3 2 � i,j (pi · pj)2 (pi · q)(pj · q), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='19) valid for massless partons, we obtain hC(η, φ) = −3 3 � i=1 � (l · pi)2 l · q pi · q − (l+ · pi)2 l+ · q pi · q − (l− · pi)2 l− · q pi · q � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='20) where q = � i pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The negative terms in the square bracket of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='20) are there to cancel the divergent rapidity behaviour of the positive term, and are clearly linear in the momentum components of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the heavy-jet mass we find hM2 H(η, φ) =θ(t0 · l) �q · l Q (2 − T0) − T0 t0 · l � − θ(t0 · l+) �q · l+ Q (2 − T0) − T0 t0 · l+ � − θ(t0 · l−) �q · l− Q (2 − T0) − T0 t0 · l− � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='21) where T0 stands for the value of the thrust at Born level, and, as before, the vector t0 is obtained by adding a zero time-component to the thrust three-vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Also in this case the subtraction terms are clearly identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Notice that the theta functions involving l+ and l− are actually independent upon l, since θ(±t0 · l+/−) = θ(±t0 · p1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='22) The light jet mass is given by hM2 l (η, φ) =θ(−t0 · l) T0 � t0 · l + q · l Q � − θ(−t0 · l+) T0 � t0 · l+ + q · l+ Q � − θ(t0 · l−) T0 � t0 · l− + q · l− Q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='23) From the heavy- and light-jet mass we also obtain the mass difference hM2 D(η, φ) = hM2 H(η, φ) − hM2 l (η, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='24) The wide jet broadening is given by hBW (η, φ) =θ(t0 · l)1 2 ��l · q Q �2 − (t0 · l)2 − θ(t0 · l+)1 2 ��l+ · q Q �2 − (t0 · l+)2 − θ(t0 · l−)1 2 ��l− · q Q �2 − (t · l−)2 + 3 � i=1 θ(t · pi)θ(−t · l) (⃗l · ⃗pi)t · pi − l · t(t · pi)2 T0 � (pi · q)2 − Q2(pi · t)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25) – 10 – The first two lines in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25) represent the variation in BW at fixed thrust axis, and the last line is the contribution due to the fact that if the emission is in the hemisphere of the hardest parton, the thrust axis is tilted, and this affects BW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Notice also that while the first term requires a subtraction (the two following terms), the last term does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact, l cannot be collinear with the partons opposite to the hardest one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus, assuming for example that parton p1 is in the hemisphere opposite to the hardest parton, there will be a cut-off for large values of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Therefore, large values of η will only be allowed if l is collinear to the hardest parton, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the one aligned with the thrust axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In this case however, it is easy to check that the numerator in the last line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the calculation of y3, we assume first that the two soft partons arising from gluon splitting are not the first pair to be clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Under this assumption, also y3 becomes additive in the soft partons, and the calculation can be done in analogy with the other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Assume for definiteness that, at the Born level, the hard partons pair yielding the smallest y3 is given by the parton labels j, k, that the remaining parton is labeled i, and that p0 k < p0 j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Then the change in y3 is given by hy3(η, φ) = � θ(dk,l < min(dj,l, di,l))2 � 2p0 kl0(1 − cos ψkj) − (p0 k)2 � ⃗l · ⃗pj |⃗pj||⃗pk| − ⃗pj · ⃗pk ⃗pk ·⃗l |⃗pj||⃗pk|3 � � + θ(dj,l < min(dk,l, di,l))2 � − (p0 k)2 � ⃗l · ⃗pk |⃗pk||⃗pj| − ⃗pj · ⃗pk ⃗pj ·⃗l |⃗pk||⃗pj|3 � � − θ(dk,l+ < min(dj,l+, di,l+))2(2p0 k(l+)0)(1 − cos ψkj) − θ(dk,l− < min(dj,l−, di,l−))2(2p0 k(l−)0)(1 − cos ψkj) � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='26) where dh,l = 1 − ⃗ph ·⃗l |⃗ph||⃗l| , and cos ψkj = ⃗pk · ⃗pj |⃗pk||⃗pj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='27) The term proportional to p0 kl0 is due the change in the energy of parton k when it combines with l, while the terms proportional to (p0 k)2 are due to the change in the angle between parton k combined with l and parton j (the first instance), and between parton j combined with l and parton k (second instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Notice that there are no subtractions associated with the change in angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact, because of the theta functions, the only collinear singularity that can arise in this case is when l is collinear to j or k, but then the angle does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As stated earlier, the y3 variable is really not additive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the y3 modification due to several soft emissions is not the sum of the modifications due to each emission since partons can be clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 In order to estimate the magnitude of the error associated with this assumption, it is interesting to compute the non-perturbative correction to y3 also in the case when the two partons are always clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In this case formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='26) still holds, with l equal to the total momentum of the pair of partons, l2 = λ2, and, in the 3This is also discussed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We thank Andrea Banfi for pointing this out to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 11 – left hand side, hy3(η, φ) is replaced by hy3(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The non-perturbative correction can then be written as [Σ(v)]NP = � � � � dσB(ΦB)δ(v(ΦB) − v) � dip � 4αsCdip 2π 1 Q � dy dφ 4π dl2 ⊥ l2 ⊥ + λ2 hy3(l) �� � � λ × INP, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='28) where the suffix λ in the closing curly bracket indicates that we should extract the coefficient of the term proportional to λ in the enclosed expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We will use formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='28) in the following to assess the error due to our approximation in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 The shift in the cumulative cross section In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) we have given the formula for the non-perturbative correction to the leading order 3-jet cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is customary to express the non-perturbative correction as a shift in Σ(v), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' to write ΣB+NP(v) − ΣB(v) = ΣB (v − δv) − ΣB(v) = −dσB dv δv, δv = HNPζ(v), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='29) where ΣB(v) is the Born level value ΣB(v) = � v 0 dσB dv dv, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='30) and using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) for the left-hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='29), we thus define ζ(v) = �dσB dv �−1 � � � � dσB(ΦB)δ(v(ΦB) − v) � �� dip Cdip CF � dη dφ 2π hv(η, φ) � � � � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='31) HNP = M × 4αsCF 2π × INP Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='32) With the above normalization, the shift function ζ in the two-jet case assumes the values 3π for the C parameter, 2 for τ = 1 − T, 1 for M2 H and 0 for M2 D and y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the wide jet broadening in the two-jet limit the linear λ term is actually accompanied by a log λ/Q, and thus the linear term does not have a finite coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We computed the functions ζ(v) for the variables listed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' With an angular-ordering argument, one can show that the limit ζ(v) for v → 0 should tend to the corresponding two-jet limit values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact, in this limit, the emitted hard gluon becomes collinear to either the quark or the antiquark, let us say to the quark for sake of discussion, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Because of coherence, the soft gluon associated with the power corrections sees the collinear quark-gluon pair as a single colour source, with the same colour of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus the emission pattern is the same as that of a quark-antiquark pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Alternatively, one may consider the emissions of the three dipoles q¯q, qg, and ¯qg, that carry the colour factors CF − CA/2 for q¯q and CA/2 for qg and ¯qg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The qg dipole does not emit in the small angle limit (the eikonal formula vanishes there), and the ¯qg dipole becomes equal to the q¯q dipole, giving CF −CA/2+CA/2 = CF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the same soft radiation – 12 – 0 1 2 3 4 5 6 7 8 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 ζ(C) C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ζ(τ) τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ζ(y3) y3 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ζ(M2H) M2H 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ζ(M2D) M2D 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ζ(BW) BW Figure 1: The function ζ plotted for C, 1 − T, y3, M2 H, M2 D and BW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' of a q¯q dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This must happen, however, when the logarithm of the shape variable is so large that it clearly prevails over single logs and constant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the case of C and 1 − T, one finds that for values of the shape variable v ≈ 10−3 the ζ function differs from the two-jet limit value by roughly 10%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' of the order of 1/ log(v), that is the natural size of single-log corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The case of M2 H and M2 D, however, are much more extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In this case, in order to – 13 – Figure 2: Dominant double logarithmic region near the two jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The qg dipole does not radiate, while the q¯q and ¯qg dipoles differ only by their colour factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ζ(M2H) M2H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 10-12 10-10 10-8 10-6 10-4 10-2 M2H Figure 3: The ζ(M2 H) function at very small value of its argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The dots are obtained by performing a quadruple precision calculation and binning the results uniformly in a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The left/right plot use a linear/logarithmic scale for the x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' check that the two jet limit of 1 and 0 respectively are actually reached, we had to perform a dedicated calculation in quadruple precision in the small v region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As an example, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 3 the result of this calculation for M2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is evident that M2 H changes sign and reaches the value 1 very near zero, varying by about 2 units in a very narrow neighbourhood around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' M2 D undergoes an even stronger variation, changing by three units, and reaching zero from negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Such an abrupt change in the three-jet distribution as we approach the two-jet limit suggests that subleading soft terms in the two-jet limit remain more important than double logarithms all the way down to very small values of the shape variable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' questioning on one side the possibility to associate the two-jet limit non-perturbative correction to the resummation of soft radiation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' on the other side,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the application of our newly computed non perturbative correction as we approach the two-jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 Numerical checks As a numerical check of the above calculations we also computed the ζ functions by directly generating the phase space comprising the three hard partons and the soft one, fixing its transverse momentum to a value λ0 = Q0/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' More explicitly, we first generate the underlying Born momenta pi, i = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 3, choose λ0 = 1 GeV and Q0 = 100 GeV, and construct the momentum of the radiated parton as in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Assuming for – 14 – sake of argument that p1 and p2 are the momenta of the radiating dipole, we construct the recoil-corrected momenta as P1 = p1 − l+ − 1 2l⊥ , P2 = p2 − l− − 1 2l⊥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='33) In this way the total momentum is conserved, and the on-shell property of P1/2 are main- tained up to terms of order λ2/Q2 = 1/104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The event comprising P1, P2, p3 and l is then used to compute directly the values of the shape variables, and its difference with respect to the value obtained for momenta p1, p2 and p3 is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Using this method, we find good agreement with the λ → 0 calculations described in the previous section, except near the zero value of the shape variable and, in the case of the C parameter, near the upper end-point of 3/4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the 3-jet symmetric limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We will make use of this method to give an estimate of corrections suppressed by higher powers of λ, as illustrated later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 5 Calculation of the observable distributions We are interested in fitting αs from event shapes in the three-jet region, where the novel results for the non-perturbative corrections can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Furthermore, in the three-jet region the relation between the observables and the value of αs is more direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For this reason, at the perturbative level we consider here only fixed-order predictions and, when determining the fit range, we will make sure that all-order resummed predictions, not included here, have a small effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Perturbative predictions for e+e− → 3 jets are available up to next-to-next-to-leading order (NNLO) accuracy and are implemented in the public code EERAD3 [1–3], which is based on the antenna subtraction formalism [49] and in a private code [4], which is based on the CoLoRFulNNLO subtraction method [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We have used here predictions from EERAD3 up to NNLO and have checked that they agree with predictions using the CoLoRFulNNLO subtraction method up to NLO accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 Denoting by v a generic event shape, the normalized integrated distribution at center- of-mass energy Q and at the renormalization scale µR can be written as ΣNNLO(v) = � v 0 dv′ 1 σNNLO dσNNLO(v′, Q) dv′ = αs(µR) 2π dA(v) dv + �αs(µR) 2π �2 dB(v, xµR) dv + �αs(µR) 2π �3 dC(v, xµR) dv , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) where xR = µR/Q and B(v, xµR) = B(v, 1) + A(v) (β0 ln xR − σ1) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2) C(v, xµR) = C(v, 1) + B(v, 1) (2β0 ln xR − σ1) + A(v) �1 2β1 ln xR + β2 0 ln2 xR + σ2 1 − σ2 � , 4We thank Adam Kardos for providing results up to NLO using the CoLoRFulNNLO subtraction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 15 – with β0 = (11CA − 4nfTR)/3, β1 = (34C2 A − 20CAnfTR − 12CFTRnf)/3, and where the expansion of the total cross section reads σNNLO = σ0 � 1 + αs(µR) 2π σ1 + �αs(µR) 2π �2 σ2 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3) with σ1 = 3CF/2 and σ2 = CF ((123/8 − 11ζ3)CA − 3/8CF + (4ζ3 − 11/2) nf TR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For our central predictions we choose xR = 1/2, and we estimate the error due to missing higher- order terms by varying this scale up and down by a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The choice of xR = 1/2 for the central value is motivated by the fact that the scale entering in the production of the third jet is somewhat lower than Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The non-perturbative corrections discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4 can be included as a shift in the argument of the cumulative cross section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='29), but using instead the full NNLO cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We now depart from the large-nf parameterization of the shift, and switch instead to the dispersive model of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [27], where the role of the effective coupling of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5) is played by a parameter α0(µ2 I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' So, rather than using the definition of eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='31) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='32), the shift (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='29)) can be written as δv = ζ(v)MµI Q 4CF π2 � α0(µ2 I) − αs(µ2 R) − α2 s(µ2 R)β0 π � 2 ln µR µI + K(1) 2β0 + 2 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4) − α3 s(µ2 R)β2 0 π2 � 4 ln2 µR µI + 4 � ln µR µI + 1 �� × � 2 + β1 2β2 0 + K(1) 2β0 � + K(2) 4β2 0 � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5) where the observable dependent part ζ(v) has been discussed in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4, the Milan factor is given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4) and α0(µ2 I) is defined as a mean value of the strong coupling in the CMW [51] scheme below an infrared scale µI which is conventionally taken equal to 2 GeV: α0(µ2 I) = 1 µI � µI 0 dµ ˜αs(µ2) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6) where in the perturbative region the MS and CMW couplings are related as ˜αs(µ2) = αs(µ2) � 1 + αs(µ2) 2π K(1) + �αs(µ2) 2π �2 K(2) + O(α3 s) � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7) with [52, 53] K(1) = CA �67 18 − π2 6 � − 5 9nf , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8) K(2) = C2 A �245 24 − 67 9 ζ2 + 11 6 ζ3 + 11 5 ζ2 2 � + CFnf � −55 24 + 2ζ3 � (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9) + CAnf � −209 108 + 10 9 ζ2 − 7 3ζ3 � − 1 27n2 f + β0 2 � CA �808 27 − 28ζ3 � − 224 54 nf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The last terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5) are subtraction terms of contributions already accounted for in the perturbative calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This assumes that non-inclusive corrections are described by – 16 – the same multiplicative Milan factor M, that applies to all observables we consider with the exception of y3, as discussed at the end of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Notice that in the large nf limit we found (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='32)) δv = HNPζ(v) = M × 4αsCF 2π ζ(v) × INP Q , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='10) where INP can be interpreted as the integral of the large-nf, CMW effective coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact, expanding the second line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5) for small αs we find INP = 1 αs(µ) � µC 0 dλ αs(λe−5/6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='11) and αs(µe−5/6) ≈ αs(µ) + 5 3b0,nf αs(µ)2 = αs(µ) � 1 − 5 9nf αs(µ) 2π � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='12) consistently with eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, formula (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='10) differs by a factor π/2 with respect to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the factor CF/(2π) is replaced by CF/π2 in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This replacement (for more details see ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [27] near formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3)) is irrelevant for the purposes of this work, but we follow this prescription in order to fit values of α0 that can be compared to those found in previous publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is possible to implement the non-perturbative corrections in different ways, leading to results that differ by terms of order O(αs/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We use this ambiguity to assign an uncertainty related to our treatment of non-perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For this purpose we define four schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Our default predictions, scheme “(a)”, are obtained by shifting the perturbative distribution ΣNNLO(v) by the non-perturbative correction computed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4 Σ(a)(v) = ΣNNLO(v − δv) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='13) Furthermore, in scheme (a), we also add to δv an approximate estimate of quadratic cor- rections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' These are obtained from the difference between the numerical evaluation at finite transverse momentum described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 with respect to our standard calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' More specifically, calling ˜ζ(v) the evaluation of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4, we correct δ(v) as follows δ(v) = ζ(v)HNP + � ˜ζ(v) × Q0 λ0 − ζ(v) � × Q0 λ0 × H2 NP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='14) Alternatively, instead of shifting the full NNLO distribution, one can shift only in the leading order term ΣB of the integrated distribution (scheme (b)): Σ(b) FULL(v) = ΣB(v − δv) + Σ(v) − ΣB(v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15) Yet another option is to expand the integrated distribution around the perturbative value (scheme (c)): Σ(c) FULL(v) = Σ(v) − δvΣB(v) dv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='16) Scheme (d) is defined as scheme (a) but without the quadratic correction of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='14) included in the other schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 17 – 6 Fit to ALEPH data We now compare the theoretical predictions including power corrections to the ALEPH data of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [42], where several shape variables were analyzed in the centre-of-mass energy range from 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 to 206 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Here we focus upon the 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 GeV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Including higher energy data does not lead to noticeable differences in the results, as we will discuss briefly in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Our goal is to fit several observables at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We need to select observables such that the power corrections in the three jet region can be computed with our methods, and that are at the same time available in ALEPH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' These are the C-parameter, τ = 1 − T, y3 in the Durham scheme, the heavy-jet mass M2 H, the mass difference M2 D, and the wide jet broadening BW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Since the y3 variable is not really additive, we need to provide an estimate of the error associated with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We will do so along the lines discussed at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The non-perturbative corrections to M2 H, M2 D and BW have a common feature: in the 3-jet regime they differ drastically from their value in the two-jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Such an abrupt change is quite worrisome, and may be taken as an indication that higher-order emissions may be associated with large corrections to the non-perturbative coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For this reason, initially we leave these variables out of the fit, and only fit the C-parameter, τ and y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We fit the value of αs(MZ) and the non-perturbative parameter α0, defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 Treatment of uncertainties 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 Statistical and systematic errors, and correlations The ALEPH data (available at the site https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='hepdata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='net/record/ins636645) includes statistical and systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Our method of choice for computing the error is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Calling Ri the statistical error, Si the systematic error, Ti the theoretical error relative to bin i, Cij the statistical correlation matrix, and Cov(Sys) ij the covariance matrix for the systematic errors, we compute the full covariance matrix as Vij = δij(R2 i + T 2 i ) + (1 − δij)CijRiRj + Cov(Sys) ij , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) where the indices i and j run over all the bins of all observables that have been included in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The ALEPH data quotes two kinds of systematic errors for the data taken at the Z pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We add these two errors in quadrature to obtain the global systematic error that we use in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We computed the statistical correlation coefficients Cij using Pythia8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Calling Ni and Nj the number of events that fall into bin i and bin j, and Nij the number of events that contribute to both bins, we have Cij = Nij N − NiNj N2 � Ni N − N2 i N2 � Nj N − N2 j N2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2) where N is the total number of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Note that Nij is zero for different bins of the same observable, so that a negative correlation is expected for all pairs of bins in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 18 – Statistical, systematic and theoretical errors are assumed to be uncorrelated among each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the covariance of the systematic errors we adopt the so called “minimum overlap” assumption (denoted in the following as MO), and set them equal to the minimum of the square of the systematic errors for the bins in question, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Cov(Sys) ij = δijS2 i + (1 − δij) min(S2 i , S2 j ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3) As an alternative, we computed the covariance matrix for the case of C, T and y3, by using the 24 systematic variations of the resulting distributions that were obtained by ALEPH in order to determine the systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 We compute the covariance matrix and the central value as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We call v(r) i the value of a shape variable for the bin i, where again i denotes both the bin and the observable, and where r labels the 25 replicas (a central value plus 24 variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We then define ¯vi = 1 Nr � r v(r) i , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4) ¯vij = 1 Nr � r v(r) i v(r) j , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5) Cov(Sys) ij = � r � v(r) i − ¯vi � � v(r) j − ¯vj � = Nr (¯vij − ¯vi¯vj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6) We use Cov(Sys) ij as covariance matrix, and for the central value we use either the replica corresponding to the ALEPH default setup, or the average over all replicas ¯vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Some of the variations provided are double sided (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' they are associated with a positive and negative variation of a parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For these variations we have included a factor 1/2 in the computation of ¯vij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the following we call this the “replica method”, and denote it with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The covariance matrix is used to compute the χ2 value according to the standard formula χ2 = � ij � vi − v(th) i � Vij � vj − v(th) j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 Perturbative theory uncertainties As a consequence of the high precision of the LEPI data, in order to obtain reasonable χ2 values when performing the fits we add the theoretical uncertainty in quadrature to the experimental one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We do not assume any correlations for the theoretical errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We define the perturbative theoretical error by considering three values for the renor- malization scale: Q, Q/2 and Q/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Calling O(µr) the value of a shape variable in a bin, we define the perturbative central value Ocv and the associated perturbative error Oerr of the theoretical prediction as follows, Ocv = max(O(Q), O(Q/2), O(Q/4)) + min(O(Q), O(Q/2), O(Q/4)) 2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8) Oerr = max(O(Q), O(Q/2), O(Q/4)) − min(O(Q), O(Q/2), O(Q/4)) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9) 5We thank Hasko Stenzel for providing these data to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 19 – The perturbative theoretical error is quite small at the NNLO level we are working with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 Non-perturbative theory uncertainty Non-perturbative corrections can be sizeable, up to the order of 10%, and thus we must also include an error associated with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As seen in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4, there is evidence that power suppressed corrections of second order are not negligible, especially near the two jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We have estimated them and used them to correct the central value, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We thus associated an uncertainty equal to twice the quadratic correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As a further point, we expect that the ζ functions may receive perturbative corrections of order αs ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We thus define the following associated error to δ(v) δerr(v) = 2 · ����˜ζ(v) × Q0 λ0 − ζ(v) ���� × Q0 λ0 × H2 NP + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 · δ(v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='10) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 Correction for heavy-quark mass effects Our NNLO calculation deals with massless quarks, while the data includes primary charm and bottom pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We correct the data by multiplying, for each bin of each observable denoted globally as i, the ratio of the Monte Carlo predictions for the corresponding ob- servables vi evaluated without and with the c and b primary production processes v(corr) i = vi × vMC,uds i vMC,udscb i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='11) The correction factors obtained using Pythia8 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Notice that corrections are quite modest, although not totally negligible in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 Hadron mass-effects corrections As already discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 3, the theoretical calculation of shape-variable distributions deals with massless particles and the massless definition can be extended to deal with massive particles using different schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Since full particle identification is not available in an experimental context, this lack of information is filled by the Monte Carlo simulation when correcting from the detector level to the generator level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We also use a Monte Carlo generator to compute shape variables in the different schemes, and then construct migration matrices to correct from the scheme adopted by the experiment to any another scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' More specifically, for each Monte Carlo event, we compute the shape variable in the standard scheme (the one adopted by the experiment, as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 2) and another scheme S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Assuming that the shape variable in the standard scheme falls into bin i, and the same shape variable in scheme S falls into bin j, we increase by one unit a migration matrix Tij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This matrix is used to correct the real data according to the formula n (S) j = � i n (data) i Tij � k Tik , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='12) designed in such a way that if one replaces the n (data) i with its Monte Carlo prediction, one obtains by construction the Monte Carlo prediction for n (S) j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the following, we use this method to assess the hadron-mass sensitivity of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 20 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='96 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='04 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 C/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='96 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='04 T Shape variable 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='96 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='04 MH2 (1/σ dσ(u,d,s)/dv) / (1/σdσ(u,d,s,c,b)/dv) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='96 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='04 y3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='96 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='04 MD2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='96 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='04 BW Figure 4: Heavy-flavours correction factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The coloured band marks the range where fits are usually performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Notice that we plot C/2 rather than C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 7 Fit results Our default fit is based on the ALEPH data of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [42] at 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 GeV, and includes the thrust variable τ = 1 − T, the C-parameter and the Durham 3-jet resolution variable y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In our perturbative predictions we fix the renormalization scale to µR = Q/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Non-perturbative effects are included as a shift of the total integrated distribution, corresponding to scheme (a) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Our default mass scheme is the E scheme discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 3, since it yields intermediate results with respect to the other schemes, and is also closer to the result obtained in the standard scheme (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' the scheme used by ALEPH, as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The treatment of correlations is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In particular, we chose the minimum-overlap method as our default choice, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We apply the heavy- to-light correction factors described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We use Pythia8 as our standard Monte Carlo to compute the heavy-to-light correction factor and, when using a different mass scheme, to compute the migration matrix to be used to correct from the scheme adopted by the experiment to any another scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' To perform our fit we use the default fit ranges listed in the second column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The lower edges of the ranges are determined in such a way that the impact of the resummation remains small (see Appendix A), while the upper edge is close to the three-particle kinematic bound of the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The result of the simultaneous fit of αs and α0, together with the total χ2 and χ2 per degree of freedom is shown in the first line of Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the same table we illustrate how the fit results change if we vary any of the default choice made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In particular, we show the fit results when fixing the central value of the – 21 – observable default Fit ranges (2) Fit ranges (3) C [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 ] [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='17 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 ] [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='375 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 ] τ [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ] [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='067 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ] [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ] y3 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ] [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='033 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ] [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='075 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 ] Table 1: Default fit range used (second column), and alternative choices (obtained by multiplying the default lower bound by 2/3 and 3/2) used to estimate the impact of the choice of the fit range (third and forth column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Variation αs(MZ) α0 χ2 χ2/Ndeg Default setup 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='64 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='17 Renormalization scale Q/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='60 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='21 Renormalization scale Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='68 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='20 NP scheme (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='77 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='16 NP scheme (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='80 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='12 NP scheme (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='66 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='13 P-scheme 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='62 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='24 D-scheme 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='79 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='13 Standard scheme 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='58 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='21 No heavy-to-light correction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='67 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='16 Herwig6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='59 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='36 Herwig7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='60 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='27 Ranges (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='62 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='23 Ranges (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='69 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='08 Replica method (around average) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='61 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='16 Replica method (around default) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='61 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='16 y3 clustered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='66 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='19 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='07 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='04 y3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1214 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='02 C, τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='07 Table 2: Default fit result for αs(MZ) and α0 (first line) and other fit results obtained by varying the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' See text for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' renormalization scale to µR = Q/4 or Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We investigate the impact of the way in which non- perturbative corrections are implemented, using the alternative schemes (b, c, d) presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 5 (near Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We also present the result obtained using the P- and D- scheme to define the observables, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3, and the result obtained in the standard scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' To assess the impact of the heavy-to-light correction factor we switch it completely off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We vary the Monte Carlo used to compute the migration matrix for the scheme and – 22 – the heavy-to-light correction factor, and consider Herwig6 [54] and Herwig7 [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We vary the fit ranges adopted, as detailed in columns three and four of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Since correlations play an important role, we also use the replica method, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6), using variations either around the average values of the replicas, or around the values of the default replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In the case of y3 there is one further uncertainty, associated with the fact that we computed the non-perturbative correction assuming that the two soft partons from the splitting of the soft gluon are not clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In order to estimate an associated uncertainty, we also computed the non-perturbative correction assuming that the two soft partons are always clustered together, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The ratio of the latter to the former results ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='85 in the fit window adopted for y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We have therefore performed the fit using alternatively the approximation where the soft partons are always clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The corresponding result is reported in the table labeled as y3-clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The central value for αs in the simultaneous fit of y3, C and T is reduced by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Given the fact that we have chosen the lowest extreme of the variation, and that the correct result must lie between the always-clustered and the never-clustered cases, our estimate of this uncertainty is very conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 Finally, we examine how the fit results change if we consider one observable at the time, or if we exclude y3 from the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 Including higher energy data In the ALEPH publication [42], data are also available for centre-of-mass energies of 133, 161, 172, 183, 189, 200 and 206 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Including these data does not appreciably change the result of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For our default setup we get αs(MZ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1184 and α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='64, compared to αs(MZ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1182 and α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='64 of the fit on the Z peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We get a χ2/Ndeg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='70, larger than the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='17 of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This is easily understood, since higher energy data have dominant statistical errors, and thus the χ2/Ndeg is more in line with the expectation from statistical dominated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 Discussion of the results Our findings can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For all results presented in the table, we observe an excellent χ2 of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In particular the χ2 over number of degrees of freedom is always well below one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This is a consequence of our treatment of the theoretical error, that has been added bin-by-bin to the experimental one without correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Because of this, the theoretical prediction has considerable flexibility to adapt to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The choice of renormalization scale changes the fit by about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5%, the largest change driven by the variation to lower scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' A similar change can be observed when examining alternative schemes to implement non-perturbative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The mass-scheme defini- tions bring in an effect of about 2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The heavy-to-light correction factor changes αs by just about one permille, hence the uncertainty associated to this correction seems negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' A few permille differences are found when using a different Monte Carlo to change from the standard definition to the E-scheme and to perform the heavy-to-light correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' These 6Notice that the anti-kt algorithms [56] are such that the softest particles are never clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 23 – small differences are not surprising since all the Monte Carlos we use are tuned to these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The choice of the fit range has an impact on the result of about one percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This confirms that the range chosen is such that the impact of the resummation is modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The choice of how to treat statistical correlations has also a similar impact, and confirms that our minimal overlap approach provides a sensible description of the correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For y3, the difference between the two limiting cases (where soft emissions are always-clustered or never-clustered) amounts also to about a one percent effect on the full fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Finally, we note that if one fits αs and α0 from the three observables considered separately, one tends to get a larger value of the strong coupling, but with very different values of α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Indeed, there is a tension in the fitted value of α0, where both thrust and C- parameter prefer a lower value, while y3 prefers a higher one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' When fitting all observables at the same time, the overall effect is that one finds an intermediate value for α0 and a lower value of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The χ2 of the fits remain excellent, which justifies a simultaneous fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The role of each variable in the common fit is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As one can see, for C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 α0 αs(MZ) C τ y3 C+τ C+τ+y3 Figure 5: Contours at ∆χ2 = 1 for fitting, C, τ and y3 individually, and then in the combinations C + τ and C + τ + y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' and τ, α0 and αS are strongly anti-correlated, and with a similar anti-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' On the other hand, y3 has a ζ function that is small and of opposite sign, and thus α0 and αS are only weakly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The combined fit is then strongly constrained leading to an intermediate value of α0 and a smaller value of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Altogether, we conclude by remarking that our fit results agree very well with the world average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In particular, we do not find low values of αs for the thrust or C-parameter which are included in the current PDG average [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, our results also clearly show that a fit of αs from event shapes with an overall uncertainty below the percent level seems today not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In particular, by changing certain choices that we have made, like the central renormalization scale or the mass scheme, one can easily obtain higher values of αS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 24 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 Comparison to results obtained by setting ζ(v) = ζ2J(v) It is natural now to ask what the results of the fits would have been if we had used the non-perturbative correction as estimated in the two-jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For C, τ, y3, M2 H and M2 D this amounts to setting the ζ(v) functions plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 1 to a constant value, according to the table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 For BW the function ζ2J(v) can be found in Appendix F of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The v C τ y3 M2 H M2 D BW ζ2J(v) 3π 2 0 1 0 App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' F of [43] Table 3: The non-perturbative coefficients in the two jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' complete results are reported in table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As shown there, the values of αs found in this way are consistently lower than those of table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For example, for our default setup we have αs(MZ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1182 and α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='64, while using ζ2J we get 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1132 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='55 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' On the other hand, the χ2 values are also quite acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 A more detailed comparison of our default fit with the newly calculated ζ functions, and with the ζ2J functions corresponding to what has been available until now is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As mentioned earlier, both fits look plausible, was it not for the fact that the ζ2J result favours values of αs lower than the world average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The quality of the fits is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As one can see, the fit with the full ζ(v) dependence seems slightly better, while the one with the ζ2J functions exhibits some tensions among the different observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, on the basis of the χ2/Ndeg values, both fits are quite acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is now interesting to see what happens to the remaining shape variables, M2 H, M2 D and BW evaluated with the same parameters used for our default fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The result is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' There we see distinctly that the full ζ(v) fit works very well towards the three jet region for all the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The ζ2J fit, on the other hand, does not work in the three- jet limit, while its description of data improves in the two-jet region, with the noticeable exception of M2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 On the structure of αsλ/Q corrections Higher-order corrections to the linear λ term are certainly present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The important issue is whether these corrections are of order αs(Q) or rather αs(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In this work we are implicitly assuming that they are suppressed by a power of αs(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We do not have a solid argument to prove this assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, by examining the structure of the linear power corrections near the two-jet limit we gain some insight into how this may actually work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact one can write schematically the correction of order λ to a shape variable v in the two-jet limit 7 For the case of y3, the coefficient is known to be zero [25], since y3 is quadratic in the transverse momentum for soft emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' As for the case of M 2 D, a colour coherence argument would lead to the conclusion that in the dominant collinear limit the corrections to the two hemispheres are identical, leading again to a null value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the remaining variables, see for example table 1 of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 8We do not ascribe any significance to the larger χ2 values in the two-jet limit, because in this case in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='10) we have assumed rather arbitrarily ζ(2)/ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 25 – Variation αs(MZ) α0 χ2 χ2/Ndeg Default setup 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='55 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='36 Renormalization scale Q/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='53 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='19 Renormalization scale Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='57 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='50 NP scheme (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='63 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='58 NP scheme (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='72 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='37 NP scheme (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='55 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='36 P-scheme 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='53 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='50 D-scheme 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='66 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='37 Standard scheme 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='51 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='36 No heavy-to-light correction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='58 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='36 Herwig6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='51 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='71 Herwig7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='52 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='49 Ranges (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='54 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='55 Ranges (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='58 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='33 Replica method (around average) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='53 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='31 Replica method (around default) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='53 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='31 y3 clustered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='55 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='36 C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='08 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='06 y3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1160 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='18 C, τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='08 Table 4: Default fit result for αs(MZ) and α0 (first line) and other fit results obtained by varying the setup, using the ζ2J values of table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' See text for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' as9 dσ dv ���� λ = Nλ � δ′(v)ζ2j + (δ′(v)V1 + δ(v)V2)αs + d dv �dσq¯qg dv ζ(v) �� , (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) where the first term is the correction to the leading (two-parton) configuration, the second term is the virtual correction of order αs, and the third term is the correction we have computed, and where we implicitly assume some regularization of the v → 0 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The derivative of the delta function in the first term is necessary to guarantee that upon in- tegration in v there are no linear corrections left at order zero in αs, since we know that they are absent in the total cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The terms V1 and V2 incorporate corrections where the hard gluon is virtual and the gluer is real or virtual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In this case, besides the derivative of the δ-function, we also include an explicit δ-function to indicate that terms that do not vanish upon integration in v must exist and are in fact divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We do not include virtual corrections to the q¯qg process for the exchange of a virtual gluon of mass λ, since it was shown in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [40] that these do not lead to linear terms in λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The absence 9This holds for all the observables that we are considering with the exception of BW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 26 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='122 ζ(v) ζ2J(v) α0 αs(MZ) Figure 6: Central values and δχ2 = 4 (dashes) and 1 (solid) contours for our default fit of table 2 (blue) and the fit obtained with the ζ2J functions, corresponding to the default fit of table 4 (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' of linear corrections to the total cross section leads us to conclude that the integral of the above formula from v = 0 up to any finite value of v must be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact, if that was not the case, such divergence could not be canceled when performing the integral in the whole range of the shape variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus the argument of αs must be taken equal to the hard scale (that in this case is not quite Q, but is related to the typical transverse mo- mentum of the perturbative gluon that sets the value of v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We have thus shown that the singular contributions of the hard gluon (hard relative to the scale λ) in the real emission and virtual exchanges cancel each other also in the coefficient of the linear term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The argument given above also suggests a possible way to match the linear corrections in the three-jet limit to those in the two-jet limit, that are entangled with resummation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' If we recall that the two-jet limit of the functions ζ(v) for C, τ, M2 H and M2 D approach the value ζ2j, we could conclude that the part of the last term in the square bracket of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) that is singular in the two jet limit must combine with the virtual correction to yield a finite result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This combined result is precisely what one gets when expanding in powers of αs the Sudakov form factor, including the shift for the two-jet non- perturbative correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Thus, it is tempting to conclude that the singular part of the last term function should be combined with the resummation component of the cross section, while only the regular part should be applied to the 3-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is unlikely, however, that this approach will work for observables like M2 H and M2 D, since in their case the limiting value is approached extremely slowly, and in the first case it has even opposite sign with respect to the average value of the ζ function in the fit range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is however reassuring to see that if we restrict ourselves to regions far away the two-jet region, all shape variables – 27 – 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1/σ dσ/dC ζ(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='35 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 data/theory τ 0 2 4 6 8 10 12 ζ2J(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='951 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 τ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 3 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 data/theory y3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 ζ2J(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 y3 Figure 7: Theoretical predictions compared to data for our default setup on the left side, and the default setup with the ζ2J functions on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The gray band represents the theoretical errors, while the red bars indicate the experimental ones, with the smaller one representing the statistical error, and the green lines show the pure perturbative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The highlighted region represents the fit range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' are well described with the ζ functions computed here, while this is not the case with the values of table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 28 – 0 1 2 3 4 5 6 7 8 1/σ dσ/dM2h ζ(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 data/theory M2h 0 1 2 3 4 5 6 7 8 9 ζ2J(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 M2h 0 1 2 3 4 5 6 1/σ dσ/dM2d ζ(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 data/theory M2d 0 1 2 3 4 5 6 ζ2J(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 M2d 0 2 4 6 8 10 1/σ dσ/dBw ζ(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 data/theory Bw 0 2 4 6 8 10 ζ2J(v) no NP corr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 Bw Figure 8: Theoretical predictions compared to data for our default setup on the left side, and the default setup with the ζ2J functions on the right side, for the M2 H, M2 D and BW shape variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The gray band represents the theoretical errors, while the red bars indicate the experimental ones, with the smaller one representing the statistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The green lines show the pure perturbative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 8 Conclusions In this work, we study the effect of power corrections in e+e− observables in comparison to data, under the light of the new findings of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [40, 41], where it was shown that power corrections can be computed directly in the three-jet configuration, rather than – 29 – extrapolating them from the two-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [40, 41] these power corrections were computed for the C-parameter and for thrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Here we also computed them for the three- jet resolution parameter in the Durham scheme y3, for the squared mass of the heavy hemisphere M2 H, for the squared-mass difference of heavy-light hemispheres M2 D, and for the wide jet broadening BW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The observables we considered are those that can be computed in the approach of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [40, 41], and that are included in the ALEPH data of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For simplicity we stick to a single data set, and we perform our calculation using the NNLO results for e+e− hadronic observables, plus the newly computed power corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We do not attempt to include resummation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Rather, we stick to ranges of the observables that are far enough from the two jet region so that no visible depletion of the resummed result with respect to the fixed-order one is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We stress that in this work we are assuming that the non-perturbative corrections as estimated according to the results of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [40, 41] are not drastically modified by the inclusion of soft radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Our argument concerning the two-jet limit region near eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1) seems to indicate that this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, we are unable to provide a solid argument for the three-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Our main results can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' First of all, for all the shape variables that we considered, with the exclusion of the wide-jet broadening, the function that pa- rameterised the non-perturbative correction, called ζ(v), approaches its two jet-limit value when its argument approaches the two-jet limit value (set conventionally to v = 0), as one expects according to simple physics arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, with the exception of y3, the limit, is approached only for exponentially small values of the shape variable, so that, in practice, one sees an effective jump of the function near v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This jump is not very important for C and for the thrust τ = 1 − T, where it is around 10-20% of the two-jet limit value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' It is instead quite large for M2 H and M2 D, where it is such that the two-jet limit value cannot be considered representative of the value of the function even very close to the two-jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In view of these observations, we exclude these observables from our fit, and also exclude BW that is positive and divergent in the two-jet limit, and is instead negative in the three-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We thus fitted C, τ and y3, extracting a value for the strong coupling constant on the Z peak, and for the non-perturbative parameter α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The result of the fits yield a value of αs in acceptable agreement with the world average, although we find that a number of variations of our procedure can lead easily to differences of the order of a percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Using the same value of αs and α0, we see that we can describe quite well also the remaining observables M2 H, M2 D and BW , as long as we remain far enough from the two-jet limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Conversely, with the traditional implementation of power corrections, good fits to C, τ and y3 can also be obtained, however the description of M2 H, M2 D and BW in the three-jet region is totally unacceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We stress again that the inclusion of resummation effects in the bulk of the three jet region leads smaller values of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='10 10In particular, for fits to the C-parameter one finds values of αs smaller by about ten percent (private communication by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Monni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 30 – We are aware that the present work should only be considered as a preliminary explo- ration of the implications of the results of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In fact, there are few directions that need further exploration in order to fully exploit these new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' First of all, it would be interesting and important to also include resummation effects in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Some ideas regarding this are discussed in the text, suggesting that perhaps the two-jet limit shift should be applied to the resummed component of the cross section, while the full ζ(v) dependent part should be applied to the finite part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Yet, whether this approach is sensible also when including resummation effects far from the two-jet region is a question that needs to be examined more closely, since for most observables ζ(0) differs considerably from ζ(v) in the three-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' A second direction of improvement regards the choice of the hadron mass-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Lacking a theoretically sound treatment of this problem, a possible development would be to see if there is a scheme that is preferred by data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This in turn would require considering enough observables that display different behaviour regarding the mass-scheme choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This brings us to consider a third extension of this work, which is to examine more variables, and find a sufficiently large set such that the requirements for the applicability of the results of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [40, 41] are met, and such that their behaviour near the two-jet limit are closer to that of the thrust and the C-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' These new variables, could also be analyzed at present using preserved LEP data [58], while waiting for the beginning of operation of new e+e− colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Acknowledgments P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' would like to thank the Max Planck Institute for hospitality while part of this work was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We thank Andrea Banfi, Adam Kardos, Stephan Kluth, Pier Francesco Monni, Silvia Ferrario Ravasio, Gavin Salam, Hasko Stenzel, Roberto Tenchini, and Andrii Verbytskyi for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 31 – A Impact of resummation The fits of αs carried out in this work rely on fixed order NNLO predictions, rather than on all-order (NNLL) predictions matched to fixed order, as computed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [9, 10, 13– 15, 59–61] for event-shapes and in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [12] for the Durham three-jet resolution parameter y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Although it is customary to include resummation effects also far away from the two- jet region, in this work we made the assumption that resummation effects should not be included when the logarithm of the shape variable is not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In order to determine a range for the fit, we thus compare in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 9 NNLO and NNLO+NNLL predictions for the thrust variable τ = 1 − T, the C-parameter, and the Durham three jet resolution variable y3 and exclude in our fits the regions where matched predictions clearly depart from the fixed order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Each plot shows the ratio to the NLO prediction obtained with central renormalization scale µR,0 = Q/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The green band shows the uncertainty of the NLO and the blue band of the NNLO, and are obtained by varying µR up and down by a factor two around the central value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the NNLO+NNLL matched predictions we fix our default setup as follows: we set the central renormalization scale to µR,0 = Q/2, the resummation scale to µQ,0 = Q/2, we use the modified logarithm L = 1/p ln � 1/vp − 1/vp lim + 1 � , where vlim denotes the kinematic limit of the event shapes, with p=3, and we use the log-R matching scheme (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The uncertainty band is then obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Around the above described default setup, we vary, one at the time, µR,0/2 ≤ µR ≤ 2µR,0, µQ,0/2 ≤ µQ ≤ 2µQ,0, we vary p to p = 2 and p = 5, and, finally, we use the R-matching scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This gives a total of eight matched predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The red uncertainty band shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 9 is obtained by taking the envelope of all these predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' The onset of resummation effects is signalled both by a drop of the distribution of the resummed result and by an increase of the NLO result with respect to the NNLO one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' We choose the lower bound of our fit ranges to be to the right of this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Furthermore, for the three observables used in the fit we observe the following features: for the thrust, the uncertainty bands of the NNLO and matched predictions overlap, with the resummation band being a few percent higher, which would lead to slightly smaller values of αs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For the C-parameter one observes a somewhat similar behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, the difference between the center of the resummed and NNLO bands now reach up to 10% and the resummed band has a slightly different shape compared to the NNLO one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' For y3 one observes small effects, at the level of a 2%, however in this case the uncertainty bands do not overlap since the NNLO band is extremely small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' From all three plots it is also clear that the difference between NNLO and matched predictions does not vanish even for large values of the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' This is due to the fact that, even with the modified logarithms, the resummation is not switched off fast enough even close to the end-point of the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' From the figures it can be seen that in the case of the thrust, the resummed predic- tion seems to follow the trend of the NLO and NNLO corrections, possibly approximating higher-order results if they follow the same trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' However, in the case of the C-parameter the resummed result has a slope that is not present in the NLO and NNLO results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Fur- thermore, in the case of y3, the trend is to have the NNLO distribution smaller than the NLO one, while the resummed result is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' In conclusion, although it has become – 32 – common practice, we see no reason in principle to include resummation effects also in the three-jet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 Ratio to dσNLO/dτ at µR=Q/2 τ NLO NNLO NNLL+NNLO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='7 Ratio to dσNLO/dC at µR=Q/2 C NLO NNLO NNLL+NNLO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='95 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content='3 Ratio to dσNLO/dy3 at µR=Q/2 y3 NLO NNLO NNLL+NNLO Figure 9: Comparison between NLO (green bands), NNLO (blue bands) and NNLO+NNLL (red bands) predictions for the thrust (left), C-parameter (central), y3 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' See text for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' – 33 – References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E2T4oBgHgl3EQfBwbf/content/2301.03607v1.pdf'} +page_content=' Gehrmann-De Ridder, T.' metadata={'source': 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Korzyuk1,2, Jan V. Rudzko1 +1Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, Belarus +2Belarusian State University, Minsk, Belarus +CLASSICAL SOLUTION OF THE INITIAL-VALUE PROBLEM FOR A QUASILINEAR +WAVE EQUATION WITH DISCONTINUOUS INITIAL CONDITIONS +Abstract. For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, +we consider the Cauchy problem. The initial conditions have discontinuity of the first kind at one point. +We construct the solution using the method of characteristics in an implicit analytical form as a solution of +some integro-differential equations. The solvability of these equations, as well the smoothness of their +solutions, is studied. For the problem in question, we prove the uniqueness of the solution, and establish +the conditions under which its classical solution exists. +Keywords: nonlinear wave equation, Cauchy problem, method of characteristics, classical +solution, discontinuous initial conditions. + +Introduction. Partial differential equations with discontinuous initial conditions are quite common +in various applications, e. g. the propagation of shock waves in a medium [1]. We usually model this +phenomenon using the Cauchy problem with discontinuous conditions. This leads to difficulties in the +definitions and interpretations of solutions [2]. +We often solve such problems by functional methods, e. g. the Fourier transform and the Laplace +transform. However, these methods usually do not cover all possible cases of giving the Cauchy conditions +[3] since inverse integral transformations converge, as a rule, to the average of the left and right limits [4]. +Therefore, various methods have been developed to solve such problems, e. g. the contour integral method +[5]. Although this method has a lot of disadvantages, some of which are also inherent in the Fourier method +[6], such as increased requirements for smoothness of functions and matching of functions, i.e., functions +must satisfy some additional matching and smoothness conditions, it has allowed us to consider a large +number of problems of dynamic impact theory, see [7–10] and cited literature. Also, we note the classical +d’Alembert method (the method of characteristics), which is not as powerful as functional methods. But we +can use it to obtain a qualitative description of impact phenomena [11; 12], to solve some boundary-value +problems [13; 14] of impact theory, and it allows us not to identify functions that differ on a set of Lebesgue +measure zero. +This work is a continuation of our studies of the Cauchy problem for a mildly quasilinear wave +equation [15] and mixed problems with discontinuous initial and boundary conditions [13; 14; 16; 17]. In +this article, we consider the Cauchy problem a one-dimensional mildly quasilinear wave equation given in +the upper half-plane. The initial conditions of this problem have discontinuity of the first kind at one point +of the real axis. We use the method of characteristics to solve this problem. We build the piecewise-smooth +solution, which satisfies additional matching conditions, in an implicit analytical form as a solution of some +integro-differential equations. We study the solvability of these equations and the smoothness of their +solutions. We prove the existence and the uniqueness of the solution to the Cauchy problem under some +smoothness conditions of the initial data. The article proposes an approach to constructing solutions with +discontinuous initial conditions. +Statement of the problem. In the domain +(0, ) +Q  +  + of two independent variables +2 +( , ) +t x +Q + + +, consider the one-dimensional nonlinear equation +( , ) +( , , ( , ), +( , ), +( , )) +( +( , +, +, +) +, ) +x +t +u t x +f t x u t x +u t x +x +u t x +t x +Q +F +t + + + + + + (1) + +2 + +where +2 +2 +2 +t +x +a +   + is the d’Alembert operator ( +0 +a  + for definiteness), F is a function given on the set +Q , and f is a function given on the set +3 +Q +. Equation (1) is equipped with the initial condition +. +(0, ) +(0 +( ), +( ), +, ) +t +u +x +u +x +x +x +x +  + + + + + (2) +where φ and ψ are some real-valued functions defined on the real axis. +The functions  and  are piecewise smooth and defined by the formula +1 +0 +1 +0 +0 +2 +0 +2 +0 +( ), +( +, +), +( ), +( +, +), +( ) +, +, +( ) +( ), +( +, +), +( ) +( +, +), +x +x +x +x +x +x +x +A +x +x +x +x +x +x +x +x +x + +  + + +  + + + + + + + + + + + + + + + + + +where x0 and A are arbitrary real numbers, +1 + and +1 + are functions given on the set +0 +( +, +] +x + +, and +2 + and +2 + are functions given on the set +0 +[ +, +) +x  . +In the case of sufficiently smooth data, namely, +2( ) +C + +, +1( ) +C + +, +1( ) +F +C Q + +, and +1 +3 +( +) +f +C Q + + +, we considered the problem (1), (2) in the article [15]. +For the linear wave equation, i.e., +0 +f  +, the problem (1), (2) was studied in the works [16] and +[13; 14; 17] in the cases of +1( ) +C + + and +2( ) +C + + respectively. +Following [18], we investigate two main questions: 1) the exact statement of the initial value +problem (1), (2); 2) the smoothness of the solution. The cause for the first question is the non-existence of +the classical solution of the problem due to the condition +2( ) +C + + or +1( ) +C + +. +Constructing the solution of the Cauchy problem. We divide the domain Q by the characteristics +0 +x +at +x + + + and +0 +x +at +x + + + into three subdomains +1 +0 +{( , ) ( , ) +}, +Q +t x +t x +Q +x +at +x + + + + + +∣ + +2 +0 +{( , ) ( , ) +}, +Q +t x +t x +Q +x +at +x + + + + + +∣ + +3 +0 +0 +{( , ) ( , ) +}. +Q +t x +t x +Q +x +at +x +x +at +x + + + + + + + + +∣ + +On the closure Q of the domain Q, we define a function u as the one coinciding with the solution +u(j) of the partial differential equation (1) + +( ) +( ) +* +( , ) +( , ), +( , ) +, +j +j +u t x +u +t x +t x +Q + + + (3) +on the domain +( ) +* +j +Q +, where +(1) +(1) +(3) +* +\ +Q +Q +Q + +, +(2) +(2) +(3) +* +\ +Q +Q +Q + +, and +(3) +(3) +* +Q +Q + +. + + + + + + +Fig. The partition of the domain Q into three subdomains. +By virtue of discontinuous initial conditions, the problem (1), (2) does not have a classical solution +defined on the set Q . Nevertheless, we can define a classical solution on a smaller set +\ +Q  so that it +belongs to the class +2( +\ +) +C +Q  and satisfies the equation (1), the initial conditions (2), and additional +matching conditions given on the set  . +Q(2) +t +x +x0 +O +Q(3) +Q(1) +x − a t = x0 +x + a t = x0 + +3 + +Definition 1. A function u is called a classical solution of the problem (1), (2) if the following +conditions are satisfied: 1) the function u(j) belongs to the class + + +2 +( ) +* +j +C +Q + for each +{1,2,3} +j +; 2) the +function u(j) satisfies Eq. (1) in the domain +( ) +* +j +Q + for each +{1,2,3} +j +; 3) the initial condition (0, ) +( ) +u +x +x +  + +is met on the whole real axis; 4) the initial condition +(0, ) +( ) +tu +x +x + +  + is met on the whole real axis, except +one point +0 +x +x + +; 5) the Goursat conditions +(3) +(1) +0 +0 +1 +0 +( , +) +( , +) +( +), +0, +u +t x +x +at +u +t x +at +A +x +t + + + + + + + +(3) +(2) +0 +0 +2 +0 +( , +) +( , +) +( +), +0, +u +t x +x +at +u +t x +at +A +x +t + + + + + + + +(4) +hold. +It turns out that finding a classical solution of the problem (1), (2) in the sense of Definition 1 is +equivalent to solving the following problem. +Problem (1), (2) with matching conditions on characteristics. Find a classical solution of Eq. (1) +with the Cauchy conditions (2) and the matching conditions + +0 +1 +0 +0 +2 +0 +[( ) +( ) ]( , +) +( +), [( ) +( ) ]( , +) +( +) +, +0. +u +u +t x +x +at +A +x +u +u +t x +x +at +x +A t + + + + + + + + + + + + +  + + +Here by () we have denoted the limit values of the function u and its partial derivatives calculated +on different sides of the characteristics +0 +x +at +x + + +; i.e., +0 +( +) ( , +( )) +lim +( , ( ) +) +p +p +t +t +u +t x +r t +u t r t + +  + + + + +  . +The functions u(1) and u(2) are determined from the Cauchy problems + +( ) +( ) +( ) +( ) +( ) +( ) +( ) +0 +( , ) +( , , +( , ), +( , ), +( , )) +( , ), ( , ) +, +(0, ) +( ), +(0, ) +( ), ( 1) +( 1) +, +j +j +j +j +j +t +x +j +j +j +j +j +t +j +u +t x +f t x u +t x +u +t x +u +t x +F t x +t x +Q +u +x +x +u +x +x +x +x + + + + + + + +  + +  + + + + (5) +for each +1,2 +j  +, and under some conditions have the representations + + + + + + + + + + + + +( ) +( +) +( ) +( ) +( ) +( ) +* +0 +( +) +( +) +( +) +1 +( , ) +( ) +2 +2 +1 +, +, , +, +, +, +, +, +, ( , ) +. +2 +x at +j +j +j +j +x at +x a t +t +j +j +j +j +t +x +x a t +x +at +x +at +u +t x +d +a +d +F +f +u +u +u +d +t x +Q +a + + + + + + + + +  + + + + + +  + + +   +  +   +   +  + + + + + + (6) +Lemma 1. Let the conditions +1( ) +F +C Q + +, +1 +3 +( +)) +f +C Q + + +, +2( +( +)) +j +j +C +  + +, and +1( +( +)) +j +C + + + +be satisfied, and let the function f satisfy the Lipschitz condition with constant L with respect to the three +last variables, i. e., + + +1 +2 +3 +1 +2 +3 +1 +1 +2 +2 +3 +3 +( , , +, +, +) +( , , +, +) + +, +f t x z z +z +f t x w w w +L z +w +z +w +z +w + + + + + + + +. Then for each +1,2 +j  +, the Cauchy problem (5) has a unique solution in the class +2 +( ) +* +( +) +j +C +Q + determined by the formula (6). +Proof. See [15]. +Now the function u(3) is determined from the Goursat problem +(3) +(3) +(3) +(3) +(3) +(3) +(1) +0 +0 +1 +0 +(3) +(2) +0 +0 +2 +0 +( , ) +( , , +( , ), +( , ), +( , )) +( , ), ( , ) +, +( , +) +( , +) +( +), +0, +( , +) +( , +) +( +), +0. +t +x +u +t x +f t x u +t x +u +t x +u +t x +F t x +t x +Q +u +t x +at +u +t +at +A +t +u +t x +at +u +A +x +x +x +x +x +t +at +t +x + + + + + + + + + + + + +  + + + + + + + +  + + (7) +Lemma 2. Let the conditions +1( ) +F +C Q + +, +1 +3 +( +)) +f +C Q + + +, +2( +( +)) +j +j +C +  + +, and +1( +( +)) +j +C + + + +be satisfied, and let the function f satisfy the Lipschitz condition with constant L with respect to the three +last variables, i. e., + + +1 +2 +3 +1 +2 +3 +1 +1 +2 +2 +3 +3 +( , , +, +, +) +( , , +, +) + +, +f t x z z +z +f t x w w w +L z +w +z +w +z +w + + + + + + + +. Then the +Goursat problem (7) has a unique solution in the class +2 +(3) +* +( +) +C +Q +. + +4 + +Proof. According to Lemma 1, the conditions +1( ) +F +C Q + +, +1 +3 +( +)) +f +C Q + + +, +2( +( +)) +j +j +C +  + +, and +1( +( +)) +j +C + + + imply existence and uniqueness of the twice continuously differentiable functions u(1) and +u(2). +This +means +that +mappings +(1) +1 +0 +1 +0 +:[0, ) +( , +( +) +) +t +u +t x +at +A +x + + + + + + + +and +(2) +2 +0 +2 +0 +:[0, ) +( , +) +( +) +t +u +t x +at +A +x + + + + + + are twice continuously differentiable too and coincide at the +point +0 +t  +, i.e., +1 +2 +(0) +(0) + +  +. Now, we use the results of the work [19] to finally prove this lemma. +Let us derive an integro-differential equation for the function u(3). We select four points + + +0 +0, +A +x +, +0 +0 +, +2 +2 +x +at +x x +at +x +B +a + + + + + + + + + + +, + + +, +C t x , +0 +0 +, +2 +2 +at +x +x +at +x +x +D +a + + + + + + + + + + + from the domain +(3) +* +Q +, apply the +curvilinear parallelogram identity [20] and obtain +(3) +(1) +(2) +0 +0 +0 +0 +1 +0 +2 +0 +( , ) +, +( +) +( +) +, +2 +2 +2 +2 +x +at +x x +at +x +at +x +x +at +x +x +u +t x +u +A +x +x +u +a +a + + + + + + + + + + + + + + +  +  + + + + + + + + + + + +* +* +(3) +2 +1 +, +, +, +, +, +4 +2 +2 +2 +2 +2 +2 +x at +x at +x +x +z +y z +y +z +y z +y +z +y z +y +dy +F +f +u +a +a +a +a +a +a +a + +  + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +(3) +(3) +(3) +* +, +, +, +, ( , ) +. +2 +2 +2 +2 +t +x +z +y z +y +z +y z +y +u +u +dz +t x +Q +a +a +a +a + + + + + + + + + + + + + + + + + + + + + + + + (8) +We state the result as the following assertion. +Theorem 1. Let the conditions +1( ) +F +C Q + +, +1 +3 +( +)) +f +C Q + + +, +2( +( +)) +j +j +C +  + +, and +1( +( +)) +j +C + + + hold, and let the function f satisfy the Lipschitz condition with constant L with respect to +the three last variables, i. e., + + +1 +2 +3 +1 +2 +3 +1 +1 +2 +2 +3 +3 +( , , +, +, +) +( , , +, +) + +, +f t x z z +z +f t x w w w +L z +w +z +w +z +w + + + + + + + +. The +initial-value problem (1), (2) has a unique classical solution in the sense of Definition 1. This solution is +determined by formulas (3), (6), and (8). +Analysis of the solution of the Cauchy problem. Taking into account twice continuous +differentiability of the functions u(j) for each +1,2,3 +j  +, independence of Lebesgue integral on the behavior +of the function on a set of measure zero, the expressions (3), (6), and (8), we can formally rewrite (3) in the +form + + + + + + + + + + + + +(3) +* +1 +0 +2 +0 +( +) +0 +( +) +( +) +( +) +( +) +( +) +1 +( , ) +( ) +( , ) +2 +2 +2 +1 +, +, , +, +, +, +, +, +, ( , ) +. +2 +x at +Q +x at +x a t +t +t +x +x a t +x +x +x +at +x +at +u t x +d +A +t x +a +d +F +f +u +u +u +d +t x +Q +a + + + + + + + +  + + +  + + + + + +  +  + + + + + + + + + +   +  +  + +  + +  + + + + + + (9) +where χA is an indicator function of a set A. We note that if +(3) +* +1 +0 +2 +0 +( +) +( +) +( , ) +0 +2 +Q +x +x +A +t x + +  + + + + + + + + + + (10) +then the representation (9) is sometimes called the (‘generalized’) d’Alembert formula [21]. +Following [16], we consider three cases of specifying the discontinuous initial conditions under the +conditions of Theorem 1. +1. +0 +0 +( +0) +( +0) +x +x +A + + +  + + +, i.e., +( ) +C + +. From the Goursat conditions (4), and the fact +( ) +2 +( ) +( +) +j +j +u +C +Q + + for each +{1,2,3} +j + we see that in this case the solution u belongs the class +( ) +C Q and +satisfies the ‘generalized’ d'Alembert formula. + +5 + +2. +0 +0 +( +0) +( +0) +x +x + + +  + + and +0 +0 +( +0) +( +0) +2 +x +x +A + + +  + + +. In this case, the solution u is no longer +continuous, but the condition (10) holds, and the solution u satisfies the ‘generalized’ d'Alembert formula +too. +3. +0 +0 +( +0) +( +0) +x +x + + +  + + and +0 +0 +( +0) +( +0) +2 +x +x +A + + +  + + +. The solution u is discontinuous and does +not satisfy the ‘generalized’ d'Alembert formula. +These results are consistent with the conclusions of the work [16]. +We note that the integro-differential equation (9) can be used to define a mild solution of the +problem (1), (2). +Conclusions. In the present paper, we have obtained the necessary and sufficient conditions under +which there exists a unique classical solution (in an extended sense) of the initial value problem for the +mildly quasilinear wave equation with discontinuous initial conditions. And we have proposed an approach +to constructing solutions with discontinuous initial conditions, even for nonlinear equations. +Acknowledgements. The article was financially supported by the Ministry of Science and Higher +Education of the Russian Federation in the framework of implementing the program of the Moscow Center +for Fundamental and Applied Mathematics by Agreement no. 075-15-2022-284. + +References + +1. Zhuravkov M. A., Starovoytov E. I., Mathematical Models of Solid Mechanics. Minsk, BSU, 2021. 535 p. +(in Russian). +2. Hadamard J. Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Moscow, Nauka +Publ., 1978. 352 p. (in Russian). + +3. Bateman H. Physical problems with discontinuous initial conditions. Proceedings of the National Academy +of Sciences of the United States of America, 1930, vol. 16, no. 3, pp. 205–211. https://doi.org/10.1073/pnas.16.3.205 + +4. Polyanin A. D. Handbook of linear partial differential equations for engineers and scientists. New York, +Chapman & Hall/CRC, 2001. 800 p. https://doi.org/10.1201/9781420035322 + +5. Rasulov M. L. Methods of Contour Integration. 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Waves induced by the longitudinal impact of a rod against a stepped +rod in contact with a rigid barrier. Journal of Applied Mathematics and Mechanics. 2009, vol. 73, no. 2, pp. 162–168. +https://doi.org/10.1016/j.jappmathmech.2009.04.006 +12. Koshlyakov N. S., Gliner E. B., Smirnov M. M. Differential Equations of Mathematical Physics. +Amsterdam, North Holland Publishing Co., 1964. 701 p. +13. Korzyuk V. I., Rudzko J. V. The classical solution of one problem of an absolutely inelastic impact on a +long elastic semi-infinite bar. Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics +series, 2021, vol. 57, no. 4, pp. 417–427 (in Russian). https://doi.org/10.29235/1561-2430-2021-57-4-417-427 +14. Korzyuk V. I., Rudzko J. V. Classical solution of one problem of a perfectly inelastic impact on a long +elastic semi-infinite bar with a linear elastic element at the end. Journal of the Belarusian State University. + +6 + +Mathematics and Informatics, 2022, vol. 2, pp. 34–46 (in Russian). https://doi.org/10.33581/2520-6508-2022-2-34- +46 +15. Korzyuk V. I., Rudzko J. V. Classical solution of the initial-value problem for a one-dimensional +quasilinear wave equation. Erugin Readings–2022, Navapołack, Polotsk State University, 2022. Part 2. pp. 38–39. +16. Korzyuk V. I., Puzyrnyi S. I. Classical solution of mixed problems for the one-dimensional wave equation +with Cauchy nonsmooth conditions. Proceedings of the National Academy of Sciences of Belarus. Physics and +Mathematics series, 2016, no. 2, pp. 22–31 (in Russian). +17. Korzyuk V. I., Rudzko J. V. The classical solution of the mixed problem for the one-dimensional wave +equation with the nonsmooth second initial condition. Proceedings of the National Academy of Sciences of Belarus. +Physics and Mathematics series, 2021, vol. 57, no. 1, pp. 23–32 (in Russian). https://doi.org/10.29235/1561-2430- +2021-57-1-23-32 +18. Chekhlov V. I. A mixed problem with discontinuous boundary conditions for the wave equation. Dokl. +Akad. Nauk SSSR, 1968, vol. 183, no. 4, pp. 787–790 (in Russian). +19. Korzyuk V. I., Kovnatskaya O. A., Sevastyuk V. A. Goursat’s problem on the plane for a quasilinear +hyperbolic equation. Doklady of the National Academy of Sciences of Belarus, 2022, vol. 66, no. 4, pp. 391–396 +(in Russian). https://doi.org/10.29235/1561-8323-2022-66-4-391-396 +20. Korzyuk V. I., Rudzko J. V. Curvilinear parallelogram identity and mean-value property for a semilinear +hyperbolic equation of second-order. arXiv:2204.09408. https://doi.org/10.48550/arXiv.2204.09408 +21. Kharibegashvili S. S., Jokhadze O. M. Solvability of a mixed problem with nonlinear boundary condition +for a one-dimensional semilinear wave equation. Mathematical Notes, 2020, vol. 108, no. 1, pp. 123–136. +https://doi.org/10.1134/S0001434620070123 + + +Information about the authors +Viktor I. Korzyuk – Academician, Professor, Dr. Sc. (Physics and Mathematics), Institute of Mathematics +of the National Academy of Sciences of Belarus (11, Surganov Str., 220072, Minsk, Republic of Belarus), Belarusian +State University (4, Nezavisimosti Ave., 220030, Minsk, Republic of Belarus). E-mail: korzyuk@bsu.by +Jan V. Rudzko – M. Sc. (Mathematics and Computer Sciences). Postgraduate student, Institute of +Mathematics of the National Academy of Sciences of Belarus (11, Surganov Str., 220072, Minsk, Republic of +Belarus). E-mail: janycz@yahoo.com. https://orcid.org/0000-0002-1482-9106 + diff --git a/U9AzT4oBgHgl3EQflv3p/content/tmp_files/load_file.txt b/U9AzT4oBgHgl3EQflv3p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b51308d1d429c5b67411c3ef6424021a1052cc88 --- /dev/null +++ b/U9AzT4oBgHgl3EQflv3p/content/tmp_files/load_file.txt @@ -0,0 +1,425 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf,len=424 +page_content='1 UDC 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='35 Academician Viktor I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Korzyuk1,2, Jan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Rudzko1 1Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk, Belarus 2Belarusian State University, Minsk, Belarus CLASSICAL SOLUTION OF THE INITIAL-VALUE PROBLEM FOR A QUASILINEAR WAVE EQUATION WITH DISCONTINUOUS INITIAL CONDITIONS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The initial conditions have discontinuity of the first kind at one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We construct the solution using the method of characteristics in an implicit analytical form as a solution of some integro-differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The solvability of these equations, as well the smoothness of their solutions, is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' For the problem in question, we prove the uniqueness of the solution, and establish the conditions under which its classical solution exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Keywords: nonlinear wave equation, Cauchy problem, method of characteristics, classical solution, discontinuous initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Partial differential equations with discontinuous initial conditions are quite common in various applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' the propagation of shock waves in a medium [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We usually model this phenomenon using the Cauchy problem with discontinuous conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' This leads to difficulties in the definitions and interpretations of solutions [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We often solve such problems by functional methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' the Fourier transform and the Laplace transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' However, these methods usually do not cover all possible cases of giving the Cauchy conditions [3] since inverse integral transformations converge, as a rule, to the average of the left and right limits [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Therefore, various methods have been developed to solve such problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' the contour integral method [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Although this method has a lot of disadvantages, some of which are also inherent in the Fourier method [6], such as increased requirements for smoothness of functions and matching of functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', functions must satisfy some additional matching and smoothness conditions, it has allowed us to consider a large number of problems of dynamic impact theory, see [7–10] and cited literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Also, we note the classical d’Alembert method (the method of characteristics), which is not as powerful as functional methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' But we can use it to obtain a qualitative description of impact phenomena [11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 12], to solve some boundary-value problems [13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 14] of impact theory, and it allows us not to identify functions that differ on a set of Lebesgue measure zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' This work is a continuation of our studies of the Cauchy problem for a mildly quasilinear wave equation [15] and mixed problems with discontinuous initial and boundary conditions [13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' In this article, we consider the Cauchy problem a one-dimensional mildly quasilinear wave equation given in the upper half-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The initial conditions of this problem have discontinuity of the first kind at one point of the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We use the method of characteristics to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We build the piecewise-smooth solution, which satisfies additional matching conditions, in an implicit analytical form as a solution of some integro-differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We study the solvability of these equations and the smoothness of their solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We prove the existence and the uniqueness of the solution to the Cauchy problem under some smoothness conditions of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The article proposes an approach to constructing solutions with discontinuous initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Statement of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' In the domain (0, ) Q \uf03d \uf0a5 \uf0b4 of two independent variables 2 ( , ) t x Q \uf0ce \uf0cc , consider the one-dimensional nonlinear equation ( , ) ( , , ( , ), ( , ), ( , )) ( ( , , , ) , ) x t u t x f t x u t x u t x x u t x t x Q F t \uf0b6 \uf0b6 \uf02b \uf0ce \uf03d (1) 2 where 2 2 2 t x a \uf03d \uf0b6 \uf02d \uf0b6 is the d’Alembert operator ( 0 a \uf03e for definiteness), F is a function given on the set Q , and f is a function given on the set 3 Q\uf0b4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Equation (1) is equipped with the initial condition .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' (0, ) (0 ( ), ( ), , ) t u x u x x x x \uf03d \uf06a \uf0b6 \uf079 \uf0ce \uf03d (2) where φ and ψ are some real-valued functions defined on the real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The functions \uf06a and \uf079 are piecewise smooth and defined by the formula 1 0 1 0 0 2 0 2 0 ( ), ( , ), ( ), ( , ), ( ) , , ( ) ( ), ( , ), ( ) ( , ), x x x x x x x A x x x x x x x x x \uf06a \uf0ce \uf02d\uf0a5 \uf0ec \uf079 \uf0ce \uf02d\uf0a5 \uf0ec \uf0ef \uf06a \uf03d \uf03d \uf079 \uf03d \uf0ed \uf0ed\uf079 \uf0ce \uf02b\uf0a5 \uf0ee \uf0ef\uf06a \uf0ce \uf02b\uf0a5 \uf0ee where x0 and A are arbitrary real numbers, 1 \uf06a and 1 \uf079 are functions given on the set 0 ( , ] x \uf02d\uf0a5 , and 2 \uf06a and 2 \uf079 are functions given on the set 0 [ , ) x \uf02b\uf0a5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' In the case of sufficiently smooth data, namely, 2( ) C \uf06a\uf0ce , 1( ) C \uf079\uf0ce , 1( ) F C Q \uf0ce , and 1 3 ( ) f C Q \uf0ce \uf0b4 , we considered the problem (1), (2) in the article [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' For the linear wave equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', 0 f \uf0ba , the problem (1), (2) was studied in the works [16] and [13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 17] in the cases of 1( ) C \uf079\uf0ce and 2( ) C \uf06a\uf0ce respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Following [18], we investigate two main questions: 1) the exact statement of the initial value problem (1), (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2) the smoothness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The cause for the first question is the non-existence of the classical solution of the problem due to the condition 2( ) C \uf06a\uf0cf or 1( ) C \uf079\uf0cf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Constructing the solution of the Cauchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We divide the domain Q by the characteristics 0 x at x \uf02d \uf03d and 0 x at x \uf02b \uf03d into three subdomains 1 0 {( , ) ( , ) }, Q t x t x Q x at x \uf03d \uf0ce \uf0d9 \uf02b \uf03c ∣ 2 0 {( , ) ( , ) }, Q t x t x Q x at x \uf03d \uf0ce \uf0d9 \uf02d \uf03e ∣ 3 0 0 {( , ) ( , ) }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Q t x t x Q x at x x at x \uf03d \uf0ce \uf0d9 \uf02b \uf03e \uf0d9 \uf02d \uf03c ∣ On the closure Q of the domain Q, we define a function u as the one coinciding with the solution u(j) of the partial differential equation (1) ( ) ( ) ( , ) ( , ), ( , ) , j j u t x u t x t x Q \uf03d \uf0ce (3) on the domain ( ) j Q , where (1) (1) (3) \\ Q Q Q \uf03d , (2) (2) (3) \\ Q Q Q \uf03d , and (3) (3) Q Q \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The partition of the domain Q into three subdomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' By virtue of discontinuous initial conditions, the problem (1), (2) does not have a classical solution defined on the set Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Nevertheless, we can define a classical solution on a smaller set \\ Q \uf047 so that it belongs to the class 2( \\ ) C Q \uf047 and satisfies the equation (1), the initial conditions (2), and additional matching conditions given on the set \uf047 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Q(2) t x x0 O Q(3) Q(1) x − a t = x0 x + a t = x0 3 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' A function u is called a classical solution of the problem (1), (2) if the following conditions are satisfied: 1) the function u(j) belongs to the class \uf028 \uf029 2 ( ) j C Q for each {1,2,3} j\uf0ce ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2) the function u(j) satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' (1) in the domain ( ) j Q for each {1,2,3} j\uf0ce ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 3) the initial condition (0, ) ( ) u x x \uf03d \uf06a is met on the whole real axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 4) the initial condition (0, ) ( ) tu x x \uf0b6 \uf03d \uf079 is met on the whole real axis, except one point 0 x x \uf03d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 5) the Goursat conditions (3) (1) 0 0 1 0 ( , ) ( , ) ( ), 0, u t x x at u t x at A x t \uf03d \uf02d \uf03d \uf02d \uf02b \uf02d\uf06a (3) (2) 0 0 2 0 ( , ) ( , ) ( ), 0, u t x x at u t x at A x t \uf03d \uf02b \uf03d \uf02b \uf02b \uf02d\uf06a (4) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' It turns out that finding a classical solution of the problem (1), (2) in the sense of Definition 1 is equivalent to solving the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Problem (1), (2) with matching conditions on characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Find a classical solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' (1) with the Cauchy conditions (2) and the matching conditions 0 1 0 0 2 0 [( ) ( ) ]( , ) ( ), [( ) ( ) ]( , ) ( ) , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' u u t x x at A x u u t x x at x A t \uf02b \uf02d \uf02b \uf02d \uf02d \uf03d \uf02d \uf03d \uf02d\uf06a \uf02d \uf03d \uf02b \uf03d \uf06a \uf02d Here by ()\uf0b1 we have denoted the limit values of the function u and its partial derivatives calculated on different sides of the characteristics 0 x at x \uf0b1 \uf03d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', 0 ( ) ( , ( )) lim ( , ( ) ) p p t t u t x r t u t r t \uf0b1 \uf064\uf0ae \uf02b \uf0b6 \uf03d \uf03d \uf0b6 \uf0b1 \uf064 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The functions u(1) and u(2) are determined from the Cauchy problems ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ) ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' )) ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ) ( ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ) ( ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ( 1) ( 1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' j j j j j t x j j j j j t j u t x f t x u t x u t x u t x F t x t x Q u x x u x x x x \uf0ec \uf02b \uf0b6 \uf0b6 \uf03d \uf0ce \uf0ef\uf0ed \uf03d \uf06a \uf0b6 \uf03d \uf079 \uf02d \uf02d \uf0ef\uf0ee (5) for each 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='2 j \uf03d ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' and under some conditions have the representations \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 ( ) ( ) ( ) ( ) ( ) ( ) 0 ( ) ( ) ( ) 1 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ) ( ) 2 2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2 x at j j j j x at x a t t j j j j t x x a t x at x at u t x d a d F f u u u d t x Q a \uf02b \uf02d \uf02b \uf02d\uf074 \uf02d \uf02d\uf074 \uf06a \uf02d \uf02b \uf06a \uf02b \uf03d \uf02b \uf079 \uf078 \uf078 \uf02b \uf02b \uf074 \uf074 \uf078 \uf02d \uf074 \uf078 \uf074 \uf078 \uf0b6 \uf074 \uf078 \uf0b6 \uf074 \uf078 \uf078 \uf0ce \uf0f2 \uf0f2 \uf0f2 (6) Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Let the conditions 1( ) F C Q \uf0ce , 1 3 ( )) f C Q \uf0ce \uf0b4 , 2( ( )) j j C \uf06a \uf0ce \uf06a , and 1( ( )) j C \uf079\uf0ce \uf079 be satisfied, and let the function f satisfy the Lipschitz condition with constant L with respect to the three last variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', \uf028 \uf029 1 2 3 1 2 3 1 1 2 2 3 3 ( , , , , ) ( , , , ) , f t x z z z f t x w w w L z w z w z w \uf02d \uf0a3 \uf02d \uf02b \uf02d \uf02b \uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Then for each 1,2 j \uf03d , the Cauchy problem (5) has a unique solution in the class 2 ( ) ( ) j C Q determined by the formula (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' See [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Now the function u(3) is determined from the Goursat problem (3) (3) (3) (3) (3) (3) (1) 0 0 1 0 (3) (2) 0 0 2 0 ( , ) ( , , ( , ), ( , ), ( , )) ( , ), ( , ) , ( , ) ( , ) ( ), 0, ( , ) ( , ) ( ), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' t x u t x f t x u t x u t x u t x F t x t x Q u t x at u t at A t u t x at u A x x x x x t at t x \uf0ec \uf02b \uf0b6 \uf0b6 \uf03d \uf0ce \uf0ef \uf03d \uf02d \uf03d \uf02d \uf02b \uf02d \uf06a \uf0ed \uf0ef \uf03d \uf02b \uf03d \uf02b \uf02b \uf02d \uf06a \uf0ee (7) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Let the conditions 1( ) F C Q \uf0ce , 1 3 ( )) f C Q \uf0ce \uf0b4 , 2( ( )) j j C \uf06a \uf0ce \uf06a , and 1( ( )) j C \uf079\uf0ce \uf079 be satisfied, and let the function f satisfy the Lipschitz condition with constant L with respect to the three last variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', \uf028 \uf029 1 2 3 1 2 3 1 1 2 2 3 3 ( , , , , ) ( , , , ) , f t x z z z f t x w w w L z w z w z w \uf02d \uf0a3 \uf02d \uf02b \uf02d \uf02b \uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Then the Goursat problem (7) has a unique solution in the class 2 (3) ( ) C Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 4 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' According to Lemma 1, the conditions 1( ) F C Q \uf0ce , 1 3 ( )) f C Q \uf0ce \uf0b4 , 2( ( )) j j C \uf06a \uf0ce \uf06a , and 1( ( )) j C \uf079\uf0ce \uf079 imply existence and uniqueness of the twice continuously differentiable functions u(1) and u(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' This means that mappings (1) 1 0 1 0 :[0, ) ( , ( ) ) t u t x at A x \uf067 \uf0a5 \uf02d \uf02b \uf02d\uf06a \uf0ce and (2) 2 0 2 0 :[0, ) ( , ) ( ) t u t x at A x \uf067 \uf0a5 \uf02b \uf02b \uf02d\uf06a are twice continuously differentiable too and coincide at the point 0 t \uf03d , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', 1 2 (0) (0) \uf067 \uf03d \uf067 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Now, we use the results of the work [19] to finally prove this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Let us derive an integro-differential equation for the function u(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We select four points \uf028 \uf029 0 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' A x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 0 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2 2 x at x x at x B a \uf02b \uf02d \uf02d \uf02b \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' \uf028 \uf029 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' C t x ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 0 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2 2 at x x at x x D a \uf02b \uf02d \uf02b \uf02b \uf0e6 \uf0f6 \uf0e7 \uf0f7 \uf0e8 \uf0f8 from the domain (3) Q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' apply the curvilinear parallelogram identity [20] and obtain (3) (1) (2) 0 0 0 0 1 0 2 0 ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ( ) ( ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2 2 2 2 x at x x at x at x x at x x u t x u A x x u a a \uf02b \uf02d \uf02d \uf02b \uf02b \uf02d \uf02b \uf02b \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf03d \uf02b \uf02d \uf06a \uf02d \uf06a \uf02b \uf02d \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 (3) 2 1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 4 2 2 2 2 2 2 x at x at x x z y z y z y z y z y z y dy F f u a a a a a a a \uf02d \uf02b \uf0e6 \uf02d \uf02b \uf02d \uf02b \uf02d \uf02b \uf0e6 \uf0f6 \uf0e6 \uf0e6 \uf0f6 \uf02d \uf02d \uf0e7 \uf0e7 \uf0f7 \uf0e7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0e8 \uf0f8 \uf0e8 \uf0f2 \uf0f2 (3) (3) (3) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ( ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2 2 2 2 t x z y z y z y z y u u dz t x Q a a a a \uf0f6 \uf0f6 \uf02d \uf02b \uf02d \uf02b \uf0e6 \uf0f6 \uf0e6 \uf0f6 \uf0b6 \uf0b6 \uf0ce \uf0f7 \uf0f7 \uf0e7 \uf0f7 \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8\uf0f8\uf0f8 (8) We state the result as the following assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Let the conditions 1( ) F C Q \uf0ce , 1 3 ( )) f C Q \uf0ce \uf0b4 , 2( ( )) j j C \uf06a \uf0ce \uf06a , and 1( ( )) j C \uf079\uf0ce \uf079 hold, and let the function f satisfy the Lipschitz condition with constant L with respect to the three last variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', \uf028 \uf029 1 2 3 1 2 3 1 1 2 2 3 3 ( , , , , ) ( , , , ) , f t x z z z f t x w w w L z w z w z w \uf02d \uf0a3 \uf02d \uf02b \uf02d \uf02b \uf02d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The initial-value problem (1), (2) has a unique classical solution in the sense of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' This solution is determined by formulas (3), (6), and (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Analysis of the solution of the Cauchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Taking into account twice continuous differentiability of the functions u(j) for each 1,2,3 j \uf03d , independence of Lebesgue integral on the behavior of the function on a set of measure zero, the expressions (3), (6), and (8), we can formally rewrite (3) in the form \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029 (3) 1 0 2 0 ( ) 0 ( ) ( ) ( ) ( ) ( ) 1 ( , ) ( ) ( , ) 2 2 2 1 , , , , , , , , , ( , ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2 x at Q x at x a t t t x x a t x x x at x at u t x d A t x a d F f u u u d t x Q a \uf02b \uf02d \uf02b \uf02d\uf074 \uf02d \uf02d\uf074 \uf06a \uf02b \uf06a \uf06a \uf02d \uf02b \uf06a \uf02b \uf0e6 \uf0f6 \uf03d \uf02b \uf079 \uf078 \uf078 \uf02b \uf02d \uf063 \uf02b \uf0e7 \uf0f7 \uf0e8 \uf0f8 \uf02b \uf074 \uf074 \uf078 \uf02d \uf074 \uf078 \uf074 \uf078 \uf0b6 \uf074 \uf078 \uf0b6 \uf074 \uf078 \uf078 \uf0ce \uf0f2 \uf0f2 \uf0f2 (9) where χA is an indicator function of a set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We note that if (3) 1 0 2 0 ( ) ( ) ( , ) 0 2 Q x x A t x \uf06a \uf02b \uf06a \uf0e6 \uf0f6 \uf02d \uf063 \uf0ba \uf0e7 \uf0f7 \uf0e8 \uf0f8 (10) then the representation (9) is sometimes called the (‘generalized’) d’Alembert formula [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Following [16], we consider three cases of specifying the discontinuous initial conditions under the conditions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 0 0 ( 0) ( 0) x x A \uf06a \uf02d \uf03d \uf06a \uf02b \uf03d , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', ( ) C \uf06a\uf0ce .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=" From the Goursat conditions (4), and the fact ( ) 2 ( ) ( ) j j u C Q \uf0ce for each {1,2,3} j\uf0ce we see that in this case the solution u belongs the class ( ) C Q and satisfies the ‘generalized’ d'Alembert formula." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 0 0 ( 0) ( 0) x x \uf06a \uf02d \uf0b9 \uf06a \uf02b and 0 0 ( 0) ( 0) 2 x x A \uf06a \uf02d \uf02b \uf06a \uf02b \uf03d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=" In this case, the solution u is no longer continuous, but the condition (10) holds, and the solution u satisfies the ‘generalized’ d'Alembert formula too." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 0 0 ( 0) ( 0) x x \uf06a \uf02d \uf0b9 \uf06a \uf02b and 0 0 ( 0) ( 0) 2 x x A \uf06a \uf02d \uf02b \uf06a \uf02b \uf0b9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=" The solution u is discontinuous and does not satisfy the ‘generalized’ d'Alembert formula." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' These results are consistent with the conclusions of the work [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' We note that the integro-differential equation (9) can be used to define a mild solution of the problem (1), (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' In the present paper, we have obtained the necessary and sufficient conditions under which there exists a unique classical solution (in an extended sense) of the initial value problem for the mildly quasilinear wave equation with discontinuous initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' And we have proposed an approach to constructing solutions with discontinuous initial conditions, even for nonlinear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' The article was financially supported by the Ministry of Science and Higher Education of the Russian Federation in the framework of implementing the program of the Moscow Center for Fundamental and Applied Mathematics by Agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 075-15-2022-284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Zhuravkov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', Starovoytov E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='205 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Polyanin A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Handbook of linear partial 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problem of transversal impact on a rectangular viscoelastic plate with supported edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 1995, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 75–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Gaiduk S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' A mathematical discussion of some problems connected with the theory of longitudinal shock along finite rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Uravn.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Certain problems that are connected with the theory of a transversal shock along rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Uravn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 1977, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 13, no.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Waves induced by the longitudinal impact of a rod against a stepped rod in contact with a rigid barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Journal of Applied Mathematics and Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2009, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 73, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 162–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='jappmathmech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='006 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Koshlyakov N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', Gliner E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', Smirnov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Differential Equations of Mathematical Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Amsterdam, North Holland Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 701 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Korzyuk V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' I.' metadata={'source': 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Mathematics and Informatics, 2022, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 34–46 (in Russian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='33581/2520-6508-2022-2-34- 46 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Korzyuk V.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='09408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='09408 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Kharibegashvili S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', Jokhadze O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Solvability of a mixed problem with nonlinear boundary condition for a one-dimensional semilinear wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Mathematical Notes, 2020, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 108, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' 123–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='1134/S0001434620070123 Information about the authors Viktor I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Korzyuk – Academician, Professor, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' (Physics and Mathematics), Institute of Mathematics of the National Academy of Sciences of Belarus (11, Surganov Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', 220072, Minsk, Republic of Belarus), Belarusian State University (4, Nezavisimosti Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', 220030, Minsk, Republic of Belarus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' E-mail: korzyuk@bsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='by Jan V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Rudzko – M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' (Mathematics and Computer Sciences).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' Postgraduate student, Institute of Mathematics of the National Academy of Sciences of Belarus (11, Surganov Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=', 220072, Minsk, Republic of Belarus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' E-mail: janycz@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content=' https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} +page_content='org/0000-0002-1482-9106' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9AzT4oBgHgl3EQflv3p/content/2301.01554v1.pdf'} diff --git a/UNE1T4oBgHgl3EQfawQW/vector_store/index.pkl b/UNE1T4oBgHgl3EQfawQW/vector_store/index.pkl new file mode 100644 index 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sha256:37a281b201bbb49937858e7f4c46f3822b5118545a141aafba1521843e86ee08 +size 11319816 diff --git a/X9E1T4oBgHgl3EQfwAUv/content/tmp_files/2301.03405v1.pdf.txt b/X9E1T4oBgHgl3EQfwAUv/content/tmp_files/2301.03405v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..57686fbcd2ad53341fa0260a1606b2711d65ad9f --- /dev/null +++ b/X9E1T4oBgHgl3EQfwAUv/content/tmp_files/2301.03405v1.pdf.txt @@ -0,0 +1,1132 @@ +Inverting multiple quantum many-body scars via disorder +Qianqian Chen1 and Zheng Zhu1, 2, ∗ +1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China +2CAS Center for Excellence in Topological Quantum Computation, +University of Chinese Academy of Sciences, Beijing, 100190, China +(Dated: January 10, 2023) +The recent observations of persistent revivals in the Rydberg atom chain have revealed a weak ergodicity +breaking mechanism known as quantum many-body scars, which is typically a collection of states with low +entanglement embedded in otherwise thermal spectra. Here, by applying a generic formalism, we propose a +direct evolution from the quantum many-body scars to their inverse case and construct multiple inverted quantum +many-body scars, i.e., different sets of excited states with volume-law entanglement embedded in a sea of the +many-body localized spectrum. When increasing the disorder strength, a tower of exact eigenstates remain +intact, acting as conventional quantum many-body scars at weak disorder, and each residing inside narrow +energy windows with the emerged inverted quantum many-body scar at strong disorder. Moreover, the strong +disorder also induces additional sets of inverted quantum many-body scars with their energies concentrating +in the middle of the exact eigenstates. As a result, all the multiple inverted quantum many-body scars are +approximately equidistant in energy. We further examine the stability of the conventional and the inverted +quantum many-body scars against the external random field. Our findings expand the variety of nonthermal +systems and connect the weak ergodicity breaking with the weak violation of many-body localization. +Introduction.— Quantum ergodicity dominates the process +where most isolated quantum many-body systems locally +evolve into an equilibrium statistical ensemble [1–4]. Due +to the quest to realize long-lived coherent dynamics, tremen- +dous attempts have been made to develop ergodicity-breaking +mechanisms. +Among the very few exceptions of quantum +ergodicity, a weak ergodicity-breaking system with the so- +called quantum many-body scar (QMBS) states [5–10] was +first discovered in the Rydberg atom chain [11] and has re- +cently garnered intense interest. Having only a few conserved +quantities and being typically disorder-free, QMBS is char- +acterized by certain initial states that periodically revive and +is comprised of isolated nonthermal eigenstates embedded +in a sea of thermal states. These features are significantly +distinguished from the previously known strong ergodicity- +breaking mechanisms, i.e., integrable systems with an exten- +sive number of conserved quantities [12–15] and many-body +localization (MBL) [3, 16–20] with low entangled eigenstates +in the presence of strong disorder. +More recently, there have been several attempts [21–23] +to construct the inverse situation of QMBS, namely, highly +entangled excited states with volume-law entanglement em- +bedded in the rest of the MBL spectra. +Such phenomena +are dubbed inverted QMBS and they enrich the categories of +nonthermal systems. Previous studies [21–23] of the inverted +QMBS focused on a single narrow energy window, and it is +unclear whether the inverted QMBS can be realized in multi- +ple energy windows with approximately or exactly equal en- +ergy spacing. Additionally, unlike the unified formalisms of +QMBS [10, 24–35], the systematic formalism to construct the +inverted QMBS is still elusive. +On the other hand, the connections between distinct +ergodicity-breaking mechanisms lie at the core of under- +standing thermalization and its absence. Indeed, the disor- +der, which is ubiquitous across the realistic quantum sim- +ulators [36–38], can bring integrability, MBL, and QMBS +together [26, 39–45]. According to recent studies [41, 42] +of QMBS in PXP models, in the process of increasing the +disorder, the system is always first deprived of the original +QMBS and becomes fully thermal, and then the possible tran- +sition/crossover to MBL emerges. The mechanisms causing +scars in the PXP model are only approximately understood +[46–54], then it is fundamentally important to explore the +exact QMBS that is analytically tractable [25–33, 55–58] in +the presence of the disorder with the interplay of different +ergodicity-breaking mechanisms. In particular, the direct evo- +lution from a system with exact QMBS to the one with an +MBL background has not been revealed. +Since both conventional and inverted QMBS are a small +fraction of states that have very different thermalization prop- +erties from other excited states, here we realize them under the +same formalism and invert them directly through disorder. We +study a typical disordered model that represents a large class +of Hamiltonians with a tower of exact QMBS states at weak +disorder. Then we increase the disorder strength and drive the +majority of the states to be many-body localized, while the +original exact QMBS eigenstates are invariable in the whole +process. As the disorder strength increases, the multiple in- +verted QMBS states embedded in an MBL spectrum emerge at +finite energy density, as characterized by a collection of states +with volume-law entanglement entropy (EE) while the whole +spectrum follows Poisson statistics. In particular, the multiple +sets of many highly entangled states concentrate in distinct en- +ergy windows with approximately equal energy spacing. We +also apply the onsite random field to examine the stability of +the scarring states, and find both the original exact QMBS +and the inverted QMBS states disappear with increasing on- +site randomness. +Model.— We consider a generic framework for QMBS [27, +34] and also apply it to construct the inverted QMBS. Such a +arXiv:2301.03405v1 [cond-mat.dis-nn] 9 Jan 2023 + +2 +framework constructs a Hamiltonian +H = Hsym + HSG + HA. +(1) +Here, Hsym is G-symmetric with [Hsym, Q+] += +0 and +[Hsym, HSG] = 0, where G is a non-Abelian symmetry and +Q+ is the spectrum-generating “ladder” operator. The second +term HSG is a linear combination of generators in the Car- +tan subalgebra of G, and meets a spectrum-generating algebra +(SGA) +� +HSG, Q+� += ωQ+ +(2) +that leads to a tower of states with energy spacing ω. A par- +ticular set of eigenstates {|Sk⟩}k is labeled by the eigenvalue +under the Casimir operators of G, and states in the set are +distinguished by their eigenvalues under Cartan generators of +G. The term HA breaks the G-symmetry, and is immaterial +to the dynamics of the eigenstates {|Sk⟩}k since it annihilates +them HA |Sk⟩ = 0. In the following, we will apply a similar +framework to realize both conventional and multiple inverted +QMBS states. +We consider Hsym as the S = 1/2 XX Heisenberg chain +that is realizable in Rydberg quantum simulators [38, 59–61] +Hsym = +L +� +j=1 +S+ +j S− +j+1 + S− +j S+ +j+1. +(3) +Hsym is integral [62] and has the Onsager symmetry [40, 63], +i.e., Hsym commutes with all the Onsager-algebra elements, +including +Q = +L +� +j=1 +Sz +j , +Q+ = +L +� +j=1 +(−1)j+1S+ +j S+ +j+1. +(4) +Since we have the SGA +[Q, Q+] = 2Q+, +(5) +it is natural to set HSG = Q and let Q+ be the spectrum- +generating operator. Due to (5), the set of degenerate states +|Sk⟩ = (Q+)k |⇓⟩ +(k = 0, . . . , ⌊L/2⌋) +(6) +of Hsym can be lifted and promoted to the evenly spaced ex- +act tower of eigenstates with energies ES = −L/2 + 2k. +Here, |⇓⟩ denotes a polarized spin-down state. Finally, HA +is added to destroy the integrability and annihilate each of +the {|Sk⟩}k. Furthermore, we choose a disordered term HA +which can drive the majority states to be MBL when increas- +ing the disorder strength, +HA = ∆ +L +� +j=1 +{cj|010⟩⟨010|}j−1,j,j+1 , +(7) +where cj are the uniform random numbers cj ∈ [−1, 1], and ∆ +-20 +-10 +0 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +En +SvN/SPage +0 +5 +10 +15 +20 +25 +30 +1 +10 +20 +30 +40 +t +Δ +0 +0.2 +0.4 +0.6 +0.8 +1.0 +[|〈ψ(t)|ψ0〉 2] +(a) +(b) +∆ = 4 +Stot +z =-1 +Fig. 1. +Typical features of exact quantum many-body scar. (a) +SvN/SPage with respect to all eigenenergies at weak disorder for +L = 18. The black circle denotes the scarred state |S4⟩. Darker col- +ors imply a higher density of the states. (b) The disorder-averaged +fidelity dynamics [| ⟨ψ(t)|ψ(0)⟩ |2] of the initial state |ψ(0)⟩ ≡ |ψ0⟩ +in scar subspace as a function of disorder strength ∆ when L = 18. +denotes the disorder strength. In the following, we will show +that HA preserves not only the exact special states {|Sk⟩}k +but also a set of states with higher EE than the MBL states. To +summarize, the total Hamiltonian reads +H = +L +� +j=1 +� +S+ +j S− +j+1 + S− +j S+ +j+1 + Sz +j +� ++ ∆ +L +� +j=1 +� +c(1) +j |010⟩⟨010| +� +j−1,j,j+1 . +(8) +We use the exact diagonalization (ED) approach to examine +the whole spectrum of the model (8). In the following, we +mainly focus on the bulk Sz +tot sectors, and we average data +over 10 ∼ 100 disorder realizations (denoted as [·]) depending +on the system size L and the Sz +tot sectors. +Exact quantum many-body scarred states.— The exact +tower of eigenstates |Sk⟩ exhibit exactly equal energy spac- +ing and persevere at any disorder strength ∆. They are con- +ventional exact QMBS states embedded in otherwise thermal +spectra at smaller ∆, while at larger ∆, they are embedded in +MBL spectra. Below we reveal their nature from the eigen- +state EE and the fidelity dynamics. +A wealth of thermalization information on physical states +can be obtained from the EE. We consider the density matrix +ρn of the nth eigenstate |φn⟩ defined by ρn = |φn⟩ ⟨φn|, and +study the EE +SvN = −TrA (ρA,n ln ρA,n) , +(9) +where ρA,n is the reduced density matrix for subsystem A +(chosen as half chain here) after tracing out the rest of the +system. Figure 1(a) shows one typical example of EE at small +∆ for Sz +tot = −1. The majority of the bulk eigenstates have +EE approaching the Page value for a random pure state [64] +SPage ≈ ln(DA)−0.5DA/DB, where DA (DB) is the Hilbert +space dimensions of subsystem A (B), while the scarred state + +3 +L=12 +L=16 +L=14 +L=18 +1 +10 +20 +30 +40 +0.35 +0.40 +0.45 +0.50 +0.55 +Δ +[〈rE〉] +Δ=1 +Δ=38 +WD +P +0 +1 +2 +3 +4 +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +sE +P(sE) +L=8 +L=12 +L=16 +1 +10 +20 +30 +40 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Δ +[SvN SPage] +1 +10 +20 +30 +40 +0 +5 +10 +15 +20 +25 +30 +Δ +t +0 +0.2 +0.4 +0.6 +0.8 +1.0 +[|〈Z2(t)|Z2〉 2] +(a) +(b) +(c) +(d) +Fig. 2. The nature of bulk states as functions of disorder strength ∆. +(a) Mean level spacing ratios [⟨rE⟩] for eigenenergies in the middle +60% of the spectrum with Sz +tot = 0 for L = 12, 16 and Sz +tot = −1 for +L = 14, 18. As a comparison, Wigner-Dyson (WD) statistics of the +GOE ⟨rE⟩ ≈ 0.536 (dashed black lines) and Poisson (P) statistics +⟨rE⟩ ≈ 0.38 (dashed gray lines) are plotted. (b) The energy level +spacing statistics for one particular disorder realization in Sz = −1 +sector with L = 18, after performing the spectrum unfolding. (b) +[SvN/SPage] as a function of ∆ with Sz +tot = 0. The data are averaged +over 100 disorder realizations and over 1/2 (but not 1/12) of all the +eigenstates that are around the state |S4⟩. (c) The disorder-averaged +fidelity dynamics [| ⟨Z2(t)|Z2⟩ |2] of the initial state |Z2⟩ with L = +14 at different disorder strength ∆. +|S4⟩ (marked by a black circle) exhibits anomaly low EE. Sz +tot +sectors with other eigenstates |Sk⟩ exhibit similar behavior +at small disorder strength. In the following, we will show +that, with the increase of the disorder strength, the energy- +level statistics of the bulk energy spectra change from Wigner- +Dyson to Poisson, while the EE of the exact tower of states +|Sk⟩ remains the same for any disorder strength ∆. +The existence of exact QMBS can also be inferred by the fi- +delity dynamics for specific initial states in the scar subspace. +Figure 1(b) shows perfectly periodic revivals in the fidelity of +the initial state |ψ0⟩ = +1 +N +�⌊L/2⌋ +k=0 +|Sk⟩, where N is the nor- +malization factor. The revival period T = π corresponds to +the energy interval ω = 2 of the scarred states |Sk⟩, as ex- +pected from the SGA (5). Since the disorder term HA annihi- +lates |Sk⟩, the exact states |Sk⟩ and the fidelity dynamics are +preserved regardless of the disorder strength ∆, as Fig. 1(b) +shows. +Thermalized background to MBL background.— Although +increasing the disorder strength in HA does not influence the +eigenstates |Sk⟩, most of the other bulk states alter from er- +godic to non-ergodic. To show this, we examine the energy +level statistics, half-chain EE, and the imbalance dynamics as +Stot +z =0 +-20 +-10 +0 +10 +20 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +E +[SvN SPage] +-30 -20 -10 +0 +10 +20 +30 +1 +10 +20 +30 +40 +E +Δ +0 +0.2 +0.4 +0.6 +0.8 +[SvN SPage] +Stot +z =-2 +Stot +z =0 +Stot +z =2 +8 +12 +16 +20 +1 +2 +3 +4 +5 +L +[SvN +max] +-20 +-10 +0 +10 +20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +E +[Σα ϕα +HA ϕn 2] +(a) +(b) +(d) +(c) +Δ=46 +Δ=46 +Stot +z =0 +Stot +z =-1 +Δ=46 +Fig. 3. +Energy-resolved features of states in Sz +tot sectors. (a) The +energy-resolved [SvN/SPage] for L = 16 in Sz +tot = 0 sector where +the state |S4⟩ with ES = 0 resides. The peak appears at the energy +window that has 722 eigenenergies on average, including ES = 0. +(b) The energy-resolved [SvN/SPage] as a function of ∆ for L = 18, +Sz +tot = −1. The bright vertical line resides in the energy window that +includes ES = −1 of the state |S4⟩. (c) The scaling of [Smax +vN ] with +L, where [Smax +vN ] are averaged over the maximum entropies Smax +vN of +eigenstates in each disorder realization, with the corresponding aver- +aged energies very close to ES. (d) The overlap between eigenstates +of H (i.e., |φn⟩) and |φHA +α ⟩ for L = 16, where HA|φHA +α ⟩ = 0. +a function of disorder strength ∆, as shown in Figs. 2. +The energy-level spacing ratios are defined by [65] +rEn = min +� +sEn, sEn−1 +� +max +� +sEn, sEn−1 +�, +(10) +where En is an increasing-ordered set of energy levels and +sEn = En+1 − En the nearest-neighbor energy-level spac- +ings. We eliminate 20% of the eigenenergies at the spectrum’s +edges when calculating the statistics of energy-level spacings. +The mean energy-level spacing ratios [⟨rE⟩] as functions of +∆ are depicted in Fig. 2(a), with ⟨·⟩ denoting the average +over the spectrum. In the Fig. 2(a), we find that [⟨rE⟩] con- +verges to the value of Wigner-Dyson statistics of the Gaussian +orthogonal ensemble (GOE) when ∆ is small, implying the +thermalization of the bulk states, and [⟨rE⟩] approaches to the +value of Poisson statistics at larger ∆, indicating that the sys- +tem is localized [66]. For a single disorder realization in each +of these two different regimes, typical profiles of the energy- +level spacing distributions in Fig. 2(b) are consistent with the +disorder-averaged energy-level spacing ratios [⟨rE⟩]. +We then look at the characteristics of the state- and +disorder-averaged EE SvN that distinguishes thermalization +from MBL [67]. Figure 2(c) show divided by SPage for var- + +4 +ious system size L in the Sz +tot = 0 sector, where we denote +the state average as ¯·. With increasing disorder strength ∆, +[SvN/SPage] goes from 1 of the thermalized states to 0 of the +many-body localized states. +We also choose the initial product state |Z2⟩ ≡| 101010 . . .⟩ +to represent the imbalance and calculate its dynamics at differ- +ent ∆. As shown in Fig. 2(d), the fidelity rapidly approaches +zero as time evolves at small ∆, demonstrating ergodic behav- +ior. In contrast, at larger ∆, the persistent imbalance dynamics +indicate a non-ergodic evolution, consistent with MBL. +Multiple inverted quantum many-body scar.— We have +demonstrated that the majority of the bulk states change from +ergodic to non-ergodic when increasing disorder strength, be- +low we will show the inverted QMBS states with anomaly +high entanglement in the MBL background. +We examine the energy-resolved EE [SvN/SPage] that is av- +eraged over disorder realizations and states in the targeted +Sz +tot sector. We first look into the Sz +tot sectors with states +|Sk⟩. As illustrated in Figs. 3(a,b), we find highly entangled +states located very close to |Sk⟩ in energy, in sharp contrast +to other localized states with low entanglement. For instance, +in Fig. 3(a), the state |S4⟩ residing in the sector Sz +tot = 0 +for L = 16 has its energy ES = 0, besides which many +highly entangled states jointly give a [SvN/SPage] peak at the +energy window of E = 0. In another example, in the sector +Sz +tot = −1 of L = 18 (c.f. Fig. 3(b)), the state |S4⟩ has en- +ergy ES = −1, and [SvN/SPage] also exhibits sharp peak at +E = −1 in a narrow energy window for large ∆. Although +states |Sk⟩ have a sub-volume-law EE [40], the disorder- +averaged maximum entropies [Smax +vN ] exhibit a volume-law be- +havior, as shown in Fig. 3(c). In the bulk Sz +tot sectors without +states |Sk⟩, we also find anomaly high entanglement states, +and they concentrate in the middle of the energies of the ex- +act eigenstates |Sk⟩. Therefore, the Hamiltonian (8) realizes +multiple inverted QMBS concentrating in different narrow en- +ergy windows with approximately equal energy spacing ≈ 1, +which is the half of the energy spacing of states |Sk⟩ [68]. We +remark that the number of high entanglement states in every +energy window is much larger than one (as detailed in the cap- +tion of Fig. 3(a)), and the narrow energy windows with highly +entangled states also exhibit peaks of the energy density of +states (DOS) [68]. +We further understand the behavior of the inverted QMBS. +Indeed, we find these highly entangled states have a large +overlap with the states +��φHA +α +� +in the null space of HA (c.f. +Fig. 3(d)), where HA +��φHA +α +� += 0. As a result, such states +stay delocalized and remain largely unaffected by the disorder +strength, similar to the exact tower of eigenstates |Sk⟩. +Stability to onsite random field.— Now we consider the sta- +bility of the aforementioned exact QMBS |Sk⟩ and inverted +QMBS to the onsite random z fields that break the formal- +ism H (1). To be more specific, we modify H in (8) to be +H′ = H + h �L +j=1 δjSz +j , where δj are the uniform random +numbers in the range δj ∈ [−1, 1]. Unlike the disordered +HA, the disorder term in H′ can drive all eigenstates to the lo- +calization. In Figs. 4, with a localized spectrum background, +5 +10 +15 +20 +25 +30 +0.01 +0.05 +0.10 +0.50 +1 +t +h +0 +1.0 +[|〈ψ(t)|ψ0〉 2] +-20 -10 +0 +10 +20 +30 +0.01 +0.05 +0.10 +0.50 +1 +E +h +0 +0.4 +[SvN SPage] +(a) +(b) +Δ=46 +Δ=46 +Stot +z =0 +Fig. 4. +Stability of |Sk⟩ and the inverted QMBS at large dis- +order strength ∆. +(a) The disorder-averaged fidelity dynamics +[| ⟨ψ(t)|ψ(0)⟩ |2] of the initial state |ψ(0)⟩ ≡ |ψ0⟩ in scar subspace +with L = 14. (b) The energy-resolved [SvN/SPage] as a function of +h for L = 16, Sz +tot = 0. The peak resides in the energy window that +includes ES = 0 of the state |S4⟩. +both the periodic revival of the fidelity for the initial state |ψ0⟩ +and the peak of [SvN/SPage] show certain stability of both ex- +act tower of eigenstates |Sk⟩ and inverted QMBS against the +onsite random z field, though they eventually disappear for a +large disorder strength h. In a thermalizing background, with +the increase of h, the conventional QMBS |Sk⟩ first disappears +and the system becomes thermal before the final localization +of all the states [68], consistent with previous scenarios of the +QMBS in the disordered PXP models [41, 42]. +Summary and outlook.— In this work, we have realized a +direct evolution from a thermal spectrum background with a +tower of exact QMBS to the MBL background with multi- +ple inverted QMBS in a disordered spin-1/2 XX Heisenberg +chain. The forms of the exact tower of states are independent +of the disorder, while the energy-level statistics of the bulk +eigenstates changes from Wigner-Dyson to Poisson when in- +creasing the disorder, despite the existence of the embedded +highly entangled states at strong disorder. Embedded in the +otherwise MBL spectra with low entanglement, the multiple +sets of many highly entangled states are located within differ- +ent narrow energy windows that are approximately equidis- +tant in energy. We also show certain stability of the highly +entangled states to the onsite random field. To the best of our +knowledge, such a scenario that inverts multiple QMBS di- +rectly is not constructed before. Our model can also be gener- +alized to other non-Abelian symmetry, and to the large classes +of QMBS Hamiltonian that resort to the annihilating term HA. +The proposal to invert QMBS in this work may also stimulate +more experimental activities to realize setups that weakly vi- +olate the quantum ergodicity or the MBL [38, 59–61]. +Note added.— When finalizing the manuscript, we became +aware of one recent work [69] on related topics. +We are grateful to Yi-Zhuang You and Shuai A. Chen +for the fruitful discussions. +This work was supported by +the National Natural Science Foundation of China (Grant +No.12074375), the Fundamental Research Funds for the Cen- +tral Universities and the Strategic Priority Research Program +of CAS (Grant No.XDB33000000). + +5 +∗ zhuzheng@ucas.ac.cn +[1] M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E +50, 888 (1994). +[2] M. Rigol, V. Dunjko, and M. 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Huse, Two universality +classes for the many-body localization transition, Phys. Rev. +Lett. 119, 075702 (2017). +[68] Further details of results and calculation are available as sup- +plementary material. +[69] M. +Iversen +and +A. +E. +B. +Nielsen, +Tower +of +quan- +tum +scars +in +a +partially +many-body +localized +system +10.48550/arxiv.2301.01681 (2023). + +7 +Supplementary Materials for +“Inverting quantum many-body scar via disorder” +Multiple inverted QMBS in different symmetry sectors. +Stot +z =2 +Stot +z =1 +Stot +z =0 +Stot +z =-1 +Stot +z =-2 +-10 +-5 +0 +5 +10 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +E +[SvN SPage] +Stot +z =2 +Stot +z =1 +Stot +z =0 +Stot +z =-1 +Stot +z =-2 +-10 +-5 +0 +5 +10 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +E +[Σα ϕα +HA ϕn 2] +Δ=1 +Δ=11 +Δ=46 +-13 -9 -5 -1 +3 +7 +11 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +E +[ρ(E)] +(a) +(b) +(c) +Stot +z =-1 +Δ=46 +Δ=46 +Fig. S1. +Features of multiple inverted QMBS in different Sz +tot sectors. (a) The energy-resolved [SvN/SPage] for L = 16. (b) The overlap +between eigenstates of H (i.e.,|φn⟩) and |φHA +α ⟩ for L = 16, where HA|φHA +α ⟩ = 0. (c) Density of states for L = 18, Sz +tot = −1. At ∆ = 46, +the peak of DOS appears in the energy window of inverted QMBS. For comparison, the DOS for smaller ∆ are also plotted. +At strong disorder, multiple sets of highly entangled states concentrating in equidistant energy windows emerge in different +Sz +tot sectors, as shown by the peaks of the energy-resolved [SvN/SPage] in Fig. S1(a). We remark that these energy windows +with peaks of [SvN/SPage] are indeed very narrow compared to the large width of the whole energy spectrum. For example, for +Sz +tot = 2 sector of L = 16 in Fig. S1(a), the width of the whole energy spectrum is ∼ 336, while the peak of [SvN/SPage] is only +∼ 8. The spacing between these energy windows is roughly 1. Figure S1(b) shows that the highly entangled states are indeed +almost annihilated by the term HA and thus remain largely undisturbed by the disorder. Moreover, at large ∆, we also find the +peak of the averaged density of states [ρ(E)] appears at the narrow energy window where the highly entangled states locate, as +shown by the typical sector Sz +tot = −1 of L = 18 in Fig. S1(c). +Fate of QMBS in the presence of onsite random field +In this section, we study the fate of QMBS in the presence of onsite random z field h. Here we consider a different annihilating +disorder with more terms +H′ +A = ∆ +� +j +{c(1) +j |010⟩⟨010| + +c(2) +j +2 (|011⟩ + |110⟩)(⟨011| + ⟨110|) ++ c(3) +j [|010⟩(⟨011| + ⟨110|) + h.c.]}j−1,j,j+1, +where c(α) +j +with α = 1, 2, 3 are the uniform random numbers c(α) +j +∈ [−1, 1]. We remark that H′ +A breaks U(1) symmetry. The +total Hamiltonian reads +H = +L +� +j=1 +� +S+ +j S− +j+1 + S− +j S+ +j+1 + Sz +j +� ++ H′ +A + h +L +� +j=1 +δjSz +j +(S1) +The last term in (S1) can be regarded as the onsite random fields that break the symmetry-based formalism mentioned in the +main text. Some characteristic features of QMBS, such as the slow relaxation from certain initial states, are still existent in the +presence of a modest disorder strength h, as shown by Fig. S2(a). As h is increased, however, the model (S1) loses the QMBS +features before switching to MBL (c.f. Figs. S2(b-d)). Remarkably, in the MBL spectrum background, there is no peak of +[SvN/SPage], since the random z fields can affect every eigenstate. + +8 +L=12 +L=14 +L=16 +0 +5 +10 +15 +20 +25 +30 +0.35 +0.40 +0.45 +0.50 +0.55 +h +[〈rE〉] +0 +5 +10 +15 +20 +25 +30 +-30 +-20 +-10 +0 +10 +20 +30 +h +E +0 +0.2 +0.8 +[SvN SPage] +(b) +(d) +Δ=1 +Δ=1 +101 +102 +0 +5 +10 +15 +20 +25 +30 +h +t +0 +0.2 +0.8 +1.0 +[|〈Z2(t)|Z2〉 2] +(c) +Δ=1 +0.01 +0.10 +100 +101 +0 +5 +10 +15 +20 +25 +30 +h +t +0 +0.2 +0.8 +1.0 +|〈ψ(t)|ψ0〉 2 +(a) +] +[ +Δ=1 +Fig. S2. Stability of QMBS |Sk⟩ at weak disorder strength ∆. (a) The disorder-averaged fidelity dynamics [f(t)] = [| ⟨ψ(t)|ψ(0)⟩ |2] of the +initial state |ψ(0)⟩ ≡ |ψ0⟩ (defined in the main text) with L = 12. (b) Mean level spacing ratios [⟨rE⟩] as a function of h. As a comparison, +Wigner-Dyson statistics of the GOE ⟨rE⟩ ≈ 0.536 (dashed black lines) and Poisson statistics ⟨rE⟩ ≈ 0.38 (dashed gray lines) are plotted. +The [⟨rE⟩] are averaged over 100 disorder realizations for L = 12, 14 and between 10 and 40 for L = 16. (c) The disorder-averaged fidelity +dynamics f(t) = | ⟨Z2(t)|Z2⟩ |2 of the initial state |Z2⟩ with L = 14 at different disorder strength h. (d) The energy-resolved [SvN/SPage] as +a function of h for L = 12. + diff --git a/X9E1T4oBgHgl3EQfwAUv/content/tmp_files/load_file.txt b/X9E1T4oBgHgl3EQfwAUv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e191e549edc6676d45d1922fe8af3bad07dcc81f --- /dev/null +++ b/X9E1T4oBgHgl3EQfwAUv/content/tmp_files/load_file.txt @@ -0,0 +1,855 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf,len=854 +page_content='Inverting multiple quantum many-body scars via disorder Qianqian Chen1 and Zheng Zhu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' ∗ 1Kavli Institute for Theoretical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' China 2CAS Center for Excellence in Topological Quantum Computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' China (Dated: January 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 2023) The recent observations of persistent revivals in the Rydberg atom chain have revealed a weak ergodicity breaking mechanism known as quantum many-body scars,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' which is typically a collection of states with low entanglement embedded in otherwise thermal spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Here, by applying a generic formalism, we propose a direct evolution from the quantum many-body scars to their inverse case and construct multiple inverted quantum many-body scars, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=', different sets of excited states with volume-law entanglement embedded in a sea of the many-body localized spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' When increasing the disorder strength, a tower of exact eigenstates remain intact, acting as conventional quantum many-body scars at weak disorder, and each residing inside narrow energy windows with the emerged inverted quantum many-body scar at strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Moreover, the strong disorder also induces additional sets of inverted quantum many-body scars with their energies concentrating in the middle of the exact eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' As a result, all the multiple inverted quantum many-body scars are approximately equidistant in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We further examine the stability of the conventional and the inverted quantum many-body scars against the external random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Our findings expand the variety of nonthermal systems and connect the weak ergodicity breaking with the weak violation of many-body localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='— Quantum ergodicity dominates the process where most isolated quantum many-body systems locally evolve into an equilibrium statistical ensemble [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Due to the quest to realize long-lived coherent dynamics, tremen- dous attempts have been made to develop ergodicity-breaking mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Among the very few exceptions of quantum ergodicity, a weak ergodicity-breaking system with the so- called quantum many-body scar (QMBS) states [5–10] was first discovered in the Rydberg atom chain [11] and has re- cently garnered intense interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Having only a few conserved quantities and being typically disorder-free, QMBS is char- acterized by certain initial states that periodically revive and is comprised of isolated nonthermal eigenstates embedded in a sea of thermal states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' These features are significantly distinguished from the previously known strong ergodicity- breaking mechanisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=', integrable systems with an exten- sive number of conserved quantities [12–15] and many-body localization (MBL) [3, 16–20] with low entangled eigenstates in the presence of strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' More recently, there have been several attempts [21–23] to construct the inverse situation of QMBS, namely, highly entangled excited states with volume-law entanglement em- bedded in the rest of the MBL spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Such phenomena are dubbed inverted QMBS and they enrich the categories of nonthermal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Previous studies [21–23] of the inverted QMBS focused on a single narrow energy window, and it is unclear whether the inverted QMBS can be realized in multi- ple energy windows with approximately or exactly equal en- ergy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Additionally, unlike the unified formalisms of QMBS [10, 24–35], the systematic formalism to construct the inverted QMBS is still elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' On the other hand, the connections between distinct ergodicity-breaking mechanisms lie at the core of under- standing thermalization and its absence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Indeed, the disor- der, which is ubiquitous across the realistic quantum sim- ulators [36–38], can bring integrability, MBL, and QMBS together [26, 39–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' According to recent studies [41, 42] of QMBS in PXP models, in the process of increasing the disorder, the system is always first deprived of the original QMBS and becomes fully thermal, and then the possible tran- sition/crossover to MBL emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The mechanisms causing scars in the PXP model are only approximately understood [46–54], then it is fundamentally important to explore the exact QMBS that is analytically tractable [25–33, 55–58] in the presence of the disorder with the interplay of different ergodicity-breaking mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In particular, the direct evo- lution from a system with exact QMBS to the one with an MBL background has not been revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Since both conventional and inverted QMBS are a small fraction of states that have very different thermalization prop- erties from other excited states, here we realize them under the same formalism and invert them directly through disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We study a typical disordered model that represents a large class of Hamiltonians with a tower of exact QMBS states at weak disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Then we increase the disorder strength and drive the majority of the states to be many-body localized, while the original exact QMBS eigenstates are invariable in the whole process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' As the disorder strength increases, the multiple in- verted QMBS states embedded in an MBL spectrum emerge at finite energy density, as characterized by a collection of states with volume-law entanglement entropy (EE) while the whole spectrum follows Poisson statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In particular, the multiple sets of many highly entangled states concentrate in distinct en- ergy windows with approximately equal energy spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We also apply the onsite random field to examine the stability of the scarring states, and find both the original exact QMBS and the inverted QMBS states disappear with increasing on- site randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='— We consider a generic framework for QMBS [27, 34] and also apply it to construct the inverted QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Such a arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='03405v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='dis-nn] 9 Jan 2023 2 framework constructs a Hamiltonian H = Hsym + HSG + HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (1) Here, Hsym is G-symmetric with [Hsym, Q+] = 0 and [Hsym, HSG] = 0, where G is a non-Abelian symmetry and Q+ is the spectrum-generating “ladder” operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The second term HSG is a linear combination of generators in the Car- tan subalgebra of G, and meets a spectrum-generating algebra (SGA) � HSG, Q+� = ωQ+ (2) that leads to a tower of states with energy spacing ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' A par- ticular set of eigenstates {|Sk⟩}k is labeled by the eigenvalue under the Casimir operators of G, and states in the set are distinguished by their eigenvalues under Cartan generators of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The term HA breaks the G-symmetry, and is immaterial to the dynamics of the eigenstates {|Sk⟩}k since it annihilates them HA |Sk⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In the following, we will apply a similar framework to realize both conventional and multiple inverted QMBS states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We consider Hsym as the S = 1/2 XX Heisenberg chain that is realizable in Rydberg quantum simulators [38, 59–61] Hsym = L � j=1 S+ j S− j+1 + S− j S+ j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (3) Hsym is integral [62] and has the Onsager symmetry [40, 63], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=', Hsym commutes with all the Onsager-algebra elements, including Q = L � j=1 Sz j , Q+ = L � j=1 (−1)j+1S+ j S+ j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (4) Since we have the SGA [Q, Q+] = 2Q+, (5) it is natural to set HSG = Q and let Q+ be the spectrum- generating operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Due to (5), the set of degenerate states |Sk⟩ = (Q+)k |⇓⟩ (k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' , ⌊L/2⌋) (6) of Hsym can be lifted and promoted to the evenly spaced ex- act tower of eigenstates with energies ES = −L/2 + 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Here, |⇓⟩ denotes a polarized spin-down state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Finally, HA is added to destroy the integrability and annihilate each of the {|Sk⟩}k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Furthermore, we choose a disordered term HA which can drive the majority states to be MBL when increas- ing the disorder strength, HA = ∆ L � j=1 {cj|010⟩⟨010|}j−1,j,j+1 , (7) where cj are the uniform random numbers cj ∈ [−1, 1], and ∆ 20 10 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 En SvN/SPage 0 5 10 15 20 25 30 1 10 20 30 40 t Δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 [|〈ψ(t)|ψ0〉 2] (a) (b) ∆ = 4 Stot z =-1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Typical features of exact quantum many-body scar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (a) SvN/SPage with respect to all eigenenergies at weak disorder for L = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The black circle denotes the scarred state |S4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Darker col- ors imply a higher density of the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (b) The disorder-averaged fidelity dynamics [| ⟨ψ(t)|ψ(0)⟩ |2] of the initial state |ψ(0)⟩ ≡ |ψ0⟩ in scar subspace as a function of disorder strength ∆ when L = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' denotes the disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In the following, we will show that HA preserves not only the exact special states {|Sk⟩}k but also a set of states with higher EE than the MBL states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' To summarize, the total Hamiltonian reads H = L � j=1 � S+ j S− j+1 + S− j S+ j+1 + Sz j � + ∆ L � j=1 � c(1) j |010⟩⟨010| � j−1,j,j+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (8) We use the exact diagonalization (ED) approach to examine the whole spectrum of the model (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In the following, we mainly focus on the bulk Sz tot sectors, and we average data over 10 ∼ 100 disorder realizations (denoted as [·]) depending on the system size L and the Sz tot sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Exact quantum many-body scarred states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='— The exact tower of eigenstates |Sk⟩ exhibit exactly equal energy spac- ing and persevere at any disorder strength ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' They are con- ventional exact QMBS states embedded in otherwise thermal spectra at smaller ∆, while at larger ∆, they are embedded in MBL spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Below we reveal their nature from the eigen- state EE and the fidelity dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' A wealth of thermalization information on physical states can be obtained from the EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We consider the density matrix ρn of the nth eigenstate |φn⟩ defined by ρn = |φn⟩ ⟨φn|, and study the EE SvN = −TrA (ρA,n ln ρA,n) , (9) where ρA,n is the reduced density matrix for subsystem A (chosen as half chain here) after tracing out the rest of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Figure 1(a) shows one typical example of EE at small ∆ for Sz tot = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The majority of the bulk eigenstates have EE approaching the Page value for a random pure state [64] SPage ≈ ln(DA)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='5DA/DB, where DA (DB) is the Hilbert space dimensions of subsystem A (B), while the scarred state 3 L=12 L=16 L=14 L=18 1 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='55 Δ [〈rE〉] Δ=1 Δ=38 WD P 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 sE P(sE) L=8 L=12 L=16 1 10 20 30 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 Δ [SvN \uf00cSPage] 1 10 20 30 40 0 5 10 15 20 25 30 Δ t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 [|〈Z2(t)|Z2〉 2] (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The nature of bulk states as functions of disorder strength ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (a) Mean level spacing ratios [⟨rE⟩] for eigenenergies in the middle 60% of the spectrum with Sz tot = 0 for L = 12, 16 and Sz tot = −1 for L = 14, 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' As a comparison, Wigner-Dyson (WD) statistics of the GOE ⟨rE⟩ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='536 (dashed black lines) and Poisson (P) statistics ⟨rE⟩ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='38 (dashed gray lines) are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (b) The energy level spacing statistics for one particular disorder realization in Sz = −1 sector with L = 18, after performing the spectrum unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (b) [SvN/SPage] as a function of ∆ with Sz tot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The data are averaged over 100 disorder realizations and over 1/2 (but not 1/12) of all the eigenstates that are around the state |S4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (c) The disorder-averaged fidelity dynamics [| ⟨Z2(t)|Z2⟩ |2] of the initial state |Z2⟩ with L = 14 at different disorder strength ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' |S4⟩ (marked by a black circle) exhibits anomaly low EE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Sz tot sectors with other eigenstates |Sk⟩ exhibit similar behavior at small disorder strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In the following, we will show that, with the increase of the disorder strength, the energy- level statistics of the bulk energy spectra change from Wigner- Dyson to Poisson, while the EE of the exact tower of states |Sk⟩ remains the same for any disorder strength ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The existence of exact QMBS can also be inferred by the fi- delity dynamics for specific initial states in the scar subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Figure 1(b) shows perfectly periodic revivals in the fidelity of the initial state |ψ0⟩ = 1 N �⌊L/2⌋ k=0 |Sk⟩, where N is the nor- malization factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The revival period T = π corresponds to the energy interval ω = 2 of the scarred states |Sk⟩, as ex- pected from the SGA (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Since the disorder term HA annihi- lates |Sk⟩, the exact states |Sk⟩ and the fidelity dynamics are preserved regardless of the disorder strength ∆, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 1(b) shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Thermalized background to MBL background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='— Although increasing the disorder strength in HA does not influence the eigenstates |Sk⟩, most of the other bulk states alter from er- godic to non-ergodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' To show this, we examine the energy level statistics, half-chain EE, and the imbalance dynamics as Stot z =0 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 E [SvN \uf00cSPage] 30 -20 -10 0 10 20 30 1 10 20 30 40 E Δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 [SvN \uf00cSPage] Stot z =-2 Stot z =0 Stot z =2 8 12 16 20 1 2 3 4 5 L [SvN max] 20 10 0 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 E [Σα \uf0b3ϕα HA ϕn\uf0b6 2] (a) (b) (d) (c) Δ=46 Δ=46 Stot z =0 Stot z =-1 Δ=46 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Energy-resolved features of states in Sz tot sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (a) The energy-resolved [SvN/SPage] for L = 16 in Sz tot = 0 sector where the state |S4⟩ with ES = 0 resides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The peak appears at the energy window that has 722 eigenenergies on average, including ES = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (b) The energy-resolved [SvN/SPage] as a function of ∆ for L = 18, Sz tot = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The bright vertical line resides in the energy window that includes ES = −1 of the state |S4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (c) The scaling of [Smax vN ] with L, where [Smax vN ] are averaged over the maximum entropies Smax vN of eigenstates in each disorder realization, with the corresponding aver- aged energies very close to ES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (d) The overlap between eigenstates of H (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=', |φn⟩) and |φHA α ⟩ for L = 16, where HA|φHA α ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' a function of disorder strength ∆, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The energy-level spacing ratios are defined by [65] rEn = min � sEn, sEn−1 � max � sEn, sEn−1 �, (10) where En is an increasing-ordered set of energy levels and sEn = En+1 − En the nearest-neighbor energy-level spac- ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We eliminate 20% of the eigenenergies at the spectrum’s edges when calculating the statistics of energy-level spacings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The mean energy-level spacing ratios [⟨rE⟩] as functions of ∆ are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 2(a), with ⟨·⟩ denoting the average over the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 2(a), we find that [⟨rE⟩] con- verges to the value of Wigner-Dyson statistics of the Gaussian orthogonal ensemble (GOE) when ∆ is small, implying the thermalization of the bulk states, and [⟨rE⟩] approaches to the value of Poisson statistics at larger ∆, indicating that the sys- tem is localized [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' For a single disorder realization in each of these two different regimes, typical profiles of the energy- level spacing distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 2(b) are consistent with the disorder-averaged energy-level spacing ratios [⟨rE⟩].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We then look at the characteristics of the state- and disorder-averaged EE SvN that distinguishes thermalization from MBL [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Figure 2(c) show divided by SPage for var- 4 ious system size L in the Sz tot = 0 sector, where we denote the state average as ¯·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' With increasing disorder strength ∆, [SvN/SPage] goes from 1 of the thermalized states to 0 of the many-body localized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We also choose the initial product state |Z2⟩ ≡| 101010 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='⟩ to represent the imbalance and calculate its dynamics at differ- ent ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 2(d), the fidelity rapidly approaches zero as time evolves at small ∆, demonstrating ergodic behav- ior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In contrast, at larger ∆, the persistent imbalance dynamics indicate a non-ergodic evolution, consistent with MBL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Multiple inverted quantum many-body scar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='— We have demonstrated that the majority of the bulk states change from ergodic to non-ergodic when increasing disorder strength, be- low we will show the inverted QMBS states with anomaly high entanglement in the MBL background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We examine the energy-resolved EE [SvN/SPage] that is av- eraged over disorder realizations and states in the targeted Sz tot sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We first look into the Sz tot sectors with states |Sk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' As illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 3(a,b), we find highly entangled states located very close to |Sk⟩ in energy, in sharp contrast to other localized states with low entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' For instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 3(a), the state |S4⟩ residing in the sector Sz tot = 0 for L = 16 has its energy ES = 0, besides which many highly entangled states jointly give a [SvN/SPage] peak at the energy window of E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In another example, in the sector Sz tot = −1 of L = 18 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 3(b)), the state |S4⟩ has en- ergy ES = −1, and [SvN/SPage] also exhibits sharp peak at E = −1 in a narrow energy window for large ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Although states |Sk⟩ have a sub-volume-law EE [40], the disorder- averaged maximum entropies [Smax vN ] exhibit a volume-law be- havior, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In the bulk Sz tot sectors without states |Sk⟩, we also find anomaly high entanglement states, and they concentrate in the middle of the energies of the ex- act eigenstates |Sk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Therefore, the Hamiltonian (8) realizes multiple inverted QMBS concentrating in different narrow en- ergy windows with approximately equal energy spacing ≈ 1, which is the half of the energy spacing of states |Sk⟩ [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We remark that the number of high entanglement states in every energy window is much larger than one (as detailed in the cap- tion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 3(a)), and the narrow energy windows with highly entangled states also exhibit peaks of the energy density of states (DOS) [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We further understand the behavior of the inverted QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Indeed, we find these highly entangled states have a large overlap with the states ��φHA α � in the null space of HA (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 3(d)), where HA ��φHA α � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' As a result, such states stay delocalized and remain largely unaffected by the disorder strength, similar to the exact tower of eigenstates |Sk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Stability to onsite random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='— Now we consider the sta- bility of the aforementioned exact QMBS |Sk⟩ and inverted QMBS to the onsite random z fields that break the formal- ism H (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' To be more specific, we modify H in (8) to be H′ = H + h �L j=1 δjSz j , where δj are the uniform random numbers in the range δj ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Unlike the disordered HA, the disorder term in H′ can drive all eigenstates to the lo- calization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 4, with a localized spectrum background, 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='50 1 t h 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 [|〈ψ(t)|ψ0〉 2] 20 -10 0 10 20 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='50 1 E h 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 [SvN \uf00cSPage] (a) (b) Δ=46 Δ=46 Stot z =0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Stability of |Sk⟩ and the inverted QMBS at large dis- order strength ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (a) The disorder-averaged fidelity dynamics [| ⟨ψ(t)|ψ(0)⟩ |2] of the initial state |ψ(0)⟩ ≡ |ψ0⟩ in scar subspace with L = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (b) The energy-resolved [SvN/SPage] as a function of h for L = 16, Sz tot = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The peak resides in the energy window that includes ES = 0 of the state |S4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' both the periodic revival of the fidelity for the initial state |ψ0⟩ and the peak of [SvN/SPage] show certain stability of both ex- act tower of eigenstates |Sk⟩ and inverted QMBS against the onsite random z field, though they eventually disappear for a large disorder strength h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' In a thermalizing background, with the increase of h, the conventional QMBS |Sk⟩ first disappears and the system becomes thermal before the final localization of all the states [68], consistent with previous scenarios of the QMBS in the disordered PXP models [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Summary and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='— In this work, we have realized a direct evolution from a thermal spectrum background with a tower of exact QMBS to the MBL background with multi- ple inverted QMBS in a disordered spin-1/2 XX Heisenberg chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The forms of the exact tower of states are independent of the disorder, while the energy-level statistics of the bulk eigenstates changes from Wigner-Dyson to Poisson when in- creasing the disorder, despite the existence of the embedded highly entangled states at strong disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Embedded in the otherwise MBL spectra with low entanglement, the multiple sets of many highly entangled states are located within differ- ent narrow energy windows that are approximately equidis- tant in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We also show certain stability of the highly entangled states to the onsite random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' To the best of our knowledge, such a scenario that inverts multiple QMBS di- rectly is not constructed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Our model can also be gener- alized to other non-Abelian symmetry, and to the large classes of QMBS Hamiltonian that resort to the annihilating term HA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The proposal to invert QMBS in this work may also stimulate more experimental activities to realize setups that weakly vi- olate the quantum ergodicity or the MBL [38, 59–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Note added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='— When finalizing the manuscript, we became aware of one recent work [69] on related topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We are grateful to Yi-Zhuang You and Shuai A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Chen for the fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='12074375), the Fundamental Research Funds for the Cen- tral Universities and the Strategic Priority Research Program of CAS (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='XDB33000000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 5 ∗ zhuzheng@ucas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='cn [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Srednicki, Chaos and quantum 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 7 Supplementary Materials for “Inverting quantum many-body scar via disorder” Multiple inverted QMBS in different symmetry sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Stot z =2 Stot z =1 Stot z =0 Stot z =-1 Stot z =-2 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 E [SvN \uf00cSPage] Stot z =2 Stot z =1 Stot z =0 Stot z =-1 Stot z =-2 10 5 0 5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 E [Σα \uf0b3ϕα HA ϕn\uf0b6 2] Δ=1 Δ=11 Δ=46 13 -9 -5 -1 3 7 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='12 E [ρ(E)] (a) (b) (c) Stot z =-1 Δ=46 Δ=46 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Features of multiple inverted QMBS in different Sz tot sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (a) The energy-resolved [SvN/SPage] for L = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (b) The overlap between eigenstates of H (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=',|φn⟩) and |φHA α ⟩ for L = 16, where HA|φHA α ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (c) Density of states for L = 18, Sz tot = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' At ∆ = 46, the peak of DOS appears in the energy window of inverted QMBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' For comparison, the DOS for smaller ∆ are also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' At strong disorder, multiple sets of highly entangled states concentrating in equidistant energy windows emerge in different Sz tot sectors, as shown by the peaks of the energy-resolved [SvN/SPage] in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' S1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We remark that these energy windows with peaks of [SvN/SPage] are indeed very narrow compared to the large width of the whole energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' For example, for Sz tot = 2 sector of L = 16 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' S1(a), the width of the whole energy spectrum is ∼ 336, while the peak of [SvN/SPage] is only ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The spacing between these energy windows is roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Figure S1(b) shows that the highly entangled states are indeed almost annihilated by the term HA and thus remain largely undisturbed by the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Moreover, at large ∆, we also find the peak of the averaged density of states [ρ(E)] appears at the narrow energy window where the highly entangled states locate, as shown by the typical sector Sz tot = −1 of L = 18 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' S1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Fate of QMBS in the presence of onsite random field In this section, we study the fate of QMBS in the presence of onsite random z field h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Here we consider a different annihilating disorder with more terms H′ A = ∆ � j {c(1) j |010⟩⟨010| + c(2) j 2 (|011⟩ + |110⟩)(⟨011| + ⟨110|) + c(3) j [|010⟩(⟨011| + ⟨110|) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=']}j−1,j,j+1, where c(α) j with α = 1, 2, 3 are the uniform random numbers c(α) j ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' We remark that H′ A breaks U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The total Hamiltonian reads H = L � j=1 � S+ j S− j+1 + S− j S+ j+1 + Sz j � + H′ A + h L � j=1 δjSz j (S1) The last term in (S1) can be regarded as the onsite random fields that break the symmetry-based formalism mentioned in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Some characteristic features of QMBS, such as the slow relaxation from certain initial states, are still existent in the presence of a modest disorder strength h, as shown by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' S2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' As h is increased, however, the model (S1) loses the QMBS features before switching to MBL (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' S2(b-d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Remarkably, in the MBL spectrum background, there is no peak of [SvN/SPage], since the random z fields can affect every eigenstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' 8 L=12 L=14 L=16 0 5 10 15 20 25 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='55 h [〈rE〉] 0 5 10 15 20 25 30 30 20 10 0 10 20 30 h E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 [SvN \uf00cSPage] (b) (d) Δ=1 Δ=1 101 102 0 5 10 15 20 25 30 h t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 [|〈Z2(t)|Z2〉 2] (c) Δ=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='10 100 101 0 5 10 15 20 25 30 h t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='0 |〈ψ(t)|ψ0〉 2 (a) ] [ Δ=1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' Stability of QMBS |Sk⟩ at weak disorder strength ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (a) The disorder-averaged fidelity dynamics [f(t)] = [| ⟨ψ(t)|ψ(0)⟩ |2] of the initial state |ψ(0)⟩ ≡ |ψ0⟩ (defined in the main text) with L = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (b) Mean level spacing ratios [⟨rE⟩] as a function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' As a comparison, Wigner-Dyson statistics of the GOE ⟨rE⟩ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='536 (dashed black lines) and Poisson statistics ⟨rE⟩ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content='38 (dashed gray lines) are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' The [⟨rE⟩] are averaged over 100 disorder realizations for L = 12, 14 and between 10 and 40 for L = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (c) The disorder-averaged fidelity dynamics f(t) = | ⟨Z2(t)|Z2⟩ |2 of the initial state |Z2⟩ with L = 14 at different disorder strength h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} +page_content=' (d) The energy-resolved [SvN/SPage] as a function of h for L = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E1T4oBgHgl3EQfwAUv/content/2301.03405v1.pdf'} diff --git a/X9E3T4oBgHgl3EQfGAla/content/tmp_files/load_file.txt b/X9E3T4oBgHgl3EQfGAla/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7edb18d3197064225ec1e5f37bd5c8085b349b8 --- /dev/null +++ b/X9E3T4oBgHgl3EQfGAla/content/tmp_files/load_file.txt @@ -0,0 +1,1078 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf,len=1077 +page_content='Thermodynamic Properties of the Mott Insulator-Metal Transition in a Triangular Lattice System Without Magnetic Order Emre Yesil1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Shusaku Imajo2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='∗ Satoshi Yamashita1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Hiroki Akutsu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Yohei Saito3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Andrej Pustogow4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Atsushi Kawamoto5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' and Yasuhiro Nakazawa1† 1Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Osaka University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Toyonaka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Osaka 560-0043,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Japan 2Institute for Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Kashiwa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Chiba 277-8581,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Japan 3Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Goethe-University Frankfurt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 60438 Frankfurt (M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Germany 4Institute of Solid State Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' TU Wien,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1040 Vienna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Austria 5Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Hokkaido University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Sapporo 060-0810,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Japan (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2023) The organic system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' κ-[(BEDT-TTF)1−x(BEDT-STF)x]2Cu2(CN)3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' showing the Mott transition between a nonmagnetic Mott insulating (NMI) state and a Fermi liquid (FL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' is systematically studied by calorimetric measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' An increase of the electronic heat capacity at the transition from the NMI state to the FL state which keeps the triangular dimer lattice demonstrates that the charge sector lost in the Mott insulating state is recovered in the FL state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' We observed that the remaining low-energy spin excitations in the Mott insulating state show unique temperature dependence, and that the NMI state has a larger lattice entropy originating from the frustrated lattice, which leads to the Pomeranchuk-like effect on the electron localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Near the Mott boundary, an unexpected enhancement and magnetic-field dependence of heat capacity are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This anomalous heat capacity is different from the behavior in the typical first-order Mott transition and shows similarities with quantum critical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' To reconcile our results with previously reported scenarios about a spin gap and the first-order Mott transition, further studies are desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' INTRODUCTION The dimer-Mott compounds with the chemical for- mula of κ-(BEDT-TTF)2X, where BEDT-TTF is bis(ethylenedithio)tetrathiafulvalene and X is a mono- valent counter anion, provide extensive possibilities for understanding physical phenomena induced by electron correlations of π-electrons[1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Electrons in them form relatively narrow electron bands governed by overlaps of molecular orbitals, and the spin, charge, and lattice de- grees of freedom appear in various manners in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The electronic states of the dimer-Mott system can be de- scribed in the frame of the Mott-Hubbard physics with on-site Coulomb repulsion U and bandwidth W (pro- portional to transfer integral t)[1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Additionally, the dimer lattice of the κ-type molecular arrangement has geometrical frustration depending on the ratio of t and t′, nearest-neighbor and second-nearest-neighbor trans- fer integrals, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Using the two pa- rameters U/t vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' t′/t, the electronic phase diagram has been understood, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(b)[6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' For the less-frustrated salts (t′/t<1), the discontinuity and hysteresis in the electrical transport indicate that the superconductivity-antiferromagnetic insulator (SC-AFI) transition dominated by a change in U/t (the blue ar- row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(b)) is first-order[8–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' As schematically described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(c), the 1st-order Mott boundary dis- appears at a critical endpoint of ∼35 K, and the nature of the Mott physics around the endpoint has been discussed ∗ imajo@issp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='jp † nakazawa@chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='osaka-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='jp in terms of high-energy criticality caused by the competi- tion between the large U and W >1000 K[10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' From the AF magnetic order induced by antiferromagnetic in- teractions, the SC with relatively high-T c has been ex- tensively discussed in terms of unconventional pairing re- lated to antiferromagnetic spin fluctuations in κ-(BEDT- TTF)2X and also in β′-, λ-type compounds[13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The variation in physical parameters near the Mott bound- ary has been studied by various measurements across the boundary[18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Based on the variation in the electronic heat capacity coefficient γ of the normal state shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(c)[14, 23–26], the low-temperature FL state can be understood by the electron-mass enhancement with increasing electron correlations (the green arrow) and the decrease in the metallic portion due to the growth of phase coexistence near the Mott boundary (the or- ange arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Although slight percolative superconduc- tivity is left in the AF Mott insulating salts very near the boundary, its γ is almost zero because the volume fraction of the FL is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' It should be noted that the information contains the magnetic entropy change re- lated to the AFI ground state of π-electrons and that the change is not a genuine feature expected in the Hubbard model because no symmetry breaking is assumed in this framework[27–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' When t′/t=1, the AFI ground state should be destabilized by the geometrical frustration, and non-ordered states may be stable even at low temper- atures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Indeed, κ-(BEDT-TTF)2Cu2(CN)3, which has been considered a prime candidate showing the quantum spin liquid (QSL) state, does not show long-range mag- netic orders down to extremely low temperatures because t′/t is almost unity[32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' However, recently, Miksch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' suggested that its ground state might be a gapped valence bond solid (VBS) with a spin gap from observa- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04310v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='str-el] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='11 Jan 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='γ (mJK-2mol-1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='AFI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='SC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='Mott ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='insulator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='Fermi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='liquid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='cross ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='over ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='T (K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='U/t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='① ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='② ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='③ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='④ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='⑤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='X = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='① : Cu[N(CN)2]Cl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='② : Cu[N(CN)2]Br ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='dn : n = number of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='deuterium in BEDT-TTF ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='③ : Cu(NCS)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='④ : Ag(CN)2H2O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='⑤ : I3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='κ-(BEDT-TTF)2X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='d8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='d6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='d4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='d0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='Mott ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='transition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='mass ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='enhancement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='ρ = AT2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='Cele = γT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='χ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='= χP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='(c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='nonmagnetic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='Mott ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='insulator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='(NMI) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='dT/dx>0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='quantum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='critical ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='regime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='Fermi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='liquid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='(FL) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='crossover ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='(d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='T (K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1 0 x FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (Color online) (a) Molecular arrangement in the conducting plane of the present system and BEDT-TTF and BEDT- STF molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' t and t′ indicate nearest-neighbor and second-nearest-neighbor transfer integrals in the dimer lattice, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (b) Electronic ground states of dimer-Mott system with parameters of electron correlations U/t (U/W) and frustration factor t′/t[6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The red arrow indicates the route controlled by substitution in the present system, whereas the light blue arrow represents the numerous previous studies on the less-frustrated dimer-Mott system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (c) Electronic phase diagram of the less-frustrated κ-type salts with the shown counter anions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The AFI and FL phases are divided by the first-order Mott transition, which terminates at a critical endpoint ∼35 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Inside the FL, SC with relatively high-T c∼10 K occurs near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The lower panel shows the variation in γ of the normal state depending on electron correlations U/t and chemical pressure of the counter anion P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (d) Schematic phase diagram of κ-[(BEDT-TTF)1−x(BEDT-STF)x]2Cu2(CN)3 deduced from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' [36–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' An increase in the mixing ratio x corresponds to a decrease in U/t shown by the red arrow in the phase diagram (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The NMI and electronic FL states exist across the quantum critical regime, which is located around x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The dashed curve indicates the phase boundary between the NMI and FL phases, its slope dT/dx is positive at low temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The black dots in the NMI region represent the x dependence of the so-called 6 K anomaly, in which error bars are determined by the present heat capacity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' tion of a drop of spin susceptibility below 6 K[31, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Although the controversy still persists because of the re- maining discrepancy with the gapless spin excitations in heat capacity[35], we hereafter use NMI for describing the Mott insulating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Recently, Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' reported that a donor al- loying system of (BEDT-TTF)1−x(BEDT-STF)x, where BEDT-TTF and BEDT-STF are the ab- breviations of bis(ethylenedithio)tetrathiafulvalene and bis(ethylenedithio)diselenadithiafulvalene, with Cu2(CN)3− exhibits continuous tuning of U/W with keeping the triangularity of the dimer lattice[36–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The Se substitution into the BEDT-TTF molecule shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(a) results in a larger overlap of the wave function with the neighboring molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Since the increase in x is considered to work as positive chemical pressure without inducing large change in average t′/t, the insulating state is altered into the FL state via the genuine Mott transition at x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2[36– 39], as indicated by the red arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This variation is similar to the tuning by external pressure to κ-(BEDT-TTF)2Cu2(CN)3 where the ground state without magnetic order shifts to a FL across the Mott insulator-metal transition[8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Namely, this variation provides profound information on the Mott transition genuinely dominated by the itinerancy/localization of the charge degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The T-x electronic phase diagram of the present alloying system is predicted from the results in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' [36–39], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' High-resolution thermodynamic measurements under pressure are typically challenging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' however, using the present chemically tunable system, thermodynamic and entropic information near the metal-insulator boundary can be obtained by ambient-pressure heat capacity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In this study, we systematically inves- tigated κ-[(BEDT-TTF)1−x(BEDT-STF)x]2Cu2(CN)3 by calorimetry to unveil thermodynamics features of the Mott transition between the potential QSL and FL states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' EXPERIMENTAL Single crystals of the alloying compounds κ-[(BEDT- TTF)1−x(BEDT-STF)x]2Cu2(CN)3 are grown by elec- trochemical oxidation method[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' As shown in Table I, the crystal structural parameters were characterized by x-ray diffraction analyses, and the macroscopic homo- geneity of the alloying crystals was confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' To evalu- ate the change in t′/t with mixing BEDT-STF molecules, SEDTLSTF10 AF sc 5 F 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 103 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Crystallographic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Fw represents the formula weight for each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' V shows the cell volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Z denotes the number of formula units in the unit cell divided by the number of independent general positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' d′/d is the ratio of average dimer-dimer distance along the t′ and t directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 Fw 982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='30 997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='05 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='18 1027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='06 1057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='07 Space group P21/c P21/c P21/c P21/c P21/c P21/c a (˚A) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1080 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1054 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1136 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1569 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1580 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1761 b (˚A) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5861 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5816 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5874 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='6000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='6017 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5982 c (˚A) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='3591 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='3751 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='3550 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='3663 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='3979 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='4037 α (◦) 90 90 90 90 90 90 β (◦) 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='691 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='565 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='66 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='519 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='551 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='208 γ (◦) 90 90 90 90 90 90 V (˚A3) 1691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='9 1694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='4 1692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='6 1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='0 1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='0 1713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='4 Z 2 2 2 2 2 2 d′/d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='926 we here introduce d′/d, the ratio of average dimer-dimer distance along the t′ and t directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The small changes in the lattice parameters within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5% were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Heat capacity measurements were carried out by a typ- ical relaxation technique using a home-made thermal- relaxation-type calorimeter in a 3He refrigerator with a 15 T superconducting magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The temperature range of these measurements is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='6-10 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Magnetic fields were applied perpendicular to the conducting plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' We measured the background data with a small amount of Apiezon N grease before mounting samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' These mea- surements were performed with single crystalline samples weighing about 80-300 µg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The details of the calorimeter and experimental setup are reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' RESULTS In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2, we present the temperature dependences of the heat capacity of the alloying system in the CpT −1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' T 2 plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The data in the temperature range up to 10 K are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' There is only a subtle difference of about 6% at 10 K, mainly originating from the decrease in the Debye temperature induced by the Se substitution[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This means that the STF substitution induces only a small change of about a few percent in phonon contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(b), the entropy S as a function of temperature is shown as a logarithmic plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The entropy is calculated by integration of the exper- imentally obtained CpT −1 and the extrapolation down to 0 K estimated by polynomial fittings, and thus, the calculated S includes the electronic and phonon contri- butions together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' At higher temperatures, the x depen- dence of the entropy is small because the main portion of the total entropy is the phonon contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In the lower temperature region, the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 salt shows the larger en- tropy compared to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' To shed light on the low- temperature region, the enlarged plots of CpT −1 below about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2 K (=10 K2) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The datasets for x<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='15 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(c) while those for x>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='15 are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(d) because the Mott insulating char- acter at the low-x region changes into the metallic one across the boundary region at x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2 according to the previous reports[36, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The small change in lattice heat capacity indicates that the origin of the change ob- served in the low-temperature region should mainly come from the electronic contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In the case of typical metals having Fermi surfaces composed of itinerant elec- trons, CpT −1 at low temperatures obeys CpT −1=γ+βT 2, where γ and β represent the Sommerfeld coefficient of electronic heat capacity and the Debye coefficient of lattice heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Indeed, the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 salt, which is deep inside the metallic FL region, shows the lin- ear behavior below 2 K with γ=24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1 mJK−2mol−1 and β=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='0 mJK−4mol−1, which are comparable with those of typical BEDT-TTF-based metallic salts[14, 16, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Above 2 K, the behavior is gradually deviated by ex- cess heat capacity that may originate from librational op- tical modes Copt∼R(TE/T)exp(TE/T)/[exp(TE/T)−1]2, where TE represents the Einstein temperature, as sug- gested for the other organic charge-transfer complexes with various structures[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' On the other hand, the insulating salts shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(d) do not share this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' At first glance, it ap- pears to follow the linear behavior below 2 K, as indicated by the black dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Also, the analysis of the data for x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 using the typical CpT −1=γ+βT 2 relation leads to γ=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='6 mJK−2mol−1 and β=21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2 mJK−4mol−1, which are comparable with the previously reported γ=12 mJK−2mol−1 and β=21 mJK−4mol−1 for x=0[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' However, above 2 K, the CpT −1 is lower than this lin- ear dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Since higher-order terms of the Debye model appears only at higher temperatures, this behav- ior indicates that the Mott insulating state cannot be ex- plained by the framework of the typical FL states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Nev- ertheless, the large low-temperature heat capacity in the insulating state proves the presence of low-energy spin excitations, which have been discussed as the finite γ and/or the relatively large β specific to the organic QSL state in the previous works[35, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' As a rough estimate, we show the lattice heat capacity ClatT −1, which is sim- ply obtained by subtracting the γ term from the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 data (the thin black line), CpT −1=γ+βT 2+Copt−1, as is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The difference from this estimate (Cp−Clat)T −1, which corresponds to the contribution of the spin excitations, is displayed as a CeleT −1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' T plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This component does not appear to be a simple γ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' ESR results[34] suggest that the ground state is a gapped VBS state with a relatively large ∆/kB∼12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' However, the red curve, exp(−∆/kBT) be- havior for ∆/kB=12 K, does not describe the present results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Even if we assume that ∆/kB is a variable pa- rameter, it is difficult to reproduce the temperature de- pendence and ∆/kB must be extremely tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Including the present result, heat capacity measurements, sensi- tive to low-energy excitations, indicate the presence of low-energy spin excitations, which is puzzling in view of 4 10 1 10 2 10 3 10 4 S (mJK 1mol 1) 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 T (K) x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='10 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 ~6% ClatT-1 (Cp−Clat)T-1 (a) βT2 γ CoptT-1 γ + βT2 + CoptT-1 ClatT-1 γ superconducting transition (c) (d) (e) (f) (b) x ~exp(−Δ/kBT) Δ/kB=12 K 200 150 100 50 0 Cp/T (mJK 2mol 1) 8 6 4 2 0 T 2 (K 2) x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='10 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 80 60 40 20 0 Cele/T (mJK 2mol 1) 6 4 2 0 T (K) x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 4 3 2 1 0 T (K) x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 10 8 6 4 2 0 T 2 (K 2) x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 1500 1000 500 0 Cp/T (mJK 2mol 1) 100 80 60 40 20 0 T 2 (K 2) x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='10 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28 x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 Schottky anomaly ~lnT FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (Color online) (a) CpT −1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' T 2 below 10 K for x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (b) Logarithmis plot of S as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (c),(d) Enlarged plots of the low-temperature region below T<3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2 K for x<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='15 (c) and x>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='15 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The dotted line in (c) is a fit to CpT −1=γ+βT 2 below 2 K for x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The thin black curve in (c) is a rough estimate of lattice heat capacity ClatT −1 obtained from the FL salt (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44) by subtracting the γ term, which highlights the contribution of the low-energy excitations in the NMI state (shaded area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The black curves in (d) show the respective components of the fit of CpT −1=γ+βT 2+CoptT −1 to the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (e),(f) Electronic heat capacity CeleT −1 obtained by subtracting the lattice heat capacity ClatT −1 from the total CpT −1 for x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 (e) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The red curve in (e) shows activation-type gapped behavior when ∆/kB=12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The red curve in (f) indicates a fit to −lnT while the black and green curves represent the typical superconducting and Schottky anomalies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' a spin gap concluded from other measurements[34, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' To clarify this point, experiments at lower temperatures seem necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Although the drop of the magnetic sus- ceptibility below 6 K is observed, an exact zero suscepti- bility in a low-temperature limit has not been reported in these works[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' To reconcile these arguments based on the temperature range of the measurements (ESR mea- surement above 2 K), one possibility is that the ground state has an incomplete spin gap, yielding some low- energy excitations, even below the putative transition at 6 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Alternatively, an extrinsic origin, such as impu- rity spins or domain walls, was suggested to describe the low-temperature magnetic behavior[31, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' However, it is unclear how to model the present temperature depen- dence with the suggested local orphan spins and local domain wall fluctuations, which may give the Schottky- type heat capacity and glass-like γT heat capacity, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' For the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 salt, located in the intermediate region[36, 39], the temperature dependence (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(d)) does not obey the CpT −1=γ+βT 2 relation due to the gradual upward deviation below ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This behav- ior is more clear in the plot of (Cp-Clat)T −1, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Even though remnants of superconduc- tive components are observed near the 1st-order Mott boundary of several κ-type salts, including some STF compounds[47], this behavior is completely distinct from the superconducting transition (the black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Fur- thermore, this deviation cannot be reproduced by ex- trinsic Schottky anomaly arising from magnetic impu- rities (the green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In the case of the AFI-FL Mott transition, such behavior is absent, and the sim- ple CpT −1=γ+βT 2 relation is observed even very near the first-order Mott boundary[23, 24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The low- temperature gradual divergence is reminiscent of quan- tum critical behavior near a quantum critical point (QCP) because of the −lnT-like behavior (the red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Indeed, the present alloying system does not show signif- icant first-order-like discontinuous behavior in our heat capacity data and resistivity data[36, 38], albeit a dielec- tric catastrophe suggestive of phase inhomogeneity has been reported[39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The first-order Mott transition ob- served in other κ-type salts is less obvious in κ-(BEDT- TTF)2Cu2(CN)3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' nevertheless, the weak first-order Mott transition with the critical endpoint located at 15-20 K has been observed in transport, NMR, and dielectric 5 measurements[8, 9, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' A small difference between the alloying system and κ-(BEDT-TTF)2Cu2(CN)3 may fur- ther lead to suppression of the remaining first-order na- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Randomness effects should be also taken into ac- count, as a disorder can lower the temperature of the critical endpoint[48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Regardless of the origin of the sup- pression of the first-order nature, the low-temperature diverging heat capacity indicates that quantum fluctua- tions are developed in this temperature region (<2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Since this behavior is significant in x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 and smaller in x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28, the QCP should be located close to x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2, which is not far from the reported position of the metal- insulator transition[36, 38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The magnetic field dependences of the heat capac- ity for x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The upper panels show the low-temperature re- gion below 10 K2 while the lower ones display the data up to 120 K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The fields are applied perpendicularly to the two-dimensional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' For the NMI (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04) salt, the magnetic-field dependence is not significant even at high magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This fact indicates the robustness of these low-energy excitations against fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' However, the response to the magnetic field for the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 sam- ple, located in the quantum critical region (QCR), is distinct from those of the other salts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The upturn ob- served at 0 T disappears with increasing magnetic field, while a broad hump structure in the temperature de- pendence of CpT −1 appears at relatively high magnetic fields of 5-6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Since this behavior indicates that the low-temperature entropy shifts to higher temperatures in magnetic fields, the origin of this field dependence cannot be attributed to the 6 K anomaly and percola- tive superconductivity which is often observed near the 1st-order Mott transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Considering the magnetic field dependence of the Mott boundary[23] and the bent quan- tum phase boundary[38, 39] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(b)), the origin of the hump structure is also attributed to the critical behav- ior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' By further increasing fields, the broad hump is also suppressed, and the field dependence is diminished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This behavior suggests that the high-field electronic state at low temperatures is out of the critical regime and can be regarded as the FL state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' DISCUSSION To deepen the understanding of the variations in the low-energy excitations around the Mott transition, we here show the low-temperature heat capacity at 1 K, Cp(1 K), as a function of x in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In order to highlight the area near the Mott transition, each region is color-coded in a different color in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Based on the variation in γ depending on U/t (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(c)), Cp(1 K) should vary like the blue broken curve if the Mott tran- sition is between the AFI and FL states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Namely, the deviation from the blue broken curve is the peculiar- ity of the NMI-FL Mott transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' For the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 salt, the value of 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='8 mJK−2mol−1, much larger than β=15 mJK−4mol−1 for the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 salt, suggests that the heat capacity involves finite low-energy excitations of the spin sector (the light blue arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' With approaching Mott transition, the Cp(1 K) increases from the constant value in the NMI region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This behavior deviates from the blue broken curve because γ is constantly zero inside the AFI Mott phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Once crossing the boundary and entering the FL regime, the Cp(1 K) asymmetrically de- creases and reaches 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1 mJK−1mol−1 at x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44, which is comparable with the typical value of Cp(T=1 K)=γ+β for the BEDT-TTF-based metallic salts with γ=20- 25 mJK−2mol−1 and β=10-15 mJK−4mol−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The dif- ference between the NMI and FL regions, shown by the pink arrow, should correspond to the contribution of the charge sectors of the π-electrons, which is absent in the NMI state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' If the inhomogeneity appearing near the first- order Mott transition develops around x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2, the strong enhancement of Cp(1 K) inside the FL region should not be observed near the boundary because the inhomogene- ity significantly reduces the electronic heat capacity, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Here, we examine the slope of the phase boundary be- tween the NMI and FL states on the electronic phase diagram, dT/dx in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The positive slope in- dicates that the localization of electrons in the NMI state gives a larger entropy than the itinerancy of elec- trons in the FL state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This unusual behavior is reminis- cent of the Pomeranchuk effect observed in 3He, melting solid 3He with lowering the temperature through spin- lattice coupling[49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 4(b), we present the x-dependence of the entropy S at 1 K (left axis) and 5 K (right axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' At 1 K, the entropy of the NMI state is lower than the entropy of the FL state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' At 5 K, it is the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' As suggested by a theory[51], it is ex- pected that there is only small energy difference between the FL and Mott states because the gain in kinetic en- ergy of electrons in the FL state is compensated by the loss in potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In particular, when the frus- tration parameter t′/t is close to unity, the slope of the Mott boundary dU/dT is almost zero or a small negative value[52], and thus, the energy difference between the two states should be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This delicate energy balance makes the Mott transition winding on the phase diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In real materials involving a variety of degrees of freedom, we must consider what contribu- tion is an eventual factor determining how large or small the entropy is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The similar x dependence of S(1 K) and Cp(1 K) demonstrates that the low-temperature behavior can be explained by the electronic part and that the en- tropy of the NMI state is smaller at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' However, at higher temperatures above 5 K, the lattice part must also be considered because the lattice com- ponents account for a large portion (>90%) of the total entropy in the soft organic crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' To explain the rever- sal of the entropy appearing with elevating temperature, we need to discuss entropy originating from the phonons as well as the low-energy spin excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The charac- teristic of the present system is the confinement of the 6 100 50 0 T 2 (K 2) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 FL 100 50 0 T 2 (K 2) 0 T 4 T 8 T x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28 FL 100 50 0 T 2 (K 2) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 NMI 8 6 4 2 0 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 QCR 10 8 6 4 2 0 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 FL 8 6 4 2 0 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28 FL 100 50 0 T 2 (K 2) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 QCR 200 150 100 50 0 Cp/T (mJK 2mol 1) 8 6 4 2 0 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 NMI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='0 Cp/T (JK 2mol 1) 100 50 0 T 2 (K 2) x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 NMI 8 6 4 2 0 x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 NMI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='0 CpT 1/ JK 2mol 1 100 50 0 T 2/ K 2 0 T 2 T 4 T 5 T 6 T 8 T 10 T 12 T x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 NMI FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (Color online) CpT −1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' T 2 at various magnetic fields for x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 at the NMI, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 at the quantum critical region (QCR), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='28, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='44 at the FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The upper panels show the data below 10 K2 while the lower ones show the data up to 120 K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The arrow for the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19 data indicates a hump observed at fields of 5-6 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' electrons in the triangular lattice making the antiferro- magnetically interacting spins frustrated and disordered, which should result in lattice softening through the spin- lattice coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The lattice softening entails shifting the phonon density of states down to a lower-temperature region, as is observed in the NMI state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Low-energy phonon excitations in the non-ordered dimer-Mott tri- angular lattice system have been discussed by thermal conductivity measurements[45], and therefore, the soft- ening of phonons can be a possible reason to explain the larger entropy in the NMI state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Nevertheless, another possibility is that the spin excitations explain the evolu- tion of entropy with temperature in the NMI state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' When dU/dT is negative, the entropy of the NMI state can be- come larger than that of the FL state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The formation of the possible non-magnetic VBS ordered state[31, 34] suggests a rapid increase in spin entropy with an en- hancement of heat capacity near the transition temper- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This gap-closing behavior around 5 K may also relate to the reversal of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Although the spin entropy in the present system is not large, the relation may not be reversed even with the gain in the lattice en- tropy without the spin entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' For the NMI system, the low-temperature Pomeranchuk-like phase boundary[53] is probably related to both contributions, namely the phonon softening effect and spin contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' To dis- cuss these in more detail, the temperature dependence of entropy up to higher temperatures is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' We emphasize that the low-temperature heat ca- pacity Cp(1 K) should reflect the variation of the ground state driven by quantum fluctuations predom- inantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The gradual increase in Cp(1 K) with ap- proaching the Mott boundary in the NMI phase sug- gests the continuous change in the low-energy excita- tions as the possible VBS state is suppressed near the Mott boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' According to the correspondence be- tween the chemical pressure characterized by x and physical pressure (∆P=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 kbar roughly corresponds to ∆x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1)[39] and the slope of the metal-insulator bound- ary dx/dT∼2*10−3 K−1 (at T=5 K), the Clausius– Clapeyron relation dP/dT=∆S/∆V leads to the volume change ∆V ∼2*10−8 m3mol−1 with the entropy differ- ence ∆S(5 K)∼50 mJK−1mol−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Despite the rough es- timation, the obtained ∆V is one order of magnitude smaller than the difference of ∆V between x=0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1, ∆V ∼2*10−7 m3mol−1[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Thus, even if the boundary is a first-order transition, its discontinuity must be al- most negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Near the QCP where the charge gap is just 0 K, quantum fluctuations related to the instabil- ity of the charge itinerancy are enhanced and destabi- lize the quasiparticles characterizing the FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' It should be noticed that the quantum critical behavior is ap- parent only in the low-temperature region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' It is wor- thy to note that this energy scale is completely differ- ent from that of the high-temperature critical behav- ior induced by U and W, which is commonly observed 7 FL NMI β γ+β QCR (a) (b) 60 40 20 0 Cp(T=1 K) (mJK 1mol 1) 60 40 20 0 S(T=1 K) (mJK 1mol 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='1 0 x 1000 900 800 S(T=5 K) (mJK 1mol 1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' (Color online) Heat capacity Cp at 1 K (a) and en- tropy S at 1 K (left) and 5 K (right) (b) as a function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The blue broken curve is the behavior expected based on the variation in γ for the Mott transition between the AFI and FL states (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The light blue and pink arrows in (a) highlight the contribution of the spin and charge sectors in the electronic heat capacity of the FL state, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The violet region represents the quantum critical region (QCR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The thick translucent curve superimposed on the data points in (b) is a visual guide to make the x-dependence of S(T=5 K) clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' in all dimer-Mott systems irrespective of the geometri- cal frustration[54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Indeed, the low-temperature criti- cal behavior is absent in the less-frustrated system κ- (d[n,n]-BEDT-TTF)2Cu[N(CN)2]Br, which can access the 1st-order Mott transition between the AFI and the FL states[23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' As the peculiarity of the NMI salt is the persistence of the low-energy excitations related to the spin part, the present critical behavior may be in- duced by the instability of the fractionalization of the electron into the spin and charge sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The present result and scenario agree with the discussion of the re- cent transport experiment under pressure[9] and the ther- modynamic investigation of κ-[(BEDSe-TTF)x(BEDT- TTF)1−x]2Cu[N(CN)2]Br[55], which is also another can- didate hosting the genuine Mott transition between the NMI and FL state, as well as theoretical works[29, 30, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Finally, we briefly discuss the so-called “6K anomaly” for the NMI sample, which has been discussed in the pristine x=0 sample[31, 35, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Although the recent studies with high-quality samples and various sensitive measurements[34, 58] have allowed us to get closer to the details of this anomaly, the detail is still unclear because of some unresolved questions, such as the presence of the gapless excitations discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Since the pristine salts reported in the previous work[35] are synthesized by other methods, their sample quality may differ from that of the present alloying series and quantitative comparison may be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Nevertheless, the systematic change in the physical parameters shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 4 allows us to qualitatively compare our data with the results reported in the earlier work[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 2(e) indicates that the anomaly seems to be broadened and suppressed down to 3-4 K for the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='04 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='12 samples compared to that of the pristine sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The black dots shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 1(d) represent the x dependence of the peak tem- perature of the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Considering the relation to the charge disproportionation[59], it seems reasonable that the anomaly is smeared out by the suppression of the electron localization with approaching the Mott bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In the lower panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 3, the magnetic field dependence of the anomaly is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This feature robust against the magnetic field is consistent with the estimation of a critical field of order 60 T for the pris- tine salt[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' To elucidate this enigmatic anomaly, more detailed investigations are desired in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' CONCLUSIONS In summary, we report the low-temperature thermo- dynamic properties for the chemical pressure tuning sys- tem of the dimer-Mott compounds that show no long- range ordering even at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The present result also provides evidence that the NMI state sup- ports some gapless spin excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' However, we also found that this low-energy excitations do not seem to be described by a simple FL-like γ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The system- atic change in the heat capacity depending on x revealed that the genuine Mott transition is potentially continu- ous via the QCP, which hosts the low-energy quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' In the NMI state, the lattice softening orig- inating from the geometrical frustrated lattice gives the larger heat capacity in total, although the opening of the charge gap reduces the electronic heat capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Based on the entropy, the balance of these degrees of freedom makes the Pomeranchuk-like unique electronic phase di- agram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' The coupling of these degrees of freedom makes the low-temperature phase competition between the NMI and FL states, leading to the low-energy quantum criti- cal behavior that may be related to the instability of the fractionalization of the electron into the spin and charge sectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' This work was partially supported by JSPS KAKENHI Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content='19K22169 and 20H01862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 8 [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Kino and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Fukuyama, Phase Diagram of Two- Dimensional Organic Conductors: (BEDT-TTF)2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' 67, 2158-2169 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/X9E3T4oBgHgl3EQfGAla/content/2301.04310v1.pdf'} +page_content=' [2] K.' metadata={'source': 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https://git-lfs.github.com/spec/v1 +oid sha256:a57a17b9963fd6a9ca7da7837f5169a7ae644b12b57a3ee67c01140c2b147cd1 +size 87802 diff --git a/XdE3T4oBgHgl3EQfFwl_/content/tmp_files/2301.04308v1.pdf.txt b/XdE3T4oBgHgl3EQfFwl_/content/tmp_files/2301.04308v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3b4913667dbd208ceec40e5beec77e1e1be028a2 --- /dev/null +++ b/XdE3T4oBgHgl3EQfFwl_/content/tmp_files/2301.04308v1.pdf.txt @@ -0,0 +1,955 @@ +arXiv:2301.04308v1 [hep-ph] 11 Jan 2023 +A self-consistent thermodynamic potential for a magnetized QCD matter +Gaoqing Cao +School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519088, China +Jianing Li +Physics Department, Tsinghua University, Beijing 100084, China +(Dated: January 12, 2023) +Within the two-flavor Nambu–Jona-Lasinio model, we derive a self-consistent thermodynamic +potential Ω for a QCD matter in an external magnetic field B. To be consistent with Schwinger’s +renormalization spirit, counter terms with vacuum quark mass are introduced into Ω and then the +explicit B-dependent parts can be regularized in a cutoff-free way. Following that, explicit expres- +sions of gap equation and magnetization can be consistently obtained according to the standard +thermodynamic relations. The formalism is able to reproduce the paramagnetic feature of a QCD +matter without ambiguity. For more realistic study, a running coupling constant is also adopted +to account for the inverse magnetic catalysis effect. It turns out that the running coupling would +greatly suppress magnetization at large B and is important to reproduce the temperature enhance- +ment effect to magnetization. The case with finite baryon chemical potential is also explored: no +sign of first-order transition is found by varying B for the running coupling and the de Haas-van +Alphen oscillation shows up in the small B region. +PACS numbers: 11.30.Qc, 05.30.Fk, 11.30.Hv, 12.20.Ds +I. +INTRODUCTION +Extremely strong magnetic field could be produced in +peripheral relativistic heavy ion collisions (HICs) [1, 2] +and is also expected to exist in magnetars [3–5] and the +early Universe [6–8]. +For that considerations, a lot of +work has been carried out to understand the systematic +features of quantum chromodynamics (QCD) matter un- +der external magnetic field. One important aspect is to +study QCD phase transition in strong magnetic field: as +the magnitude of magnetic field is of the order of the +QCD energy scale ΛQCD ∼ 0.2 GeV, the effect is expected +to be considerable. In the end of 20th century, experts +took the magnetic field into account in the chiral effec- +tive Nambu–Jona-Lasinio model and established the ba- +sic notion of ”magnetic catalysis effect” to chiral conden- +sate [9–11]. However, in 2012, the first-principle lattice +QCD (LQCD) simulations [12, 13] showed that the chi- +ral condensate could decrease with large magnetic field at +the pseudo-critical temperature T ∼ 0.155 GeV, known +as ”inverse magnetic catalysis effect”. Such anomalous +feature had drawn most attentions of researchers inter- +ested in the thermodynamic properties of QCD matter +and the QCD phase has been widely explored in the cir- +cumstances where magnetic field is involved, refer to the +reviews Ref. [14–16] and the literatures therein. +Besides, magnetization is also an important thermo- +dynamic quantity to understand QCD matter. In 2013, +both the hadron resonance gas model [17] and 2 + 1 +LQCD [18] had been adopted to study the magnetiza- +tion and the results turned out that the QCD matter is +consistently paramagnetic at zero temperature. The 2+1 +LQCD simulations had been extended to finite temper- +ature the next year and the magnetization was found to +be enhanced by thermal motions [19]. In the following +years, only few works concerned the magnetization fea- +ture in chiral models such as the two-flavor chiral pertur- +bation theory [20, 21], three-flavor Polyakov-linear-sigma +(PLS) model [22], and two- and three-flavor (Polyakov- +)NJL model [23, 24]. The studies in PLS and (P)NJL +models seem more realistic as chiral symmetry breaking +and restoration was self-consistently taken into account +for the evaluation of magnetization. However, compared +to previous thermodynamic potential [25], it is unsatis- +fied that one had to introduce cutoff for the explicitly +magnetic field dependent terms to evaluate magnetiza- +tion in the PNJL model [23]. Furthermore, the definition +of magnetization seemed ambiguous as one must apply +the renormalization scheme of LQCD simulations [18] to +get the correct paramagnetic feature [23]. +This work is devoted to solving the regularization prob- +lem of (P)NJL model in a self-consistent way. In Sec.II, +we will derive a self-consistent thermodynamic potential +for finite magnetic field, temperature, and baryon chemi- +cal potential. From that, expressions of gap equation and +magnetization can be given explicitly. Then, numerical +calculations will be carried out in Sec.III, where we com- +pare the results with different regularization schemes or +different forms of coupling constants. Finally, we sum- +marize in Sec.IV. +II. +THE SELF-CONSISTENT FORMALISM +The Lagrangian density of the two-flavor NJL model +with baryon chemical potential µB can be given as [9, 26] +L = ¯ψ +� +i/D−iγ4 µB +3 −m0 +� +ψ+G(eB) +�� ¯ψψ +�2+ +� ¯ψiγ5τψ +�2� +(1) + +2 +in Euclidean space, where ψ = (u, d)T represents the two- +flavor quark field, m0 is its current mass, and τ are Pauli +matrices in flavor space. In minimal coupling scheme, the +covariant derivative is defined as Dµ ≡ ∂µ−iqAµ with the +electric charge matrix q ≡ diag(qu, qd) = diag( 2 +3, − 1 +3)e +and the magnetic effect introduced through the vector +potential Aµ. For more general consideration, we have in- +troduced a coupling constant G(eB) that could run with +the magnetic field B here. +To obtain the analytic form of the basic thermody- +namic potential, we take Hubbard-Stratonovich transfor- +mation with the help of the auxiliary fields σ = −2G ¯ψψ +and π = −2G ¯ψiγ5τψ [9] and the Lagrangian becomes +L = ¯ψ +� +i/D−iγ4 µB +3 −iγ5τ · π−σ−m0 +� +ψ − σ2 + π2 +4G(eB) .(2) +We assume ⟨σ⟩ ≡ m − m0 ̸= 0 and ⟨π⟩ = 0 in mean +field approximation, and then the quark degrees of free- +dom can be integrated out to give the thermodynamic +potential formally as +Ω = (m − m0)2 +4G(eB) +− T +V Tr ln +� +i/D − m − iγ4 µB +3 +� +with the trace Tr over the coordinate, spinor, fla- +vor +and +color spaces. +Recalling +that +the +quark +propagator in a magnetic field takes the form S += +− +� +i/D − m − iγ4 µB +3 +�−1, Ω can be alternatively presented +as +Ω = (m − m0)2 +4G(eB) +− T +V +� +d m Tr S. +(3) +At zero temperature and chemical potential, the full +fermion propagator in a magnetic field had been well eval- +uated with the help of proper time by Schwinger in 1951. +In coordinate space, it takes the from [27]: +Sf(x, x′) = −i qfB +(4π)2 +� ∞ +0 +ds +s e−iqf +� x +x′ A·dx exp +� +− im2s + i +4 +� +qfB +tan(qfBs)(y2 +1 + y2 +2) + 1 +s(y2 +3 + y2 +4) +� � +� +m− qfB +2 +�� +cot(qfBs)γ1+γ2� +y1+ +� +cot(qfBs)γ2 − γ1� +y2 +� +− 1 +2s +� +γ3y3 + γ4y4 +�� � +cot(qfBs) + γ1γ2� +(4) +with yµ = xµ −x′ +µ and s the proper time. For the calculation of Ω, the Schwinger phase term e−iqf +� x +x′ A·dx is irrelevant +since we would take the limit x → x′. After dropping this term, the left effective propagator becomes translation +invariant and can be conveniently presented in energy-momentum space as +ˆSf(p) = i +� ∞ +0 +ds exp +� +− i(m2 + p2 +4 + p2 +3)s − itan(qfBs) +qfB +(p2 +1 + p2 +2) +� � +m − γ4p4−γ3p3−γ2(p2 + tan(qfBs)p1) +−γ1(p1 − tan(qfBs)p2) +� � +1 + γ1γ2 tan(qfBs) +� +. +(5) +In vanishing B limit, the well-known fermion propagator S(p) = +1 +m−/p can be reproduced by completing the integration +over s, hence the effective propagator is helpful for the discussion of regularization. Then, the bare thermodynamic +potential follows directly as +Ω0 =(m − m0)2 +4G(eB) ++ Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 e−m2s +qfBs +tanh(qfBs) +(6) +after substituting the propagator Eq.(4) into Eq.(3). +The last term of Eq.(6) is divergent and must be regularized for exploring physics. If we formally expand it as a +serial sum of B2k (k ∈ N) around B ∼ 0, we would find that only the B0 and B2 terms are divergent. According to +Schwinger’s initial proposal [27], the B0 term is physics irrelevant and the B2 terms can be absorbed by performing +renormalizations of electric charges and magnetic field. Then, the finite form of Eq.(6) would be +Ω0 = (m − m0)2 +4G(eB) ++ Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 e−m2s +� +qfBs +tanh(qfBs) − 1 − 1 +3(qfBs)2 +� +. +This is correct when the magnetic field is much smaller than the current mass square m2 in QED systems. But for +QCD systems, the dynamical mass m is itself determined by the minimum of the thermodynamic potential, the B0 +term can not be dropped at all [25]. Moreover, the dynamical mass m is also B-dependent due to magnetic catalysis +effect [11], the term e−m2s 1 +3(qfBs)2 actually contains o(B4) terms which can not be absorbed by the renormalizations +of electric charges and magnetic field. + +3 +The solutions could be the following. Firstly, the B0 term can be recovered with three momentum cutoff according +to the discussions in Ref. [25], then we have +Ω0 = (m − m0)2 +4G(eB) ++ Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 e−m2s +� +qfBs +tanh(qfBs) − 1 +� +− 4Nc +� Λ d3p +(2π)3 Ep(m) +(7) +with Ep(m) = (p2 + m2)1/2. Next, to absorb the B2 divergent term but not o(B4) terms, we could refer to the term +with vacuum quark mass mv for help. Then, a thermodynamic potential consistent with Schwinger’s renormalization +spirit can be given as +Ω0 = (m − m0)2 +4G(eB) +− 4Nc +� Λ d3p +(2π)3 Ep(m) + Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 +� +e−m2s − e−m2 +vs� � +qfBs +tanh(qfBs) − 1 +� ++ Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 e−m2 +vs +� +qfBs +tanh(qfBs) − 1 − 1 +3(qfBs)2 +� +. +(8) +Note that the subtracted term with integrand e−m2 +vs 1 +3(qfBs)2 only contains B2 term as mv is a constant. +Eventually, to make sure the pressure to be consistent with the one given in Ref. [27] when m = mv for any B, +m-independent terms can be subtracted to get the physical thermodynamic potential as +Ω0 = (m − m0)2 − (mv − m0)2 +4G(eB) +− 4Nc +� Λ d3p +(2π)3 [Ep(m) − Ep(mv)] + Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 +� +e−m2s − e−m2 +vs� +× +� +qfBs +tanh(qfBs) − 1 +� ++ Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 e−m2 +vs +� +qfBs +tanh(qfBs) − 1 − 1 +3(qfBs)2 +� +. +(9) +This form of Ω0 would be adopted for analytic derivations in the following and numerical calculations in next section. +Finite temperature and chemical potential usually do not induce extra divergence and the corresponding terms of +thermodynamic potential can be easily evaluated with the help of Landau levels as +ΩT µ = −2NcT +t=± +� +f=u,d +|qfB| +2π +∞ +� +n=0 +αn +� ∞ +−∞ +dp3 +2π ln +� +1 + e− 1 +T (En +f (p3,m)+t µB +3 )� +, +(10) +where αn = 1 − δn0/2 and En +f (p3, m) = (2n|qfB|+ p2 +3 + m2)1/2. So the total thermodynamic potential of a magnetized +QCD matter is Ω = Ω0 + ΩT µ, and the expressions of gap equation and magnetization follow the thermodynamic +relations ∂Ω/∂m = 0 and M = −∂Ω/∂eB as +0 = m − m0 +2G(eB) − 4Nc +� Λ d3p +(2π)3 +m +Ep(m) − Ncm +4π2 +� +f=u,d +� ∞ +0 +ds +s2 e−m2s +� +qfBs +tanh(qfBs) − 1 +� ++ 2Nc +t=± +� +f=u,d +|qfB| +2π +∞ +� +n=0 +αn +� ∞ +−∞ +dp3 +2π +m +En +f (p3, m) +1 +1 + e +1 +T [En +f (p3,m)+t µB +3 ] , +(11) +M = (m − m0)2 − (mv − m0)2 +4 +G′(eB) +G2(eB) − Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 +� +e−m2s − e−m2 +vs� � +˜qfs +tanh(qfBs) − +˜qfqfBs2 +sinh2(qfBs) +� +− +Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 e−m2 +vs +� +˜qfs +tanh(qfBs) − +˜qfqfBs2 +sinh2(qfBs) − 2 +3 ˜qfqfBs2 +� ++ 2NcT +t=± +� +f=u,d +|˜qf| +2π +∞ +� +n=0 +αn +� ∞ +−∞ +dp3 +2π +ln +� +1 + e− 1 +T (En +f (p3,m)+t µB +3 )� +− 2Nc +t=± +� +f=u,d +|qfB| +2π +∞ +� +n=0 +αn +� ∞ +−∞ +dp3 +2π +n|˜qf| +En +f (p3, m) +1 +1 + e +1 +T [En +f (p3,m)+t µB +3 ] +(12) +with ˜qf = qf/e. +For comparison, the gap equation and magnetization in the so-called vacuum magnetic regularization (VMR) [23] + +4 +are +0 = m − m0 +2G(0) − 4Nc +� Λ d3p +(2π)3 +m +Ep(m) − Ncm +4π2 +� +f=u,d +� ∞ +0 +ds +s2 e−m2s +� +qfBs +tanh(qfBs) − 1 − 1 +3(qfBs)2 +� +−Ncm +12π2 +� +f=u,d +� ∞ +1 +Λ2 +ds +s2 e−m2s(qfBs)2, +(13) +M0 = − Nc +8π2 +� +f=u,d +� ∞ +0 +ds +s3 e−m2s +� +˜qfs +tanh(qfBs) − +˜qfqfBs2 +sinh2(qfBs) − 2 +3 ˜qfqfBs2 +� +− Nc +12π2 +� +f=u,d +� ∞ +1 +Λ2 +ds +s +� +e−m2s − e−m2 +vs� +˜qfqfB +(14) +at zero temperature for a constant coupling G(0). But instead of proper-time regularization [23], we regularize the +explicitly B-independent term with three momentum cutoff for better comparison here. Note that the mv-dependent +term in Eq.(14) is important to reproduce the paramagnetic feature of QCD matter though they did not manage to +give the explicit form [23]. +III. +NUMERICAL RESULTS +To carry out numerical calculations, the model param- +eters are fixed as m0 = 5 MeV, Λ = 653 MeV, G(0)Λ2 = +2.10 by fitting to the vacuum values: chiral condensate +⟨ ¯ψψ⟩ = −2 × (250 MeV)2, pion mass mπ = 135 MeV, +and pion decay constant fπ = 93 MeV [28, 29]. For fi- +nite magnetic field, the explicit form of G(eB) should be +given. In Ref. [16], a form of G(eB) had been determined +by fitting to the data of π0 mass from LQCD simulations, +and we were able to explain inverse magnetic catalysis +effect for larger B with that form. However, there was +nonphysical increasing of G(eB) around eB ∼ 0; to avoid +that, we choose to fit to the region eB ≥ 0.6 GeV2 here +and get a monotonic form G(eB) = +G(0) +1+0.524 eB2 . Hence, +G′(eB) +G2(eB) = − 1.048 eB +G(0) . +For a constant coupling G(0), we compare the results +of our self-consistent regularization scheme with those +of VMR scheme in Fig. 1 at zero temperature. +Both +results are consistent with the LQCD data [18] for the +region 0 ≤ eB ≤ 0.6 GeV2, but they diverge quite much +for larger B. In our opinion, the cutoff to the explicitly +B-dependent term in VMR would introduce artifact at +larger B – the non-monotonic feature of m is a reflection +of that. +In the following, we would explore how a running cou- +pling constant could affect the dynamical mass and the +corresponding magnetization in the self-consistent regu- +larization. At zero temperature, the results with G(0) +and G(eB) are shown together in Fig. 2. +Due to the +running of coupling constant, m shows a non-monotonic +feature though the absolute value of chiral condensate +m/2G(eB) increases with B almost linearly [16]. +Ac- +cordingly, the second term in Eq.(12) demonstrates a +non-monotonic feature and becomes negative at larger +B. Such feature is responsible for the strong suppression +of magnetization with the running coupling at larger B +compared to the constant coupling case. +At finite temperature, the results are illustrated in +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +m +(GeV) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +e� (GeV2) +ℳ (GeV2) +FIG. 1. The dynamical mass m (upper panel) and magne- +tization M (lower panel) with the self-consistent regulariza- +tion (blue dashed lines) and vacuum magnetic regularization +(black dotted lines) schemes at zero temperature. +Fig. 3. +As we can see, the temperature tends to sup- +press magnetization in the case with G(0) but enhance +magnetization in the case with G(eB). +In their book, +Landau and Lifshitz had calculated magnetic suscepti- +bility χ ≡ eM +NB of a non-relativistic dilute electronic gas +at high temperature and found it decreases as 1/T [? +]. +To be concrete, the situations they considered are +√ +B ≪ T ≪ me and the electric chemical potential +−µe(≳ me) changes with T to keep the total number +N constant. If we keep −µe(≳ me) a constant, then the + +5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +m (GeV) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +eB (GeV2) +ℳ (GeV2) +FIG. 2. The dynamical mass m (upper panel) and magneti- +zation M (lower panel) with the constant coupling G(0) (blue +dashed lines) and the running coupling G(eB) (red lines) at +zero temperature. The dotted lines correspond to the corre- +sponding contributions of the second term in Eq.(12). +total electronic number N could be easily evaluated to +increase with temperature as T 3/2. Therefore, the mag- +netization M = χNB/e would increase with tempera- +ture as +√ +T, and the result with G(eB) is qualitatively +consistent with the non-relativistic study. +That is not +the end of story: when we keep m = mv for G(0), M +would increase with T for a given B; so it is adequate +chiral symmetry restoration induced by T that reduces +the contribution of second term in Eq.(12) and thus re- +verses the trend. One can refer to Fig.2 for the dynami- +cal mass effect on magnetization. For G(eB), m changes +mildly with B for a given T , that is, the large mass gaps +induced by T at vanishing B sustain to strong magnetic +field. According to our analysis, it is the great enhance- +ment of the forth T -dependent term in Eq.(12) that helps +to recover the trend of naiive expectation. In fact, the +result with G(eB) is qualitatively consistent with that +found in LQCD simulations at finite temperature [19], so +we conclude that the running coupling is able to consis- +tently explain both inverse magnetic catalysis effect and +magnetization enhancement with temperature. +At finite baryon chemical potential, the results are il- +lustrated in Fig. 4. For G(0), m always changes discon- +tinuously with B for µB > mv, which signals a first-order +transition. But for G(eB), m only changes slightly at +µB = 0 and no sign of first-order transition could be +identified for a given µB. The de Haas-van Alphen oscil- +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +m (GeV) +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.02 +0.04 +0.06 +eB (GeV2) +ℳ (GeV2) +T=0 +0.15 GeV +0.2 GeV +FIG. 3. The dynamical mass m (upper panel) and magneti- +zation M (lower panel) as functions of the magnetic field B +at temperature T = 0, 0.15, and 0.2 GeV. The dashed, dot- +ted, and dashdotted lines correspond to the results with the +constant coupling G(0) and the solid lines correspond to the +results with the running coupling G(eB). +lation [30] can be found both in the evolutions of m and +M with B: the effect is significant to m only when µB is +a little larger than 3mv but is significant to M for any +µB > 3mv. According to the mechanism of de Haas-van +Alphen oscillation [30], the last non-analytic points of M +can be roughly determined by +� +2qd|B| ≈ µB/3, that is, +eB ≈ 0.167 GeV2 for µB = 1 GeV and eB ≈ 0.375 GeV2 +for µB = 1.5 GeV. That is consistent with the numerical +results shown in the lower panel of Fig. 4. Moreover, at +larger B, M does not depend on µB for G(0) due to the +”Silver braze” property but increases with µB for G(eB) +due to the strong suppression of m. +IV. +SUMMARY +In this work, a self-consistent thermodynamic poten- +tial has been obtained for a magnetized QCD matter in +two-flavor NJL model by following Schwinger’s renormal- +ization spirit. +The thermodynamic potential is free of +cutoff for the explicitly magnetic field dependent terms +and explicit expressions of gap equation and magneti- +zation could be derived from that according to thermo- +dynamic relations. Compared to the VMR scheme, the +numerical calculations showed that magnetic catalysis ef- +fect persists to very large magnetic field at zero temper- + +6 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +m (GeV) +0.0 +0.5 +1.0 +1.5 +2.0 +0.00 +0.02 +0.04 +0.06 +eB (GeV2) +ℳ (GeV2) +μB=0 +1 GeV +1.5 GeV +FIG. 4. The dynamical mass m (upper panel) and magne- +tization M (lower panel) as functions of the magnetic field +B at baryon chemical potential µB = 0, 1, and 1.5 GeV. The +dashed, dotted, and dashdotted lines correspond to the results +with the constant coupling G(0) and the solid lines correspond +to the results with the running coupling G(eB). +ature when adopting the self-consistent scheme, and the +magnetization is strongly affected accordingly. +Keeping in the self-consistent scheme, results with the +constant coupling G(0) and running coupling G(eB) are +compared with each other. +At zero temperature and +chemical potential, the running coupling greatly sup- +presses the dynamical mass m at large magnetic field +B and thus reduces the magnetization M a lot. At finite +temperature T , M decreases with T for G(0) due to ad- +equate suppression of m but increases with T for G(eB) +due to the persistance of large mass gaps at large B. At +finite baryon chemical potential µB, no sign of first-order +transition could be identified for G(eB) by varying B +and de Haas-van Alphen oscillation could be found both +in the evolutions of m and M with B. +Since we found that the regularization scheme could af- +fect the result greatly in the large magnetic field region, +we would try to perform similar study in three-flavor NJL +or PNJL model. Then, we could compare the magneti- +zation with the LQCD data for finite temperature in the +region 0 ≤ eB ≤ 1 GeV2 [19] and give further predictions +for much larger magnetic field. The situation with finite +baryon chemical potential could also be explored for com- +pleteness, which might help to understand the properties +of magnetars. +Acknowledgments G.C. is supported by the National +Natural Science Foundation of China with Grant No. +11805290. J. Li is supported by the National Natural +Science Foundation of China with Grant No. 11890712. +[1] V. Skokov, A. Y. Illarionov and V. 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Vshivtsev, “Magnetic oscillations in dense cold +quark matter with four fermion interactions,” Phys. Rev. +D 61, 025005 (2000). +[26] T. Hatsuda and T. Kunihiro, “QCD phenomenology +based on a chiral effective Lagrangian,” Phys. Rept. 247, +221 (1994). +[27] J. S. Schwinger, “On gauge invariance and vacuum po- +larization,” Phys. Rev. 82, 664-679 (1951). +[28] P. Zhuang, J. Hufner and S. P. Klevansky, “Thermody- +namics of a quark - meson plasma in the Nambu-Jona- +Lasinio model,” Nucl. Phys. A 576, 525 (1994). +[29] P. Rehberg, S. P. Klevansky and J. Hufner, “Hadroniza- +tion in the SU(3) Nambu-Jona-Lasinio model,” Phys. +Rev. C 53, 410 (1996). +[30] L.D. Landau and E.M. Lifshitz, Statistical physics. Pt.1 +(Pergamon Press, Oxford, 1999). + diff --git a/XdE3T4oBgHgl3EQfFwl_/content/tmp_files/load_file.txt b/XdE3T4oBgHgl3EQfFwl_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..57760d5e5ee0abc8a6bef05917bda404b8517d4a --- /dev/null +++ b/XdE3T4oBgHgl3EQfFwl_/content/tmp_files/load_file.txt @@ -0,0 +1,506 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf,len=505 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='04308v1 [hep-ph] 11 Jan 2023 A self-consistent thermodynamic potential for a magnetized QCD matter Gaoqing Cao School of Physics and Astronomy, Sun Yat-sen University, Zhuhai 519088, China Jianing Li Physics Department, Tsinghua University, Beijing 100084, China (Dated: January 12, 2023) Within the two-flavor Nambu–Jona-Lasinio model, we derive a self-consistent thermodynamic potential Ω for a QCD matter in an external magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' To be consistent with Schwinger’s renormalization spirit, counter terms with vacuum quark mass are introduced into Ω and then the explicit B-dependent parts can be regularized in a cutoff-free way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Following that, explicit expres- sions of gap equation and magnetization can be consistently obtained according to the standard thermodynamic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The formalism is able to reproduce the paramagnetic feature of a QCD matter without ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For more realistic study, a running coupling constant is also adopted to account for the inverse magnetic catalysis effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' It turns out that the running coupling would greatly suppress magnetization at large B and is important to reproduce the temperature enhance- ment effect to magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The case with finite baryon chemical potential is also explored: no sign of first-order transition is found by varying B for the running coupling and the de Haas-van Alphen oscillation shows up in the small B region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' PACS numbers: 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='Qc, 05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='Fk, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='Hv, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='Ds I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' INTRODUCTION Extremely strong magnetic field could be produced in peripheral relativistic heavy ion collisions (HICs) [1, 2] and is also expected to exist in magnetars [3–5] and the early Universe [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For that considerations, a lot of work has been carried out to understand the systematic features of quantum chromodynamics (QCD) matter un- der external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' One important aspect is to study QCD phase transition in strong magnetic field: as the magnitude of magnetic field is of the order of the QCD energy scale ΛQCD ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 GeV, the effect is expected to be considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In the end of 20th century, experts took the magnetic field into account in the chiral effec- tive Nambu–Jona-Lasinio model and established the ba- sic notion of ”magnetic catalysis effect” to chiral conden- sate [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' However, in 2012, the first-principle lattice QCD (LQCD) simulations [12, 13] showed that the chi- ral condensate could decrease with large magnetic field at the pseudo-critical temperature T ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='155 GeV, known as ”inverse magnetic catalysis effect”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Such anomalous feature had drawn most attentions of researchers inter- ested in the thermodynamic properties of QCD matter and the QCD phase has been widely explored in the cir- cumstances where magnetic field is involved, refer to the reviews Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' [14–16] and the literatures therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Besides, magnetization is also an important thermo- dynamic quantity to understand QCD matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In 2013, both the hadron resonance gas model [17] and 2 + 1 LQCD [18] had been adopted to study the magnetiza- tion and the results turned out that the QCD matter is consistently paramagnetic at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The 2+1 LQCD simulations had been extended to finite temper- ature the next year and the magnetization was found to be enhanced by thermal motions [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In the following years, only few works concerned the magnetization fea- ture in chiral models such as the two-flavor chiral pertur- bation theory [20, 21], three-flavor Polyakov-linear-sigma (PLS) model [22], and two- and three-flavor (Polyakov- )NJL model [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The studies in PLS and (P)NJL models seem more realistic as chiral symmetry breaking and restoration was self-consistently taken into account for the evaluation of magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' However, compared to previous thermodynamic potential [25], it is unsatis- fied that one had to introduce cutoff for the explicitly magnetic field dependent terms to evaluate magnetiza- tion in the PNJL model [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Furthermore, the definition of magnetization seemed ambiguous as one must apply the renormalization scheme of LQCD simulations [18] to get the correct paramagnetic feature [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' This work is devoted to solving the regularization prob- lem of (P)NJL model in a self-consistent way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='II, we will derive a self-consistent thermodynamic potential for finite magnetic field, temperature, and baryon chemi- cal potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' From that, expressions of gap equation and magnetization can be given explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Then, numerical calculations will be carried out in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='III, where we com- pare the results with different regularization schemes or different forms of coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Finally, we sum- marize in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' THE SELF-CONSISTENT FORMALISM The Lagrangian density of the two-flavor NJL model with baryon chemical potential µB can be given as [9, 26] L = ¯ψ � i/D−iγ4 µB 3 −m0 � ψ+G(eB) �� ¯ψψ �2+ � ¯ψiγ5τψ �2� (1) 2 in Euclidean space, where ψ = (u, d)T represents the two- flavor quark field, m0 is its current mass, and τ are Pauli matrices in flavor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In minimal coupling scheme, the covariant derivative is defined as Dµ ≡ ∂µ−iqAµ with the electric charge matrix q ≡ diag(qu, qd) = diag( 2 3, − 1 3)e and the magnetic effect introduced through the vector potential Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For more general consideration, we have in- troduced a coupling constant G(eB) that could run with the magnetic field B here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' To obtain the analytic form of the basic thermody- namic potential, we take Hubbard-Stratonovich transfor- mation with the help of the auxiliary fields σ = −2G ¯ψψ and π = −2G ¯ψiγ5τψ [9] and the Lagrangian becomes L = ¯ψ � i/D−iγ4 µB 3 −iγ5τ · π−σ−m0 � ψ − σ2 + π2 4G(eB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (2) We assume ⟨σ⟩ ≡ m − m0 ̸= 0 and ⟨π⟩ = 0 in mean field approximation, and then the quark degrees of free- dom can be integrated out to give the thermodynamic potential formally as Ω = (m − m0)2 4G(eB) − T V Tr ln � i/D − m − iγ4 µB 3 � with the trace Tr over the coordinate, spinor, fla- vor and color spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Recalling that the quark propagator in a magnetic field takes the form S = − � i/D − m − iγ4 µB 3 �−1, Ω can be alternatively presented as Ω = (m − m0)2 4G(eB) − T V � d m Tr S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (3) At zero temperature and chemical potential, the full fermion propagator in a magnetic field had been well eval- uated with the help of proper time by Schwinger in 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In coordinate space, it takes the from [27]: Sf(x, x′) = −i qfB (4π)2 � ∞ 0 ds s e−iqf � x x′ A·dx exp � − im2s + i 4 � qfB tan(qfBs)(y2 1 + y2 2) + 1 s(y2 3 + y2 4) � � � m− qfB 2 �� cot(qfBs)γ1+γ2� y1+ � cot(qfBs)γ2 − γ1� y2 � − 1 2s � γ3y3 + γ4y4 �� � cot(qfBs) + γ1γ2� (4) with yµ = xµ −x′ µ and s the proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For the calculation of Ω, the Schwinger phase term e−iqf � x x′ A·dx is irrelevant since we would take the limit x → x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' After dropping this term, the left effective propagator becomes translation invariant and can be conveniently presented in energy-momentum space as ˆSf(p) = i � ∞ 0 ds exp � − i(m2 + p2 4 + p2 3)s − itan(qfBs) qfB (p2 1 + p2 2) � � m − γ4p4−γ3p3−γ2(p2 + tan(qfBs)p1) −γ1(p1 − tan(qfBs)p2) � � 1 + γ1γ2 tan(qfBs) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (5) In vanishing B limit, the well-known fermion propagator S(p) = 1 m−/p can be reproduced by completing the integration over s, hence the effective propagator is helpful for the discussion of regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Then, the bare thermodynamic potential follows directly as Ω0 =(m − m0)2 4G(eB) + Nc 8π2 � f=u,d � ∞ 0 ds s3 e−m2s qfBs tanh(qfBs) (6) after substituting the propagator Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (4) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The last term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (6) is divergent and must be regularized for exploring physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' If we formally expand it as a serial sum of B2k (k ∈ N) around B ∼ 0, we would find that only the B0 and B2 terms are divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' According to Schwinger’s initial proposal [27], the B0 term is physics irrelevant and the B2 terms can be absorbed by performing renormalizations of electric charges and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Then, the finite form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (6) would be Ω0 = (m − m0)2 4G(eB) + Nc 8π2 � f=u,d � ∞ 0 ds s3 e−m2s � qfBs tanh(qfBs) − 1 − 1 3(qfBs)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' This is correct when the magnetic field is much smaller than the current mass square m2 in QED systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' But for QCD systems, the dynamical mass m is itself determined by the minimum of the thermodynamic potential, the B0 term can not be dropped at all [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Moreover, the dynamical mass m is also B-dependent due to magnetic catalysis effect [11], the term e−m2s 1 3(qfBs)2 actually contains o(B4) terms which can not be absorbed by the renormalizations of electric charges and magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 3 The solutions could be the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Firstly, the B0 term can be recovered with three momentum cutoff according to the discussions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' [25], then we have Ω0 = (m − m0)2 4G(eB) + Nc 8π2 � f=u,d � ∞ 0 ds s3 e−m2s � qfBs tanh(qfBs) − 1 � − 4Nc � Λ d3p (2π)3 Ep(m) (7) with Ep(m) = (p2 + m2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Next, to absorb the B2 divergent term but not o(B4) terms, we could refer to the term with vacuum quark mass mv for help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Then, a thermodynamic potential consistent with Schwinger’s renormalization spirit can be given as Ω0 = (m − m0)2 4G(eB) − 4Nc � Λ d3p (2π)3 Ep(m) + Nc 8π2 � f=u,d � ∞ 0 ds s3 � e−m2s − e−m2 vs� � qfBs tanh(qfBs) − 1 � + Nc 8π2 � f=u,d � ∞ 0 ds s3 e−m2 vs � qfBs tanh(qfBs) − 1 − 1 3(qfBs)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (8) Note that the subtracted term with integrand e−m2 vs 1 3(qfBs)2 only contains B2 term as mv is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Eventually, to make sure the pressure to be consistent with the one given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' [27] when m = mv for any B, m-independent terms can be subtracted to get the physical thermodynamic potential as Ω0 = (m − m0)2 − (mv − m0)2 4G(eB) − 4Nc � Λ d3p (2π)3 [Ep(m) − Ep(mv)] + Nc 8π2 � f=u,d � ∞ 0 ds s3 � e−m2s − e−m2 vs� × � qfBs tanh(qfBs) − 1 � + Nc 8π2 � f=u,d � ∞ 0 ds s3 e−m2 vs � qfBs tanh(qfBs) − 1 − 1 3(qfBs)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (9) This form of Ω0 would be adopted for analytic derivations in the following and numerical calculations in next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Finite temperature and chemical potential usually do not induce extra divergence and the corresponding terms of thermodynamic potential can be easily evaluated with the help of Landau levels as ΩT µ = −2NcT t=± � f=u,d |qfB| 2π ∞ � n=0 αn � ∞ −∞ dp3 2π ln � 1 + e− 1 T (En f (p3,m)+t µB 3 )� , (10) where αn = 1 − δn0/2 and En f (p3, m) = (2n|qfB|+ p2 3 + m2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' So the total thermodynamic potential of a magnetized QCD matter is Ω = Ω0 + ΩT µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' and the expressions of gap equation and magnetization follow the thermodynamic relations ∂Ω/∂m = 0 and M = −∂Ω/∂eB as 0 = m − m0 2G(eB) − 4Nc � Λ d3p (2π)3 m Ep(m) − Ncm 4π2 � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d � ∞ 0 ds s2 e−m2s � qfBs tanh(qfBs) − 1 � + 2Nc t=± � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d |qfB| 2π ∞ � n=0 αn � ∞ −∞ dp3 2π m En f (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' m) 1 1 + e 1 T [En f (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='m)+t µB 3 ] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (11) M = (m − m0)2 − (mv − m0)2 4 G′(eB) G2(eB) − Nc 8π2 � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d � ∞ 0 ds s3 � e−m2s − e−m2 vs� � ˜qfs tanh(qfBs) − ˜qfqfBs2 sinh2(qfBs) � − Nc 8π2 � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d � ∞ 0 ds s3 e−m2 vs � ˜qfs tanh(qfBs) − ˜qfqfBs2 sinh2(qfBs) − 2 3 ˜qfqfBs2 � + 2NcT t=± � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d |˜qf| 2π ∞ � n=0 αn � ∞ −∞ dp3 2π ln � 1 + e− 1 T (En f (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='m)+t µB 3 )� − 2Nc t=± � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d |qfB| 2π ∞ � n=0 αn � ∞ −∞ dp3 2π n|˜qf| En f (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' m) 1 1 + e 1 T [En f (p3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='m)+t µB 3 ] (12) with ˜qf = qf/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For comparison,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' the gap equation and magnetization in the so-called vacuum magnetic regularization (VMR) [23] 4 are 0 = m − m0 2G(0) − 4Nc � Λ d3p (2π)3 m Ep(m) − Ncm 4π2 � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d � ∞ 0 ds s2 e−m2s � qfBs tanh(qfBs) − 1 − 1 3(qfBs)2 � −Ncm 12π2 � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d � ∞ 1 Λ2 ds s2 e−m2s(qfBs)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (13) M0 = − Nc 8π2 � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d � ∞ 0 ds s3 e−m2s � ˜qfs tanh(qfBs) − ˜qfqfBs2 sinh2(qfBs) − 2 3 ˜qfqfBs2 � − Nc 12π2 � f=u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='d � ∞ 1 Λ2 ds s � e−m2s − e−m2 vs� ˜qfqfB (14) at zero temperature for a constant coupling G(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' But instead of proper-time regularization [23], we regularize the explicitly B-independent term with three momentum cutoff for better comparison here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Note that the mv-dependent term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (14) is important to reproduce the paramagnetic feature of QCD matter though they did not manage to give the explicit form [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' NUMERICAL RESULTS To carry out numerical calculations, the model param- eters are fixed as m0 = 5 MeV, Λ = 653 MeV, G(0)Λ2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='10 by fitting to the vacuum values: chiral condensate ⟨ ¯ψψ⟩ = −2 × (250 MeV)2, pion mass mπ = 135 MeV, and pion decay constant fπ = 93 MeV [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For fi- nite magnetic field, the explicit form of G(eB) should be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' [16], a form of G(eB) had been determined by fitting to the data of π0 mass from LQCD simulations, and we were able to explain inverse magnetic catalysis effect for larger B with that form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' However, there was nonphysical increasing of G(eB) around eB ∼ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' to avoid that, we choose to fit to the region eB ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='6 GeV2 here and get a monotonic form G(eB) = G(0) 1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='524 eB2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Hence, G′(eB) G2(eB) = − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='048 eB G(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For a constant coupling G(0), we compare the results of our self-consistent regularization scheme with those of VMR scheme in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 1 at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Both results are consistent with the LQCD data [18] for the region 0 ≤ eB ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='6 GeV2, but they diverge quite much for larger B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In our opinion, the cutoff to the explicitly B-dependent term in VMR would introduce artifact at larger B – the non-monotonic feature of m is a reflection of that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In the following, we would explore how a running cou- pling constant could affect the dynamical mass and the corresponding magnetization in the self-consistent regu- larization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' At zero temperature, the results with G(0) and G(eB) are shown together in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Due to the running of coupling constant, m shows a non-monotonic feature though the absolute value of chiral condensate m/2G(eB) increases with B almost linearly [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Ac- cordingly, the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (12) demonstrates a non-monotonic feature and becomes negative at larger B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Such feature is responsible for the strong suppression of magnetization with the running coupling at larger B compared to the constant coupling case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' At finite temperature, the results are illustrated in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='4 m (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='14 e� (GeV2) ℳ (GeV2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The dynamical mass m (upper panel) and magne- tization M (lower panel) with the self-consistent regulariza- tion (blue dashed lines) and vacuum magnetic regularization (black dotted lines) schemes at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' As we can see, the temperature tends to sup- press magnetization in the case with G(0) but enhance magnetization in the case with G(eB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In their book, Landau and Lifshitz had calculated magnetic suscepti- bility χ ≡ eM NB of a non-relativistic dilute electronic gas at high temperature and found it decreases as 1/T [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' To be concrete, the situations they considered are √ B ≪ T ≪ me and the electric chemical potential −µe(≳ me) changes with T to keep the total number N constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' If we keep −µe(≳ me) a constant, then the 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='4 m (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='14 eB (GeV2) ℳ (GeV2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The dynamical mass m (upper panel) and magneti- zation M (lower panel) with the constant coupling G(0) (blue dashed lines) and the running coupling G(eB) (red lines) at zero temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The dotted lines correspond to the corre- sponding contributions of the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' total electronic number N could be easily evaluated to increase with temperature as T 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Therefore, the mag- netization M = χNB/e would increase with tempera- ture as √ T, and the result with G(eB) is qualitatively consistent with the non-relativistic study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' That is not the end of story: when we keep m = mv for G(0), M would increase with T for a given B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' so it is adequate chiral symmetry restoration induced by T that reduces the contribution of second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (12) and thus re- verses the trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' One can refer to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 for the dynami- cal mass effect on magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For G(eB), m changes mildly with B for a given T , that is, the large mass gaps induced by T at vanishing B sustain to strong magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' According to our analysis, it is the great enhance- ment of the forth T -dependent term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' (12) that helps to recover the trend of naiive expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' In fact, the result with G(eB) is qualitatively consistent with that found in LQCD simulations at finite temperature [19], so we conclude that the running coupling is able to consis- tently explain both inverse magnetic catalysis effect and magnetization enhancement with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' At finite baryon chemical potential, the results are il- lustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' For G(0), m always changes discon- tinuously with B for µB > mv, which signals a first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' But for G(eB), m only changes slightly at µB = 0 and no sign of first-order transition could be identified for a given µB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The de Haas-van Alphen oscil- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 m (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='06 eB (GeV2) ℳ (GeV2) T=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='15 GeV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The dynamical mass m (upper panel) and magneti- zation M (lower panel) as functions of the magnetic field B at temperature T = 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='15, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The dashed, dot- ted, and dashdotted lines correspond to the results with the constant coupling G(0) and the solid lines correspond to the results with the running coupling G(eB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' lation [30] can be found both in the evolutions of m and M with B: the effect is significant to m only when µB is a little larger than 3mv but is significant to M for any µB > 3mv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' According to the mechanism of de Haas-van Alphen oscillation [30], the last non-analytic points of M can be roughly determined by � 2qd|B| ≈ µB/3, that is, eB ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='167 GeV2 for µB = 1 GeV and eB ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='375 GeV2 for µB = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' That is consistent with the numerical results shown in the lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Moreover, at larger B, M does not depend on µB for G(0) due to the ”Silver braze” property but increases with µB for G(eB) due to the strong suppression of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' SUMMARY In this work, a self-consistent thermodynamic poten- tial has been obtained for a magnetized QCD matter in two-flavor NJL model by following Schwinger’s renormal- ization spirit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The thermodynamic potential is free of cutoff for the explicitly magnetic field dependent terms and explicit expressions of gap equation and magneti- zation could be derived from that according to thermo- dynamic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Compared to the VMR scheme, the numerical calculations showed that magnetic catalysis ef- fect persists to very large magnetic field at zero temper- 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 m (GeV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='06 eB (GeV2) ℳ (GeV2) μB=0 1 GeV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 GeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The dynamical mass m (upper panel) and magne- tization M (lower panel) as functions of the magnetic field B at baryon chemical potential µB = 0, 1, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='5 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The dashed, dotted, and dashdotted lines correspond to the results with the constant coupling G(0) and the solid lines correspond to the results with the running coupling G(eB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' ature when adopting the self-consistent scheme, and the magnetization is strongly affected accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Keeping in the self-consistent scheme, results with the constant coupling G(0) and running coupling G(eB) are compared with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' At zero temperature and chemical potential, the running coupling greatly sup- presses the dynamical mass m at large magnetic field B and thus reduces the magnetization M a lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' At finite temperature T , M decreases with T for G(0) due to ad- equate suppression of m but increases with T for G(eB) due to the persistance of large mass gaps at large B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' At finite baryon chemical potential µB, no sign of first-order transition could be identified for G(eB) by varying B and de Haas-van Alphen oscillation could be found both in the evolutions of m and M with B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Since we found that the regularization scheme could af- fect the result greatly in the large magnetic field region, we would try to perform similar study in three-flavor NJL or PNJL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Then, we could compare the magneti- zation with the LQCD data for finite temperature in the region 0 ≤ eB ≤ 1 GeV2 [19] and give further predictions for much larger magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' The situation with finite baryon chemical potential could also be explored for com- pleteness, which might help to understand the properties of magnetars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Acknowledgments G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' is supported by the National Natural Science Foundation of China with Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' 11805290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE3T4oBgHgl3EQfFwl_/content/2301.04308v1.pdf'} +page_content=' Li is supported by the National Natural Science Foundation of China with Grant No.' metadata={'source': 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diff --git a/YdE5T4oBgHgl3EQfCw6a/content/tmp_files/2301.05399v1.pdf.txt b/YdE5T4oBgHgl3EQfCw6a/content/tmp_files/2301.05399v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5df00a7aac5c29b3a83b6f66bf630c8e53059d44 --- /dev/null +++ b/YdE5T4oBgHgl3EQfCw6a/content/tmp_files/2301.05399v1.pdf.txt @@ -0,0 +1,1932 @@ +arXiv:2301.05399v1 [math.AG] 13 Jan 2023 +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC +JOHNSON HOMOMORPHISMS +MA LUO AND TATSUNARI WATANABE +Abstract. A Collino cycle is a higher cycle on the Jacobian of a hyperelliptic +curve. The universal family of Collino cycles naturally gives rise to a normal +function, whose induced monodromy relates to the hyperelliptic Johnson ho- +momorphism. Colombo computed this monodromy explicitly and made this +relation precise. We recast this in the perspective of relative completion. In +particular, we use Colombo’s result to construct Collino classes, which are co- +homology classes of hyperelliptic mapping class groups with coefficients in a +certain symplectic representation. We also determine the dimension of their +span in the case of the level two hyperelliptic mapping class group. +Contents +1. +Introduction +1 +2. +Mapping class groups +3 +3. +Hyperelliptic mapping class groups +4 +4. +Moduli spaces of hyperelliptic curves +6 +5. +Relative completions of hyperelliptic mapping class groups +7 +6. +Hyperelliptic Johnson homomorphisms +10 +7. +Normal functions and Ceresa cycles +15 +8. +Collino cycles and their associated normal functions +18 +9. +Hyperelliptic Johnson homomorphisms and Collino classes +24 +References +27 +1. Introduction +Denote the mapping class group of a compact topological surface S of genus g +with n distinct marked points by Γg,n. For g ≥ 1, there is a surjective natural rep- +resentation Γg,n → Sp(H1(S, Z)) and its kernel is called the Torelli group, denoted +by Tg,n. +For a compact Riemann surface C (which we call a curve in this paper), let Jac C +be its jacobian. Then using a point x in C, we have the algebraic cycle Cx − C− +x in +Jac C called the Ceresa cycle. Let Tg,n be the Torelli space of marked, n-pointed +curves of genus g. There is a bundle J1 → Tg,n over Tg,n of the intermediate jaco- +bians whose fiber over [C] is J1(Jac C). The cycle Cx−C− +x determines a point eC,x in +The first author is supported partly by Science and Technology Commission of Shanghai Mu- +nicipality (No. +22DZ2229014), and partly by National Natural Science Foundation of China, +Grant No. 12201217. +1 + +2 +MA LUO AND TATSUNARI WATANABE +the intermediate jacobian J1(Jac C). This construction extends to families and de- +fines a section eg,1 : Tg,1 → J1 of the bundle. This section is by definition a normal +function (see Definition 7.1). It induces a homomorphism of fundamental groups +ξg,1 : Tg,1 → H3(Jac C, Z) = Λ3H1(C), which is equal to twice the Johnson homo- +morphism (see [12, §6]). On the other hand, the Johnson homomorphism yields a +nontrivial class in H1(Γg,1, H1(S, Q)) and H1(Γg,1, Λ3H1(S, Q)/θ ∧ H1(S, Q)) (see +[26], [21, §5]). +When C is hyperelliptic and x is a Weierstrass point, its corresponding Ceresa cy- +cle Cx−C− +x is trivial, and in general its image in the primitive jacobian J1(Jac C)prim +is trivial. So instead, we will consider a canonical higher cycle (Z, q1, q2) associ- +ated to C with two ordered distinct Weierstrass points q1 and q2 constructed in +[7] by Collino. The higher cycle Z can be viewed as a degeneration of the Ceresa +cycle for the stable curve obtained from C by gluing q1 and q2. Collino proves +in [7] that the regulator image, reg(Z), of Z is nontrivial for general hyperelliptic +curves. In [8], Colombo constructs an extension class Pe associated to C with q1 +and q2, which is equal to (2g + 1)reg(Z) in the primitive intermediate jacobian +I2(Jac C)prim. Denote the hyperelliptic Torelli space by Hg[0]. There is a normal +function PE : Hg[0] → I2prim extending the class Pe and Colombo computes its +monodromy action using higher Johnson homomorphisms, which we call hyperel- +liptic Johnson homomorphisms in this paper. +For a Weierstrass point q of a hyperelliptic curve C, denote the Lie algebra of the +unipotent completion of π1(C, q) over Q by p and its derivation algebra by Der p. +The Lie algebras p and Der p admit weight filtrations W•p and W• Der p from Hodge +theory (see [14]). Fix a hyperelliptic involution σ of S. The hyperelliptic mapping +class group ∆g is defined as the subgroup of Γg consisting of elements that com- +mute with the class [σ]. The hyperelliptic Torelli group denoted by T ∆g is given by +the intersection ∆g ∩ Tg in Γg. As a higher Johnson homomorphism, we have the +hyperelliptic Johnson homomorphism T ∆g → GrW +−2 Der p, denoted by τ hyp +q +. Com- +posing τ hyp +q +with a certain Sp(H1(C, Q))-equivariant projection of GrW +−2 Der p onto +Λ2H1(C, Q)/⟨θ⟩, we obtain an Sp(H1(C, Q))-equivariant homomorphism, which we +denote by ˜τhyp +q +. +Denote the normal function extending reg(Z) by �RZ and the +projection onto the fiber by pI2prim. Their composition pI2prim ◦ �RZ induces a ho- +momorphism of fundamental groups T ∆g → Λ2H1(C, Z)/⟨θ⟩ by πZ. As a remark +on Colombo’s work, we prove +Theorem 1. With notation as above, if g ≥ 2, then +˜τ hyp +q2 +− ˜τ hyp +q1 += (g + 1)πZ. +The homomorphism ˜τ hyp +q +is Sp(H1(C, Q))-equivariant and yields a nontrivial +class, denoted by [q], in H1(∆g[2], Λ2H1(C, Q)/⟨θ⟩), where ∆g[2] is the level 2 +hyperelliptic mapping class group. We call the class [q] a Weierstrass class. Due +to Theorem 1, the homomorphism (g + 1)πZ produces a nontrivial class given by +[q2] − [q1], which we call a Collino class in H1(∆g[2], Λ2H1(C, Q)/⟨θ⟩). The fact +that reg(Z) is nontrivial for general hyperelliptic curves is equivalent to that the +corresponding Collino class is nontrivial. +Our second main result is concerned +with the subspace, denoted by Xζ, of H1(∆g[2], Λ2H1(C, Q)/⟨θ⟩) spanned by all +Collino classes. Denote the subspace of H1(∆g[2], Λ2H1(C, Q)/⟨θ⟩) spanned by all +Weierstrass classes by Xω. + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS +3 +Theorem 2. With notation as above, if g ≥ 2, then Xζ = Xω and dim Xζ = 2g+1. +In fact, the 2g + 2 Weierstrass classes satisfy a single equation �2g+2 +i=1 [qi] = 0, +and each class (2g + 2)[qi] is an integral combination of (2g + 1) Collino classes +(see Remark 9.5). From the Teichm¨uller theory, it is known that, for g = 2 by +Hubbard in [22] and for g > 2 by Earle and Kra in [9], the universal curve over +a branch of the hyperelliptic locus in the Teichm¨uller space admits exactly 2g + 2 +Weierstrass sections. Our result implies that Weierstrass sections satisfy an alge- +braic relation via the hyperelliptic Johnson homomorphisms. On the other hand, +the tautological sections of the universal curve yield linearly independent classes in +H1(Γg,n, H1(S, Q)) (cf. [18, Prop. 12.1]). It is of our interest to investigate further +how the relative completion of the level 2 hyperelliptic mapping class group ∆g[2] +determines the Weierstrass sections (cf. [31]). +Acknowledgments: We are grateful to Richard Hain for helpful discussions on +the relative completions of hyperelliptc mapping class groups and letting us use a +part of an unpublished notes on the hyperelliptic mapping class groups [20]. +2. Mapping class groups +Fix a smooth, compact oriented surface S of genus g and a subset P of S con- +sisting of n distinct points of S. Assume that 2g − 2 + n > 0. The mapping class +group ΓS,P is defined to be the group of isotopy classes of orientation-preserving +diffeomorphisms of S that fix P pointwise. By the classification of surfaces, ΓS,P +is independent of a choice of the pair (S, P), and hence we denote ΓS,P by Γg,n. +When n = 0, we denote Γg,0 by Γg. +2.1. Level structures. Denote H1(S; R) by HR where R is a commutative ring. +Denote the algebraic intersection pairing on HR by ⟨ , ⟩ : H⊗2 +R +→ R. +It is a +unimodular symplectic form. Set +Sp(HR) = Aut(HR, ⟨ , ⟩). +Fixing a symplectic basis a1, . . . , ag, b1, . . . , bg of HR gives an isomorphism Sp(HR) +with the classical symplectic group Spg(R) consisting of 2g × 2g symplectic matri- +ces. The principal congruence subgroup Sp(HZ)[m] of Sp(HZ) of level m ∈ Z is the +kernel of the reduction mod m mapping: +Sp(HZ)[m] := ker{Sp(HZ) → Sp(HZ/mZ)}. +Fix a point q in S. The action of Γg,n on π1(S, q) induces an action on HZ that +preserves the intersection pairing. Therefore, there is a representation +ρ : Γg,n → Sp(HZ). +This is well known to be surjective when R = Z (e.g. [10, Thm. 6.4]). For each +integer m ≥ 0, we define the level m subgroup of Γg,n to be the kernel of the +reduction of ρ mod m: +Γg,n[m] = ker{Γg,n → Sp(HZ/mZ)}. +When m = 1, we omit the level notation, so Γg,n[1] = Γg,n. +The Torelli group Tg,n is defined to be the kernel of ρ and it is the the level 0 +subgroup of Γg,n: +Tg,n = Γg,n[0] = ker{Γg,n → Sp(HZ)}. + +4 +MA LUO AND TATSUNARI WATANABE +The Torelli groups are torsion free (e.g. +[10, Thm. 6.12]). +Since Sp(HZ)[m] is +torsion free for all m ≥ 3, it follows that Γg,n[m] is torsion free for all m ≥ 3. +3. Hyperelliptic mapping class groups +The content of this section comes from an unpublished notes on the completions +of hyperelliptic mapping class groups [20]. +Fix a hyperelliptic involution σ : S → S. It is an orientation-preserving diffeo- +morphism of order 2 of S with exactly 2g +2 fixed points, which we call Weierstrass +points. The quotient space S/⟨σ⟩ is a sphere by the Riemann-Hurwicz formula. +Therefore, it then follows that all hyperelliptic involutions are conjugate in the +group of orientation preserving diffeomorphisms of S, denoted by Diff+S. +An orientation-preserving diffeomorphism of S is said to be symmetric if it com- +mutes with σ. The hyperelliptic mapping class group, denoted by ∆g, is defined to +be the group of isotopy classes of orientation-preserving symmetric diffeomorphisms +of S: +∆g := π0(centralizer of σ in Diff+S) +The following result by Birman and Hilden allows us to consier ∆g as a subgroup +of Γg. +Theorem 3.1 (Birman-Hilden [3]). The natural homomorphism ∆g → Γg is in- +jective. Its image is the centralizer of the isotopy class of σ in Γg. +3.1. Level 2 hyperelliptic mapping class group. Denote the set of fixed points +of σ by W. The action of ∆ on W yields a homomorphism ρW : ∆g → AutW. It +follows from [3, Thm. 1] that the homomorphism ρW is surjective and that there is +an extension: +(1) +1 → ⟨σ⟩ → ker ρW → Γ0,2g+2 → 1. +To identify the kernel, we need to recall the connection between labeling of the +Weierstrass points and level 2 structures on hyperelliptic curves. (Cf. [27, p. 3.31].) +Denote the F2-valued characteristic function W → F2 of a subset T of the set of +Weierstrass points by eT . There is an F2 linear mapping +q : {eT : T ⊆ W, |T | is even} → H1(S, F2) = HF2. +It is defined as follows: let f : S → S2 be the 2-to-1 mapping whose Galois group +is generated by σ. This induces an isomorphism of W with the set of critical values +of f. If T is an even subset of W then there is a 1-chain γ in S2 whose mod 2 +boundary is f(T ). Any two such chains differ by a boundary mod 2. Define q(eT ) +to be the class in H1(S, F2) of f −1(γ). Its homology class is well defined mod 2. +Lemma 3.2. The mapping q induces a linear isomorphism +q : {eT : T ⊆ W, |T | ≡ 0 mod 2}/{eT − eT c : |T | ≡ 0 mod 2} → HF2 +where T c denotes the complement W − T of T . +□ +The intersection pairing on HF2 corresponds to the pairing +eS · eT = #(S ∩ T ) mod 2 +on the even subsets of W. Thus every automorphism of W induces a symplectic +automorphism of HF2. +Corollary 3.3. There is a natural faithful representation Aut W ֒→ Sp(HF2). + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS +5 +Denote the restriction of Γg to ∆g by +ρhyp : ∆g → Sp(HZ). +For each m ≥ 0, we define the level m subgroup ∆g[m] of ∆g by the reduction of +ρhyp mod m: +∆g[m] = ker(∆g → Sp(HZ/mZ)). +Remark 3.4. Since every curve of genus 2 is hyperelliptic, Γ2[m] ∼= ∆2[m] for all +m ≥ 0. +Corollary 3.5. The image of the natural homomorphism ∆g → Sp(HF2) is Aut W. +Consequently, the kernel of ρW : ∆g → Aut W is ∆g[2]. +□ +One therefore has extensions +1 −−−−→ +⟨σ⟩ +−−−−→ ∆g[2] −−−−→ Γ0,2g+2 −−−−→ 1 +1 −−−−→ ∆g[2] −−−−→ +∆g +−−−−→ Aut W −−−−→ 1 +(2) +Since Γg[m] is torsion free for all m ≥ 3, the same is true for ∆g[m]. The group +∆g[0] is the kernel of ρhyp. We shall call it the hyperelliptic Torelli group and denote +it by T ∆g. +3.2. Image of ρhyp. Consider Aut W as a subgroup of Sp(HZ/2Z) via the above +faithful representation. Denote the inverse image of Aut W under the reduction +map Sp(HZ) → Sp(HZ/2Z) by Gg. We have the following diagram. +Gg +−−−−→ +Sp(HZ) +� +� +Aut W −−−−→ Sp(HZ/2Z) +Here we recall the following result of A’Campo (see also [27, §8, Lemma 8.12]). +Theorem 3.6 ([1]). If g ≥ 2, then the image of ρhyp : ∆g → Sp(HZ) is Gg. In +particular, the image of the restriction of ρhyp to ∆g[2] is Sp(HZ)[2]. +3.3. Subgroups ∆g,n. Fixing a labeling W ∼= {1, 2, . . ., 2g + 2} determines an +isomorphism of Aut W with the symmetric group S2g+2. Fix a Weierstrass point +q ∈ W. +Denote its stabilizer in Aut W by Autq W. +This is isomorphic to the +symmetric group S2g+1. The group ∆g,1 is defined by +∆g,1 := (ρW )−1(Autq W). +It contains ∆g[2] and is an extension +(3) +1 → ∆g[2] → ∆g,1 → Autq W → 1. +More generally, let Q be a subset of W with |Q| = n. Denote the subgroup of +Aut W fixing Q pointwise by AutQ W. This group is isomorphic to the symmetric +group S2g+2−n. By abuse of notation, for any such subset Q of W, the subgroup +∆g,n of ∆g is defined to be the inverse image of AutQ W under ρW : +∆g,n = (ρW )−1(AutQ W). +Similarly, there is an extension +1 → ∆g[2] → ∆g,n → AutQ W → 1. + +6 +MA LUO AND TATSUNARI WATANABE +Similarly, denote the image of ∆g,n under ρhyp in Sp(HZ) by Gg,n. +Then, for +0 ≤ n ≤ 2g + 2, there is a commutative diagram of extensions: +1 +� T ∆g +� ∆g,n +� +� +Gg,n +� +� +1 +1 +� T ∆g +� ∆g +� Gg +� 1. +When n = 2g + 2, we denote ∆g,2g+2 by ∆g[2]. +3.4. Group cohomology of ∆g and ∆g[2]. Here we state basic results about +cohomology of hyperelliptic mapping class groups ∆g and ∆g[2]. +Proposition 3.7. If V is a rational ∆g-module, then H•(∆g[2], V ) is an Aut W- +module and there are natural isomorphisms +H•(∆g, V ) ∼= H•(∆g[2], V )Aut W and H•(∆g,1, V ) ∼= H•(∆g[2], V )Autq W . +If σ acts as − id on V , then H•(∆g, V ) = H•(∆g[2], V ) = 0. If σ acts trivially on +V , then V is a Γ0,2g+2-module and there is a natural isomorphism +H•(∆g[2], V ) ∼= H•(Γ0,2g+2, V ). +Proof. The first assertions follow from the Hochschild-Serre spectral sequence of +the extensions (2) and (3). The second assertion follows from the fact that the +centralizer of a group acts trivially on the cohomology. Since σ is central in ∆g, +it acts as a ∆g-linear automorphism of V as − id, and therefore of H•(∆g, V ) and +H•(∆g[2], V ). +But since the centralizer acts trivially on the cohomology of the +group, it follows that σ also acts trivially. Thus multiplication by 2 annihilates +both of these cohomology groups. Since V is a rational representation, our claim +follows. The third assertion follows from the Hochschild-Serre spectral sequence of +the extension (1). +□ +4. Moduli spaces of hyperelliptic curves +By a complex curve, or a curve for short, we shall mean a Riemann surface. +Suppose that 2g − 2 + n > 0. Let S be a reference surface of genus g as in §2 and +P a subset of S consisting of n points. Denote the Teichm¨uller space of marked, +n-pointed, compact genus g curves by Xg,n. As a set Xg,n is +� +orientation-preserving +diffeomorphisms +f : S → C to a complex curve +� � +isotopies, constant on P. +This is a contractible complex manifold of dimension 3g − 3 + n. When n = 0, +Xg,0 is denoted by Xg. The mapping class group Γg,n acts on Xg,n as a group of +biholomorphisms via its action on the markings: +φ : f �→ f ◦ φ−1, +φ ∈ Γg,n, f ∈ Xg,n. +This action is properly discontinuous and virtually free. +The isotropy group of +[(S, P) → (C; x1, . . . , xn)] ∈ Xg,n is isomorphic to the set of automorphisms of C +that fix {x1, . . . , xn} pointwise. +For each m ≥ 0, the moduli space of n-pointed smooth projective curves of genus +g with a level m structure is the quotient of Xg,n by the level m subgroup of Γg,n: +Mg,n[m] = +� +Γg,n[m]\Xg,n +�orb. +It will be regarded as a complex analytic orbifold. It is a model of the classifying +space of Γg,n[m]. When the group Γg,n[m] is torsion free, Mg,n[m] is a smooth + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS +7 +variety and also a fine moduli space for n-pointed smooth projective curves of +genus g with a level m structure. This occurs, for example, when m ≥ 3, m = 0, or +when n > 2g + 2. Note that Mg,n[1] = Mg,n and that Mg,n[0] is the quotient of +Teichm¨uller space by the Torelli group Tg,n. It is known as Torelli space. We shall +denote it by Tg,n. +Fix a hyperelliptic involution σ of S. Let Yg be the set of points of Xg fixed by +σ: +Yg = Xσ +g +The set Yg consists of points [f : S → C] such that f ◦ σ ◦ f −1 is an automorphism +of C. It is connected and contractible of dimension 2g −1, since it is biholomorphic +to X0,2g+2. Note that ∆g is the stabilizer of Yg. For each m ≥ 0, as an analytic +orbifold, the moduli stack Hg[m] of smooth projective hyperelliptic curves of genus +g with a level m structure is the orbifold quotient +Hg[m] = (∆g[m]\Yg)orb. +When m = 0, Hg[0] is the hyperelliptic Torelli space corresponding to σ. In fact, +the hyperelliptic locus of Xg is the disjoint uniton of translates of Yg. Since Yg is +contractible, Hg[m] is a model of the classifying space of ∆g[m]. Hence we have an +isomorphism +πorb +1 +(Hg[m]) ∼= ∆g[m], +where πorb +1 +denotes the orbifold fundamental group. Similarly, for ∆g,n with 0 ≤ +n ≤ 2g + 2, we denote the orbifold quotient of Yg by ∆g,n by +Hg,n = (∆g,n\Yg)orb. +Note that Hg,2g+2 = Hg[2]. The orbifold fundamental group πorb +1 +(Hg,n) is isomor- +phic to ∆g,n. +4.1. Universal family over Hg,n. As an analytic orbifold, Hg,n admits the uni- +versal family πg,n : CHg,n → Hg,n. It is a proper smooth morphism of orbifolds. +Since n Weierstrass points are trivialized, the family πg,n admits n distinct sections, +which we call Weierstrass sections. Let x = [C] be in Hg,n. The fiber of πg,n over +x is C. Let q be a Weierstrass point of C. Then associated to πg,n, there is a +homotopy exact sequence +(4) +1 → π1(C, q) → πorb +1 +(CHg,n, q) → πorb +1 +(Hg,n, x) → 1. +Denote the fundamental group πorb +1 +(CHg,n) by ∆C +g,n. +The monodromy action of +Hg,n on C yields natural symplectic representations +ρg,n : ∆g,n → Sp(HZ) +and +ρC +g,n : ∆C +g,n → Sp(HZ). +By Theorem 3.6, the images of ρg,n and ρC +g,n contain Sp(HZ)[2] and hence are +Zariski dense in Sp(HQ). +5. Relative completions of hyperelliptic mapping class groups +In this section, we review basics for hyperelliptic mapping class groups and rel- +ative completions, while setting up all the notations. + +8 +MA LUO AND TATSUNARI WATANABE +5.1. Relative completion. A detailed summary of relative completion can be +found in [19, §3]. Let F be a field of characteristic zero. Suppose that +(i) Γ is a discrete group, +(ii) R is a reductive F-group, and +(iii) ρ : Γ → R(F) is a Zariski-dense homomorphism. +The relative completion of Γ with respect to ρ consists of a proalgebraic F-group +G that is an extension +(5) +1 → U → G → R → 1 +of R by a prounipotent F-group U and a Zariski-dense homomorphism ˜ρ : Γ → G(F) +such that the diagram +Γ +˜ρ � +ρ +�❖ +❖ +❖ +❖ +❖ +❖ +❖ +G(F) +� R(F) +commutes. It satisfies the following universal property. Let G be a proalgebraic +F-group that is an extension +1 → U → G → R → 1 +of R by a prounipotent F-group U. If ρG : Γ → G(F) is a homomorphism that lifts +ρ, then there exists a unique homomorphism φ : G → G such that the diagram +Γ +ρG � +˜ρ +� G(F) +� +φ +�♣♣♣♣♣ +G(F) +� R(F) +commutes. +5.2. Key properties. Denote the Lie algebra of U by u. It is a pro-nilpotent Lie +algebra. Relative completions are to some degree computable, since it is controlled +by cohomology. The extension (5) splits by a generalization of Levi’s Theorem and +any two splittings are conjugate by an element of U. Therefore, the Lie algebra u +is an R-module and there is an isomorphism +G ∼= R ⋉ U ∼= R ⋉ exp u +that is unique up to conjugation by an element of U ∼= exp u. The following result +relates the Lie algebra u and the group cohomology of Γ. +Theorem 5.1 ([17, Thm. 3.8]). For all finite dimensional R-modules V , +(i) there is a natural isomorphism +HomR(H1(u), V ) ∼= H1(Γ, V ), +and +(ii) there is a natural injection +HomR(H2(u), V ) ֒→ H2(Γ, V ). + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS +9 +5.3. Relative completions of hyperelliptic mapping class groups. Assume +that g ≥ 2. Recall that Gg,n is the image of ρg,n : ∆g,n → Sp(HZ). Note also that +the hyperelliptic Torelli group T ∆g is the kernel of ρg = ρg,0 and that ker ρg,n = +T ∆g for each n = 1, . . . , 2g + 2. The representation ρC +g,n : ∆C +g,n → Sp(HZ) factors +through ρg,n, and therefore has the same image Gg,n. With abuse of notation, we +denote the maps restricted to this image also by ρg,n and ρC +g,n. +Set H = HQ. Denote the relative completion of ∆g,n with respect to ρg,n by +Dg,n equipped with ˜ρg,n : ∆g,n → Dg(Q). There is an extension +(6) +1 → Ug,n → Dg,n → Sp(H) → 1 +of Sp(H) by a prounipotent radical Ug,n of Dg,n. Denote the Lie algebras of Dg,n +and Ug,n by dg,n and ug,n respectively. When n = 2g + 2, we denote Dg,n, Ug,n, +dg,n, and ug,n by Dg[2], Ug[2], dg[2], and ug[2], respectively. The exact sequence 6 +fits in the commutative diagram +1 +� T ∆g +� +˜ρg,n|T ∆g � +∆g,n +� +˜ρg,n +� +Gg,n +� +� � +� +1 +1 +� Ug,n +� Dg,n +�� Sp(H) +� 1. +Denote the prounipotent completion of T ∆g over Q by T ∆un +g/Q. The right exactness +of relative completions [19, Prop. 3.7] gives the exact sequence +T ∆un +g/Q → Dg → Sp(H) → 1. +This implies that the homomorphism T ∆un +g/Q → Ug is surjective. Since the image +of T ∆g in T ∆un +g/Q is Zariski dense, it follows that the image of ˜ρg,n|T ∆g is Zariski +dense. +Similarly, the relative completion of ∆C +g,n with respect to ρC +g,n : ∆C +g,n → Gg,n is +denoted by DC +g,n and ˜ρC +g,n : ∆C +g,n → DC +g,n(Q) and its prounipotent radical by UC +g,n. +Denote their Lie algebras by dC +g,n and uC +g,n respectively. +Denote the unipotent completion of π1(C, q) over Q and its Lie algebra by P +and p, respectively. The center freeness of P imply that the sequence 4 gives exact +sequences +(7) +1 → P → DC +g,n → Dg,n → 1 +and +(8) +0 → p → dC +g,n → dg,n → 0. +5.4. Mixed Hodge structures on the relative completions. Let V be the +dual of R1πg,n∗Z. +It is a polarized variation of Hodge structure of weight −1. +Its monodromy representation can be identified with ρg,n. By the main theorem +of Hain in [16], the Lie algebras of the relative completions dg,n and dC +g,n admit +natural Q-MHSs, where their Lie brackets are morphisms of MHSs. The Lie algebra +p admits a natural Q-MHS, being the Lie algebra of the unipotent completion of +π1(C, q). On the other hand, it also admits a Q-MHS as the kernel of the surjection +dC +g,n → dg,n. By the naturality properties of the MHS of relative completions [16, +Thm. 13.12], these MHSs on p are equal. Denote the Lie algebra of Sp(H) by s. +The basic properties of the weight filtrations of these Lie algebras follow from [16, +Cor. 13.2] and listed below. + +10 +MA LUO AND TATSUNARI WATANABE +Proposition 5.2. The weight filtrations W•dg,n, W•dC +g,n, and W•p satisfy the prop- +erties: +(1) +W−1dg,n = ug,n = W−1ug,n, +W−1dC +g,n = uC +g,n = W−1uC +g,n, +and p = W−1p, +(2) +GrW +0 dg,n = GrW +0 dC +g,n = s, +and +(3) for m ≥ 1, each graded quotients GrW +−m dg,n, GrW +−m dC +g,n, and GrW +−m p are +Sp(H)-modules, and the induced maps +GrW +• p → GrW +• dC +g,n and GrW +• dC +g,n → GrW +• dg,n +are Sp(H)-equivariant graded Lie algebra homomorphisms. +6. Hyperelliptic Johnson homomorphisms +Fix a Weierstrass point q in S. Denote the fundamental group π1(S, q) by Π, +and denote the pronilpotent Lie algebra of the unipotent completion (Malcev com- +pletion, see [25]) of Π over Q by p as well (The unipotent completions of π1(C) +and π1(S) are isomorphic). For a group G, denote the lower central series of G by +L•G, where L1G = G and LkG = [Lk−1G, G] for k ≥ 2. For each k ≥ 1, denote +the associated graded quotient LkG/Lk+1G by GrL +k G and the quotient G/LkG by +NkG. There is an exact sequence +1 → GrL +k G → Nk+1G → NkG → 1. +When k = 2, we have the exact sequence +1 → GrL +2 G → N3G → H1(G) → 1. +The mapping class group Γg,1 acts NkΠ and so there is a homomorphism +ρk : Γg,1 → Aut(NkΠ). +The action of Γg,1 on GrL +k Π factors through the natural representation ρ : Γg,1 → +Sp(HZ). Note that when k = 2, N2Π = H1(S, Z) = HZ, ρ2 is the homomorphism +ρ, and ker ρ2 = Tg,1. The Johnson homomorphism +τ : ker ρ2 = Tg,1 → Hom(HZ, GrL +2 Π) +is defined by setting φ �→ (u �→ φ(˜u)˜u−1 ∈ GrL +2 Π), where ˜u is any lift of u in +N3Π. +It is a Γg,1-equivariant homomorphism, and the induced homomorphism +H1(Tg,1) → Hom(HZ, GrL +2 Π) is Sp(HZ)-equivariant. It is well known that as an +Sp(HZ)-module, GrL +2 Π is isomorphic to Λ2HZ/⟨θ⟩. +In fact, Johnson proved in +[23, 24] that the image of τ is contained in Λ3HZ ⊂ Hom(HZ, Λ2HZ/⟨θ⟩) and τ +induces an isomorphism H1(Tg,1) → Λ3HZ mod 2-torsion. +In the case of the hyperelliptic mapping class group ∆g,1, the corresponding +Johnson homomorphism is trivial as follows. Since the hyperelliptic involution σ +commutes with the elements of T ∆g, it acts trivially on T ∆g. On the other hand, +σ acts as − id on Hom(HZ, GrL +2 Π). The triviality of the homomorphism implies +that elements of T ∆g act trivially on N3Π. From the exact sequence +1 → GrL +3 Π → N4Π → N3Π → 1, + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 11 +we obtain the hyperelliptic Johnson homomorphism +τ hyp +q +: T ∆g → Hom(HZ, GrL +3 Π) φ �→ (u �→ φ(˜u)˜u−1 ∈ GrL +3 Π), +where ˜u is any lift of u in N4Π. The homomorphism τ hyp +q +is ∆g,1-equivariant. +6.1. Key Sp(H)-representations. Suppose that g ≥ 2. +Set H = HQ and fix +a symplectic basis a1, b1, . . . , ag, bg of H. +The isomorphism class of each finite +dimensional irreducible representation of Sp(H) can be indexed by a partition λ of +a nonnegative integer n into less than g parts: +n = λ1 + λ2 + · · · + λl with l ≤ g and λ1 ≥ λ2 ≥ · · · ≥ λl. +Denote the irreducible Sp(H)-representation whose isomorphism class corresponds +to a partition λ by Vλ. Let θ = �g +i=1 ai ∧ bi. Then for example, we have +V[1] ∼= H, +V[12] ∼= Λ2H/⟨θ⟩, and V[13] ∼= Λ3H/θ ∧ H. +6.2. The Lie algebras p and Der p as Sp(H)-representations. Since H1(p) is +pure of weight −1, it follows that the natural weight filtration on p coincides with +the lower central series, that is, W−mp = Lmp for each m ≥ 1. Each associated +graded quotient GrW +−m p = W−mp/W−m−1p denoted by p(−m) is a finite dimen- +sional Sp(H)-module. Let V be a vector space over a field. Denote the free Lie +algebra generated by V by L(V ). It is the direct sum ⊕k≥1Lk(V ) of components +Lk(V ) of bracket length k. It is well known that the graded Lie algebra GrW +• p has +a minimal presentation +GrW +• p ∼= L(H)/⟨θ⟩. +The following result shows some of the Sp(H) representations appearing in GrW +• p +for small values of m. +Proposition 6.1 ([17, Prop. 8.4, Cor. 8.5]). For all g ≥ 2, the irreducible decom- +position of p(−m) as an Sp(H)-representation for 1 ≤ m ≤ 3 is given by +p(−m) = + + + + + +V[1] +for m = 1 +V[12] +for m = 2 +V[2+1] +for m = 3. +The derivation algebra Der p is also a MHS induced by that of p. Another key +object we consider is GrW +−2 Der p. The derivation algebra Der L(H) of L(H) is given +Der L(H) = ⊕k≥0 Hom(H, Lk(H)). +Furthermore, the derivation GrW +• Der p = Der GrW +• p is an Sp(H)-submodule of +Der L(H) and is given by +GrW +• Der p = ⊕m≥0 Der−m p, +where Der−m p is the Sp(H)-submodule of Hom(H, Lm+1(H)) consisting of deriva- +tions that annihilate θ. By [17, Prop. 9.1], in the representation ring of Sp(H), we +have +Der−m p = p(−1) ⊗ p(−1 − m) − p(−2 − m). +In this paper, the weight −2 component Der−2 p plays an essential role. +Proposition 6.2 ([17, Prop. 9.1, Cor. 9.2]). For all g ≥ 2, the irreducible decom- +position of Der−2 p as an Sp(H)-module is given by +Der−2 p = V[22] + V[12]. + +12 +MA LUO AND TATSUNARI WATANABE +6.3. Projection of Der−2 p onto V[12]. Here we will explain how to identify the +V[12]-component of Der−2 p, using the projection used in [8, Cor.5.1] and give an +explicit formula. +Define an Sp(H)-equivariant map φ : S2Λ2H → Hom(H, L3(H)) by +φ : (u1 ∧ v1)(u2 ∧ v2) �→ +x �→ ⟨u1, x⟩[v1, [u2, v2]]+⟨v1, x⟩[[u2, v2], u1]+⟨u2, x⟩[v2, [u1, v1]]+⟨v2, x⟩[[u1, v1], u2], +where ⟨·, ·⟩ : Λ2H → Z is the algebraic intersection paring. +The following result will be useful to describe the image of a Dehn twist under +a hyperelliptic Johnson homomorphism. An easy computation gives +Lemma 6.3. For I ⊂ {1, . . ., g}, set HI = Span{ai, bi|i ∈ I} and +Hc +I = Span{ai, bi|i ̸∈ I}. Let θI = � +i∈I ai ∧ bi. Then +φ(θ2 +I)(x) = +� +0 +x ∈ Hc +I +−2[θI, x] +x ∈ HI +. +Furthermore, we have +Lemma 6.4. The image of φ is in Der−2 p. +Proof. Let θ = �g +i=1 ai ∧ bi. Then for each (u1 ∧ v1)(u2 ∧ v2), we have +φ((u1 ∧ v1)(u2 ∧ v2))(θ) = +g +� +i=1 +[−⟨u1, ai⟩bi + ⟨u1, bi⟩ai, [v1, [u2, v2]]] ++ [−⟨v1, ai⟩bi + ⟨v1, bi⟩ai, [[u2, v2], u1]] ++ [−⟨u2, ai⟩bi + ⟨u2, bi⟩ai, [v2, [u1, v1]]] ++ [−⟨v2, ai⟩bi + ⟨v2, bi⟩ai, [[u1, v1], u2]] += [u1, [v1, [u2, v2]]] + [v1, [[u2, v2], u1]] ++ [u2, [v2, [u1, v1]]] + [v2, [[u1, v1], u2]] += −[[u2, v2], [u1, v1]] − [[u1, v1], [u2, v2]] += [[u1, v1], [u2, v2]] − [[u1, v1], [u2, v2]] = 0. +Hence, our claim follows. +□ +In fact, it is not difficult to see that φ is a surjection onto Der−2 p by Schur’s +Lemma. +We define the explicit projection of Der−2 p onto the copy of V[12] as +follows. First, define an Sp(H)-equivariant map pH : ⊗3H → H by +u ⊗ v ⊗ w �→ ⟨u, v⟩w. +Note that the map pH is the dual of the injection θ ⊗ · : H ֒→ ⊗3H and that +the composition pH ◦ (θ ⊗ ·) = 2g idH. Secondly, define an Sp(H)-equivariant map +pΛ2H : Hom(H, ⊗3H) → Λ2H by +γ �→ +g +� +i=1 +ai ∧ pHγ(bi) − bi ∧ pHγ(ai). +Now, we have the identification L3(H) = (Λ2H ⊗ H)/Λ3H and then using the +inclusion Λ2H → ⊗2H, u ∧ v �→ u ⊗ v − v ⊗ u, we get the inclusion L3(H) → ⊗3H. + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 13 +Therefore, we have the Sp(H)-equivariant projection Der−2 p → Λ2H given by the +composition +Der−2 p ֒→ Hom(H, L3(H)) ֒→ Hom(H, ⊗3H) +pΛ2H +→ Λ2H, +which we denote by πΛ2H. The following result is useful when computing the V[12]- +component in Der−2 p of the image of an element of T ∆g under τ hyp. +Lemma 6.5. The composition πΛ2H ◦ φ : S2Λ2H → Λ2H is given by +(u1 ∧ v1)(u2 ∧ v2) �→ 4[⟨u1, v1⟩v2 ∧ u2 + ⟨v2, u2⟩u1 ∧ v1]+ +2[⟨u1, v2⟩v1 ∧ u2 + ⟨v1, u2⟩u1 ∧ v2 + ⟨u1, u2⟩v2 ∧ v1 + ⟨v2, v1⟩u1 ∧ u2]. +Proof. A direct computation suffices. +□ +Denote the projection Λ2H → Λ2H/⟨θ⟩ by ˜θ. On the other hand, we may view +V[12] as a submodule of Λ2H and there is the Sp(H)-projection ˆθ : Λ2H → V[12] +given by u ∧ v �→ u ∧ v − ⟨u,v⟩ +g +θ. Denote the composition ˆθ ◦ πΛ2H ◦ φ by ˆπ. Denote +the map V[12] → S2Λ2H given by multiplication by θ by jθ. An easy computation +together with Lemma 6.5 gives +Lemma 6.6. The composition +V[12] +jθ→ S2Λ2H +ˆπ→ V[12] +is given by −4(g + 1) times the identity map idV[12]. +Lemma 6.7. With notation as above, for each x ∈ S2Λ2H, the vector +x − +1 +−4(g + 1)(jθ ◦ ˆπ)(x) +lands in the V[22]-component of Der−2 p via φ. +Proof. Let s = φ +� +x − +1 +−4(g+1)(jθ ◦ ˆπ)(x) +� +. +By Lemma 6.6, (ˆθ ◦ πΛ2H)(s) = 0, +and hence by Proposition 6.2 and Schur’s Lemma, s is in the V[22]-component of +Der−2 p. +□ +6.4. Nontriviality of τ hyp. Suppose that g ≥ 2. Let ci be a separating simple +closed curve in S that divides S into two subsurfaces S′ +i and S′′ +i of genus i and +g − i, respectively. Fix a symplectic basis a1, b1, . . . , ag, bg for HZ such that for +each i = 1, . . . , g − 1, the sets {al, bl|1 ≤ l ≤ i} and {al, bl|i + 1 ≤ l ≤ g} form +symplectic bases for H1(S′ +i) and H1(S′′ +i ), respectively. Let θ′ +i = �i +l=1 al ∧ bl and +θ′′ +i = �g +l=i+1 al∧bl. Then we have θ = θ′ +i+θ′′ +i . Denote the isotopy class of the Dehn +twist along ci by di. It is an element of the hyperelliptic Torelli group T ∆g. For +simplicity, assume that the Weierstrass point q lies in the subsurface S′ +i. Lemma +6.3 implies that we have +φ((θ′′ +i )2)(x) = +� +0 +x ∈ H1(S′ +i) +−2[θ′′ +i , x] +x ∈ H1(S′′ +i ) . +It then follows from the construction of τhyp +q +with the base point q that we have +Proposition 6.8 ([31, Prop. 6.2]). If g ≥ 2, then +τ hyp +q +(di) = 1 +2φ((θ′′ +i )2). + +14 +MA LUO AND TATSUNARI WATANABE +S′ +i +S′′ +i +ci +q +Figure 1. Subsurfaces S′ +i and S′′ +i separated by ci +The following result of Brendle, Margalit, and Putman is a key to understand +the hyperelliptic Johnson homomorphisms. +A separating curve d is said to be +symmetric if σ(d) = d. +Theorem 6.9 ([5, Thm. A]). For g ≥ 0, the group T ∆g is generated by Dehn +twists about symmetric separating curves. +As immediate consequences of Proposition 6.8 and Theorem 6.9, we have +Corollary 6.10. If g ≥ 2, τ hyp +q +is nontrivial and the image of τhyp +q +is contained in +Der−2 p. +6.5. The cohomology class of τ hyp +q +. Here we associate a cohomology class to +τ hyp +q +via the relative completion as follows. Assume that n ≥ 1 and ∆g,n fixes q. +Consider the homotopy exact sequence +1 +� π1(C, q) +� πorb +1 +(CHg,n, q) πg,n∗ � πorb +1 +(Hg,n, x) +sq∗ +� +� 1. +The Weierstrass section sq of the universal curve πg,n : CHg,n → Hg,n given by q +induces a section sq∗ of πg,n∗. Taking the relative completion of ∆C +g,n and ∆g,n +produces the exact sequence 7, which yields the exact sequence of prounipotent +groups over Q +1 +� P +� UC +g,n +� Ug,n +˜sq +� +� 1. +By the universal property of relative completions, sq∗ induces a section ˜sq of DC +g,n → +Dg,n, which restricts to a section of UC +g,n → Ug,n. We denote this restriction by ˜sq +as well. Applying the log map to the sequence, we obtain the exact sequence of +pronilpotent Lie algebras +0 +� p +� uC +g,n +� ug,n +d˜sq +� +� 0. +The section ˜sq induces a Lie algebra section d˜sq of uC +g,n → ug,n. Therefore, the Lie +algebra ug,n acts on p via the adjoint action of uC +g,n on p, and hence we have the +adjoint map adjq : ug,n → Der p. Since adjq preservers the weight filtrations, we +obtain an Sp(H) equivariant graded Lie algebra homomorphism +GrW +• adjq : GrW +• ug,n → GrW +• Der p. +The proof of [31, Prop. 7.7] implies that H1(ug,n) is pure of weight −2. Therefore, +we have +GrW +−2 ug,n = GrW +−2 H1(ug,n) = H1(ug,n), + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 15 +and so GrW +−2 adjq can be expressed as an Sp(H)-equivariant map +GrW +−2 adjq : H1(ug,n) → Der−2 p. +On the other hand, recall that the restriction of the relative completion ˜ρg,n : +∆g,n → Gg,n to T ∆g induces the map T ∆g → Ug,n, whose image is Zariski dense +in Ug,n. Composing with the log map Ug,n → ug,n and ug,n → H1(ug,n), we obtain +the map rg,n : T ∆g → H1(ug,n). Now, since the adjoint map is induced by the +conjugation action of ∆C +g,n on π1(C, q), the construction of the hyperelliptic Johnson +homomorphism implies that there is a commutative diagram +T ∆g +rg,n � +τ hyp +q +�❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +H1(ug,n) +GrW +−2 adjq +� Der−2 p. +Denote the composition +˜θ ◦ πΛ2H ◦ τ hyp +q +: T ∆g → Λ2H/⟨θ⟩ = V[12] +by ˜τ hyp +q +and the composition +˜θ ◦ πΛ2H ◦ GrW +−2 adjq : H1(ug,n) → Λ2H/⟨θ⟩ +by ˜τ adj +q +. Since the map rg,n has a Zariski dense image, it follows that the homo- +morphism ˜τ adj +q +is a unique Sp(H)-equivariant map that makes the diagram +T ∆g +rg,n � +˜τ hyp +q +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +H1(ug,n) +˜τ adj +q +� Λ2H/⟨θ⟩ +commute. Now the homomorphism ˜τ adj +q +corresponds to a class in H1(∆g,n, V[12]), +denoted by [q], via the natural isomorphism +H1(∆g,n, V[12]) ∼= HomSp(H)(H1(ug,n), V[12]) +induced by relative completion in Theorem 5.1. Since ∆g[2] is a subgroup of ∆g,n, +there is a natural homomorphism H1(∆g,n, V[12]) → H1(∆g[2], V[12]). It is easy to +see that the image of [q] in H1(∆g[2], V[12]) is equal to the class of q produced by +the above construction when n = 2g + 2. +7. Normal functions and Ceresa cycles +Normal functions are holomorphic sections of families of intermediate Jacobians +that satisfy certain asymptotic conditions. This notion naturally arise when study- +ing families of algebraic varieties. In this section, we recall the construction of the +normal function associated to a family of homologically trivial algebraic cycles in a +family of smooth projective varieties. Then we review Ceresa cycles and how their +associated normal function relates to the Johnson homomorphism. More details +can be found in Hain [12, 13]. + +16 +MA LUO AND TATSUNARI WATANABE +7.1. Intermediate Jacobians. Suppose that X is a smooth projective variety and +that Z is an algebraic d-cycle in X. We have an exact sequence of mixed Hodge +structures +0 → H2d+1(X) → H2d+1(X, Z) → H2d(Z) → H2d(X). +The class of the cycle Z defines a morphism of mixed Hodge structures +cZ : Z(d) → H2d(Z). +If Z is homologically trivial, we can pull back the previous sequence along cZ to +obtain an extension +0 → H2d+1(X) → EZ → Z(d) → 0 +in the category MHS of mixed Hodge structures. Tensoring with Z(−d) gives an +extension +0 → H2d+1(X, Z(−d)) → EZ(−d) → Z → 0 +and thus a class eZ in +Ext1 +MHS(Z, H2d+1(X, Z(−d))). +Note that H2d+1(X) has weight −(2d + 1), and thus H2d+1(X, Z(−d)) has weight +−1. +For a mixed Hodge structure V whose weights are all negative, there is a natural +isomorphism +J(V ) ∼= Ext1 +MHS(Z, V ) +where +J(V ) := +VC +F 0VC + VZ +. +In general, by Carlson [6], we have +Ext1 +MHS(B, A) ∼= J(Hom(B, A)) +where Ext1 +MHS(B, A) is the set of congruence classes of extensions of B by A for +separated mixed Hodge structures A and B, i.e. the highest weight of A is less than +the lowest weight of B. +The class eZ of a homologically trivial d-cycle Z in X can thus be viewed as a +class +eZ ∈ J(H2d+1(X, Z(−d))), +which, in turn, can be viewed as a class in the d-th intermediate Jacobian +Jd(X) := (F d+1H2d+1(X))∗/H2d+1(X, Z) +as Jd(X) ∼= J(H2d+1(X, Z(−d))). This class can be described explicitly by Griffiths’ +generalization of the Abel-Jacobi construction as follows. Write Z = ∂Γ, where Γ is +a topological (2d + 1)-chain. Each class in F d+1H2d+1(X) can be represented by a +closed form in the Hodge filtration F d+1 of the de Rham complex of X. By Stokes’ +theorem, integrating these representatives over Γ gives a well defined functional +� +Γ +: F d+1H2d+1(X) → C. +The choice of Γ is unique up to a topological (2d + 1)-cycle. So � +Γ determines a +point in Jd(X) that corresponds to eZ. + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 17 +7.2. Normal functions. Suppose that T is a smooth variety and that V → T is +a variation of mixed Hodge structures of negative weights over T . Let V be the +corresponding bundle whose fiber over t ∈ T is Vt. Denote by J (V) the bundle +over T whose fiber over t is +J(Vt) ∼= Ext1 +MHS(Z, Vt). +Definition 7.1. A holomorphic section s : T → J (V) of J (V) → T is a normal +function if it defines an extension +0 → V → E → ZT → 0 +in the category of admissible variations of mixed Hodge structure over T . +Remark 7.2. Admissibility characterizes good variations in the sense of Steenbrink– +Zucker[29] and Saito[28]. It is satisfied in the geometric situations, Guill´en et al[11] +and Hain[14]. An important characterization is the existence of a relative weight +filtration at the infinity, which amounts to certain asymptotic conditions. +Example 7.3. Families of homologically trivial algebraic cycles give rise to such +extensions of variations of mixed Hodge structures and thus normal functions. Sup- +pose that X → T is a family of smooth projective varieties over a smooth base T . +Suppose that Z is an algebraic cycle in X, which is proper over T of relative di- +mension d. Denote the fibers of X and Z over t ∈ T by Xt and Zt respectively. +Suppose that Zt is homologically trivial in Xt for all t. +The Hodge structures H2d+1(Xt, Z(−d)) form a variation of Hodge structure V +over T of weight −1. The intermediate Jacobians Jd(Xt) ∼= J(H2d+1(Xt, Z(−d))) +form the relative intermediate Jacobian +Jd → T. +The family of cycles Z defines a section of this bundle. This is called the normal +function of the cycle Z. +7.3. Ceresa cycles and their associated normal functions. Let C be a com- +pact Riemann surface of genus g, and JC its Jacobian. When g ≥ 1, for each +x ∈ C, the Abel-Jacobi map +νx : C → JC +y �→ y − x +is an embedding. Denote the image by Cx, which is an algebraic 1-cycle in JC. +There is an involution on JC by D �→ −D. Denote the image of Cx under this +involution by C− +x . +Since the involution acts trivially on H2(JC), the algebraic +1-cycle +ZC,x := Cx − C− +x +is homologically trivial. By §7.1, this ZC,x determines a point in the intermediate +Jacobian +eC,x ∈ J1(JC) ∼= J(H3(JC, Z(−1))). +The primitive decomposition +H3(JC, Q) = H1(JC, Q) ⊕ PH3(JC, Q) + +18 +MA LUO AND TATSUNARI WATANABE +is the decomposition of H3(JC, Q) into irreducible Sp(H1(C))-modules, the highest +weights of the pieces being λ1 and λ3. Pontrjagin product with the class of C induces +a homomorphism +Φ : JC → J1(JC). +Denote the cokernel of Φ by JQ(JC). By [12, Prop. 6.1], we have +eC,x − eC,y = Φ(x − y). +It follows that the image of eC,x in JQ(JC) is independent of x. The image will be +denoted by eC. +Now suppose that the genus g ≥ 3. Denote by J1 and J1prim the bundles over +Mg,n whose fiber over [C; {x1, · · · , xn}] is J1(JC) and JQ(JC) respectively. +We construct sections eg,1 : [C, x] �→ eC,x and eg : [C] �→ eC of J1 → Mg,1 and +J1prim → Mg respectively. +J1 +J1prim +Mg,1 +Mg +eg,1 +eg +By Definition 7.1 and Remark 7.2, we have the following result by construction. +Theorem 7.4. The sections eg,1 and eg are normal functions. +Fix base points [C, x] ∈ Mg,1 and [C] ∈ Mg respectively. So H := H1(C, Z) is +fixed. Taking fundamental groups, the normal functions eg,1 and eg induce Sp(H)- +module homomorphisms +ξg,1 : H1(Tg,1, Z) → H1(J1(JC), Z) ∼= Λ3H1(C, Z) +and +ξg : H1(Tg, Z) → H1(JQ(JC), Z) ∼= Λ3H1(C, Z)/H1(C, Z) +respectively. The following result is shown in Hain [12, Prop. 6.3]. +Theorem 7.5. The map ξg,n is twice the Johnson homomorphism τg,n for n = 0, 1. +8. Collino cycles and their associated normal functions +In this section, we review Colombo’s results on Collino cycles [8]. Since these +can be viewed as degenerations of Ceresa cycles, they give rise to elements in higher +Chow groups of the Jacobian JC of a hyperelliptic curve C [7, 1.1]. Their regulator +images, defined in [2, 4], can be expressed in terms of iterated integrals. They +are thus identified with some specific extensions constructed from the fundamental +group of C. As in the case of Ceresa cycles, we can construct the normal functions +associated to Collino cycles and compute their induced monodromies. +8.1. Regulators. Let X be a smooth projective variety of dimension n. An ele- +ment in the first higher Chow group CHn(X, 1) is defined by +A := +� +i +(Ci, fi) +where Ci is an irreducible curve on X and fi a rational function on Ci such that +� +i +[div(fi)] = 0. + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 19 +Let γi := f −1 +i +([0, ∞]). The above condition tells us that +η := +� +i +γi +is a loop. In fact, it is homologically trivial, so that +η = ∂D +where D is a 2-chain. Then the regulator map +reg : CHn(X, 1) → I2(X) := (F 1H2(X))∗/H2(X, Z(1)) +A �→ reg(A) +where reg(A) denotes the current, i.e. linear functional on F 1H2(X), taking +α �→ +� +i +� +Ci−γi +log(fi)α + 2πi +� +D +α +for every α ∈ F 1H2(X). A similar definition of regulator is given by +reg : CHn(X, 1) → I2(X)prim := (F 1H2(X)prim)∗/H2(X, Z(1))prim. +using the primitive part H2 +prim of H2. +8.2. Collino cycles. A Collino cycle Z is a canonical higher cycle, depending on +the choice of two Weierstrass points of a genus g hyperelliptic curve C, on the +Jacobian JC of C [7]. Fix Weierstrass points q1, q2, let h be a degree 2 morphism +h : C → P1 +such that h(q1) = 0 and h(q2) = ∞. Recall that for each x ∈ C, we denote by Cx +the image under the Abel-Jacobi map +νx : C → JC +y �→ y − x. +Let hs := h ◦ ν−1 +qs be a function on Cs := Cqs = νqs(C) for s = 1, 2. Then +Z := (C1, h1) + (C2, h2) ∈ CHg(JC, 1) +is called a Collino cycle. Its regulator is nonzero for general C. In fact, it can be +computed using iterated integrals. +Theorem 8.1 (Thm 1.1 [8]). Let φ and ψ be harmonic 1-forms on JC with ψ of +type (1, 0). We use the same notation for their pullback to C. Then +reg(Z)(φ ∧ ψ) = 2 +� +C−γ +log(h)φ ∧ ψ + 2πi +� +γ +(φψ − ψφ) +where γ := h−1([0, ∞]). +Remark 8.2. One of the forms ψ being type (1, 0) makes φ ∧ ψ a representative of +an element in F 1H2(JC). + +20 +MA LUO AND TATSUNARI WATANABE +8.3. Colombo’s construction of extension class from fundamental groups. +Colombo relates the regulator image reg(Z) to an extension class Pe, primitive part +of an extension class e constructed from the fundamental groups of the punctured +curves C −{q1} and C −{q2} with the same base point p, another Weierstrass point +of C. More precisely, as being done for the regulator, these extension classes are +expressed in terms of iterated integrals on the Jacobian JC of the curve C, and +Colombo shows that Pe is a rational multiple of reg(Z). +We first review her construction1 of extensions e and Pe from the MHS on the +fundamental groups. Fix the base point p for the fundamental group π1(C −{q}, p). +Denote by Lq the augmentation ideal2 of the group algebra Zπ1(C − {q}, p). The +powers of Lq gives a natural filtration +· · · ⊆ Lk+1 +q +⊆ Lk +q ⊆ · · · ⊆ Lq ⊆ Zπ1(C − {q}, p). +So we have natural extensions +(9) +0 → (Lq/Lk +q)∗ → (Lq/Lk+1 +q +)∗ → (Lk +q/Lk+1 +q +)∗ → 0. +The graded pieces has pure Hodge weights as +(Lk +q/Lk+1 +q +)∗ ≃ ⊗kH1(C, Z). +To simplify notation, we denote H1(C, Z) by H1. If k = 2, the above sequence (9) +becomes +0 → H1 → (Lq/L3 +q)∗ → ⊗2H1 → 0. +In particular, when q = qs (s = 1, 2) is a Weierstrass point, this extension splits +and has a natural retraction +rs : (Lqs/L3 +qs)∗ → H1. +Pushing along this retraction on the sequence (9) for k = 3, q = qs +0 +(Lqs/L3 +qs)∗ +(Lqs/L4 +qs)∗ +(Lqs/L3 +qs)∗ +0 +0 +H1 +Es +⊗3H1 +0 +rs +≃ +we get extension class es ∈ ExtMHS(⊗3H1, H1) represented by Es for s = 1, 2. +We introduce several natural morphisms of MHS: +(1) Tensoring with the polarization Ω: +JΩ : H1(−1) → ⊗3H1. +(2) The surjection: +Π : ⊗2H1 → Z(−1), +which is the composition of the cup product with the isomorphism H2(C, Z) ≃ +Z(−1). +(3) The standard inclusion: +ι : Λ2H1 → ⊗2H1. +1We have another construction for these same extensions, but since Colombo’s construction is +already in the literature, our construction is not necessary here. +2i.e. the kernel of the augmentation map Zπ1(C − {q}, p) → Z that sends each element of the +fundamental group to 1. + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 21 +(4) The integration over C: +� +C +: Λ2H1 → Z +which is the map Π ◦ ι up to a Tate twist Z(−1). Since we have natural +isomorphism Λ2H1 ≃ H2(JC, Z), we can identify the kernel +ker +� +C +∼= H2(JC)prim +with the primitive part of H2(JC, Z). By Carlson, we can identify +Ext1 +MHS(Λ2H1, Z) ∼= I2(JC) +and +Ext1 +MHS(ker +� +C +, Z) ∼= I2(JC)prim. +Now we are ready to construct the extension classes e and Pe, represented by +the extensions E and PE respectively in the following diagram. +0 +H1 +E2 − E1 +⊗3H1 +0 +0 +H1 +EΩ +H1(−1) +0 +0 +⊗2H1 +EΩ ⊗ H1 +⊗2H1(−1) +0 +0 +Z(−1) +�E(−1) +⊗2H1(−1) +0 +0 +Z +�E +⊗2H1 +0 +0 +Z +E +Λ2H1 +0 +0 +Z +PE +ker +� +C +0 +⊗H1 +⊗H1 +JΩ +⊗H1 +Π +⊗Z(1) +⊗Z(1) +⊗Z(1) +ι +Theorem 8.3 (Thm 2.1 [8]). Let C be a hyperelliptic curve with Weierstrass points +q1, q2 and p. Let h be a degree 2 morphism h : C → P1 such that h(q1) = 0 and +h(q2) = ∞. Then we have +e = (2g + 1) +� +reg(Z) + log(h(p)) +� +C +� +∈ I2(JC) +and +Pe = (2g + 1)reg(Z) ∈ I2(JC)prim. +Remark 8.4. Although the computation of Pe involves the base point p, it only +depends on q1 and q2, since it is a rational multiple of reg(Z), whose construction +does not involve p. + +22 +MA LUO AND TATSUNARI WATANABE +8.4. The normal functions and their induced monodromies. One can extend +the constructions of both reg(Z) and Pe for a hyperelliptic curve to families of +hyperelliptic curves [8, §4, §5]. They give rise to normal functions, i.e. sections on +variations of mixed Hodge structures over the hyperelliptic Torelli space Hg[0]. We +make this precise in the next paragraph. +Recall that the hyperelliptic Torelli space Hg[0] is the moduli space of hyperel- +liptic curves of genus g with a fixed symplectic basis of homology. Let +π : CHg[0] → Hg[0] +and +I2prim → Hg[0] +be the universal hyperelliptic curve and the bundle over the hyperelliptic Torelli +space Hg[0], whose fiber over the moduli point [C; {aj, bj}g +j=1] ∈ Hg[0] is I2(JC)prim. +After choosing sections of Weierstrass points +C +Hg[0] +q1 +q2 +p +we can construct reg(Z) and Pe for each fiber, and they assemble to form sections +rZ and PE of the family I2prim. +I2prim +I2prim +Hg[0] +Hg[0] +rZ +P E +Note that by Hain[14], I2prim is an admissible variation of mixed Hodge structures, +since each of its fiber I2(JC)prim is naturally constructed from fundamental group. +Each fiber I2(JC)prim can be identified with +I2(JC)prim ∼= Ext1(H2(JC)prim, Z) ∼= Ext1(Z, H2(JC)prim(2)) ∼= J(H2(JC)prim(2)) +using Poincar´e duality in the middle, where H2(JC)prim(2) := H2(JC)prim ⊗ Z(2). +By definition 7.1, the sections rZ and PE are normal functions. +These normal functions induce monodromies which we now compute. Note that +the monodromy action of the hyperelliptic Torelli group T ∆g is trivial, so the as- +sociated bundle I2prim is a trivial bundle. Fix a base point [C; {aj, bj}g +j=1]. Denote +by pI2prim the projection of I2prim to its fiber I2(JC)prim. The fundamental group +of the fiber I2(JC)prim is Λ2H/⟨θ⟩ where H is the first homology of C generated +by the fixed basis {aj, bj}g +j=1, and θ = �g +j=1 aj ∧ bj. Denote by pZ and pE the +compositions of the projection pI2prim with normal functions rZ and PE, and they +respectively induce homomorphisms of fundamental groups +πZ = (pZ)∗ : T ∆g → Λ2H/⟨θ⟩ +and +πE = (pE)∗ : T ∆g → Λ2H/⟨θ⟩. +Colombo compared these monodromies on a particular element di ∈ T ∆g. The +element is chosen to be the class of a Dehn twist of a simple closed curve ci on C +seperating q1 and q2, where ci is invariant under the hyperelliptic involution. +Theorem 8.5 (Cor. 5.1 [8]). +πE(di) = (2g + 1)πZ(di). +This theorem can easily be improved, as indicated by Colombo. + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 23 +Proposition 8.6. +πE = (2g + 1)πZ. +Proof. By Theorem 8.3, Pe and (2g+1)reg(Z) are identified on each fiber I2(JC)prim +of I2prim → Hg[0], so they induce the same homomorphisms of fundamental groups +(PE)∗ = (2g + 1)(rZ)∗. +The result follows from composing this with (pI2prim)∗, which is the identity map. +□ +Moreover, using Colombo’s computation the monodromies on particular gener- +ators di of the hyperelliptic Torelli group T ∆g [8, Cor. 4.2], we have the following +result. +Theorem 8.7. Let di ∈ T ∆g be the class of the Dehn twist of a separating simple +closed curve ci on C, where ci is invariant under the hyperelliptic involution, then +πZ(di) = +� +0 +if ci does not separate q1 and q2 +4θ′′ +i +if ci separates q1 and q2 +Proof. We follow the same steps as in [8, Prop. 4.1] to compute. We provide full +details here so that interested readers can compare it with Colombo’s computation. +First, we set some notations. Pick a base point [C] ∈ H. Let d ∈ T ∆g be the +class of a Dehn twist Dd of a separating simple closed curve c on C, invariant under +the hyperelliptic involution. Let λd be the loop in H based at [C], that corresponds +to the Dehn twist Dd. This loop lifts to a path ˜λd : [0, 1] → ˜H in the universal +covering ˜H of H with ˜λd(0) = [C] and ˜λd(1) = [DdC]. There is a universal family +of hyperelliptic curves over the path ˜λd and we denote the fiber over ˜λd(t) by Ct. In +particular, C0 = C. The sections q1 and q2 over λd lift to ˜q1 and ˜q2 over ˜λd, which +on each fiber Ct correspond respectively to the zero and the pole of a degree 2 map +ht : Ct → P1 that we chose to construct the Collino cycle. Let γt := h−1 +t ([0, ∞]) +be the path on Ct, then h0 = h and γ0 = γ are the same as those defined in Thm. +8.1. The section, i.e. normal function, rZ restricts to the loop λd in H, and it can +be lifted to a normal function ˜rZ along ˜λd in ˜H. +Now we compute the monodromy. For any t ∈ [0, 1], ˜rZ(t) is the regulator of the +Collino cycle constructed from Weierstrass points ˜q1(t) and ˜q2(t). By Thm. 8.1, +we have +˜rZ(t)(φt ∧ ψt) = 2 +� +Ct−γt +log(ht)φt ∧ ψt + 2πi +� +γt +(φtψt − ψtφt) +for closed 1-forms φt and ψt on Ct, with ψt of type (1, 0). By covering theory, we +have +πZ(d) = 1 +2π [˜rZ(1) − ˜rZ(0)] ∈ Λ2H/⟨θ⟩. +We can choose φ1 = φ0 =: φ and ψ1 = ψ0 =: ψ. So +[˜rZ(1) − ˜rZ(0)](φ ∧ ψ) = 2 +�� +C1−γ1 +log(h1)φ ∧ ψ − +� +C−γ +log(h)φ ∧ ψ +� ++ 2πi +�� +γ1 +(φψ − ψφ) − +� +γ +(φψ − ψφ) +� + +24 +MA LUO AND TATSUNARI WATANABE +Case(i): If c = ci does not separate q1 and q2, then it is easy to see that we can +choose the same branch for the logarithm +log(h1) = log(h) +because q1 and q2 are on the same subsurface. Moreover, γ does not intersect with +ci, so that the Dehn twist does not change γ and +γ1 = γ. +Therefore, on the right hand side of the above equation, both terms vanish and we +have +πZ(d) = 0. +Case(ii): If c = ci separates q1 and q2, then the result follows directly from [8, +Cor. 4.2]. The essential changes are that of choosing a different branch of log(h) +and that the Dehn twist carrying γ to γ1 = γ + 2d. +□ +9. Hyperelliptic Johnson homomorphisms and Collino classes +In this section, we relate the results in the previous sections, with a point of view +from relative completion. Recall that by Proposition 6.2, we have the decomposition +Der−2 p = V[22] + V[12] as an Sp(H)-module. Set V = V[12] and V ′ = V[22] for +simplicity. +Recall that ˜θ is the projection Λ2H → Λ2H/⟨θ⟩ ∼= V . +Denote the +composition ˜θ ◦ πΛ2H (see §4) by ˜πΛ2H. +9.1. A Collino class in H1(∆g,2, V ). Let q1 and q2 be distinct Weierstrass points +in S. Recall from 6.5 that we have the classes [q1] and [q2] in H1(∆g,2, V ) given by +their corresponding hyperelliptic Johnson homomorphisms τ hyp +qi +. Let ζ = [q2] − [q1] +in H1(∆g,2, V ). Via the isomorphism H1(∆g,2, V ) ∼= HomSp(H)(H1(ug,2), V ), the +class ζ corresponds to the Sp(H)-equivariant map +˜τ adj +ζ +:= ˜τ adj +q2 − ˜τ adj +q1 +: H1(ug,2) → V. +Denote the map ˜τ hyp +q2 +− ˜τ hyp +q1 +: T ∆g → V by ˜τ hyp +ζ +. Note that there is a commutative +diagram +T ∆g +˜τ hyp +ζ +�● +● +● +● +● +● +● +● +● +rab +� +H1(ug,2) +˜τ adj +ζ +� V, +where the map rab is induced by the relative completion of ∆g,2. +Theorem 9.1. With notation as above, if g ≥ 2, we have +˜τ hyp +ζ += (g + 1)πZ +Proof. Suppose that g ≥ 2. Let D in T ∆g be the class of the Dehn twist along a +simple separating curve d. In the case where d does not separate q1 and q2, it follows +from Proposition 6.8 and 8.7 that both ˜τhyp +ζ +(D) and πZ(D) are zero. So consider +the case where d separates q1 and q2. Say S is divided by d into two subsurfaces +S′ +i of genus i and S′′ +i of genus g − i, which contain q1 and q2, respectively, as in +6.4. Fix symplectic bases a1, . . . , ai, b1, . . . , bi and ai+1, . . . , ag, bi+1, . . . , bg for S′ +i +and S′′ +i , respectively. Let θ′ +i = �i +ℓ=1 aℓ ∧ bℓ and θ′′ +i = �g +ℓ=i+1 aℓ ∧ bℓ. We have + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 25 +θ = θ′ +i+θ′′ +i . Then by Proposition 6.8, the image (τ hyp +q2 +−τ hyp +q1 )(D) can be represented +as +ζD := 1 +2φ((θ′ +i)2 − (θ′′ +i )2). +An easy computation using Lemma 6.5 shows that the projection of the derivation +ζD onto V by ˜πΛ2H is given by +˜πΛ2H(ζD) = 2(2g + 2)θ′′ +i +mod θ. +By Theorem 8.7, we have πZ(D) = 4θ′′ +i , and hence +˜τ hyp +ζ +(D) = (g + 1)πZ(D). +Our claim follows from Theorem 6.9 stating that T ∆g is generated by the classes +of Dehn twists along symmetric separating simple closed curves. +□ +Remark 9.2. Consequently, the Collino cycle (Z, q1, q2) determined by q1 and q2 +yeilds a nontrivial class in H1(∆g,2, V ), where ∆g,2 fixes q1 and q2. +9.2. Weierstrass subspace of H1(∆g[2], V ). Recall from 3.1 that ∆g[2] fixes all +of the Weierstrass points. +Therefore for each Weierstrass point q, we have the +hyperelliptic Johnson homomorphism τ hyp +q +. +Lemma 9.3. For two distinct Weierstrass points q1 and q2, the image of τ hyp +q2 −τ hyp +q1 +is contained in V . +Proof. With notation from the proof of Theorem 9.1, we will show that the projec- +tion of ζD in the V ′-component of Der−2 p is trivial. Let ζ′ +D be the V ′-part of ζD +in Der−2 p. We have +(ˆθ ◦ πΛ2H)(ζD) = 2(2g + 2) +� +θ′′ +i − g − i +g +θ +� +, +where ˆθ : Λ2H → V is the projection given by u ∧ v − ⟨u,v⟩ +g +θ (see §4). Then by +Lemma 6.7, the vector +1 +2((θ′ +i)2 − (θ′′ +i )2) + θ′′ +i θ − g − i +g +θ2 +maps into the V ′-component of Der−2 p under φ. By Proposition 6.3, the image of +θ2 is trivial because φ(θ2) = −2adjθ, and since θ = θ′ +i + θ′′ +i , we have +ζ′ +D = φ +�1 +2((θ′ +i)2 − (θ′′ +i )2) + θ′′ +i θ − g − i +g +θ2 +� += φ +�1 +2((θ′ +i)2 − (θ′′ +i )2) + θ′′ +i θ +� += φ +�1 +2(θ′ +i)2 + 1 +2(θ′′ +i )2 + θ′ +iθ′′ +i +� += 1 +2φ((θ′ +i)2 + 2θ′ +iθ′′ +i + (θ′′ +i )2) += 1 +2φ(θ2). +This implies that ζ′ +D is zero in Der−2 p. Therefore, ζD is in V . Since D is arbitrary, +it follows that the image of τhyp +q2 +− τ hyp +q1 +is contained in V . Note that when D is +the Dehn twist along a simple separating curve that does not separate q1 and q2, +τ hyp +q1 (D) = τ hyp +q2 (D), and so (τ hyp +q2 +− τ hyp +q1 )(D) = 0. +□ + +26 +MA LUO AND TATSUNARI WATANABE +We call the image of ζ in H1(∆g[2], V ) via the homomorphism H1(∆g,2, V ) → +H1(∆g[2], V ) as a Collino class. +Since ˜τ hyp +ζ +is a nontrivial homomorphism, It +follows that each Collino class in H1(∆g[2], V ) is nontrivial. Similarly, for each +Weierstrass point q, we call the class [q] in H1(∆g,1, V ) and its image under the +homomorphism H1(∆g,1, V ) → H1(∆g[2], V ) as a Weierstrass class. Let Xζ be the +subspace of H1(∆g[2], V ) spanned by all Collino classes and let Xω be the subspace +of H1(∆g[2], V ) spanned by all Weierstrass classes. +Theorem 9.4. With notation as above, if g ≥ 2, then Xζ = Xω and dim Xω = +2g + 1. +Proof. Let q1, . . . , q2g+2 be the Weierstrass points of S and W = {q1, . . . , q2g+2}. +Then we have the Weierstrass classes [qi] in H1(∆g[2], V ). Set p = q2g+2. For each +i = 1, . . . , 2g + 1, define ζi by ζi = [qi] − [p]. By Theorem 9.1, each ζi is a Collino +class. Suppose that +(*) +2g+1 +� +i=1 +ciζi = 0, +where each ci is in Q. Then we have �2g+1 +i=1 ci˜τ adj +ζi += 0. Recall that the subgroup +Autp W of Aut W fixing p is isomorphic to S2g+1 and that it acts on H1(∆g[2], V ) +and hence on HomSp(H)(H1(ug[2]), V ). Let t be a cycle of length 2g + 1 in Autp W. +Then we have +2g+1 +� +j=1 +tj +�2g+1 +� +i=1 +ci˜τ adj +ζi +� += +2g+1 +� +j=1 +2g+1 +� +i=1 +ci˜τ adj +tj(ζi) += +2g+1 +� +j=1 +�2g+1 +� +i=1 +ci +� +˜τ adj +ζj += +�2g+1 +� +i=1 +ci +� 2g+1 +� +j=1 +˜τ adj +ζj += 0. +Suppose that �2g+1 +i=1 ci ̸= 0. Then we have �2g+1 +i=1 +˜τ adj +ζi += 0. Now for every D in +T ∆g, we have +˜θ ◦ πΛ2H +�2g+1 +� +i=1 +(τ hyp +qi +− τhyp +p +)(D) +� += +2g+1 +� +i=1 +˜τ adj +ζi (rab(D)) = 0, +where rab is the homomorphism T ∆g → H1(ug[2]) induced by the relative comple- +tion of ∆g[2]. By Lemma 9.3, it then follows that +2g+1 +� +i=1 +(τ hyp +qi +− τhyp +p +)(D) = 0. +Thus we have �2g+1 +i=1 (τ hyp +qi +− τ hyp +p +) = 0. This last equation becomes +2g+1 +� +i=1 +τ hyp +qi += (2g + 1)τhyp +p +. +Now, as in 6.4, let D in T ∆g be the Dehn twist about a simple separating closed +curve d separating q1 and p such that d separates S into two subsurfaces S′ +k of genus + +REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 27 +k and S′′ +k of genus g − k, which contain q1 and p, respectively. Fix symplectic bases +a1, . . . , ak, b1, . . . , bk and ak+1, . . . , ag, bk+1, . . . , bg for S′ +k and S′′ +k, respectively. Let +θ′ +k = �k +ℓ=1 aℓ ∧ bℓ and θ′′ +k = �g +ℓ=k+1 aℓ ∧ bℓ. Set x = ak+1. Then +2g+1 +� +i=1 +τ hyp +qi +(D)(x) = lτ hyp +q1 (D)(x) = −l[θ′′ +k, x], +where l is the number of Weierstrass points contained in S′ +k. Since l ≥ 1, this +is a nontrivial element. On the other hand, (2g + 1)τ hyp +p +(D)(x) = 0, which is a +contradiction. Therefore, �2g+1 +i=1 ci = 0. So substituting c2g+1 = − �2g +i=1 ci into +the equation (*), we obtain +(**) +2g +� +i=1 +ci(ζi − ζ2g+1) = 0. +Setting p = q2g+1 and replacing ζi with [qi] − [p] for each i = 1, . . . , 2g, the eqation +(∗∗) becomes +2g +� +i=1 +ciζi = 0. +Inductively, we get c1([q1] − [q2]) = 0. The class [q1] − [q2] is nontrivial, and so +c1 = 0. From the inductive steps, it follows that each ci = 0 for i = 1, . . . , 2g + 1. +Therefore, dim Xζ ≥ 2g + 1. +On the other hand, we observe that the vector �2g+2 +i=1 [qi] is fixed by the action +of S2g+2. Thus, by Proposition 3.7, it is in H1(∆g, V ) = H1(∆g[2], V )S2g+2. It +follows from a result of Tanaka [30] that H1(∆g, V ) = 0, and hence �2g+2 +i=1 [qi] = 0 +and dim Xω ≤ 2g + 1. Since Xζ ⊂ Xω and dim Xζ ≥ 2g + 1, our claim follows. +□ +Remark 9.5. In H1(∆g[2], V ), each Weierstrass class is a linear combination of +Collino classes. In fact, for each i = 1, . . . , 2g + 2, we have +1 +2g + 2 +2g+2 +� +j=1 +([qi] − [qj]) = [qi] − +1 +2g + 2 +2g+2 +� +j=1 +[qj] = [qi], +and so (2g+2)[qi] is an integral combination of 2g+1 Collino classes in H1(∆g[2], V ). +References +[1] N. A’Campo: Tresses, monodromie et le groupe symplectique, Comment. Math. Helv. 54 +(1979), 318–327. +[2] A. Beilinson: Higher regulators and values of L-functions, Current problems in mathematics, +Vol. 24, 181–238, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. +Inform., Moscow, 1984. +[3] J. Birman, H. Hilden: On the mapping class groups of closed surfaces as covering spaces, +in “Advances in the theory of Riemann surfaces,” 81–115, Ann. of Math. Studies, No. 66. +Princeton Univ. Press, 1971. +[4] S. Bloch: Algebraic cycles and higher K-theory, Adv. in Math. 61 (1986), 267–304. +[5] T. Brendle, D. Margalit, A. Putman Generators for the hyperelliptic Torelli group and the +kernel of the Burau representation at t = −1, Invent. Math. 200 (2015), no. 1, 263-310. +[6] J. Carlson: +Extensions of mixed Hodge structures, Journ´ees de G´eometrie Alg´ebrique +d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, pp. 107–127, Sijthoff & Noord- +hoff, Alphen aan den Rijn–Germantown, Md., 1980. + +28 +MA LUO AND TATSUNARI WATANABE +[7] A. Collino: Griffiths’ infinitesimal invariant and higher K-theory on hyperelliptic Jacobians, +J. Algebraic Geom. 6 (1997), 393–415. +[8] E. Colombo: The mixed Hodge structure on the fundamental group of hyperelliptic curves +and higher cycles, J. Algebraic Geom. 11 (2002), 761–790. +[9] C. Earle, I. Kra: On sections of some holomorphic families of closed Riemann surfaces, Acta +Math. 137 (1976), 49-79. +[10] B. Farb and D. Margalit: A primer on mapping class groups, vol. 49, Princeton Math. Series, +Princeton University Press, Princeton, NJ, 2012. +[11] F. Guill´en, V. Navarro Aznar, P. Pascual Gainza, F. Puerta: Hyperr´esolutions cubiques et de- +scente cohomologique, Papers from the Seminar on Hodge-Deligne Theory held in Barcelona, +1982. Lecture Notes in Mathematics, 1335. Springer-Verlag, Berlin, 1988. +[12] R. Hain: Completions of mapping class groups and the cycle C − C−, Mapping class groups +and moduli spaces of Riemann surfaces (G¨ottingen, 1991/Seattle, WA, 1991), Contemp. +Math. 150 (1993) 75–105. +[13] R. Hain: Normal functions and the geometry of moduli spaces of curves, Handbook of moduli. +Vol. I, 527–578, Adv. Lect. Math. (ALM), 24, Int. Press, Somerville, MA, 2013. +[14] R. Hain: The de Rham homotopy theory of complex algebraic varieties. I, K-Theory 1 (1987), +no. 3, 271–324. +[15] R. Hain: The Hodge-de Rham theory of modular groups, in Recent Advances in Hodge Theory, +arxiv [arXiv:1403.6443] (2015), Cambridge University Press. +[16] R. Hain: The Hodge De Rham theory of relative Malcev completion, Annales Scientifiques de +l’Ecole Normale Superieure, vol. 31 no. 1 (1998), pp. 47-92. +[17] R. Hain: Infinitesimal presentations of Torelli groups, J. Amer. Math. Soc. 10 (1997), 597- +651. +[18] R. Hain: Rational points of universal curves, J. Amer. Math. Soc. 24 (2011), 709-769. +[19] R. Hain: Relative weight filtrations on completions of mapping class groups, in Groups of +Diffeomorphisms, Advanced Studies in Pure Mathematics, vol. 52 (2008), pp. 309-368, Math- +ematical Society of Japan. +[20] R. Hain, K. Kordek, and T. Watanabe: Completions of hyperelliptic mapping class groups +unpublished. +[21] R. Hain and M. Matsumoto: Galois actions on fundamental groups of curves and the cycle +C − C−. J. Inst. Math. Jussieu, 4(3):363–403, 2005. +[22] J. Hubbard: +Sur la non-existence de sections analytiques `a la courbe univerelle de +Teichm¨uller, C. R. Acad. Sci. Paris S´er. A-B 274 (1972), A978-A979. +[23] D. Johnson: An abelian quotient of the mapping class group Ig, Math. Ann. 249 (1980), +225–242. MR 87a:57008. +[24] D. Johnson: The structure of the Torelli group—III: The abelianization of I, Topology 24 +(1985), 127–144. MR 87a:57016. +[25] A. Malcev: On a class of homogeneous spaces, Izv. Akad. Nauk SSSR Ser. Mat. 13 (1949), +9–32; English translation, Amer. Math. Soc. Transl. 39 (1962). MR 10:507e. +[26] S. Morita: The extension of Johnson’s homomorphism from the Torelli group to the mapping +class group, Invent. Math. 111 (1993), 197–224. +[27] D. Mumford: Tata lectures on theta, II: Jacobian theta functions and differential equations, +with the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura, +Progress in Mathematics, 43. Birkh¨auser Boston, Inc., Boston, MA, 1984. +[28] M. Saito: Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. +[29] J. Steenbrink, S. Zucker: Variation of mixed Hodge structure. I, Invent. Math. 80 (1985), no. +3, 489–542. +[30] A. Tanaka: The first homology group of the hyperelliptic mapping class group with twisted +coefficients, Topology and its Applications 115 (2001) 19–42. +[31] T. Watanabe: On the rational points of universal hyperelliptic curves, J. Algebra, Volume +533 (2019), 44-89. +School of Mathematical Sciences, East China Normal University, Shanghai +Email address: mluo@math.ecnu.edu.cn +Mathematics Department, Embry-Riddle Aeronautical University, Prescott +Email address: watanabt@erau.edu + diff --git a/YdE5T4oBgHgl3EQfCw6a/content/tmp_files/load_file.txt b/YdE5T4oBgHgl3EQfCw6a/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d75393fe185b2b54856a90a2c2e146eb53b9a23b --- /dev/null +++ b/YdE5T4oBgHgl3EQfCw6a/content/tmp_files/load_file.txt @@ -0,0 +1,1087 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf,len=1086 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='05399v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='AG] 13 Jan 2023 REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS MA LUO AND TATSUNARI WATANABE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A Collino cycle is a higher cycle on the Jacobian of a hyperelliptic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The universal family of Collino cycles naturally gives rise to a normal function, whose induced monodromy relates to the hyperelliptic Johnson ho- momorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Colombo computed this monodromy explicitly and made this relation precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We recast this in the perspective of relative completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In particular, we use Colombo’s result to construct Collino classes, which are co- homology classes of hyperelliptic mapping class groups with coefficients in a certain symplectic representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We also determine the dimension of their span in the case of the level two hyperelliptic mapping class group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Introduction 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Mapping class groups 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Hyperelliptic mapping class groups 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Moduli spaces of hyperelliptic curves 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Relative completions of hyperelliptic mapping class groups 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Hyperelliptic Johnson homomorphisms 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Normal functions and Ceresa cycles 15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Collino cycles and their associated normal functions 18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Hyperelliptic Johnson homomorphisms and Collino classes 24 References 27 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Introduction Denote the mapping class group of a compact topological surface S of genus g with n distinct marked points by Γg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For g ≥ 1, there is a surjective natural rep- resentation Γg,n → Sp(H1(S, Z)) and its kernel is called the Torelli group, denoted by Tg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For a compact Riemann surface C (which we call a curve in this paper), let Jac C be its jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then using a point x in C, we have the algebraic cycle Cx − C− x in Jac C called the Ceresa cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let Tg,n be the Torelli space of marked, n-pointed curves of genus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' There is a bundle J1 → Tg,n over Tg,n of the intermediate jaco- bians whose fiber over [C] is J1(Jac C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The cycle Cx−C− x determines a point eC,x in The first author is supported partly by Science and Technology Commission of Shanghai Mu- nicipality (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 22DZ2229014), and partly by National Natural Science Foundation of China, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 12201217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 1 2 MA LUO AND TATSUNARI WATANABE the intermediate jacobian J1(Jac C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This construction extends to families and de- fines a section eg,1 : Tg,1 → J1 of the bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This section is by definition a normal function (see Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It induces a homomorphism of fundamental groups ξg,1 : Tg,1 → H3(Jac C, Z) = Λ3H1(C), which is equal to twice the Johnson homo- morphism (see [12, §6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' On the other hand, the Johnson homomorphism yields a nontrivial class in H1(Γg,1, H1(S, Q)) and H1(Γg,1, Λ3H1(S, Q)/θ ∧ H1(S, Q)) (see [26], [21, §5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When C is hyperelliptic and x is a Weierstrass point, its corresponding Ceresa cy- cle Cx−C− x is trivial, and in general its image in the primitive jacobian J1(Jac C)prim is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' So instead, we will consider a canonical higher cycle (Z, q1, q2) associ- ated to C with two ordered distinct Weierstrass points q1 and q2 constructed in [7] by Collino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The higher cycle Z can be viewed as a degeneration of the Ceresa cycle for the stable curve obtained from C by gluing q1 and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Collino proves in [7] that the regulator image, reg(Z), of Z is nontrivial for general hyperelliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In [8], Colombo constructs an extension class Pe associated to C with q1 and q2, which is equal to (2g + 1)reg(Z) in the primitive intermediate jacobian I2(Jac C)prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the hyperelliptic Torelli space by Hg[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' There is a normal function PE : Hg[0] → I2prim extending the class Pe and Colombo computes its monodromy action using higher Johnson homomorphisms, which we call hyperel- liptic Johnson homomorphisms in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For a Weierstrass point q of a hyperelliptic curve C, denote the Lie algebra of the unipotent completion of π1(C, q) over Q by p and its derivation algebra by Der p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The Lie algebras p and Der p admit weight filtrations W•p and W• Der p from Hodge theory (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix a hyperelliptic involution σ of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The hyperelliptic mapping class group ∆g is defined as the subgroup of Γg consisting of elements that com- mute with the class [σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The hyperelliptic Torelli group denoted by T ∆g is given by the intersection ∆g ∩ Tg in Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' As a higher Johnson homomorphism, we have the hyperelliptic Johnson homomorphism T ∆g → GrW −2 Der p, denoted by τ hyp q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Com- posing τ hyp q with a certain Sp(H1(C, Q))-equivariant projection of GrW −2 Der p onto Λ2H1(C, Q)/⟨θ⟩, we obtain an Sp(H1(C, Q))-equivariant homomorphism, which we denote by ˜τhyp q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the normal function extending reg(Z) by �RZ and the projection onto the fiber by pI2prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Their composition pI2prim ◦ �RZ induces a ho- momorphism of fundamental groups T ∆g → Λ2H1(C, Z)/⟨θ⟩ by πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' As a remark on Colombo’s work, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' With notation as above, if g ≥ 2, then ˜τ hyp q2 − ˜τ hyp q1 = (g + 1)πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The homomorphism ˜τ hyp q is Sp(H1(C, Q))-equivariant and yields a nontrivial class, denoted by [q], in H1(∆g[2], Λ2H1(C, Q)/⟨θ⟩), where ∆g[2] is the level 2 hyperelliptic mapping class group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We call the class [q] a Weierstrass class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Due to Theorem 1, the homomorphism (g + 1)πZ produces a nontrivial class given by [q2] − [q1], which we call a Collino class in H1(∆g[2], Λ2H1(C, Q)/⟨θ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The fact that reg(Z) is nontrivial for general hyperelliptic curves is equivalent to that the corresponding Collino class is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Our second main result is concerned with the subspace, denoted by Xζ, of H1(∆g[2], Λ2H1(C, Q)/⟨θ⟩) spanned by all Collino classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the subspace of H1(∆g[2], Λ2H1(C, Q)/⟨θ⟩) spanned by all Weierstrass classes by Xω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 3 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' With notation as above, if g ≥ 2, then Xζ = Xω and dim Xζ = 2g+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In fact, the 2g + 2 Weierstrass classes satisfy a single equation �2g+2 i=1 [qi] = 0, and each class (2g + 2)[qi] is an integral combination of (2g + 1) Collino classes (see Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' From the Teichm¨uller theory, it is known that, for g = 2 by Hubbard in [22] and for g > 2 by Earle and Kra in [9], the universal curve over a branch of the hyperelliptic locus in the Teichm¨uller space admits exactly 2g + 2 Weierstrass sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Our result implies that Weierstrass sections satisfy an alge- braic relation via the hyperelliptic Johnson homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' On the other hand, the tautological sections of the universal curve yield linearly independent classes in H1(Γg,n, H1(S, Q)) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is of our interest to investigate further how the relative completion of the level 2 hyperelliptic mapping class group ∆g[2] determines the Weierstrass sections (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Acknowledgments: We are grateful to Richard Hain for helpful discussions on the relative completions of hyperelliptc mapping class groups and letting us use a part of an unpublished notes on the hyperelliptic mapping class groups [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Mapping class groups Fix a smooth, compact oriented surface S of genus g and a subset P of S con- sisting of n distinct points of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Assume that 2g − 2 + n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The mapping class group ΓS,P is defined to be the group of isotopy classes of orientation-preserving diffeomorphisms of S that fix P pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By the classification of surfaces, ΓS,P is independent of a choice of the pair (S, P), and hence we denote ΓS,P by Γg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When n = 0, we denote Γg,0 by Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Level structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote H1(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' R) by HR where R is a commutative ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the algebraic intersection pairing on HR by ⟨ , ⟩ : H⊗2 R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is a unimodular symplectic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Set Sp(HR) = Aut(HR, ⟨ , ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fixing a symplectic basis a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , ag, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , bg of HR gives an isomorphism Sp(HR) with the classical symplectic group Spg(R) consisting of 2g × 2g symplectic matri- ces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The principal congruence subgroup Sp(HZ)[m] of Sp(HZ) of level m ∈ Z is the kernel of the reduction mod m mapping: Sp(HZ)[m] := ker{Sp(HZ) → Sp(HZ/mZ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix a point q in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The action of Γg,n on π1(S, q) induces an action on HZ that preserves the intersection pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, there is a representation ρ : Γg,n → Sp(HZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This is well known to be surjective when R = Z (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For each integer m ≥ 0, we define the level m subgroup of Γg,n to be the kernel of the reduction of ρ mod m: Γg,n[m] = ker{Γg,n → Sp(HZ/mZ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When m = 1, we omit the level notation, so Γg,n[1] = Γg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The Torelli group Tg,n is defined to be the kernel of ρ and it is the the level 0 subgroup of Γg,n: Tg,n = Γg,n[0] = ker{Γg,n → Sp(HZ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 4 MA LUO AND TATSUNARI WATANABE The Torelli groups are torsion free (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' [10, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since Sp(HZ)[m] is torsion free for all m ≥ 3, it follows that Γg,n[m] is torsion free for all m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Hyperelliptic mapping class groups The content of this section comes from an unpublished notes on the completions of hyperelliptic mapping class groups [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix a hyperelliptic involution σ : S → S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is an orientation-preserving diffeo- morphism of order 2 of S with exactly 2g +2 fixed points, which we call Weierstrass points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The quotient space S/⟨σ⟩ is a sphere by the Riemann-Hurwicz formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, it then follows that all hyperelliptic involutions are conjugate in the group of orientation preserving diffeomorphisms of S, denoted by Diff+S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' An orientation-preserving diffeomorphism of S is said to be symmetric if it com- mutes with σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The hyperelliptic mapping class group, denoted by ∆g, is defined to be the group of isotopy classes of orientation-preserving symmetric diffeomorphisms of S: ∆g := π0(centralizer of σ in Diff+S) The following result by Birman and Hilden allows us to consier ∆g as a subgroup of Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 (Birman-Hilden [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The natural homomorphism ∆g → Γg is in- jective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Its image is the centralizer of the isotopy class of σ in Γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Level 2 hyperelliptic mapping class group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the set of fixed points of σ by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The action of ∆ on W yields a homomorphism ρW : ∆g → AutW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It follows from [3, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 1] that the homomorphism ρW is surjective and that there is an extension: (1) 1 → ⟨σ⟩ → ker ρW → Γ0,2g+2 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' To identify the kernel, we need to recall the connection between labeling of the Weierstrass points and level 2 structures on hyperelliptic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' (Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' [27, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=') Denote the F2-valued characteristic function W → F2 of a subset T of the set of Weierstrass points by eT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' There is an F2 linear mapping q : {eT : T ⊆ W, |T | is even} → H1(S, F2) = HF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is defined as follows: let f : S → S2 be the 2-to-1 mapping whose Galois group is generated by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This induces an isomorphism of W with the set of critical values of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If T is an even subset of W then there is a 1-chain γ in S2 whose mod 2 boundary is f(T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Any two such chains differ by a boundary mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Define q(eT ) to be the class in H1(S, F2) of f −1(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Its homology class is well defined mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The mapping q induces a linear isomorphism q : {eT : T ⊆ W, |T | ≡ 0 mod 2}/{eT − eT c : |T | ≡ 0 mod 2} → HF2 where T c denotes the complement W − T of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ The intersection pairing on HF2 corresponds to the pairing eS · eT = #(S ∩ T ) mod 2 on the even subsets of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Thus every automorphism of W induces a symplectic automorphism of HF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' There is a natural faithful representation Aut W ֒→ Sp(HF2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 5 Denote the restriction of Γg to ∆g by ρhyp : ∆g → Sp(HZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For each m ≥ 0, we define the level m subgroup ∆g[m] of ∆g by the reduction of ρhyp mod m: ∆g[m] = ker(∆g → Sp(HZ/mZ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since every curve of genus 2 is hyperelliptic, Γ2[m] ∼= ∆2[m] for all m ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The image of the natural homomorphism ∆g → Sp(HF2) is Aut W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Consequently, the kernel of ρW : ∆g → Aut W is ∆g[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ One therefore has extensions 1 −−−−→ ⟨σ⟩ −−−−→ ∆g[2] −−−−→ Γ0,2g+2 −−−−→ 1 1 −−−−→ ∆g[2] −−−−→ ∆g −−−−→ Aut W −−−−→ 1 (2) Since Γg[m] is torsion free for all m ≥ 3, the same is true for ∆g[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The group ∆g[0] is the kernel of ρhyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We shall call it the hyperelliptic Torelli group and denote it by T ∆g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Image of ρhyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Consider Aut W as a subgroup of Sp(HZ/2Z) via the above faithful representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the inverse image of Aut W under the reduction map Sp(HZ) → Sp(HZ/2Z) by Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We have the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Gg −−−−→ Sp(HZ) \uf8e6\uf8e6� \uf8e6\uf8e6� Aut W −−−−→ Sp(HZ/2Z) Here we recall the following result of A’Campo (see also [27, §8, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='6 ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If g ≥ 2, then the image of ρhyp : ∆g → Sp(HZ) is Gg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In particular, the image of the restriction of ρhyp to ∆g[2] is Sp(HZ)[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Subgroups ∆g,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fixing a labeling W ∼= {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=', 2g + 2} determines an isomorphism of Aut W with the symmetric group S2g+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix a Weierstrass point q ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote its stabilizer in Aut W by Autq W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This is isomorphic to the symmetric group S2g+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The group ∆g,1 is defined by ∆g,1 := (ρW )−1(Autq W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It contains ∆g[2] and is an extension (3) 1 → ∆g[2] → ∆g,1 → Autq W → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' More generally, let Q be a subset of W with |Q| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the subgroup of Aut W fixing Q pointwise by AutQ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This group is isomorphic to the symmetric group S2g+2−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By abuse of notation, for any such subset Q of W, the subgroup ∆g,n of ∆g is defined to be the inverse image of AutQ W under ρW : ∆g,n = (ρW )−1(AutQ W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Similarly, there is an extension 1 → ∆g[2] → ∆g,n → AutQ W → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6 MA LUO AND TATSUNARI WATANABE Similarly, denote the image of ∆g,n under ρhyp in Sp(HZ) by Gg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then, for 0 ≤ n ≤ 2g + 2, there is a commutative diagram of extensions: 1 � T ∆g � ∆g,n � � Gg,n � � 1 1 � T ∆g � ∆g � Gg � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When n = 2g + 2, we denote ∆g,2g+2 by ∆g[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Group cohomology of ∆g and ∆g[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Here we state basic results about cohomology of hyperelliptic mapping class groups ∆g and ∆g[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If V is a rational ∆g-module, then H•(∆g[2], V ) is an Aut W- module and there are natural isomorphisms H•(∆g, V ) ∼= H•(∆g[2], V )Aut W and H•(∆g,1, V ) ∼= H•(∆g[2], V )Autq W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If σ acts as − id on V , then H•(∆g, V ) = H•(∆g[2], V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If σ acts trivially on V , then V is a Γ0,2g+2-module and there is a natural isomorphism H•(∆g[2], V ) ∼= H•(Γ0,2g+2, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The first assertions follow from the Hochschild-Serre spectral sequence of the extensions (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The second assertion follows from the fact that the centralizer of a group acts trivially on the cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since σ is central in ∆g, it acts as a ∆g-linear automorphism of V as − id, and therefore of H•(∆g, V ) and H•(∆g[2], V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' But since the centralizer acts trivially on the cohomology of the group, it follows that σ also acts trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Thus multiplication by 2 annihilates both of these cohomology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since V is a rational representation, our claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The third assertion follows from the Hochschild-Serre spectral sequence of the extension (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Moduli spaces of hyperelliptic curves By a complex curve, or a curve for short, we shall mean a Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that 2g − 2 + n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let S be a reference surface of genus g as in §2 and P a subset of S consisting of n points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the Teichm¨uller space of marked, n-pointed, compact genus g curves by Xg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' As a set Xg,n is � orientation-preserving diffeomorphisms f : S → C to a complex curve � � isotopies, constant on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This is a contractible complex manifold of dimension 3g − 3 + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When n = 0, Xg,0 is denoted by Xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The mapping class group Γg,n acts on Xg,n as a group of biholomorphisms via its action on the markings: φ : f �→ f ◦ φ−1, φ ∈ Γg,n, f ∈ Xg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This action is properly discontinuous and virtually free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The isotropy group of [(S, P) → (C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , xn)] ∈ Xg,n is isomorphic to the set of automorphisms of C that fix {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , xn} pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For each m ≥ 0, the moduli space of n-pointed smooth projective curves of genus g with a level m structure is the quotient of Xg,n by the level m subgroup of Γg,n: Mg,n[m] = � Γg,n[m]\\Xg,n �orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It will be regarded as a complex analytic orbifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is a model of the classifying space of Γg,n[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When the group Γg,n[m] is torsion free, Mg,n[m] is a smooth REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 7 variety and also a fine moduli space for n-pointed smooth projective curves of genus g with a level m structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This occurs, for example, when m ≥ 3, m = 0, or when n > 2g + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that Mg,n[1] = Mg,n and that Mg,n[0] is the quotient of Teichm¨uller space by the Torelli group Tg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is known as Torelli space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We shall denote it by Tg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix a hyperelliptic involution σ of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let Yg be the set of points of Xg fixed by σ: Yg = Xσ g The set Yg consists of points [f : S → C] such that f ◦ σ ◦ f −1 is an automorphism of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is connected and contractible of dimension 2g −1, since it is biholomorphic to X0,2g+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that ∆g is the stabilizer of Yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For each m ≥ 0, as an analytic orbifold, the moduli stack Hg[m] of smooth projective hyperelliptic curves of genus g with a level m structure is the orbifold quotient Hg[m] = (∆g[m]\\Yg)orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When m = 0, Hg[0] is the hyperelliptic Torelli space corresponding to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In fact, the hyperelliptic locus of Xg is the disjoint uniton of translates of Yg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since Yg is contractible, Hg[m] is a model of the classifying space of ∆g[m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Hence we have an isomorphism πorb 1 (Hg[m]) ∼= ∆g[m], where πorb 1 denotes the orbifold fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Similarly, for ∆g,n with 0 ≤ n ≤ 2g + 2, we denote the orbifold quotient of Yg by ∆g,n by Hg,n = (∆g,n\\Yg)orb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that Hg,2g+2 = Hg[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The orbifold fundamental group πorb 1 (Hg,n) is isomor- phic to ∆g,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Universal family over Hg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' As an analytic orbifold, Hg,n admits the uni- versal family πg,n : CHg,n → Hg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is a proper smooth morphism of orbifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since n Weierstrass points are trivialized, the family πg,n admits n distinct sections, which we call Weierstrass sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let x = [C] be in Hg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The fiber of πg,n over x is C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let q be a Weierstrass point of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then associated to πg,n, there is a homotopy exact sequence (4) 1 → π1(C, q) → πorb 1 (CHg,n, q) → πorb 1 (Hg,n, x) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the fundamental group πorb 1 (CHg,n) by ∆C g,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The monodromy action of Hg,n on C yields natural symplectic representations ρg,n : ∆g,n → Sp(HZ) and ρC g,n : ∆C g,n → Sp(HZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='6, the images of ρg,n and ρC g,n contain Sp(HZ)[2] and hence are Zariski dense in Sp(HQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Relative completions of hyperelliptic mapping class groups In this section, we review basics for hyperelliptic mapping class groups and rel- ative completions, while setting up all the notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 8 MA LUO AND TATSUNARI WATANABE 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Relative completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A detailed summary of relative completion can be found in [19, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let F be a field of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that (i) Γ is a discrete group, (ii) R is a reductive F-group, and (iii) ρ : Γ → R(F) is a Zariski-dense homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The relative completion of Γ with respect to ρ consists of a proalgebraic F-group G that is an extension (5) 1 → U → G → R → 1 of R by a prounipotent F-group U and a Zariski-dense homomorphism ˜ρ : Γ → G(F) such that the diagram Γ ˜ρ � ρ �❖ ❖ ❖ ❖ ❖ ❖ ❖ G(F) � R(F) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It satisfies the following universal property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let G be a proalgebraic F-group that is an extension 1 → U → G → R → 1 of R by a prounipotent F-group U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If ρG : Γ → G(F) is a homomorphism that lifts ρ, then there exists a unique homomorphism φ : G → G such that the diagram Γ ρG � ˜ρ � G(F) � φ �♣♣♣♣♣ G(F) � R(F) commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Key properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the Lie algebra of U by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is a pro-nilpotent Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Relative completions are to some degree computable, since it is controlled by cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The extension (5) splits by a generalization of Levi’s Theorem and any two splittings are conjugate by an element of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, the Lie algebra u is an R-module and there is an isomorphism G ∼= R ⋉ U ∼= R ⋉ exp u that is unique up to conjugation by an element of U ∼= exp u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The following result relates the Lie algebra u and the group cohomology of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 ([17, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For all finite dimensional R-modules V , (i) there is a natural isomorphism HomR(H1(u), V ) ∼= H1(Γ, V ), and (ii) there is a natural injection HomR(H2(u), V ) ֒→ H2(Γ, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Relative completions of hyperelliptic mapping class groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Assume that g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Recall that Gg,n is the image of ρg,n : ∆g,n → Sp(HZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note also that the hyperelliptic Torelli group T ∆g is the kernel of ρg = ρg,0 and that ker ρg,n = T ∆g for each n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , 2g + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The representation ρC g,n : ∆C g,n → Sp(HZ) factors through ρg,n, and therefore has the same image Gg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' With abuse of notation, we denote the maps restricted to this image also by ρg,n and ρC g,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Set H = HQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the relative completion of ∆g,n with respect to ρg,n by Dg,n equipped with ˜ρg,n : ∆g,n → Dg(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' There is an extension (6) 1 → Ug,n → Dg,n → Sp(H) → 1 of Sp(H) by a prounipotent radical Ug,n of Dg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the Lie algebras of Dg,n and Ug,n by dg,n and ug,n respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When n = 2g + 2, we denote Dg,n, Ug,n, dg,n, and ug,n by Dg[2], Ug[2], dg[2], and ug[2], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The exact sequence 6 fits in the commutative diagram 1 � T ∆g � ˜ρg,n|T ∆g � ∆g,n � ˜ρg,n � Gg,n � � � � 1 1 � Ug,n � Dg,n �� Sp(H) � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the prounipotent completion of T ∆g over Q by T ∆un g/Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The right exactness of relative completions [19, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7] gives the exact sequence T ∆un g/Q → Dg → Sp(H) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This implies that the homomorphism T ∆un g/Q → Ug is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since the image of T ∆g in T ∆un g/Q is Zariski dense, it follows that the image of ˜ρg,n|T ∆g is Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Similarly, the relative completion of ∆C g,n with respect to ρC g,n : ∆C g,n → Gg,n is denoted by DC g,n and ˜ρC g,n : ∆C g,n → DC g,n(Q) and its prounipotent radical by UC g,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote their Lie algebras by dC g,n and uC g,n respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the unipotent completion of π1(C, q) over Q and its Lie algebra by P and p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The center freeness of P imply that the sequence 4 gives exact sequences (7) 1 → P → DC g,n → Dg,n → 1 and (8) 0 → p → dC g,n → dg,n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Mixed Hodge structures on the relative completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let V be the dual of R1πg,n∗Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is a polarized variation of Hodge structure of weight −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Its monodromy representation can be identified with ρg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By the main theorem of Hain in [16], the Lie algebras of the relative completions dg,n and dC g,n admit natural Q-MHSs, where their Lie brackets are morphisms of MHSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The Lie algebra p admits a natural Q-MHS, being the Lie algebra of the unipotent completion of π1(C, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' On the other hand, it also admits a Q-MHS as the kernel of the surjection dC g,n → dg,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By the naturality properties of the MHS of relative completions [16, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='12], these MHSs on p are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the Lie algebra of Sp(H) by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The basic properties of the weight filtrations of these Lie algebras follow from [16, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2] and listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 10 MA LUO AND TATSUNARI WATANABE Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The weight filtrations W•dg,n, W•dC g,n, and W•p satisfy the prop- erties: (1) W−1dg,n = ug,n = W−1ug,n, W−1dC g,n = uC g,n = W−1uC g,n, and p = W−1p, (2) GrW 0 dg,n = GrW 0 dC g,n = s, and (3) for m ≥ 1, each graded quotients GrW −m dg,n, GrW −m dC g,n, and GrW −m p are Sp(H)-modules, and the induced maps GrW p → GrW dC g,n and GrW dC g,n → GrW dg,n are Sp(H)-equivariant graded Lie algebra homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Hyperelliptic Johnson homomorphisms Fix a Weierstrass point q in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the fundamental group π1(S, q) by Π, and denote the pronilpotent Lie algebra of the unipotent completion (Malcev com- pletion, see [25]) of Π over Q by p as well (The unipotent completions of π1(C) and π1(S) are isomorphic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For a group G, denote the lower central series of G by L•G, where L1G = G and LkG = [Lk−1G, G] for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For each k ≥ 1, denote the associated graded quotient LkG/Lk+1G by GrL k G and the quotient G/LkG by NkG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' There is an exact sequence 1 → GrL k G → Nk+1G → NkG → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When k = 2, we have the exact sequence 1 → GrL 2 G → N3G → H1(G) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The mapping class group Γg,1 acts NkΠ and so there is a homomorphism ρk : Γg,1 → Aut(NkΠ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The action of Γg,1 on GrL k Π factors through the natural representation ρ : Γg,1 → Sp(HZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that when k = 2, N2Π = H1(S, Z) = HZ, ρ2 is the homomorphism ρ, and ker ρ2 = Tg,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The Johnson homomorphism τ : ker ρ2 = Tg,1 → Hom(HZ, GrL 2 Π) is defined by setting φ �→ (u �→ φ(˜u)˜u−1 ∈ GrL 2 Π), where ˜u is any lift of u in N3Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is a Γg,1-equivariant homomorphism, and the induced homomorphism H1(Tg,1) → Hom(HZ, GrL 2 Π) is Sp(HZ)-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is well known that as an Sp(HZ)-module, GrL 2 Π is isomorphic to Λ2HZ/⟨θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In fact, Johnson proved in [23, 24] that the image of τ is contained in Λ3HZ ⊂ Hom(HZ, Λ2HZ/⟨θ⟩) and τ induces an isomorphism H1(Tg,1) → Λ3HZ mod 2-torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In the case of the hyperelliptic mapping class group ∆g,1, the corresponding Johnson homomorphism is trivial as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since the hyperelliptic involution σ commutes with the elements of T ∆g, it acts trivially on T ∆g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' On the other hand, σ acts as − id on Hom(HZ, GrL 2 Π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The triviality of the homomorphism implies that elements of T ∆g act trivially on N3Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' From the exact sequence 1 → GrL 3 Π → N4Π → N3Π → 1, REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 11 we obtain the hyperelliptic Johnson homomorphism τ hyp q : T ∆g → Hom(HZ, GrL 3 Π) φ �→ (u �→ φ(˜u)˜u−1 ∈ GrL 3 Π), where ˜u is any lift of u in N4Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The homomorphism τ hyp q is ∆g,1-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Key Sp(H)-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Set H = HQ and fix a symplectic basis a1, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , ag, bg of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The isomorphism class of each finite dimensional irreducible representation of Sp(H) can be indexed by a partition λ of a nonnegative integer n into less than g parts: n = λ1 + λ2 + · · · + λl with l ≤ g and λ1 ≥ λ2 ≥ · · · ≥ λl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the irreducible Sp(H)-representation whose isomorphism class corresponds to a partition λ by Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let θ = �g i=1 ai ∧ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then for example, we have V[1] ∼= H, V[12] ∼= Λ2H/⟨θ⟩, and V[13] ∼= Λ3H/θ ∧ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The Lie algebras p and Der p as Sp(H)-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since H1(p) is pure of weight −1, it follows that the natural weight filtration on p coincides with the lower central series, that is, W−mp = Lmp for each m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Each associated graded quotient GrW −m p = W−mp/W−m−1p denoted by p(−m) is a finite dimen- sional Sp(H)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let V be a vector space over a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the free Lie algebra generated by V by L(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is the direct sum ⊕k≥1Lk(V ) of components Lk(V ) of bracket length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is well known that the graded Lie algebra GrW p has a minimal presentation GrW p ∼= L(H)/⟨θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The following result shows some of the Sp(H) representations appearing in GrW p for small values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 ([17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For all g ≥ 2, the irreducible decom- position of p(−m) as an Sp(H)-representation for 1 ≤ m ≤ 3 is given by p(−m) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 V[1] for m = 1 V[12] for m = 2 V[2+1] for m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The derivation algebra Der p is also a MHS induced by that of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Another key object we consider is GrW −2 Der p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The derivation algebra Der L(H) of L(H) is given Der L(H) = ⊕k≥0 Hom(H, Lk(H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Furthermore, the derivation GrW Der p = Der GrW p is an Sp(H)-submodule of Der L(H) and is given by GrW Der p = ⊕m≥0 Der−m p, where Der−m p is the Sp(H)-submodule of Hom(H, Lm+1(H)) consisting of deriva- tions that annihilate θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By [17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1], in the representation ring of Sp(H), we have Der−m p = p(−1) ⊗ p(−1 − m) − p(−2 − m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In this paper, the weight −2 component Der−2 p plays an essential role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2 ([17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For all g ≥ 2, the irreducible decom- position of Der−2 p as an Sp(H)-module is given by Der−2 p = V[22] + V[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 12 MA LUO AND TATSUNARI WATANABE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Projection of Der−2 p onto V[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Here we will explain how to identify the V[12]-component of Der−2 p, using the projection used in [8, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1] and give an explicit formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Define an Sp(H)-equivariant map φ : S2Λ2H → Hom(H, L3(H)) by φ : (u1 ∧ v1)(u2 ∧ v2) �→ x �→ ⟨u1, x⟩[v1, [u2, v2]]+⟨v1, x⟩[[u2, v2], u1]+⟨u2, x⟩[v2, [u1, v1]]+⟨v2, x⟩[[u1, v1], u2], where ⟨·, ·⟩ : Λ2H → Z is the algebraic intersection paring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The following result will be useful to describe the image of a Dehn twist under a hyperelliptic Johnson homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' An easy computation gives Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For I ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=', g}, set HI = Span{ai, bi|i ∈ I} and Hc I = Span{ai, bi|i ̸∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let θI = � i∈I ai ∧ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then φ(θ2 I)(x) = � 0 x ∈ Hc I −2[θI, x] x ∈ HI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Furthermore, we have Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The image of φ is in Der−2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let θ = �g i=1 ai ∧ bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then for each (u1 ∧ v1)(u2 ∧ v2), we have φ((u1 ∧ v1)(u2 ∧ v2))(θ) = g � i=1 [−⟨u1, ai⟩bi + ⟨u1, bi⟩ai, [v1, [u2, v2]]] + [−⟨v1, ai⟩bi + ⟨v1, bi⟩ai, [[u2, v2], u1]] + [−⟨u2, ai⟩bi + ⟨u2, bi⟩ai, [v2, [u1, v1]]] + [−⟨v2, ai⟩bi + ⟨v2, bi⟩ai, [[u1, v1], u2]] = [u1, [v1, [u2, v2]]] + [v1, [[u2, v2], u1]] + [u2, [v2, [u1, v1]]] + [v2, [[u1, v1], u2]] = −[[u2, v2], [u1, v1]] − [[u1, v1], [u2, v2]] = [[u1, v1], [u2, v2]] − [[u1, v1], [u2, v2]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Hence, our claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ In fact, it is not difficult to see that φ is a surjection onto Der−2 p by Schur’s Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We define the explicit projection of Der−2 p onto the copy of V[12] as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' First, define an Sp(H)-equivariant map pH : ⊗3H → H by u ⊗ v ⊗ w �→ ⟨u, v⟩w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that the map pH is the dual of the injection θ ⊗ · : H ֒→ ⊗3H and that the composition pH ◦ (θ ⊗ ·) = 2g idH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Secondly, define an Sp(H)-equivariant map pΛ2H : Hom(H, ⊗3H) → Λ2H by γ �→ g � i=1 ai ∧ pHγ(bi) − bi ∧ pHγ(ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Now, we have the identification L3(H) = (Λ2H ⊗ H)/Λ3H and then using the inclusion Λ2H → ⊗2H, u ∧ v �→ u ⊗ v − v ⊗ u, we get the inclusion L3(H) → ⊗3H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 13 Therefore, we have the Sp(H)-equivariant projection Der−2 p → Λ2H given by the composition Der−2 p ֒→ Hom(H, L3(H)) ֒→ Hom(H, ⊗3H) pΛ2H → Λ2H, which we denote by πΛ2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The following result is useful when computing the V[12]- component in Der−2 p of the image of an element of T ∆g under τ hyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The composition πΛ2H ◦ φ : S2Λ2H → Λ2H is given by (u1 ∧ v1)(u2 ∧ v2) �→ 4[⟨u1, v1⟩v2 ∧ u2 + ⟨v2, u2⟩u1 ∧ v1]+ 2[⟨u1, v2⟩v1 ∧ u2 + ⟨v1, u2⟩u1 ∧ v2 + ⟨u1, u2⟩v2 ∧ v1 + ⟨v2, v1⟩u1 ∧ u2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A direct computation suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ Denote the projection Λ2H → Λ2H/⟨θ⟩ by ˜θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' On the other hand, we may view V[12] as a submodule of Λ2H and there is the Sp(H)-projection ˆθ : Λ2H → V[12] given by u ∧ v �→ u ∧ v − ⟨u,v⟩ g θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the composition ˆθ ◦ πΛ2H ◦ φ by ˆπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the map V[12] → S2Λ2H given by multiplication by θ by jθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' An easy computation together with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5 gives Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The composition V[12] jθ→ S2Λ2H ˆπ→ V[12] is given by −4(g + 1) times the identity map idV[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' With notation as above, for each x ∈ S2Λ2H, the vector x − 1 −4(g + 1)(jθ ◦ ˆπ)(x) lands in the V[22]-component of Der−2 p via φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let s = φ � x − 1 −4(g+1)(jθ ◦ ˆπ)(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='6, (ˆθ ◦ πΛ2H)(s) = 0, and hence by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2 and Schur’s Lemma, s is in the V[22]-component of Der−2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Nontriviality of τ hyp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let ci be a separating simple closed curve in S that divides S into two subsurfaces S′ i and S′′ i of genus i and g − i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix a symplectic basis a1, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , ag, bg for HZ such that for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , g − 1, the sets {al, bl|1 ≤ l ≤ i} and {al, bl|i + 1 ≤ l ≤ g} form symplectic bases for H1(S′ i) and H1(S′′ i ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let θ′ i = �i l=1 al ∧ bl and θ′′ i = �g l=i+1 al∧bl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then we have θ = θ′ i+θ′′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the isotopy class of the Dehn twist along ci by di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is an element of the hyperelliptic Torelli group T ∆g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For simplicity, assume that the Weierstrass point q lies in the subsurface S′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3 implies that we have φ((θ′′ i )2)(x) = � 0 x ∈ H1(S′ i) −2[θ′′ i , x] x ∈ H1(S′′ i ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It then follows from the construction of τhyp q with the base point q that we have Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='8 ([31, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If g ≥ 2, then τ hyp q (di) = 1 2φ((θ′′ i )2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 14 MA LUO AND TATSUNARI WATANABE S′ i S′′ i ci q Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Subsurfaces S′ i and S′′ i separated by ci The following result of Brendle, Margalit, and Putman is a key to understand the hyperelliptic Johnson homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A separating curve d is said to be symmetric if σ(d) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='9 ([5, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For g ≥ 0, the group T ∆g is generated by Dehn twists about symmetric separating curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' As immediate consequences of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='8 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='9, we have Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If g ≥ 2, τ hyp q is nontrivial and the image of τhyp q is contained in Der−2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The cohomology class of τ hyp q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Here we associate a cohomology class to τ hyp q via the relative completion as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Assume that n ≥ 1 and ∆g,n fixes q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Consider the homotopy exact sequence 1 � π1(C, q) � πorb 1 (CHg,n, q) πg,n∗ � πorb 1 (Hg,n, x) sq∗ � � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The Weierstrass section sq of the universal curve πg,n : CHg,n → Hg,n given by q induces a section sq∗ of πg,n∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Taking the relative completion of ∆C g,n and ∆g,n produces the exact sequence 7, which yields the exact sequence of prounipotent groups over Q 1 � P � UC g,n � Ug,n ˜sq � � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By the universal property of relative completions, sq∗ induces a section ˜sq of DC g,n → Dg,n, which restricts to a section of UC g,n → Ug,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We denote this restriction by ˜sq as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Applying the log map to the sequence, we obtain the exact sequence of pronilpotent Lie algebras 0 � p � uC g,n � ug,n d˜sq � � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The section ˜sq induces a Lie algebra section d˜sq of uC g,n → ug,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, the Lie algebra ug,n acts on p via the adjoint action of uC g,n on p, and hence we have the adjoint map adjq : ug,n → Der p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since adjq preservers the weight filtrations, we obtain an Sp(H) equivariant graded Lie algebra homomorphism GrW adjq : GrW ug,n → GrW Der p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The proof of [31, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7] implies that H1(ug,n) is pure of weight −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, we have GrW −2 ug,n = GrW −2 H1(ug,n) = H1(ug,n), REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 15 and so GrW −2 adjq can be expressed as an Sp(H)-equivariant map GrW −2 adjq : H1(ug,n) → Der−2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' On the other hand, recall that the restriction of the relative completion ˜ρg,n : ∆g,n → Gg,n to T ∆g induces the map T ∆g → Ug,n, whose image is Zariski dense in Ug,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Composing with the log map Ug,n → ug,n and ug,n → H1(ug,n), we obtain the map rg,n : T ∆g → H1(ug,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Now, since the adjoint map is induced by the conjugation action of ∆C g,n on π1(C, q), the construction of the hyperelliptic Johnson homomorphism implies that there is a commutative diagram T ∆g rg,n � τ hyp q �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ H1(ug,n) GrW −2 adjq � Der−2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the composition ˜θ ◦ πΛ2H ◦ τ hyp q : T ∆g → Λ2H/⟨θ⟩ = V[12] by ˜τ hyp q and the composition ˜θ ◦ πΛ2H ◦ GrW −2 adjq : H1(ug,n) → Λ2H/⟨θ⟩ by ˜τ adj q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since the map rg,n has a Zariski dense image, it follows that the homo- morphism ˜τ adj q is a unique Sp(H)-equivariant map that makes the diagram T ∆g rg,n � ˜τ hyp q �❘ ❘ ❘ ❘ ❘ ❘ ❘ H1(ug,n) ˜τ adj q � Λ2H/⟨θ⟩ commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Now the homomorphism ˜τ adj q corresponds to a class in H1(∆g,n, V[12]), denoted by [q], via the natural isomorphism H1(∆g,n, V[12]) ∼= HomSp(H)(H1(ug,n), V[12]) induced by relative completion in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since ∆g[2] is a subgroup of ∆g,n, there is a natural homomorphism H1(∆g,n, V[12]) → H1(∆g[2], V[12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is easy to see that the image of [q] in H1(∆g[2], V[12]) is equal to the class of q produced by the above construction when n = 2g + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Normal functions and Ceresa cycles Normal functions are holomorphic sections of families of intermediate Jacobians that satisfy certain asymptotic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This notion naturally arise when study- ing families of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In this section, we recall the construction of the normal function associated to a family of homologically trivial algebraic cycles in a family of smooth projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then we review Ceresa cycles and how their associated normal function relates to the Johnson homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' More details can be found in Hain [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 16 MA LUO AND TATSUNARI WATANABE 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Intermediate Jacobians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that X is a smooth projective variety and that Z is an algebraic d-cycle in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We have an exact sequence of mixed Hodge structures 0 → H2d+1(X) → H2d+1(X, Z) → H2d(Z) → H2d(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The class of the cycle Z defines a morphism of mixed Hodge structures cZ : Z(d) → H2d(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If Z is homologically trivial, we can pull back the previous sequence along cZ to obtain an extension 0 → H2d+1(X) → EZ → Z(d) → 0 in the category MHS of mixed Hodge structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Tensoring with Z(−d) gives an extension 0 → H2d+1(X, Z(−d)) → EZ(−d) → Z → 0 and thus a class eZ in Ext1 MHS(Z, H2d+1(X, Z(−d))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that H2d+1(X) has weight −(2d + 1), and thus H2d+1(X, Z(−d)) has weight −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For a mixed Hodge structure V whose weights are all negative, there is a natural isomorphism J(V ) ∼= Ext1 MHS(Z, V ) where J(V ) := VC F 0VC + VZ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In general, by Carlson [6], we have Ext1 MHS(B, A) ∼= J(Hom(B, A)) where Ext1 MHS(B, A) is the set of congruence classes of extensions of B by A for separated mixed Hodge structures A and B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' the highest weight of A is less than the lowest weight of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The class eZ of a homologically trivial d-cycle Z in X can thus be viewed as a class eZ ∈ J(H2d+1(X, Z(−d))), which, in turn, can be viewed as a class in the d-th intermediate Jacobian Jd(X) := (F d+1H2d+1(X))∗/H2d+1(X, Z) as Jd(X) ∼= J(H2d+1(X, Z(−d))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This class can be described explicitly by Griffiths’ generalization of the Abel-Jacobi construction as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Write Z = ∂Γ, where Γ is a topological (2d + 1)-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Each class in F d+1H2d+1(X) can be represented by a closed form in the Hodge filtration F d+1 of the de Rham complex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Stokes’ theorem, integrating these representatives over Γ gives a well defined functional � Γ : F d+1H2d+1(X) → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The choice of Γ is unique up to a topological (2d + 1)-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' So � Γ determines a point in Jd(X) that corresponds to eZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Normal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that T is a smooth variety and that V → T is a variation of mixed Hodge structures of negative weights over T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let V be the corresponding bundle whose fiber over t ∈ T is Vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote by J (V) the bundle over T whose fiber over t is J(Vt) ∼= Ext1 MHS(Z, Vt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A holomorphic section s : T → J (V) of J (V) → T is a normal function if it defines an extension 0 → V → E → ZT → 0 in the category of admissible variations of mixed Hodge structure over T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Admissibility characterizes good variations in the sense of Steenbrink– Zucker[29] and Saito[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It is satisfied in the geometric situations, Guill´en et al[11] and Hain[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' An important characterization is the existence of a relative weight filtration at the infinity, which amounts to certain asymptotic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Families of homologically trivial algebraic cycles give rise to such extensions of variations of mixed Hodge structures and thus normal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Sup- pose that X → T is a family of smooth projective varieties over a smooth base T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that Z is an algebraic cycle in X, which is proper over T of relative di- mension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the fibers of X and Z over t ∈ T by Xt and Zt respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that Zt is homologically trivial in Xt for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The Hodge structures H2d+1(Xt, Z(−d)) form a variation of Hodge structure V over T of weight −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The intermediate Jacobians Jd(Xt) ∼= J(H2d+1(Xt, Z(−d))) form the relative intermediate Jacobian Jd → T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The family of cycles Z defines a section of this bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This is called the normal function of the cycle Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Ceresa cycles and their associated normal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let C be a com- pact Riemann surface of genus g, and JC its Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' When g ≥ 1, for each x ∈ C, the Abel-Jacobi map νx : C → JC y �→ y − x is an embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the image by Cx, which is an algebraic 1-cycle in JC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' There is an involution on JC by D �→ −D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the image of Cx under this involution by C− x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since the involution acts trivially on H2(JC), the algebraic 1-cycle ZC,x := Cx − C− x is homologically trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1, this ZC,x determines a point in the intermediate Jacobian eC,x ∈ J1(JC) ∼= J(H3(JC, Z(−1))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The primitive decomposition H3(JC, Q) = H1(JC, Q) ⊕ PH3(JC, Q) 18 MA LUO AND TATSUNARI WATANABE is the decomposition of H3(JC, Q) into irreducible Sp(H1(C))-modules, the highest weights of the pieces being λ1 and λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Pontrjagin product with the class of C induces a homomorphism Φ : JC → J1(JC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the cokernel of Φ by JQ(JC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By [12, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1], we have eC,x − eC,y = Φ(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It follows that the image of eC,x in JQ(JC) is independent of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The image will be denoted by eC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Now suppose that the genus g ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote by J1 and J1prim the bundles over Mg,n whose fiber over [C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' {x1, · · · , xn}] is J1(JC) and JQ(JC) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We construct sections eg,1 : [C, x] �→ eC,x and eg : [C] �→ eC of J1 → Mg,1 and J1prim → Mg respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' J1 J1prim Mg,1 Mg eg,1 eg By Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 and Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2, we have the following result by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The sections eg,1 and eg are normal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix base points [C, x] ∈ Mg,1 and [C] ∈ Mg respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' So H := H1(C, Z) is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Taking fundamental groups, the normal functions eg,1 and eg induce Sp(H)- module homomorphisms ξg,1 : H1(Tg,1, Z) → H1(J1(JC), Z) ∼= Λ3H1(C, Z) and ξg : H1(Tg, Z) → H1(JQ(JC), Z) ∼= Λ3H1(C, Z)/H1(C, Z) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The following result is shown in Hain [12, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The map ξg,n is twice the Johnson homomorphism τg,n for n = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Collino cycles and their associated normal functions In this section, we review Colombo’s results on Collino cycles [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since these can be viewed as degenerations of Ceresa cycles, they give rise to elements in higher Chow groups of the Jacobian JC of a hyperelliptic curve C [7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Their regulator images, defined in [2, 4], can be expressed in terms of iterated integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' They are thus identified with some specific extensions constructed from the fundamental group of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' As in the case of Ceresa cycles, we can construct the normal functions associated to Collino cycles and compute their induced monodromies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let X be a smooth projective variety of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' An ele- ment in the first higher Chow group CHn(X, 1) is defined by A := � i (Ci, fi) where Ci is an irreducible curve on X and fi a rational function on Ci such that � i [div(fi)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 19 Let γi := f −1 i ([0, ∞]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The above condition tells us that η := � i γi is a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In fact, it is homologically trivial, so that η = ∂D where D is a 2-chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then the regulator map reg : CHn(X, 1) → I2(X) := (F 1H2(X))∗/H2(X, Z(1)) A �→ reg(A) where reg(A) denotes the current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' linear functional on F 1H2(X), taking α �→ � i � Ci−γi log(fi)α + 2πi � D α for every α ∈ F 1H2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A similar definition of regulator is given by reg : CHn(X, 1) → I2(X)prim := (F 1H2(X)prim)∗/H2(X, Z(1))prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' using the primitive part H2 prim of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Collino cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A Collino cycle Z is a canonical higher cycle, depending on the choice of two Weierstrass points of a genus g hyperelliptic curve C, on the Jacobian JC of C [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix Weierstrass points q1, q2, let h be a degree 2 morphism h : C → P1 such that h(q1) = 0 and h(q2) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Recall that for each x ∈ C, we denote by Cx the image under the Abel-Jacobi map νx : C → JC y �→ y − x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let hs := h ◦ ν−1 qs be a function on Cs := Cqs = νqs(C) for s = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then Z := (C1, h1) + (C2, h2) ∈ CHg(JC, 1) is called a Collino cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Its regulator is nonzero for general C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In fact, it can be computed using iterated integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 (Thm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let φ and ψ be harmonic 1-forms on JC with ψ of type (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We use the same notation for their pullback to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then reg(Z)(φ ∧ ψ) = 2 � C−γ log(h)φ ∧ ψ + 2πi � γ (φψ − ψφ) where γ := h−1([0, ∞]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' One of the forms ψ being type (1, 0) makes φ ∧ ψ a representative of an element in F 1H2(JC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 20 MA LUO AND TATSUNARI WATANABE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Colombo’s construction of extension class from fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Colombo relates the regulator image reg(Z) to an extension class Pe, primitive part of an extension class e constructed from the fundamental groups of the punctured curves C −{q1} and C −{q2} with the same base point p, another Weierstrass point of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' More precisely, as being done for the regulator, these extension classes are expressed in terms of iterated integrals on the Jacobian JC of the curve C, and Colombo shows that Pe is a rational multiple of reg(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We first review her construction1 of extensions e and Pe from the MHS on the fundamental groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix the base point p for the fundamental group π1(C −{q}, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote by Lq the augmentation ideal2 of the group algebra Zπ1(C − {q}, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The powers of Lq gives a natural filtration · · ⊆ Lk+1 q ⊆ Lk q ⊆ · · · ⊆ Lq ⊆ Zπ1(C − {q}, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' So we have natural extensions (9) 0 → (Lq/Lk q)∗ → (Lq/Lk+1 q )∗ → (Lk q/Lk+1 q )∗ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The graded pieces has pure Hodge weights as (Lk q/Lk+1 q )∗ ≃ ⊗kH1(C, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' To simplify notation, we denote H1(C, Z) by H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' If k = 2, the above sequence (9) becomes 0 → H1 → (Lq/L3 q)∗ → ⊗2H1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In particular, when q = qs (s = 1, 2) is a Weierstrass point, this extension splits and has a natural retraction rs : (Lqs/L3 qs)∗ → H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Pushing along this retraction on the sequence (9) for k = 3, q = qs 0 (Lqs/L3 qs)∗ (Lqs/L4 qs)∗ (Lqs/L3 qs)∗ 0 0 H1 Es ⊗3H1 0 rs ≃ we get extension class es ∈ ExtMHS(⊗3H1, H1) represented by Es for s = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We introduce several natural morphisms of MHS: (1) Tensoring with the polarization Ω: JΩ : H1(−1) → ⊗3H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' (2) The surjection: Π : ⊗2H1 → Z(−1), which is the composition of the cup product with the isomorphism H2(C, Z) ≃ Z(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' (3) The standard inclusion: ι : Λ2H1 → ⊗2H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 1We have another construction for these same extensions, but since Colombo’s construction is already in the literature, our construction is not necessary here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' the kernel of the augmentation map Zπ1(C − {q}, p) → Z that sends each element of the fundamental group to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 21 (4) The integration over C: � C : Λ2H1 → Z which is the map Π ◦ ι up to a Tate twist Z(−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since we have natural isomorphism Λ2H1 ≃ H2(JC, Z), we can identify the kernel ker � C ∼= H2(JC)prim with the primitive part of H2(JC, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Carlson, we can identify Ext1 MHS(Λ2H1, Z) ∼= I2(JC) and Ext1 MHS(ker � C , Z) ∼= I2(JC)prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Now we are ready to construct the extension classes e and Pe, represented by the extensions E and PE respectively in the following diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 0 H1 E2 − E1 ⊗3H1 0 0 H1 EΩ H1(−1) 0 0 ⊗2H1 EΩ ⊗ H1 ⊗2H1(−1) 0 0 Z(−1) �E(−1) ⊗2H1(−1) 0 0 Z �E ⊗2H1 0 0 Z E Λ2H1 0 0 Z PE ker � C 0 ⊗H1 ⊗H1 JΩ ⊗H1 Π ⊗Z(1) ⊗Z(1) ⊗Z(1) ι Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3 (Thm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let C be a hyperelliptic curve with Weierstrass points q1, q2 and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let h be a degree 2 morphism h : C → P1 such that h(q1) = 0 and h(q2) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then we have e = (2g + 1) � reg(Z) + log(h(p)) � C � ∈ I2(JC) and Pe = (2g + 1)reg(Z) ∈ I2(JC)prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Although the computation of Pe involves the base point p, it only depends on q1 and q2, since it is a rational multiple of reg(Z), whose construction does not involve p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 22 MA LUO AND TATSUNARI WATANABE 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The normal functions and their induced monodromies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' One can extend the constructions of both reg(Z) and Pe for a hyperelliptic curve to families of hyperelliptic curves [8, §4, §5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' They give rise to normal functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' sections on variations of mixed Hodge structures over the hyperelliptic Torelli space Hg[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We make this precise in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Recall that the hyperelliptic Torelli space Hg[0] is the moduli space of hyperel- liptic curves of genus g with a fixed symplectic basis of homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let π : CHg[0] → Hg[0] and I2prim → Hg[0] be the universal hyperelliptic curve and the bundle over the hyperelliptic Torelli space Hg[0], whose fiber over the moduli point [C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' {aj, bj}g j=1] ∈ Hg[0] is I2(JC)prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' After choosing sections of Weierstrass points C Hg[0] q1 q2 p we can construct reg(Z) and Pe for each fiber, and they assemble to form sections rZ and PE of the family I2prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' I2prim I2prim Hg[0] Hg[0] rZ P E Note that by Hain[14], I2prim is an admissible variation of mixed Hodge structures, since each of its fiber I2(JC)prim is naturally constructed from fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Each fiber I2(JC)prim can be identified with I2(JC)prim ∼= Ext1(H2(JC)prim, Z) ∼= Ext1(Z, H2(JC)prim(2)) ∼= J(H2(JC)prim(2)) using Poincar´e duality in the middle, where H2(JC)prim(2) := H2(JC)prim ⊗ Z(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1, the sections rZ and PE are normal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' These normal functions induce monodromies which we now compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that the monodromy action of the hyperelliptic Torelli group T ∆g is trivial, so the as- sociated bundle I2prim is a trivial bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix a base point [C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' {aj, bj}g j=1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote by pI2prim the projection of I2prim to its fiber I2(JC)prim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The fundamental group of the fiber I2(JC)prim is Λ2H/⟨θ⟩ where H is the first homology of C generated by the fixed basis {aj, bj}g j=1, and θ = �g j=1 aj ∧ bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote by pZ and pE the compositions of the projection pI2prim with normal functions rZ and PE, and they respectively induce homomorphisms of fundamental groups πZ = (pZ)∗ : T ∆g → Λ2H/⟨θ⟩ and πE = (pE)∗ : T ∆g → Λ2H/⟨θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Colombo compared these monodromies on a particular element di ∈ T ∆g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The element is chosen to be the class of a Dehn twist of a simple closed curve ci on C seperating q1 and q2, where ci is invariant under the hyperelliptic involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5 (Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' πE(di) = (2g + 1)πZ(di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This theorem can easily be improved, as indicated by Colombo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 23 Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' πE = (2g + 1)πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3, Pe and (2g+1)reg(Z) are identified on each fiber I2(JC)prim of I2prim → Hg[0], so they induce the same homomorphisms of fundamental groups (PE)∗ = (2g + 1)(rZ)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The result follows from composing this with (pI2prim)∗, which is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ Moreover, using Colombo’s computation the monodromies on particular gener- ators di of the hyperelliptic Torelli group T ∆g [8, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2], we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let di ∈ T ∆g be the class of the Dehn twist of a separating simple closed curve ci on C, where ci is invariant under the hyperelliptic involution, then πZ(di) = � 0 if ci does not separate q1 and q2 4θ′′ i if ci separates q1 and q2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We follow the same steps as in [8, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1] to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We provide full details here so that interested readers can compare it with Colombo’s computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' First, we set some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Pick a base point [C] ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let d ∈ T ∆g be the class of a Dehn twist Dd of a separating simple closed curve c on C, invariant under the hyperelliptic involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let λd be the loop in H based at [C], that corresponds to the Dehn twist Dd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This loop lifts to a path ˜λd : [0, 1] → ˜H in the universal covering ˜H of H with ˜λd(0) = [C] and ˜λd(1) = [DdC].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' There is a universal family of hyperelliptic curves over the path ˜λd and we denote the fiber over ˜λd(t) by Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In particular, C0 = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The sections q1 and q2 over λd lift to ˜q1 and ˜q2 over ˜λd, which on each fiber Ct correspond respectively to the zero and the pole of a degree 2 map ht : Ct → P1 that we chose to construct the Collino cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let γt := h−1 t ([0, ∞]) be the path on Ct, then h0 = h and γ0 = γ are the same as those defined in Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' normal function, rZ restricts to the loop λd in H, and it can be lifted to a normal function ˜rZ along ˜λd in ˜H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Now we compute the monodromy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For any t ∈ [0, 1], ˜rZ(t) is the regulator of the Collino cycle constructed from Weierstrass points ˜q1(t) and ˜q2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1, we have ˜rZ(t)(φt ∧ ψt) = 2 � Ct−γt log(ht)φt ∧ ψt + 2πi � γt (φtψt − ψtφt) for closed 1-forms φt and ψt on Ct, with ψt of type (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By covering theory, we have πZ(d) = 1 2π [˜rZ(1) − ˜rZ(0)] ∈ Λ2H/⟨θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We can choose φ1 = φ0 =: φ and ψ1 = ψ0 =: ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' So [˜rZ(1) − ˜rZ(0)](φ ∧ ψ) = 2 �� C1−γ1 log(h1)φ ∧ ψ − � C−γ log(h)φ ∧ ψ � + 2πi �� γ1 (φψ − ψφ) − � γ (φψ − ψφ) � 24 MA LUO AND TATSUNARI WATANABE Case(i): If c = ci does not separate q1 and q2, then it is easy to see that we can choose the same branch for the logarithm log(h1) = log(h) because q1 and q2 are on the same subsurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Moreover, γ does not intersect with ci, so that the Dehn twist does not change γ and γ1 = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, on the right hand side of the above equation, both terms vanish and we have πZ(d) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Case(ii): If c = ci separates q1 and q2, then the result follows directly from [8, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The essential changes are that of choosing a different branch of log(h) and that the Dehn twist carrying γ to γ1 = γ + 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Hyperelliptic Johnson homomorphisms and Collino classes In this section, we relate the results in the previous sections, with a point of view from relative completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Recall that by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2, we have the decomposition Der−2 p = V[22] + V[12] as an Sp(H)-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Set V = V[12] and V ′ = V[22] for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Recall that ˜θ is the projection Λ2H → Λ2H/⟨θ⟩ ∼= V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the composition ˜θ ◦ πΛ2H (see §4) by ˜πΛ2H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A Collino class in H1(∆g,2, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let q1 and q2 be distinct Weierstrass points in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Recall from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5 that we have the classes [q1] and [q2] in H1(∆g,2, V ) given by their corresponding hyperelliptic Johnson homomorphisms τ hyp qi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let ζ = [q2] − [q1] in H1(∆g,2, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Via the isomorphism H1(∆g,2, V ) ∼= HomSp(H)(H1(ug,2), V ), the class ζ corresponds to the Sp(H)-equivariant map ˜τ adj ζ := ˜τ adj q2 − ˜τ adj q1 : H1(ug,2) → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Denote the map ˜τ hyp q2 − ˜τ hyp q1 : T ∆g → V by ˜τ hyp ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that there is a commutative diagram T ∆g ˜τ hyp ζ �● rab � H1(ug,2) ˜τ adj ζ � V, where the map rab is induced by the relative completion of ∆g,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' With notation as above, if g ≥ 2, we have ˜τ hyp ζ = (g + 1)πZ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let D in T ∆g be the class of the Dehn twist along a simple separating curve d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In the case where d does not separate q1 and q2, it follows from Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='8 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7 that both ˜τhyp ζ (D) and πZ(D) are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' So consider the case where d separates q1 and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Say S is divided by d into two subsurfaces S′ i of genus i and S′′ i of genus g − i, which contain q1 and q2, respectively, as in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix symplectic bases a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , ai, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , bi and ai+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , ag, bi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , bg for S′ i and S′′ i , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let θ′ i = �i ℓ=1 aℓ ∧ bℓ and θ′′ i = �g ℓ=i+1 aℓ ∧ bℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We have REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 25 θ = θ′ i+θ′′ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='8, the image (τ hyp q2 −τ hyp q1 )(D) can be represented as ζD := 1 2φ((θ′ i)2 − (θ′′ i )2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' An easy computation using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5 shows that the projection of the derivation ζD onto V by ˜πΛ2H is given by ˜πΛ2H(ζD) = 2(2g + 2)θ′′ i mod θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7, we have πZ(D) = 4θ′′ i , and hence ˜τ hyp ζ (D) = (g + 1)πZ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Our claim follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='9 stating that T ∆g is generated by the classes of Dehn twists along symmetric separating simple closed curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Consequently, the Collino cycle (Z, q1, q2) determined by q1 and q2 yeilds a nontrivial class in H1(∆g,2, V ), where ∆g,2 fixes q1 and q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Weierstrass subspace of H1(∆g[2], V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Recall from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1 that ∆g[2] fixes all of the Weierstrass points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore for each Weierstrass point q, we have the hyperelliptic Johnson homomorphism τ hyp q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For two distinct Weierstrass points q1 and q2, the image of τ hyp q2 −τ hyp q1 is contained in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' With notation from the proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1, we will show that the projec- tion of ζD in the V ′-component of Der−2 p is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let ζ′ D be the V ′-part of ζD in Der−2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' We have (ˆθ ◦ πΛ2H)(ζD) = 2(2g + 2) � θ′′ i − g − i g θ � , where ˆθ : Λ2H → V is the projection given by u ∧ v − ⟨u,v⟩ g θ (see §4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7, the vector 1 2((θ′ i)2 − (θ′′ i )2) + θ′′ i θ − g − i g θ2 maps into the V ′-component of Der−2 p under φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3, the image of θ2 is trivial because φ(θ2) = −2adjθ, and since θ = θ′ i + θ′′ i , we have ζ′ D = φ �1 2((θ′ i)2 − (θ′′ i )2) + θ′′ i θ − g − i g θ2 � = φ �1 2((θ′ i)2 − (θ′′ i )2) + θ′′ i θ � = φ �1 2(θ′ i)2 + 1 2(θ′′ i )2 + θ′ iθ′′ i � = 1 2φ((θ′ i)2 + 2θ′ iθ′′ i + (θ′′ i )2) = 1 2φ(θ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This implies that ζ′ D is zero in Der−2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, ζD is in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since D is arbitrary, it follows that the image of τhyp q2 − τ hyp q1 is contained in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Note that when D is the Dehn twist along a simple separating curve that does not separate q1 and q2, τ hyp q1 (D) = τ hyp q2 (D), and so (τ hyp q2 − τ hyp q1 )(D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ 26 MA LUO AND TATSUNARI WATANABE We call the image of ζ in H1(∆g[2], V ) via the homomorphism H1(∆g,2, V ) → H1(∆g[2], V ) as a Collino class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since ˜τ hyp ζ is a nontrivial homomorphism, It follows that each Collino class in H1(∆g[2], V ) is nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Similarly, for each Weierstrass point q, we call the class [q] in H1(∆g,1, V ) and its image under the homomorphism H1(∆g,1, V ) → H1(∆g[2], V ) as a Weierstrass class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let Xζ be the subspace of H1(∆g[2], V ) spanned by all Collino classes and let Xω be the subspace of H1(∆g[2], V ) spanned by all Weierstrass classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' With notation as above, if g ≥ 2, then Xζ = Xω and dim Xω = 2g + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , q2g+2 be the Weierstrass points of S and W = {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , q2g+2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then we have the Weierstrass classes [qi] in H1(∆g[2], V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Set p = q2g+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , 2g + 1, define ζi by ζi = [qi] − [p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='1, each ζi is a Collino class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that (*) 2g+1 � i=1 ciζi = 0, where each ci is in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then we have �2g+1 i=1 ci˜τ adj ζi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Recall that the subgroup Autp W of Aut W fixing p is isomorphic to S2g+1 and that it acts on H1(∆g[2], V ) and hence on HomSp(H)(H1(ug[2]), V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let t be a cycle of length 2g + 1 in Autp W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then we have 2g+1 � j=1 tj �2g+1 � i=1 ci˜τ adj ζi � = 2g+1 � j=1 2g+1 � i=1 ci˜τ adj tj(ζi) = 2g+1 � j=1 �2g+1 � i=1 ci � ˜τ adj ζj = �2g+1 � i=1 ci � 2g+1 � j=1 ˜τ adj ζj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Suppose that �2g+1 i=1 ci ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then we have �2g+1 i=1 ˜τ adj ζi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Now for every D in T ∆g, we have ˜θ ◦ πΛ2H �2g+1 � i=1 (τ hyp qi − τhyp p )(D) � = 2g+1 � i=1 ˜τ adj ζi (rab(D)) = 0, where rab is the homomorphism T ∆g → H1(ug[2]) induced by the relative comple- tion of ∆g[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' By Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='3, it then follows that 2g+1 � i=1 (τ hyp qi − τhyp p )(D) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Thus we have �2g+1 i=1 (τ hyp qi − τ hyp p ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' This last equation becomes 2g+1 � i=1 τ hyp qi = (2g + 1)τhyp p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Now, as in 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='4, let D in T ∆g be the Dehn twist about a simple separating closed curve d separating q1 and p such that d separates S into two subsurfaces S′ k of genus REMARKS ON COLLINO CYCLES AND HYPERELLIPTIC JOHNSON HOMOMORPHISMS 27 k and S′′ k of genus g − k, which contain q1 and p, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Fix symplectic bases a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , ak, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , bk and ak+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , ag, bk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , bg for S′ k and S′′ k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Let θ′ k = �k ℓ=1 aℓ ∧ bℓ and θ′′ k = �g ℓ=k+1 aℓ ∧ bℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Set x = ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Then 2g+1 � i=1 τ hyp qi (D)(x) = lτ hyp q1 (D)(x) = −l[θ′′ k, x], where l is the number of Weierstrass points contained in S′ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since l ≥ 1, this is a nontrivial element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' On the other hand, (2g + 1)τ hyp p (D)(x) = 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, �2g+1 i=1 ci = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' So substituting c2g+1 = − �2g i=1 ci into the equation (*), we obtain (**) 2g � i=1 ci(ζi − ζ2g+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Setting p = q2g+1 and replacing ζi with [qi] − [p] for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , 2g, the eqation (∗∗) becomes 2g � i=1 ciζi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Inductively, we get c1([q1] − [q2]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' The class [q1] − [q2] is nontrivial, and so c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' From the inductive steps, it follows that each ci = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , 2g + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Therefore, dim Xζ ≥ 2g + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' On the other hand, we observe that the vector �2g+2 i=1 [qi] is fixed by the action of S2g+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Thus, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='7, it is in H1(∆g, V ) = H1(∆g[2], V )S2g+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' It follows from a result of Tanaka [30] that H1(∆g, V ) = 0, and hence �2g+2 i=1 [qi] = 0 and dim Xω ≤ 2g + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Since Xζ ⊂ Xω and dim Xζ ≥ 2g + 1, our claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' □ Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In H1(∆g[2], V ), each Weierstrass class is a linear combination of Collino classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' In fact, for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' , 2g + 2, we have 1 2g + 2 2g+2 � j=1 ([qi] − [qj]) = [qi] − 1 2g + 2 2g+2 � j=1 [qj] = [qi], and so (2g+2)[qi] is an integral combination of 2g+1 Collino classes in H1(∆g[2], V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' A’Campo: Tresses, monodromie et le groupe symplectique, Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content=' 54 (1979), 318–327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} 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mluo@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='cn Mathematics Department, Embry-Riddle Aeronautical University, Prescott Email address: watanabt@erau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdE5T4oBgHgl3EQfCw6a/content/2301.05399v1.pdf'} diff --git a/Z9E3T4oBgHgl3EQfcwrJ/content/tmp_files/2301.04529v1.pdf.txt b/Z9E3T4oBgHgl3EQfcwrJ/content/tmp_files/2301.04529v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..16fece112c1c688b0e694d098706a689995d5909 --- /dev/null +++ b/Z9E3T4oBgHgl3EQfcwrJ/content/tmp_files/2301.04529v1.pdf.txt @@ -0,0 +1,1319 @@ +Detecting Primordial Stochastic Gravitational Waves with Reduced Astrophysical Foregrounds +Zhen Pan1, ∗ and Huan Yang1, 2, † +1Perimeter Institute for Theoretical Physics, Ontario, N2L 2Y5, Canada +2University of Guelph, Guelph, Ontario N1G 2W1, Canada +(Dated: January 12, 2023) +One of the primary targets of third-generation (3G) ground-based gravitational wave (GW) detectors is de- +tecting the stochastic GW background (SGWB) from early universe processes. The astrophysical foreground +from compact binary mergers will be a major contamination to the background, which must be reduced to high +precision to enable the detection of primordial background. In this Letter, we revisit the limit of foreground +reduction computed in previous studies, and propose a novel cleaning method subtracting the approximate sig- +nal strain and removing the average residual power. With this method, the binary black hole foreground can be +reduced with fractional residual energy density below 10−4 for frequency f ∈ (10, 102) Hz, below 10−3 for fre- +quency f ∈ (102, 103) Hz and below the detector sensitivity limit for all relevant frequencies. Similar precision +can be achieved to clean the foreground from binary neutron stars (BNSs) that are above the detection threshold, +so that the residual foreground is dominated by sub-threshold BNSs, which will be the next critical problem to +solve for detecting the primordial SGWB in the 3G era. +Introduction. Primordial stochastic gravitational wave back- +ground (SGWB) from various physical processes from the +early universe has been investigated, including inflation [1, 2] +and preheating [3, 4], first-order phase transitions [5–10] and +cosmic strings [11–16] (see [17–22] for more complete re- +views). Measuring the primordial SGWB at different frequen- +cies will open a unique window to the universe at the earliest +moments. Therefore the primordial SGWB detection has been +one of the primary targets for gravitational wave (GW) detec- +tors in different frequency bands, including pulsar timing ar- +rays [23–26], spaceborne GW detectors [27, 28] and ground +based detectors [29–31]. +In addition to the primordial SGWB, GWs from various +astrophysical sources are much better understood and mea- +sured [32–35]. In the sensitive band of ground based detec- +tors, compact binaries, including binary black holes (BBHs), +black hole-neutron star binaries (BHNSs) and binary neutron +stars (BNSs) are the dominant source of astrophysical fore- +ground [36–39]. In order to improve the sensitivity of probing +the primordial background, the effect of astrophysical fore- +ground must be reduced. +Cleaning the astrophysical fore- +ground with the residual foreground energy density below the +detector sensitivity limit has been argued to be possible in the +era of third-generation (3G) ground-based detectors, e.g., Ein- +stein Telescope (ET) [40] and Cosmic Explorer (CE) [41–43], +which are so sensitive that almost all compact binary mergers +are expected to be detected and subtracted out [44]. However, +recently Zhou et al. [45, 46] pointed out the straightforward +event subtraction proposed in [44] only removes ∼ 50% of the +foreground noise. The resulting astrophysical foreground is +way above the detector sensitivity, which poses severe chal- +lenges of detecting primordial GWs [47]. +In this Letter, we will show that the astrophysical fore- +ground can be cleaned by orders of magnitude, with a novel +cleaning method detailed later. Applying this method to 3G +detectors, the BBH foreground and the foreground from in- +dividually resolved BNSs can be reduced to be well below +the detector sensitivity limit. As a result, the residual fore- +ground is expected to be dominated by sub-threshold BNSs. +Notice that the noise reduction of sub-threshold events may +be achieved following a Bayesian framework, as discussed in +[48–50]. One subtlety may be that this method requires enor- +mous computational cost, and it is more susceptible to non- +Gaussian noise in detectors. For space-borne detectors, it has +also been shown that astrophysical foreground cleaning may +benefit from multi-band observations [51]. +In this Letter, we use geometrical units G = c = 1 and we +assume a flat ΛCDM cosmology with H0 = 70 km/s/Mpc, +ΩΛ = 0.7 and Ωm = 0.3. +Stochastic GW Foreground from Compact Binary Merg- +ers. The energy density of stochastic GWs per logarithmic +frequency is related to its power spectrum density (PSD) by +(our notation is different from that in [52, 53] by a factor 8π) +ΩGW( f) := +1 +ρcrit +dρGW +d ln f = 4π2 +3H2 +0 +f 3H( f) , +(1) +with ρcrit := 3H2 +0/8π the critical energy density to close the +universe and the PSD (see e.g., [51, 53, 54] for derivation) +H( f) = 1 +T +� +i +� +|h+( f)|2 + |h×( f)|2� +i , +(2) +where the index i runs over all binaries in the universe that +merger within the observation time span (0, T) , and h+,× are +the two polarizations of incoming GWs. Driven by the incom- +ing GWs, the detector strain responds as +h( f) = F+(θ, φ, ψ)h+( f) + F×(θ, φ, ψ)h×( f) , +(3) +where the attena pattern F+,× depend on the source sky loca- +tion (θ, φ) and the source polarization angle ψ. In terms of the +detector strain, the PSD is expressed as +H( f) = +2 +⟨F2 ++⟩ + ⟨F2 +×⟩ +1 +T +� +i +|h( f)|2 +i = 5 +T +� +i +|h( f)|2 +i , +(4) +where ⟨⟩ is an average over the three attena pattern depen- +dent angles. +It is known that ⟨F2 ++⟩ = ⟨F2 +×⟩ = 1/5 for +arXiv:2301.04529v1 [gr-qc] 11 Jan 2023 + +2 +LIGO/Virgo/KAGRA (LVK) like L-shape interferometers and +⟨F2 ++⟩ = ⟨F2 +×⟩ = 3/20 for LISA/ET like triangle-shape interfer- +ometers (see e.g., [55] for details). We have chosen a L-shape +interferometer as the reference detector in the second equal +sign of the equation above. +A typical non-precessing BBH waveform depends on +7 model parameters h+,×(Mz, Mz, χ, tc, φc, ι, DL): the (red- +shifted) chirp mass Mz = (1 + z)M, the (redshifted) total +mass Mz = (1 + z)M, the effective spin χ, the coalescence +time tc (arriving at a detector), the coalescence phase φc, the +inclination angle ι, the luminosity distance DL. Explicitly, the +detector strain is formulated as +h( f) = +� +5 +24 +f −7/6 +π2/3 ζAphen( f)ei[2πftc+φ0+Ψphen( f)] , +(5) +where the strain phase φ0 is defined as +e2iφ0 := e2iφc F+(1 + cos2 ι)/2 − iF× cos ι +� +F2 ++( 1+cos2 ι +2 +)2 + F2 +× cos2 ι +, +(6) +and the strain amplitude +ζ := M5/6 +z +DL +� +F2 ++(1 + cos2 ι +2 +)2 + F2 +× cos2 ι . +(7) +The waveform dependence on the intrinsic binary parameters +is encoded in Aphen( f; Mz, Mz, χ) and Ψphen( f; Mz, Mz, χ), the +explicit expressions of which differ in different waveform +models. In this work, we will use the simple PhenomB wave- +form model [56] (the foreground cleaning results have little +change if other waveform models are applied instead, e.g., +PhenomC or PhenomD [57–59]). +Because of parameter degeneracies, not all the binary pa- +rameters can be well constrained even for a loud merger event, +e.g., the coalescence phase φc is in general weakly constrained +due to its degeneracy with angles {ι, θ, φ, ψ}. To mitigate the +parameter degeneracy, we instead use two detector dependent +parameters that are of weak degeneracy with other parameters +(similar parameterization was used in [60] for efficient param- +eter inference): the strain amplitude ζ [Eq. (7)] and the strain +phase φopt at the optimal frequency +φopt := 2πfopttc + φ0 + Ψphen( fopt) . +(8) +The optimal frequency fopt := +� |h(f)|2 +Pn(f) f d f +� � |h( f)|2 +Pn( f) d f is the +frequency where the waveform is best constrained, with Pn( f) +being the detector noise PSD. To summarize, we parametrized +the PhenomB waveform with the following 10 model param- +eters {Mz, Mz, χ, ζ, tc, φopt} + {ι, θ, φ, ψ}. In this parameteriza- +tion, the detector strain h( f) depends on the first 6 parameters +[see Eqs. (5,8)], and the remaining 4 can only be measured +with multiple detectors. Note that {ζ, tc, φopt} are detector de- +pendent quantities and we choose the most sensitive detec- +tor as the reference detector if multiple detectors are in use. +It turns out that this re-parametrization of parameters signif- +icantly alleviates the errors from signal subtraction, as dis- +cussed later. +Foreground Cleaning Method. +For an incoming binary +merger signal h( f; Θ) at a detector, one can infer its ML (Max- +imum Likelihood) estimate h( f; ΘML) along with the poste- +rior P(Θ|d), where d( f) = h( f) + n( f) is the detector strain +data, consisting of signal h and noise n. From the observable +d( f), the detector noise PSD Pn( f), and the inferred quanti- +ties ΘML(d) and P(Θ|d), one can construct various foreground +cleaning methods. We shall primarily discuss a method of +subtracting the signal from the detector strain and removing +the average residual power. In the supplementary material we +present an alternative approach which also allows substantial +foreground reduction. +To clean the foreground, we first subtract the ML strain +hML( f), +δh( f) = h( f) − hML( f) . +(9) +Strictly speaking, the correct subtraction should be formulated +as d( f) − hML(f) because the observable is data d( f) instead +of signal h( f). But we will use the notation of Eq. (9) for +convenience. After subtracting the ML strain, there is no way +to further clean the residual strain δh( f) [61], but the residual +power |δh( f)|2 is statistically known as +⟨|δh( f; Θ)|2⟩ = ⟨ +���h( f; Θ) − h( f; Θbst|Θ) +���2⟩ , +(10) +where ⟨⟩ is the ensemble average over different noise realiza- +tions and Θbst denotes the ML estimate of a signal h( f; Θ) +in an arbitary noise realization. +For a signal h( f; Θ) in a +random noise realization, the ML parameter Θbst can be ef- +ficiently pinned down with common optimization algorithms +(e.g., those in Python package Scipy) in the 6-dimensional +space {Mz, Mz, χ, ζ, tc, φopt} when the initial guess is well in- +formed by the posterior P(Θ|d). The average residual power +(10) is only known with some uncertainty informed by the +posterior P(Θ|d). Therefore the residual power estimator we +could construct is +⟨|δh( f; Θ|d)|2⟩ = +� +P(Θ|d) ⟨|δh( f; Θ)|2⟩ dΘ , +(11) +which is computationally more expensive than Eq. (10) since +a high-dimensional integration is involved. In this work, we +will use the approximation P(Θ|d) ≈ δ(Θ − ΘML(d)), i.e., +⟨|δh( f; Θ|d)|2⟩ ≈ ⟨|δh( f; Θ = ΘML(d))|2⟩ , +(12) +which turns out to be a very good approximation inducing a +small bias as we will show later. After removing the average +power, we arrive at the final result, +δrfn +1 H( f) = 5 +T +NO +� +i=1 +� +|δh( f; Θ)|2 − ⟨|δh( f; ΘML(d))|2⟩ +� +i , +(13) +where the 1st term on R.H.S. is the residual power after sub- +tracting the ML strain, and the 2nd is the (approximate) aver- +age residual power to be removed. +We now calculate the residual PSD δrfn +1 H( f). A simple scal- +ing analysis shows that, σ(φopt) ∼ σ(ζ)/ζ ∼ ρ−1, therefore + +3 +|δh|/|h| ∼ ρ−1 and the fractional residual power |δh|2/|h|2 ∼ +ρ−2, where ρ is the signal to noise ratio (SNR). Considering +that |h|2 ∝ ρ2, therefore |δh|2 ∼ ρ0, i.e., the residual power +|δh|2 +i is independent from the event SNR. After removing the +average residual power, the fractional residual power is fur- +ther reduced by a factor √NO until hitting the bias floor (NO +is the total number of mergers detected), i.e., +δrfn +1 H( f) = δvar +1 H( f) + δbias +1 +H(f) , +(14) +where +δvar +1 H( f) := 5 +T +NO +� +i=1 +|δh( f; Θ)|2 +i − ⟨|δh( f; Θ)|2⟩i , +δbias +1 +H( f) := 5 +T +NO +� +i=1 +⟨|δh( f; Θ)|2⟩i − ⟨|δh( f; ΘML(d))|2⟩i . +Here δvar +1 H( f) is the variance of a finite number of events, +and δbias +1 +H( f) is the bias induced by the approximation +in Eq. (12). +The fractional residuals scale with SNR as +δvar +1 H/H ∼ ρ−2N−1/2 +O +and δbias +1 +H/H ∼ ρ−3 considering that +|δh|2/|h|2 ∼ ρ−2 and δbias +1 +H/H ≈ |δh|2 +,αδΘα/|h|2 ∼ ρ−3. +Quantitatively, we expand the residual δh to O(ρ−2) as +δh = h,αδΘα + 1 +2h,αβδΘαδΘβ + O(ρ−3) . +(15) +where δΘ = Θ − ΘML. Consequently, +� +i +|δh( f; Θ)|2 +i = +� +i +|h,αδΘα|2 +i + 1 +4|h,αβδΘαδΘβ|2 +i ++ +� +i +|h,αδΘα|2 +i × O +� ρ−1 +√NO +� +i +, +⟨|δh( f; Θ)|2⟩ =Cαβ(Θ)h⋆ +,αh,β ++1 +4h⋆ +,αβh,γτ(CαβCγτ + CαγCβτ + CατCγβ) , +(16) +and similarly for ⟨|δh( f; ΘML(d))|2⟩, where we have used the +fact ⟨δΘαδΘβδΘγ⟩ = 0 and Cαβ(Θ) := ⟨δΘαδΘβ⟩ is the covari- +ance matrix. Plugging the above equations into Eq. (14), we +obtain +δrfn +1 H(f) = 5 +T +� +i +� +|h,αδΘα|2 − Cαβ(ΘML(d))h⋆ML +,α +hML +,β +� +i ++ 5 +T +� +i +� +|h,αδΘα|2 × +�O(ρ−1) +√NO ++ O(ρ−3) +�� +i +, +(17) +where the 2nd row on the R.H.S. contributes as a small cor- +rection to the 1st row. As a conservative estimate, we will +take O(ρ−1) = 10ρ−1 and O(ρ−3) = 10ρ−3 in the following +calculation. For reference, we denote the residual PSD after +subtracting the ML strain and before removing the average +power as +δ1H( f) = 5 +T +� +i +|h,αδΘα|2 +i . +(18) +At this stage, it is informative to compare the limit of fore- +ground reduction in [e.g., 45, 46, 62], where the residual PSD +after subtraction is formulated as +δH( f) = 1 +T +� +i +� +|h+( f) − hML ++ ( f)|2 + |h×(f) − hML +× ( f)|2� +i , +and the fractional residual was found to be δH( f)/H( f) ∼ +50% for BBHs detected with 3G detectors [45, 46]. +This +residual level is much higher than δ1H( f) with the ρ−2 scal- +ing (c.f. +Eq. 18), simply because |h+,×( f) − hML ++,×( f)| ≈ +|h+,×( f)(eiφc −eiφML +c )| = |h+,×( f)|×|1−eiδφc| and the coalescence +phase φc is weakly constrained with uncertainty σ(φc) = O(1) +due to the parameter degeneracy. In our subtraction method, a +different parameterization is used where the parameter degen- +eracy is largely mitigated with σ(φopt) ∼ σ(ζ)/ζ ∼ ρ−1, and +the subtraction is performed on the detector strain h( f) instead +of the polarizations h+,×( f). As a result, a much better preci- +sion δ1H( f)/H( f) ∼ |δh|2/|h|2 ∼ (δζ)2/ζ2 + (δφopt)2 ∼ ρ−2 is +achieved. +In summary, our method has achieved a two-step noise re- +duction: using a new set of binary parameters for event sub- +traction to obtain δ1H( f) and performing a further residual +power subtraction to arrive at δrfn +1 H( f). +Cleaning the BBH Foreground with 3G Detectors. We now +consider a population of BBH mergers and apply the fore- +ground cleaning method to a mock observation of 3G detec- +tors. Following the discussion in Ref. [63], we consider a +3G detector network: CE 40 (Idaho, USA) + CE 20 (New +South Wales, Australia) + ET D (Cascina, Italy) consisting of +a stage-2 40-km compact-binary optimized CE, a stage-2 20- +km compact-binary optimized CE, and an ET of type D (see +[40, 43, 63] for details about detector sensitivities, locations +and orientations). +In a population model, we need to specify the volumetric +merger rate R(z) (number of mergers per comoving volume +per cosmic time at redshift z), the mass distribution p(m1, m2) +and the effective spin distribution p(χ). The merger rate in the +observer frame is written as +˙NO = +� dVc(z) +dz +R(z) +1 + z dz , +(19) +where Vc(z) is the comoving volume up to redshift z, and the +factor 1+z comes from the time dialation due to cosmic expan- +sion. Consistent with the LVK O1-O3 observations [34, 35], +we take the BBH merger rate as RBBH(z) = R0 ×(1+z)2.9e−z2/3 +for (z ≤ 6), with the local merger rate R0 = 20 Gpc−3yr−1, a +spin distribution p(χ) as a Gaussian distribution with a mean +value 0.06 and a standard deviation 0.1, and a mass distri- +bution p(m1, m2) ∝ m−1 +1 (m1 − mmin)−1 for mmin ≤ m1 ≤ +m2 ≤ mmax, with mmin = 5M⊙ and mmax = 42M⊙. In this +population model, the total BBH merger rate turns out to be +˙NO = 4.6 × 104 yr−1, and the BBH foreground energy density +is Ωgw( f) ≈ 0.8 × 10−10 × ( f/Hz)2/3 for f ≲ 200 Hz. +We generate 16 BBH population realizations, with 6.5×104 +BBH mergers in each realization (that corresponds to approxi- +mately 1.4 years of observation with the assumed BBH merger + +4 +101 +102 +103 +f [Hz] +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +10 +9 +BBHs, SNRthr = 10 +GW +det. lim. +1 +GW +rfn +1 +GW +GW, unr +FIG. 1. Residuals after cleaning the BBH foreground. The top black +solid/dashed lines are the total energy density ΩGW of the BBH fore- +ground and the detector sensitivity limit Ωdet.lim., respectively. The +blue solid/dot-dashed lines are the energy density of the residual fore- +ground after implementing the primitive [δ1Ω, Eq. (18)] and refined +[δrfn +1 Ω, Eq. (17)] subtractions, respectively. The green dashed line is +the GW foreground energy density Ωunr of unresolved BBH mergers +with ρ < 10. +rate). The BH masses, spins, and redshifts are sampled ac- +cording to the distributions specified above, and all the angles +are sampled assuming isotropy. For each merger, we calculate +the expected SNR as ρ = +� +4 � +det. k ⟨h(k)|h(k)⟩, where the inner +production is defined as +⟨h|g⟩ = +� ∞ +0 +Real{h⋆( f)g( f)} +Pn( f) +d f , +(20) +with Pn,(k)( f) and h(k) being the noise PSD and the signal strain +of the i-th detector, respectively. We find the merger SNR dis- +tribution peaks around 30 with a long tail extending to several +hundred, and almost all the merger are of SNR > 10. +For each merger with model parameter Θ, we sample its +ML parameters ΘML from a multivariate Gaussian distribution +with a mean value Θ and a covariance matrix C(Θ), which is +calculated as the inverse matrix of Fisher matrix +Fαβ = 4 +� +det. k +⟨h(k),α|h(k),β⟩ . +(21) +We process the mergers with ρ > ρthr = 10 through the fore- +ground cleaning processes with the most sensitive CE 40 as +the reference detector, and label the remaining mergers as un- +resolved. We average all different residuals over the 16 real- +izations simulated (see Fig. 1). The black dashed line is the +detector sensitivity limit Ωdet.lim.( f) [64]. After subtracting the +ML strain, the fractional residual PSD [Eq. (18)] turns out to +be δ1H/H = δ1Ω/ΩGW ≈ 3 × 10−4 ∼ NO/ � +i ρ2 +i at f ≈ 20 +Hz. After removing the average residual power [Eq. (17)], +the fractional residual PSD further improves to ≈ 3 × 10−6 ∼ +� +i ρ−1 +i / � +i ρ2 +i at the same frequency, which shows that the +residual is dominated by the small bias δbias +1 +H( f) induced by +101 +102 +103 +f [Hz] +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +10 +9 +BNSs, SNRthr = 10 +GW +det. lim. +1 +GW +rfn +1 +GW +unr +FIG. 2. Same to Fig. 1 except for BNSs. +the approximation in Eq. (12). And the residual energy den- +sity δrfn +1 Ω turns out to be well below the detector sensitivity +Ωdet.lim. in the whole frequency range. In addition, the energy +density Ωunr of unresolved BBH foreground is always far be- +low the detector sensitivity limit Ωdet.lim.. +Cleaning the BNS Foreground with 3G Detectors. +Sim- +ilar to the BBH population, we take the BNS merger rate as +RBNS(z) = R0 × (1 + z)2.9e−z2/3 for (z ≤ 6), with the local +merger rate R0 = 160 Gpc−3yr−1, a spin distribution p(χ) as +a Gaussian distribution with a mean value 0.03 and a stan- +dard deviation 0.03. and a uniform mass distribution between +1.1M⊙ and 2.1M⊙. In this population model, the total merger +rate turns out to be ˙NO = 3.7 × 105 yr−1, and the GW fore- +ground energy density is Ωgw( f) ≈ 1.5×10−11 ×( f/Hz)2/3 for +f ≲ 2000 Hz. +We generate 16 BNS population realizations, with 5.2×105 +BNS mergers in each realization (roughly 1.4 years of obser- +vation). We find the merger SNR distribution peaks around 8 +and about half of the BNSs are sub-threshold with ρ < ρthr = +10. +Implementing the foreground cleaning method above, +the fraction residual PSD of BNSs with ρ > ρthr turns out +to be δrfn +1 H/H ≈ 10−4 ∼ � +i ρ−1 +i / � +i ρ2 +i at f ≈ 20 Hz, and +the residual energy density δrfn +1 Ω is well below the detector +sensitivity limit Ωdet.lim. in the whole frequency range. Due +to a large fraction of low-SNR BNSs in the population, the +sub-threshold BNSs turns out to dominate the residual fore- +ground with Ωunr( f) ≈ 1.4 × 10−12 × ( f/Hz)2/3, which is well +above the detector sensitivity limit in a large frequency range +(similar conclusion had also been reached in previous works +[44, 62, 65]). +Summary. +To better probe the primordial SGWB from +early universe processes, we proposed a foreground cleaning +method by first subtracting the signal strain from data using +the ML strain as a proxy, then removing the average resid- +ual power. With this method, the BBH foreground can be +reduced with residual power far below the detector sensitiv- +ity limit. This method largely improves the limits of fore- + +5 +ground obtained in previous studies [45, 46, 65], which can be +used as the benchmark for various model studies with primor- +dial GWs. Similar precision can be achieved for cleaning the +foreground from BNSs that are individually detectable, and +the residual foreground is expected to be dominated the con- +fusion foreground from sub-threshold BNSs, which will be +the next critical problem to solve for detecting the primordial +SGWB in the 3G era. One possible solution to this problem +is improving design sensitivity of 3G detectors: a rough esti- +mate shows that the unresolved BNS foreground energy den- +sity Ωunr can be reduced by O(102) if all the 3G detector noise +levels +� +Pn( f) were 3 times lower than what has been assumed +in this work. On the other hand, it is possible to measure the +unresolved BNS foreground Ωunr via the BNS merger rate at +high redshift if the delay time between BNS mergers and BBH +mergers is known. +Acknowledgement. We thank Liang Dai, Neal Dalal, Reed +Essick and Junwu Huang for valueable discussions. 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As a result, the +residual δ1H( f) or δrfn +1 H( f) is in general lowest around the +optimal frequency fopt where the phase of the detector strain +is best measured, and the fractional residual blows up at much +lower or much higher frequencies where the phase is not well +constrained (see Fig. 1). On the other hand, the foreground +energy density or the PSD depends only on the strain ampli- +tude ΩGW( f) ∝ H( f) ∝ � +i |h( f)|2 +i . Therefore it is possible to +measure the PSD using the amplitude information only. +For this purpose, an obvious estimator to use would be +ˆH( f) = 5 +T +� +i +|hML( f)|2 +i , +(A.22) +which is in fact a biased estimator with H( f) − ⟨ ˆH( f)⟩ < 0. +The above primitive estimator can be refined by compensating +the bias as +ˆHrfn(f) = 5 +T +� +i +� +|hML( f)|2 − ⟨|σh( f)|2⟩ +� +i , +(A.23) +where − ⟨|σh( f; Θ)|2⟩ := |h( f; Θ)|2 − ⟨|h( f; Θbst|Θ)|2⟩ is the +compensation term. It can be computed if the true parameters +Θ were known, while Θ is only known with some uncertainty +informed by the posterior P(Θ|d). Using the same approxi- +mation P(Θ|d) ≈ δ(Θ − ΘML(d)) as in Method 1, we have +⟨|σh( f; Θ)|2⟩ ≈ ⟨|σh( f; Θ = ΘML(d))|2⟩ . +(A.24) +As a result, the final form of the refined estimator is +ˆHrfn( f) = 5 +T +� +i +� +|hML( f)|2 − ⟨|σh( f; ΘML(d))|2⟩ +� +i , (A.25) +with residual PSD +δrfn +2 H( f) = +���H( f) − ˆHrfn( f) +��� . +(A.26) +Similar to in Method 1, the residual PSD can be computed +with the noise PSD Pn( f). Making use of the fact that the un- +certainty in amplitude |h( f)| = ζAphen( f; Mz, Mz, χ) is mainly +sourced by the uncertainty in strain amplitude parameter ζ, the +bias term can be further approximated as +⟨|σh( f; ΘML(d))|2⟩ ≈ +����� +σ(ζ) +ζML hML( f) +����� +2 +, +(A.27) +where ζML is the ML strain amplitude and σ(ζ) = +� +Cζζ(ΘML) +is its 1-σ uncertainty. With this approximation, we obtain a +conservative estimate of the residual PSD +δrfn +2 H( f) ≈ 5 +T +������� +� +i +|h( f)|2 +i − |hML( f)|2 +i + +����� +σ(ζ) +ζML hML( f) +����� +2 +i +������� . +(A.28) +For comparison use, we denote the residual PSD of the prim- +itive estimator as +δ2H( f) = +���H( f) − ˆH( f) +��� . +(A.29) +We also apply the Method 2 to the simulated BBHs and +BNSs in the maintext, and compare the preformance of the +two methods in Figs. 3 and 4. With the primitive estimate +in Method 2 [Eq. (A.29)], the residual energy density δ2Ω of + +9 +101 +102 +103 +f [Hz] +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +10 +9 +BBHs, SNRthr = 10 +GW +det. lim. +1 +GW +rfn +1 +GW +2 +GW +rfn +2 +GW +GW, unr +FIG. 3. Residuals after cleaning the BBH foreground. The top black +solid/dashed lines are the total energy density ΩGW of the BBH fore- +ground and the detector sensitivity limit Ωdet.lim., respectively. The +blue solid/dot-dashed lines are the energy density of the residual fore- +ground after implementing the primitive [δ1Ω, Eq. (18)] and refined +[δrfn +1 Ω, Eq. (17)] subtractions in Method 1, respectively. The orange +solid/dashed lines are the energy density of the residual foreground +after implementing the primitive [δ2Ω, Eq. (A.29)] and refined [δrfn +2 Ω, +Eq.(A.28)] estimates in Method 2, respectively. The green dashed +line is the energy density Ωunr of GWs from unresolved BBH merg- +ers with ρ < 10. +101 +102 +103 +f [Hz] +10 +15 +10 +14 +10 +13 +10 +12 +10 +11 +10 +10 +10 +9 +BNSs, SNRthr = 10 +GW +det. lim. +1 +GW +rfn +1 +GW +2 +GW +rfn +2 +GW +unr +FIG. 4. Same to Fig. 3 except for BNSs. +BBHs is already below Ωdet.lim. (with the fractional residual +δ2Ω/ΩGW ≈ 2 × 10−4 ∼ NO/ � +i ρ2 +i ) across the whole fre- +quency range and the refined estimate [Eq.(A.28)] further im- +proves the residual by a factor ∼ 4 at low frequency f ≲ 102 +Hz. The improvement factor is much lower than √NO be- +cause the bias term of each merger is of different magnitude +with |σh( f; ΘML(d))|2 +i ∝ ρ−2 +i , therefore the fractional resid- +ual decreases slower than the scaling N−1/2 +O +. Applying the +primitive estimate in Method 2 to the BNSs that are individu- +ally detectable, we find the fractional residual energy density +δ2Ω/ΩGW ≈ 3×10−3 ∼ NO/ � +i ρ2 +i and the refined estimate fur- +ther improves the residual by a factor ∼ 3×102, which is much +closer to √NO because the SNRs of individually detectable +BNSs are more concentrated around ρthr and therefore the bias +term of each merger |σh( f; ΘML(d))|2 +i ∝ ρ−2 +i +is of similar mag- +nitude. As a result, the fractional residual energy of BNSs +turns out to be δrfn +2 Ω/ΩGW ∼ √NO/ � +i ρ2 +i , which is lower +than the residual of Method 1 δrfn +1 Ω/ΩGW ∼ � +i ρ−1 +i / � +i ρ2 +i (see +Fig. 4). + diff --git a/Z9E3T4oBgHgl3EQfcwrJ/content/tmp_files/load_file.txt b/Z9E3T4oBgHgl3EQfcwrJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3ce4cb33c3da3cc0a8640e85700d796c133b319 --- /dev/null +++ b/Z9E3T4oBgHgl3EQfcwrJ/content/tmp_files/load_file.txt @@ -0,0 +1,1109 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf,len=1108 +page_content='Detecting Primordial Stochastic Gravitational Waves with Reduced Astrophysical Foregrounds Zhen Pan1, ∗ and Huan Yang1, 2, † 1Perimeter Institute for Theoretical Physics, Ontario, N2L 2Y5, Canada 2University of Guelph, Guelph, Ontario N1G 2W1, Canada (Dated: January 12, 2023) One of the primary targets of third-generation (3G) ground-based gravitational wave (GW) detectors is de- tecting the stochastic GW background (SGWB) from early universe processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The astrophysical foreground from compact binary mergers will be a major contamination to the background, which must be reduced to high precision to enable the detection of primordial background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In this Letter, we revisit the limit of foreground reduction computed in previous studies, and propose a novel cleaning method subtracting the approximate sig- nal strain and removing the average residual power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' With this method, the binary black hole foreground can be reduced with fractional residual energy density below 10−4 for frequency f ∈ (10, 102) Hz, below 10−3 for fre- quency f ∈ (102, 103) Hz and below the detector sensitivity limit for all relevant frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Similar precision can be achieved to clean the foreground from binary neutron stars (BNSs) that are above the detection threshold, so that the residual foreground is dominated by sub-threshold BNSs, which will be the next critical problem to solve for detecting the primordial SGWB in the 3G era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Primordial stochastic gravitational wave back- ground (SGWB) from various physical processes from the early universe has been investigated, including inflation [1, 2] and preheating [3, 4], first-order phase transitions [5–10] and cosmic strings [11–16] (see [17–22] for more complete re- views).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Measuring the primordial SGWB at different frequen- cies will open a unique window to the universe at the earliest moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Therefore the primordial SGWB detection has been one of the primary targets for gravitational wave (GW) detec- tors in different frequency bands, including pulsar timing ar- rays [23–26], spaceborne GW detectors [27, 28] and ground based detectors [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In addition to the primordial SGWB, GWs from various astrophysical sources are much better understood and mea- sured [32–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In the sensitive band of ground based detec- tors, compact binaries, including binary black holes (BBHs), black hole-neutron star binaries (BHNSs) and binary neutron stars (BNSs) are the dominant source of astrophysical fore- ground [36–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In order to improve the sensitivity of probing the primordial background, the effect of astrophysical fore- ground must be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Cleaning the astrophysical fore- ground with the residual foreground energy density below the detector sensitivity limit has been argued to be possible in the era of third-generation (3G) ground-based detectors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', Ein- stein Telescope (ET) [40] and Cosmic Explorer (CE) [41–43], which are so sensitive that almost all compact binary mergers are expected to be detected and subtracted out [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' However, recently Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [45, 46] pointed out the straightforward event subtraction proposed in [44] only removes ∼ 50% of the foreground noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The resulting astrophysical foreground is way above the detector sensitivity, which poses severe chal- lenges of detecting primordial GWs [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In this Letter, we will show that the astrophysical fore- ground can be cleaned by orders of magnitude, with a novel cleaning method detailed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Applying this method to 3G detectors, the BBH foreground and the foreground from in- dividually resolved BNSs can be reduced to be well below the detector sensitivity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' As a result, the residual fore- ground is expected to be dominated by sub-threshold BNSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Notice that the noise reduction of sub-threshold events may be achieved following a Bayesian framework, as discussed in [48–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' One subtlety may be that this method requires enor- mous computational cost, and it is more susceptible to non- Gaussian noise in detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' For space-borne detectors, it has also been shown that astrophysical foreground cleaning may benefit from multi-band observations [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In this Letter, we use geometrical units G = c = 1 and we assume a flat ΛCDM cosmology with H0 = 70 km/s/Mpc, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='7 and Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Stochastic GW Foreground from Compact Binary Merg- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The energy density of stochastic GWs per logarithmic frequency is related to its power spectrum density (PSD) by (our notation is different from that in [52, 53] by a factor 8π) ΩGW( f) := 1 ρcrit dρGW d ln f = 4π2 3H2 0 f 3H( f) , (1) with ρcrit := 3H2 0/8π the critical energy density to close the universe and the PSD (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', [51, 53, 54] for derivation) H( f) = 1 T � i � |h+( f)|2 + |h×( f)|2� i , (2) where the index i runs over all binaries in the universe that merger within the observation time span (0, T) , and h+,× are the two polarizations of incoming GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Driven by the incom- ing GWs, the detector strain responds as h( f) = F+(θ, φ, ψ)h+( f) + F×(θ, φ, ψ)h×( f) , (3) where the attena pattern F+,× depend on the source sky loca- tion (θ, φ) and the source polarization angle ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In terms of the detector strain, the PSD is expressed as H( f) = 2 ⟨F2 +⟩ + ⟨F2 ×⟩ 1 T � i |h( f)|2 i = 5 T � i |h( f)|2 i , (4) where ⟨⟩ is an average over the three attena pattern depen- dent angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' It is known that ⟨F2 +⟩ = ⟨F2 ×⟩ = 1/5 for arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='04529v1 [gr-qc] 11 Jan 2023 2 LIGO/Virgo/KAGRA (LVK) like L-shape interferometers and ⟨F2 +⟩ = ⟨F2 ×⟩ = 3/20 for LISA/ET like triangle-shape interfer- ometers (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', [55] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We have chosen a L-shape interferometer as the reference detector in the second equal sign of the equation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' A typical non-precessing BBH waveform depends on 7 model parameters h+,×(Mz, Mz, χ, tc, φc, ι, DL): the (red- shifted) chirp mass Mz = (1 + z)M, the (redshifted) total mass Mz = (1 + z)M, the effective spin χ, the coalescence time tc (arriving at a detector), the coalescence phase φc, the inclination angle ι, the luminosity distance DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Explicitly, the detector strain is formulated as h( f) = � 5 24 f −7/6 π2/3 ζAphen( f)ei[2πftc+φ0+Ψphen( f)] , (5) where the strain phase φ0 is defined as e2iφ0 := e2iφc F+(1 + cos2 ι)/2 − iF× cos ι � F2 +( 1+cos2 ι 2 )2 + F2 × cos2 ι , (6) and the strain amplitude ζ := M5/6 z DL � F2 +(1 + cos2 ι 2 )2 + F2 × cos2 ι .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (7) The waveform dependence on the intrinsic binary parameters is encoded in Aphen( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Mz, Mz, χ) and Ψphen( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Mz, Mz, χ), the explicit expressions of which differ in different waveform models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In this work, we will use the simple PhenomB wave- form model [56] (the foreground cleaning results have little change if other waveform models are applied instead, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', PhenomC or PhenomD [57–59]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Because of parameter degeneracies, not all the binary pa- rameters can be well constrained even for a loud merger event, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', the coalescence phase φc is in general weakly constrained due to its degeneracy with angles {ι, θ, φ, ψ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' To mitigate the parameter degeneracy, we instead use two detector dependent parameters that are of weak degeneracy with other parameters (similar parameterization was used in [60] for efficient param- eter inference): the strain amplitude ζ [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (7)] and the strain phase φopt at the optimal frequency φopt := 2πfopttc + φ0 + Ψphen( fopt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (8) The optimal frequency fopt := � |h(f)|2 Pn(f) f d f � � |h( f)|2 Pn( f) d f is the frequency where the waveform is best constrained, with Pn( f) being the detector noise PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' To summarize, we parametrized the PhenomB waveform with the following 10 model param- eters {Mz, Mz, χ, ζ, tc, φopt} + {ι, θ, φ, ψ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In this parameteriza- tion, the detector strain h( f) depends on the first 6 parameters [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (5,8)], and the remaining 4 can only be measured with multiple detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Note that {ζ, tc, φopt} are detector de- pendent quantities and we choose the most sensitive detec- tor as the reference detector if multiple detectors are in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' It turns out that this re-parametrization of parameters signif- icantly alleviates the errors from signal subtraction, as dis- cussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Foreground Cleaning Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' For an incoming binary merger signal h( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ) at a detector, one can infer its ML (Max- imum Likelihood) estimate h( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ΘML) along with the poste- rior P(Θ|d), where d( f) = h( f) + n( f) is the detector strain data, consisting of signal h and noise n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' From the observable d( f), the detector noise PSD Pn( f), and the inferred quanti- ties ΘML(d) and P(Θ|d), one can construct various foreground cleaning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We shall primarily discuss a method of subtracting the signal from the detector strain and removing the average residual power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In the supplementary material we present an alternative approach which also allows substantial foreground reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' To clean the foreground, we first subtract the ML strain hML( f), δh( f) = h( f) − hML( f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (9) Strictly speaking, the correct subtraction should be formulated as d( f) − hML(f) because the observable is data d( f) instead of signal h( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' But we will use the notation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (9) for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' After subtracting the ML strain, there is no way to further clean the residual strain δh( f) [61], but the residual power |δh( f)|2 is statistically known as ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2⟩ = ⟨ ���h( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ) − h( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θbst|Θ) ���2⟩ , (10) where ⟨⟩ is the ensemble average over different noise realiza- tions and Θbst denotes the ML estimate of a signal h( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ) in an arbitary noise realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' For a signal h( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ) in a random noise realization, the ML parameter Θbst can be ef- ficiently pinned down with common optimization algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', those in Python package Scipy) in the 6-dimensional space {Mz, Mz, χ, ζ, tc, φopt} when the initial guess is well in- formed by the posterior P(Θ|d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The average residual power (10) is only known with some uncertainty informed by the posterior P(Θ|d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Therefore the residual power estimator we could construct is ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ|d)|2⟩ = � P(Θ|d) ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2⟩ dΘ , (11) which is computationally more expensive than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (10) since a high-dimensional integration is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In this work, we will use the approximation P(Θ|d) ≈ δ(Θ − ΘML(d)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ|d)|2⟩ ≈ ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ = ΘML(d))|2⟩ , (12) which turns out to be a very good approximation inducing a small bias as we will show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' After removing the average power, we arrive at the final result, δrfn 1 H( f) = 5 T NO � i=1 � |δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2 − ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ΘML(d))|2⟩ � i , (13) where the 1st term on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' is the residual power after sub- tracting the ML strain, and the 2nd is the (approximate) aver- age residual power to be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We now calculate the residual PSD δrfn 1 H( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' A simple scal- ing analysis shows that, σ(φopt) ∼ σ(ζ)/ζ ∼ ρ−1, therefore 3 |δh|/|h| ∼ ρ−1 and the fractional residual power |δh|2/|h|2 ∼ ρ−2, where ρ is the signal to noise ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Considering that |h|2 ∝ ρ2, therefore |δh|2 ∼ ρ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', the residual power |δh|2 i is independent from the event SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' After removing the average residual power, the fractional residual power is fur- ther reduced by a factor √NO until hitting the bias floor (NO is the total number of mergers detected), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', δrfn 1 H( f) = δvar 1 H( f) + δbias 1 H(f) , (14) where δvar 1 H( f) := 5 T NO � i=1 |δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2 i − ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2⟩i , δbias 1 H( f) := 5 T NO � i=1 ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2⟩i − ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ΘML(d))|2⟩i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Here δvar 1 H( f) is the variance of a finite number of events, and δbias 1 H( f) is the bias induced by the approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The fractional residuals scale with SNR as δvar 1 H/H ∼ ρ−2N−1/2 O and δbias 1 H/H ∼ ρ−3 considering that |δh|2/|h|2 ∼ ρ−2 and δbias 1 H/H ≈ |δh|2 ,αδΘα/|h|2 ∼ ρ−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Quantitatively, we expand the residual δh to O(ρ−2) as δh = h,αδΘα + 1 2h,αβδΘαδΘβ + O(ρ−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (15) where δΘ = Θ − ΘML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Consequently, � i |δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2 i = � i |h,αδΘα|2 i + 1 4|h,αβδΘαδΘβ|2 i + � i |h,αδΘα|2 i × O � ρ−1 √NO � i , ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2⟩ =Cαβ(Θ)h⋆ ,αh,β +1 4h⋆ ,αβh,γτ(CαβCγτ + CαγCβτ + CατCγβ) , (16) and similarly for ⟨|δh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ΘML(d))|2⟩, where we have used the fact ⟨δΘαδΘβδΘγ⟩ = 0 and Cαβ(Θ) := ⟨δΘαδΘβ⟩ is the covari- ance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Plugging the above equations into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (14), we obtain δrfn 1 H(f) = 5 T � i � |h,αδΘα|2 − Cαβ(ΘML(d))h⋆ML ,α hML ,β � i + 5 T � i � |h,αδΘα|2 × �O(ρ−1) √NO + O(ρ−3) �� i , (17) where the 2nd row on the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' contributes as a small cor- rection to the 1st row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' As a conservative estimate, we will take O(ρ−1) = 10ρ−1 and O(ρ−3) = 10ρ−3 in the following calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' For reference, we denote the residual PSD after subtracting the ML strain and before removing the average power as δ1H( f) = 5 T � i |h,αδΘα|2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (18) At this stage, it is informative to compare the limit of fore- ground reduction in [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', 45, 46, 62], where the residual PSD after subtraction is formulated as δH( f) = 1 T � i � |h+( f) − hML + ( f)|2 + |h×(f) − hML × ( f)|2� i , and the fractional residual was found to be δH( f)/H( f) ∼ 50% for BBHs detected with 3G detectors [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' This residual level is much higher than δ1H( f) with the ρ−2 scal- ing (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 18), simply because |h+,×( f) − hML +,×( f)| ≈ |h+,×( f)(eiφc −eiφML c )| = |h+,×( f)|×|1−eiδφc| and the coalescence phase φc is weakly constrained with uncertainty σ(φc) = O(1) due to the parameter degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In our subtraction method, a different parameterization is used where the parameter degen- eracy is largely mitigated with σ(φopt) ∼ σ(ζ)/ζ ∼ ρ−1, and the subtraction is performed on the detector strain h( f) instead of the polarizations h+,×( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' As a result, a much better preci- sion δ1H( f)/H( f) ∼ |δh|2/|h|2 ∼ (δζ)2/ζ2 + (δφopt)2 ∼ ρ−2 is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In summary, our method has achieved a two-step noise re- duction: using a new set of binary parameters for event sub- traction to obtain δ1H( f) and performing a further residual power subtraction to arrive at δrfn 1 H( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Cleaning the BBH Foreground with 3G Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We now consider a population of BBH mergers and apply the fore- ground cleaning method to a mock observation of 3G detec- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Following the discussion in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [63], we consider a 3G detector network: CE 40 (Idaho, USA) + CE 20 (New South Wales, Australia) + ET D (Cascina, Italy) consisting of a stage-2 40-km compact-binary optimized CE, a stage-2 20- km compact-binary optimized CE, and an ET of type D (see [40, 43, 63] for details about detector sensitivities, locations and orientations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In a population model, we need to specify the volumetric merger rate R(z) (number of mergers per comoving volume per cosmic time at redshift z), the mass distribution p(m1, m2) and the effective spin distribution p(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The merger rate in the observer frame is written as ˙NO = � dVc(z) dz R(z) 1 + z dz , (19) where Vc(z) is the comoving volume up to redshift z, and the factor 1+z comes from the time dialation due to cosmic expan- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Consistent with the LVK O1-O3 observations [34, 35], we take the BBH merger rate as RBBH(z) = R0 ×(1+z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='9e−z2/3 for (z ≤ 6), with the local merger rate R0 = 20 Gpc−3yr−1, a spin distribution p(χ) as a Gaussian distribution with a mean value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='06 and a standard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='1, and a mass distri- bution p(m1, m2) ∝ m−1 1 (m1 − mmin)−1 for mmin ≤ m1 ≤ m2 ≤ mmax, with mmin = 5M⊙ and mmax = 42M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In this population model, the total BBH merger rate turns out to be ˙NO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='6 × 104 yr−1, and the BBH foreground energy density is Ωgw( f) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='8 × 10−10 × ( f/Hz)2/3 for f ≲ 200 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We generate 16 BBH population realizations, with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='5×104 BBH mergers in each realization (that corresponds to approxi- mately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='4 years of observation with the assumed BBH merger 4 101 102 103 f [Hz] 10 15 10 14 10 13 10 12 10 11 10 10 10 9 BBHs, SNRthr = 10 GW det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 1 GW rfn 1 GW GW, unr FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Residuals after cleaning the BBH foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The top black solid/dashed lines are the total energy density ΩGW of the BBH fore- ground and the detector sensitivity limit Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The blue solid/dot-dashed lines are the energy density of the residual fore- ground after implementing the primitive [δ1Ω, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (18)] and refined [δrfn 1 Ω, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (17)] subtractions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The green dashed line is the GW foreground energy density Ωunr of unresolved BBH mergers with ρ < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The BH masses, spins, and redshifts are sampled ac- cording to the distributions specified above, and all the angles are sampled assuming isotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' For each merger, we calculate the expected SNR as ρ = � 4 � det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' k ⟨h(k)|h(k)⟩, where the inner production is defined as ⟨h|g⟩ = � ∞ 0 Real{h⋆( f)g( f)} Pn( f) d f , (20) with Pn,(k)( f) and h(k) being the noise PSD and the signal strain of the i-th detector, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We find the merger SNR dis- tribution peaks around 30 with a long tail extending to several hundred, and almost all the merger are of SNR > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' For each merger with model parameter Θ, we sample its ML parameters ΘML from a multivariate Gaussian distribution with a mean value Θ and a covariance matrix C(Θ), which is calculated as the inverse matrix of Fisher matrix Fαβ = 4 � det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' k ⟨h(k),α|h(k),β⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (21) We process the mergers with ρ > ρthr = 10 through the fore- ground cleaning processes with the most sensitive CE 40 as the reference detector, and label the remaining mergers as un- resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We average all different residuals over the 16 real- izations simulated (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The black dashed line is the detector sensitivity limit Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ( f) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' After subtracting the ML strain, the fractional residual PSD [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (18)] turns out to be δ1H/H = δ1Ω/ΩGW ≈ 3 × 10−4 ∼ NO/ � i ρ2 i at f ≈ 20 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' After removing the average residual power [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (17)], the fractional residual PSD further improves to ≈ 3 × 10−6 ∼ � i ρ−1 i / � i ρ2 i at the same frequency, which shows that the residual is dominated by the small bias δbias 1 H( f) induced by 101 102 103 f [Hz] 10 15 10 14 10 13 10 12 10 11 10 10 10 9 BNSs, SNRthr = 10 GW det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 1 GW rfn 1 GW unr FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Same to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 1 except for BNSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' the approximation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' And the residual energy den- sity δrfn 1 Ω turns out to be well below the detector sensitivity Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' in the whole frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In addition, the energy density Ωunr of unresolved BBH foreground is always far be- low the detector sensitivity limit Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='. Cleaning the BNS Foreground with 3G Detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Sim- ilar to the BBH population, we take the BNS merger rate as RBNS(z) = R0 × (1 + z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='9e−z2/3 for (z ≤ 6), with the local merger rate R0 = 160 Gpc−3yr−1, a spin distribution p(χ) as a Gaussian distribution with a mean value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='03 and a stan- dard deviation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' and a uniform mass distribution between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='1M⊙ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='1M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' In this population model, the total merger rate turns out to be ˙NO = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='7 × 105 yr−1, and the GW fore- ground energy density is Ωgw( f) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='5×10−11 ×( f/Hz)2/3 for f ≲ 2000 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We generate 16 BNS population realizations, with 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='2×105 BNS mergers in each realization (roughly 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='4 years of obser- vation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We find the merger SNR distribution peaks around 8 and about half of the BNSs are sub-threshold with ρ < ρthr = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Implementing the foreground cleaning method above, the fraction residual PSD of BNSs with ρ > ρthr turns out to be δrfn 1 H/H ≈ 10−4 ∼ � i ρ−1 i / � i ρ2 i at f ≈ 20 Hz, and the residual energy density δrfn 1 Ω is well below the detector sensitivity limit Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' in the whole frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Due to a large fraction of low-SNR BNSs in the population, the sub-threshold BNSs turns out to dominate the residual fore- ground with Ωunr( f) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='4 × 10−12 × ( f/Hz)2/3, which is well above the detector sensitivity limit in a large frequency range (similar conclusion had also been reached in previous works [44, 62, 65]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' To better probe the primordial SGWB from early universe processes, we proposed a foreground cleaning method by first subtracting the signal strain from data using the ML strain as a proxy, then removing the average resid- ual power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' With this method, the BBH foreground can be reduced with residual power far below the detector sensitiv- ity limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' This method largely improves the limits of fore- 5 ground obtained in previous studies [45, 46, 65], which can be used as the benchmark for various model studies with primor- dial GWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Similar precision can be achieved for cleaning the foreground from BNSs that are individually detectable, and the residual foreground is expected to be dominated the con- fusion foreground from sub-threshold BNSs, which will be the next critical problem to solve for detecting the primordial SGWB in the 3G era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' One possible solution to this problem is improving design sensitivity of 3G detectors: a rough esti- mate shows that the unresolved BNS foreground energy den- sity Ωunr can be reduced by O(102) if all the 3G detector noise levels � Pn( f) were 3 times lower than what has been assumed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' On the other hand, it is possible to measure the unresolved BNS foreground Ωunr via the BNS merger rate at high redshift if the delay time between BNS mergers and BBH mergers is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We thank Liang Dai, Neal Dalal, Reed Essick and Junwu Huang for valueable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We also thank Bei Zhou for sharing the detector sensitivity limit curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' We are supported by the Natural Sciences and Engineering Research Council of Canada and in part by Perimeter Insti- tute for Theoretical Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Devel- opment Canada and by the Province of Ontario through the Ministry of Colleges and Universities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ∗ zpan@perimeterinstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='ca † hyang@perimeterinstitute.' metadata={'source': 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“Frequency-domain gravitational waves from non- precessing black-hole binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' New numerical waveforms and anatomy of the signal,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' D 93, 044006 (2016), arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='07250 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [59] Sebastian Khan, Sascha Husa, Mark Hannam, Frank Ohme, Michael P¨urrer, Xisco Jim´enez Forteza, and Alejandro Boh´e, “Frequency-domain gravitational waves from nonprecessing black-hole binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' A phenomenological model for the advanced detector era,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' D 93, 044007 (2016), 8 arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='07253 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [60] Javier Roulet, Seth Olsen, Jonathan Mushkin, Tousif Islam, Te- jaswi Venumadhav, Barak Zackay, and Matias Zaldarriaga, “Removing degeneracy and multimodality in gravitational wave source parameters,” arXiv e-prints , arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='03508 (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='03508 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [61] As claimed in [66], the residual strain δh can be further cleaned by removing its projection along the tangential space |hML ,α ⟩ ⟨hML ,β |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' This is in fact not achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The reason is that the residual data δh + n is known while δh is not, and the projec- tion ⟨δh + n|hML ,α ⟩ ⟨hML ,β | vanishes exactly, because the ML strain is defined such that ⟨d − hML|d − hML⟩ maximizes, which gives ⟨d − hML|hML ,α ⟩ = ⟨δh + n|hML ,α ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' This projection and removal procedure works only if some approximate ML strain hprox in- stead of the ML strain is known δh = h − hprox � h − hML and the detector noise vanishes as considered in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [63] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Borhanian, “GWBENCH: a novel Fisher information pack- age for gravitational-wave benchmarking,” Classical and Quan- tum Gravity 38, 175014 (2021), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='15202 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [64] The detector sensitivity limit is defined such that any SGWB with energy density ΩSGWB( f) that is tangent to the detector sensitivity limit curve at f0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', ΩSGWB( f) = Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ( f0) × ( f/f0)γ0 with the power index γ0 = d ln Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (f0) d ln f0 , could be de- tected by the detector network with 3σ confidence level in 4 years if there was no foreground contamination (see [68] for the original definition and [45, 69] for the calculation details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [65] Haowen Zhong, Rich Ormiston, and Vuk Mandic, “Detect- ing cosmological gravitational waves background after re- moval of compact binary coalescences in future gravitational wave detectors,” arXiv e-prints , arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='11877 (2022), arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='11877 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [66] Jan Harms, Christoph Mahrdt, Markus Otto, and Malte Prieß, “Subtraction-noise projection in gravitational-wave detector networks,” Phys.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' D 102, 063009 (2020), arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='16116 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [68] Eric Thrane and Joseph D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Romano, “Sensitivity curves for searches for gravitational-wave backgrounds,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' D 88, 124032 (2013), arXiv:1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='5300 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='IM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' [69] C J Moore, R H Cole, and C P L Berry, “Gravitational-wave sensitivity curves,” Classical and Quantum Gravity 32, 015014 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Foreground Cleaning Method 2: only measure the power The basic picture of the foreground cleaning method in the maintext (referred as Method 1 in the following discussion) is subtracting the unknown signal h( f) from data d( f) with the model hML( f) as a proxy, where the precision of strain phase measurement makes a big difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' As a result, the residual δ1H( f) or δrfn 1 H( f) is in general lowest around the optimal frequency fopt where the phase of the detector strain is best measured, and the fractional residual blows up at much lower or much higher frequencies where the phase is not well constrained (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' On the other hand, the foreground energy density or the PSD depends only on the strain ampli- tude ΩGW( f) ∝ H( f) ∝ � i |h( f)|2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Therefore it is possible to measure the PSD using the amplitude information only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' For this purpose, an obvious estimator to use would be ˆH( f) = 5 T � i |hML( f)|2 i , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='22) which is in fact a biased estimator with H( f) − ⟨ ˆH( f)⟩ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The above primitive estimator can be refined by compensating the bias as ˆHrfn(f) = 5 T � i � |hML( f)|2 − ⟨|σh( f)|2⟩ � i , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='23) where − ⟨|σh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2⟩ := |h( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2 − ⟨|h( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θbst|Θ)|2⟩ is the compensation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' It can be computed if the true parameters Θ were known, while Θ is only known with some uncertainty informed by the posterior P(Θ|d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Using the same approxi- mation P(Θ|d) ≈ δ(Θ − ΘML(d)) as in Method 1, we have ⟨|σh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ)|2⟩ ≈ ⟨|σh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Θ = ΘML(d))|2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='24) As a result, the final form of the refined estimator is ˆHrfn( f) = 5 T � i � |hML( f)|2 − ⟨|σh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ΘML(d))|2⟩ � i , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='25) with residual PSD δrfn 2 H( f) = ���H( f) − ˆHrfn( f) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='26) Similar to in Method 1, the residual PSD can be computed with the noise PSD Pn( f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Making use of the fact that the un- certainty in amplitude |h( f)| = ζAphen( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Mz, Mz, χ) is mainly sourced by the uncertainty in strain amplitude parameter ζ, the bias term can be further approximated as ⟨|σh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ΘML(d))|2⟩ ≈ ����� σ(ζ) ζML hML( f) ����� 2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='27) where ζML is the ML strain amplitude and σ(ζ) = � Cζζ(ΘML) is its 1-σ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' With this approximation, we obtain a conservative estimate of the residual PSD δrfn 2 H( f) ≈ 5 T ������� � i |h( f)|2 i − |hML( f)|2 i + ����� σ(ζ) ζML hML( f) ����� 2 i ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='28) For comparison use, we denote the residual PSD of the prim- itive estimator as δ2H( f) = ���H( f) − ˆH( f) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='29) We also apply the Method 2 to the simulated BBHs and BNSs in the maintext, and compare the preformance of the two methods in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' With the primitive estimate in Method 2 [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='29)], the residual energy density δ2Ω of 9 101 102 103 f [Hz] 10 15 10 14 10 13 10 12 10 11 10 10 10 9 BBHs, SNRthr = 10 GW det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 1 GW rfn 1 GW 2 GW rfn 2 GW GW, unr FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Residuals after cleaning the BBH foreground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The top black solid/dashed lines are the total energy density ΩGW of the BBH fore- ground and the detector sensitivity limit Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The blue solid/dot-dashed lines are the energy density of the residual fore- ground after implementing the primitive [δ1Ω, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (18)] and refined [δrfn 1 Ω, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (17)] subtractions in Method 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The orange solid/dashed lines are the energy density of the residual foreground after implementing the primitive [δ2Ω, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='29)] and refined [δrfn 2 Ω, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='28)] estimates in Method 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The green dashed line is the energy density Ωunr of GWs from unresolved BBH merg- ers with ρ < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 101 102 103 f [Hz] 10 15 10 14 10 13 10 12 10 11 10 10 10 9 BNSs, SNRthr = 10 GW det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 1 GW rfn 1 GW 2 GW rfn 2 GW unr FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Same to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 3 except for BNSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' BBHs is already below Ωdet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (with the fractional residual δ2Ω/ΩGW ≈ 2 × 10−4 ∼ NO/ � i ρ2 i ) across the whole fre- quency range and the refined estimate [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content='28)] further im- proves the residual by a factor ∼ 4 at low frequency f ≲ 102 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' The improvement factor is much lower than √NO be- cause the bias term of each merger is of different magnitude with |σh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ΘML(d))|2 i ∝ ρ−2 i , therefore the fractional resid- ual decreases slower than the scaling N−1/2 O .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' Applying the primitive estimate in Method 2 to the BNSs that are individu- ally detectable, we find the fractional residual energy density δ2Ω/ΩGW ≈ 3×10−3 ∼ NO/ � i ρ2 i and the refined estimate fur- ther improves the residual by a factor ∼ 3×102, which is much closer to √NO because the SNRs of individually detectable BNSs are more concentrated around ρthr and therefore the bias term of each merger |σh( f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' ΘML(d))|2 i ∝ ρ−2 i is of similar mag- nitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' As a result, the fractional residual energy of BNSs turns out to be δrfn 2 Ω/ΩGW ∼ √NO/ � i ρ2 i , which is lower than the residual of Method 1 δrfn 1 Ω/ΩGW ∼ � i ρ−1 i / � i ρ2 i (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E3T4oBgHgl3EQfcwrJ/content/2301.04529v1.pdf'} diff --git a/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf b/_dFJT4oBgHgl3EQfqSzu/content/2301.11604v1.pdf new file mode 100644 index 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[math.RA] 6 Jan 2023 +PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF +JORDAN TYPE +TOM DE MEDTS +LOUIS ROWEN +YOAV SEGEV +Abstract. We show that primitive 4-generated axial algebras of Jor- +dan type are at most 81-dimensional. +1. Introduction +Axial algebras were introduced in 2015 by Jonathan Hall, Felix Rehren +and Sergey Shpectorov [HRS15b]. They are non-associative commutative +algebras generated by axes, i.e., idempotents for which the left multiplica- +tion operator is semisimple and such that the resulting eigenspaces multiply +according to a given fusion law (see §2 for precise definitions). +In the easiest interesting case, these multiplication operators admit pre- +cisely 3 eigenvalues 0, 1 and η. A typical example is provided by Jordan +algebras, where each idempotent gives rise to a Peirce decomposition of the +algebra. In this case, we have η = 1 +2, and the fusion law is the following. +∗ +1 +0 +η +1 +{1} +∅ +{η} +0 +∅ +{0} +{η} +η +{η} +{η} +{1, 0} +Table 1. The Jordan fusion law Φ(η) +We call the axial algebras with a fusion law Φ(η) axial algebras of Jordan +type η. Other than Jordan algebras themselves, there are other interesting +examples of axial algebras of Jordan type (for arbitrary values of η ̸= 0, 1), +namely the Matsuo algebras arising from 3-transposition groups. +In this +case, the dimension of the algebra is equal to the size of the normal gen- +erating set of 3-transpositions of the group. (See Example 2.5 below for +details.) +The classification of 3-transposition groups has a long history (see [CH95, +Hal22] and the references therein). It is a highly non-trivial fact that finitely +generated 3-transposition groups are finite. In fact, this is a consequence +Date: January 9, 2023. +1 + +2 +TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV +of the classification of finite simple groups, and a direct proof of this fact +would be very valuable. (See [CH95, Theorem (1.3), p. 153].) +One possible approach for such a direct proof is precisely via the corre- +sponding Matsuo algebras. More generally, we ask the following question. +(We refer to Definition 2.3 below for the precise meaning.) +Question. Let A be a primitive axial algebra of Jordan type. Assume that +A is generated by a finite set of axes. Can we conclude that A is finite- +dimensional? +Notice that, by Corollary 2.8 below, a positive answer to this question +would show, in particular, that finitely generated 3-transposition groups are +finite. In fact, for η ̸= 1 +2, it is equivalent. +It is natural to try to answer this question for an increasing number +of axes. +For 2-generated primitive axial algebras of Jordan type, this is +almost trivial: such algebras are at most 3-dimensional. +(In fact, much +more can be said: [HRS15a, Theorem 1.1] gives a complete classification of +such algebras.) +For 3-generated algebras, this question was answered affirmatively in the +recent paper [GS20]: such algebras are at most 9-dimensional. +Our main result is the following. +Main Theorem. Primitive 4-generated axial algebras of Jordan type η are +at most 81-dimensional, for any η. Moreover, this result is best possible. +To go from 3-generated to 4-generated primitive axial algebras of Jordan +type is a large step that required substantial new ideas. In fact, in our new +setup, it is almost a triviality to recover the earlier result from [GS20] that +such 3-generated algebras are at most 9-dimensional. One of the key ideas +is that we will almost never use the actual multiplication in the algebra, but +instead, we use sequences of Miyamoto involutions (see Definition 3.3 below). +These sequences will allow us to formulate many “rewriting rules” that we +can use to systematically deal with larger and larger expressions, until we +eventually “wrap up” so that we can reduce every possible expression of +length larger than 6. The precise meaning of this will be explained below +and can be seen in Theorem 5.1, which is a more detailed version of our +Main Theorem. +It is worth pointing out that going to the next step, primitive 5-generated +axial algebras of Jordan type, is expected to be increasingly more difficult, +because the upper bound of the dimension will be at least 312 = 531441. (In +fact, this is our conjectured upper bound.) In addition, one of the examples +(of dimension 306936) arises from the largest sporadic Fischer group Fi24. +2. Primitive axial algebras of Jordan type +Throughout the paper, F will be a commutative field with char F ̸= 2. All +our algebras will be commutative but non-associative1 F-algebras. +1As usual, non-associative means “not necessarily associative”. + +PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE +3 +For the definition of fusion laws and axial algebras, we rely on [DMPSVC20]. +Definition 2.1. +(i) A fusion law is a pair (X, ∗), where X is a set and ∗ +is a map from X × X to 2X, where 2X denotes the power set of X. A +fusion law (X, ∗) is called symmetric if x ∗ y = y ∗ x for all x, y ∈ X. +(ii) The Jordan fusion law is the fusion law with X = {0, 1, η} (where η +is just a symbol) and with ∗ given by Table 1 above. +Definition 2.2. Let Φ = (X, ∗) be a fusion law. +(i) A Φ-decomposition of an algebra A is a direct sum decomposition +A = � +x∈X Ax (as vector spaces) such that AxAy ⊆ Ax∗y for all +x, y ∈ X, where AY := � +y∈Y Ay for all Y ⊆ X. +(ii) A Φ-decomposition algebra is a triple (A, I, Ω) where A is an F-algebra, +I is an index set and Ω is a tuple of Φ-decompositions of A indexed +by I. In other words, for each i ∈ I, we have a corresponding Φ-de- +composition A = � +x∈X A(i) +x +of the algebra A. +Definition 2.3. Let Φ = (X, ∗) be a fusion law with 1 ∈ X ⊆ F. +(i) For each a ∈ A, we write ada for the left multiplication by a, i.e., +ada: A → A: x �→ ax. +(ii) An element a ∈ A is called a Φ-axis if it is idempotent (i.e., a2 = a) +and the decomposition of A into the eigenspaces for ada is a Φ-decom- +position. +(iii) The algebra A is a Φ-axial algebra if it is generated by a set of Φ-axes. +This makes A into a Φ-decomposition algebra (with I identified with +the given set of axes). +(iv) A Φ-axial algebra A is primitive if for each axis a of the generating +Φ-axes of A, the 1-eigenspace A(a) +1 +is 1-dimensional, i.e., is equal to +Fa. +(v) An axial algebra of Jordan type η is a Φ-axial algebra for the fusion +law Φ = Φ(η) as in Table 1. +As we mentioned in the introduction, the two main sources of examples +of axial algebras of Jordan type are (1) Jordan algebras, and (2) Matsuo +algebras. We give some details. +Example 2.4. Let J be a Jordan algebra over F, i.e., J is a unital commu- +tative non-associative algebra such that a2(ab) = a(a2b) for all a, b ∈ J. If +e ∈ J is an idempotent, then it is an axis for the Jordan fusion law Φ(1 +2); +this is the famous Peirce decomposition for Jordan algebras (see, e.g., [Jac68, +Chapter III]). In particular, if J is generated by idempotents, then it is an +axial algebra of Jordan type 1 +2. +Example 2.5. Let (G, D) be a 3-transposition group, i.e., G is a group and +D ⊆ G is a generating set of involutions, closed under conjugation in G, +such that the product of any two elements in D has order at most 3. Let + +4 +TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV +η ∈ F\{0, 1} be arbitrary. Then the Matsuo algebra Mη(G, D) is the algebra +with basis D, with multiplication given by +de := + + + + + +e +if d = e +0 +if o(de) = 2 +η +2(d + e − f) +if o(de) = 3, where f = de = ed in G. +By [HRS15a, Theorem 6.5], Mη(G, D) is a primitive axial algebra of Jordan +type η. +Axial algebras of Jordan type, and more generally any type of decom- +position algebras where the fusion law admits a Z/2-grading, admit many +involutory automorphisms, the so-called Miyamoto involutions. +Definition 2.6. +(i) A Z/2-grading of a fusion law (X, ∗) is a map θ: X → +Z/2 such that x ∗ y ⊆ θ−1(θ(x) + θ(y)) for all x, y ∈ X. For instance, +the Jordan fusion law from Table 1 is Z/2-graded with θ(0) = θ(1) = 0 +and θ(η) = 1. +(ii) If (A, I, Ω) is a Φ-decomposition algebra for a Z/2-graded fusion law +(X, ∗), then for each i ∈ I, we define a Miyamoto involution +τi : A → A: ax �→ (−1)θ(x)ax, +when ax ∈ A(i) +x . +In other words, τi fixes the 0-graded elements and negates the 1-graded +elements with respect to the i-th decomposition of A. +Corollary 2.8 below is an important motivation for the main result of our +paper. +Proposition 2.7. Let (G, D) be a 3-transposition group. The following are +equivalent: +(a) G is finite. +(b) D is finite. +(c) Mη(G, D) is finite-dimensional. +Proof. Of course, (a) implies (b), and (b) and (c) are equivalent because +Mη(G, D) has dimension |D|. In particular, the dimension of Mη(G, D) is +independent of the choice of the base field F and of η ∈ F, so to show that +(c) implies (a), we may assume that F is a finite field. +Then A := Mη(G, D) is finite. +By [DMR17, p. 325] (which relies on +[Asc97, p. 92, Example (4)]), G/Z(G) is embedded in Aut(A), so G/Z(G) +is a finite group. By a theorem of Schur, [Asc00, (33.9), p. 168], the derived +subgroup G′ is finite. Since G/G′ is an abelian group generated by a finite +number of involutions, it is finite, so we conclude that G is finite. +□ +Corollary 2.8. The following are equivalent: +(a) Every finitely generated 3-transposition group (G, D) is finite. +(b) Every primitive axial algebra A of Jordan type η ̸= 1 +2 generated by a +finite set of axes X is finite-dimensional. + +PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE +5 +Proof. (a) ⇒ (b) Let A and X be as in (b). +For x ∈ X, let τx be the +Miyamoto involution associated with x. By [HRS15a, Theorem (5.4), +p. 105], the group G = ⟨τx | x ∈ X⟩ is a 3-transposition group. By +the assumption, G is finite. By [HRS15a, Corollary (1.2), p. 81], A is +spanned by {xg | x ∈ X, g ∈ G}, so A is finite-dimensional. +(b) ⇒ (a) Let (G, D) be a finitely generated 3-transposition group. Then G +is generated by a finite number of elements from D, hence the algebra +Mη(G, D) is finitely generated. Thus, by the assumption, it is finite- +dimensional. Proposition 2.7 then tells us that G is finite. +□ +In order to get an idea about the complexity of the primitive 4-generated +axial algebras of Jordan type, it is useful to look at the list of 4-generated +3-transposition groups first. In particular, this will provide us with an exam- +ple of such an algebra of dimension 81, which is precisely the upper bound +that we will obtain in our main result. +Theorem 2.9. Let (G, D) be a 3-transposition group generated by 4 ele- +ments from D (but not by less than 4). Then its central type is one of the +following: +(i) W(A4), the Weyl group of type A4 (with |D| = 10); +(ii) W(D4), the Weyl group of type D4 (with |D| = 12); +(iii) 33 : Sym(4) (with |D| = 18); +(iv) 21+6 : SU3(2)′ (with |D| = 36); +(v) Hall’s 3-transposition group [310]: 2 (with |D| = 81) or its affine quo- +tient 33+3 : 2 (with |D| = 27). +Proof. The definition of central type, and the proof of this fact (together +with the size of D in each case) can be found in [HS95, Proposition (4.2)], +where the authors point out that this classification has been proven indepen- +dently by Zara, Hall and Moori; the first written source seems to be Zara’s +(unpublished) thesis from 1984. +□ +The unique 3-transposition group in this list attaining the upper bound +|D| = 81 is particularly interesting because it arises as a 3-transposition +subgroup of the sporadic Fischer groups Fi23 and Fi24. We give an explicit +construction of the resulting Matsuo algebra, based on [LB83, §4.1]. +In +fact, we had implemented this example on a computer to experiment with +identities, which is how some of our ideas arose. +Example 2.10. Let D be the 4-dimensional vector space over the field F3 +(so |D| = 81). We first set +(x1, x2, x3, x4) • (y1, y2, y3, y4) +:= +� +x1 + y1, x2 + y2, x3 + y3, x4 + y4 + (x1y2 − x2y1)(x3 − y3) +� +for all xi, yi ∈ F3. Next, we set +d ∗ e := (d • e) • (d • e) + +6 +TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV +for all d, e ∈ D. For any η ∈ F \ {0, 1}—recall that F is still our arbitrary +base field of characteristic different from 2—we now define an F-algebra with +basis D, and with multiplication given by +de := +� +e +if d = e +η +2(d + e − d ∗ e) +if d ̸= e. +Then by combining [LB83] with Example 2.5, we see that this is precisely +the Matsuo algebra corresponding to Hall’s 3-transposition group [310]: 2. +3. Method +From now on, we assume that A is a primitive axial algebra of Jordan +type η generated by a finite set S of axes. +Definition 3.1. For each i ≥ 0, we set +S[i] := ⟨τa1τa2 · · · τaℓ(b) | ℓ ≤ i, a1, . . . , aℓ, b ∈ S⟩. +In particular, S[0] = ⟨S⟩, and the S[i] form an ascending chain of subspaces +of A. +Our goal is to show that S[n] = A for some n. The following proposition +tells us that we can do this by showing that the ascending chain of the S[i] +stabilizes. +Proposition 3.2. Assume that S[n] = S[n+1] for some n. Then A = S[n]. +Proof. Following [HRS15a, p. 81], we define the closure of the set S of axes +to be the smallest set C of axes of A containing S such that for each a ∈ C, +we have τa(C) ⊆ C. In fact, C = {τa1τa2 · · · τaℓ(b) | ℓ ≥ 0, a1, . . . , aℓ, b ∈ S}; +see, for instance, [KMS20, Lemma 3.5]. It now suffices to observe that if +S[n] = S[n + 1], then S[n] = S[ℓ] for all ℓ ≥ n, hence S[n] = ⟨C⟩. By +[HRS15a, Cor. (1.2), p. 81], however, A is spanned by C, and the result +follows. +□ +From now on, when we refer to an arbitrary axis of A, we will always +mean an element of the closure C of S (which is indeed always an axis for +the same fusion law). +The following two definitions will play a crucial role. +Definition 3.3. +(i) We let +�a1, a2, . . . , aℓ� := τa1τa2 · · · τaℓ +for all axes a1, . . . , aℓ ∈ A. +(ii) For all x, y ∈ A, we set +x ≡(i) y ⇐⇒ x − y ∈ S[i]. +Notice that x ≡(i) y implies x ≡(j) y for all j ≥ i, and also implies +that �a1, . . . , aℓ�x ≡(i+ℓ) �a1, . . . , aℓ�y for all a1, . . . , aℓ ∈ S. + +PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE +7 +Remark 3.4. The notation �a1, . . . , aℓ� will also be used when the ai are +axes that are not necessarily contained in S. Some care is needed with the +use of the equivalence relations ≡(i) in such a situation, as these relations +are always meant with respect to the given generating set S. +By [HSS18, Theorem 4.1], primitive axial algebras of Jordan type always +admit a (necessarily unique) normalized symmetric Frobenius form. +Definition 3.5. +(i) A bilinear form (·, ·): A×A → F is called a (normal- +ized) Frobenius form on A if (xy, z) = (x, yz) for all x, y, z ∈ A and, +in addition, (a, a) = 1 for each axis a ∈ A. +(ii) It will be useful to introduce the notation +ǫx,y := 1 − 2 +η(x, y) +for all x, y ∈ A. +Proposition 3.6. Let a ∈ A be an axis and x ∈ A be arbitrary. Then +τa(x) = x + 2 +η(a, x)a − 2 +ηax. +Proof. This is [HSS18, Lemma 3.3] combined with the statement from [HSS18, +Theorem 4.1] that (a, x) = ϕa(x). +□ +Remark 3.7. In [HSS18], their Lemma 3.3 is used, in fact, in the proof of +their Theorem 4.1 (the existence of the Frobenius form). On the other hand, +if we already assume the existence of the Frobenius form to begin with, then +there is an easy direct proof of Proposition 3.6 by simply decomposing x with +respect to the eigenspaces for the axis a. +Proposition 3.6 has the following immediate but useful consequences. +Corollary 3.8. Let a, b ∈ A be axes. Then: +(i) �a�b − �b�a = ǫa,b(b − a). +(ii) If a ∈ S and x ∈ S[i], then (− 2 +η)ax ≡(0) �a�x − x ≡(i) �a�x. +Proof. +(i) By Proposition 3.6, we have +τa(b) − τb(a) = +� +1 − 2 +η(a, b) +� +(b − a). +(ii) This follows immediately from Proposition 3.6. +□ +We recall the following important fact, which we will be using over and +over again, often without explicitly mentioning it. +Proposition 3.9. We have �a, b, a� = �τa(b)� for all axes a, b ∈ A. +Proof. This follows from [HRS15a, Lemma 5.1, p. 103] and the fact that +τa ∈ Aut(A). +□ +The following result is a first instance of how useful it is. +Proposition 3.10. Let a, b ∈ S and x ∈ A. Then: +(i) �a, b, a�x ≡(1) x − 2 +ητa(b)x. + +8 +TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV +(ii) �a, b, a�x − �b, a, b�x ≡(1) ǫa,b(�b�x − �a�x). In particular, if x ∈ S[i] +for some i ≥ 0, then �a, b, a�x ≡(i+1) �b, a, b�x. +(iii) �b, a, b, a�x ≡(2) �a, b�x + ǫa,b(x − �b, a�x). In particular, if x ∈ S[i] +for some i ≥ 2, then �b, a, b, a�x ≡(i) �a, b�x − ǫa,b�b, a�x. +Proof. +(i) We apply Proposition 3.9 to x and use Proposition 3.6 on the +right-hand side to get +�a, b, a�x = x + 2 +η(τa(b), x)�a�b − 2 +ητa(b)x. +(3.1) +Since �a�b ∈ S[1], the result follows. +(ii) Interchanging a and b in (i) and subtracting gives, using Corollary 3.8(i), +�a, b, a�x − �b, a, b�x ≡(1) (− 2 +η)ǫa,b(b − a)x. +By Corollary 3.8(ii), however, +(− 2 +η)(bx − ax) ≡(0) (�b�x − x) − (�a�x − x) = �b�x − �a�x, +and the result follows. +(iii) This follows immediately by applying τb on (ii). +□ +Lemma 3.11. Let a, b, c ∈ S. Then: +(i) �a, b�a = ǫa,ba + b − ǫa,b�a�b ≡(0) −ǫa,b�a�b. +(ii) �a, b, a�c = αc − α�a�b + �c, a�b where α = ǫτa(b),c ∈ F. +(iii) �a, b, c�a ≡(0) δ�a�b − ǫa,c�a, b�c + �c, a�b for some δ ∈ F. +Proof. +(i) By Corollary 3.8(i), +�a, b�a = �a�(�b�a − �a�b + �a�b) = ǫa,b�a�(a − b) + b. +(ii) Let α = ǫτa(b),c. +By substituting τa(b) for a and c for b in Corol- +lary 3.8(i), we get +�τa(b)�c − �c, a�b = α(c − �a�b). +The result now follows from Proposition 3.9. +(iii) By Corollary 3.8(i), we have +�a, b, c�a = �a, b, a�c + �a, b� +� +�c�a − �a�c +� += �a, b, a�c + ǫa,c�a, b�(a − c), +so (iii) follows from (i) and (ii). +□ +4. Rewriting rules +In this section, we will gradually build up “rewriting rules” that will allow +us to simplify certain expressions. As the length of the expressions increases, +the proofs become more and more involved. +Proposition 4.1. Let a, b, c, d ∈ S. Then: +(i) �a�b ≡(0) �b�a. +(ii) �a, b�a ∈ S[1]. + +PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE +9 +(iii) �a, b, a�c ≡(1) �c, a�b. +(iv) �a, b, c�a ≡(1) �c, b�a − ǫa,c�a, b�c +(v) �a, b, a, c�d ≡(1) �c, d, c, a�b. +(vi) �a, b, c, d, b�a ≡(2) �b, d, c, b�a − ǫa,τb(d)�a, b, c�d ≡(3) �b, d, c, b�a. +(vii) �a, b, c, a, b�d ≡(3) �b, a, d, b, a�c. +Proof. +(i) This follows from Corollary 3.8(i). +(ii) By (i), we have �a, b�a ≡(1) �a, a�b = b. (Of course, this also follows +from Lemma 3.11(i).) +(iii) This follows from Lemma 3.11(ii). +(iv) This follows from Lemma 3.11(iii) and (i). +(v) We have +�a, b, a, c�d − �c, d, c, a�b = �τa(b)�τc(d) − �τc(d)�τa(b), +which is contained in ⟨τa(b) − τc(d)⟩ ≤ S[1] by Corollary 3.8(i). +(vi) We have +�a, b, c, d, b�a = �a, τb(c), τb(d)�a. +Now let S′ = {a, τb(c), τb(d)} and apply (iv) with respect to this set +S′ in place of S. Notice that S′[1] ≤ S[2], because +�τb(c)�a = �b, c, b�a ∈ S[2] +(by (iii)), +�τb(c)�τb(d) = �b, c, b, b�d = �b, c�d ∈ S[2], +so we see that indeed �x�y ∈ S[2] for all x, y ∈ S′. Hence +�a, τb(c), τb(d)�a ≡(2) �τb(d), τb(c)�a − ǫa,τb(d)�a, τb(c)�τb(d) += �b, d, c, b�a − ǫa,τb(d)�a, b, c�d +so we conclude that indeed +�a, b, c, d, b�a ≡(2) �b, d, c, b�a − ǫa,τb(d)�a, b, c�d ≡(3) �b, d, c, b�a. +(vii) We start from +�b, a, c, a, b�d = �τbτa(c)�d += �d�τbτa(c) + ǫd,τbτa(c)(d − τbτa(c)) += �d, b, a�c + ǫd,τbτa(c)(d − �b, a�c) +≡(0) �d, b, a�c − ǫd,τbτa(c)�b, a�c. +In particular, �b, a, c, a, b�d ∈ S[3]. Moreover, applying �b, a� to this +equivalence yields +�b, a, b, a, c, a, b�d ≡(2) �b, a, d, b, a�c − ǫd,τbτa(c)�b, a, b, a�c +≡(2) �b, a, d, b, a�c, +(4.1) +where the last equivalence holds because by (iii) and (iv), we have +�b, a, b, a�c ≡(2) �b, c, a�b ∈ S[2]. + +10 +TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV +We now apply Proposition 3.10(iii) with x = �c, a, b�d ∈ S[3], which +gives +�b, a, b, a, c, a, b�d ≡(3) �a, b, c, a, b�d − ǫa,b�b, a, c, a, b�d +≡(3) �a, b, c, a, b�d. +The claim follows by combining this with (4.1). +□ +For our next rewriting rule in Proposition 4.4, we first need the following +lemma. +Lemma 4.2. Let a, b, c, d ∈ S and let S′ = {a, b, c, τa(d)}. Then S′[3] ⊆ +S[4]. +Proof. Let �x, y, z�w be any element with x, y, z, w ∈ S′. Of course, if none +of these four elements is equal to τa(d), then �x, y, z�w ∈ S[3] ⊆ S[4], and if +all four elements are equal to τa(d), then �x, y, z�w = τa(d) ∈ S[1] ⊆ S[4]. +Case 1. Suppose that only one of these four elements is equal to τa(d). +If w = τa(d), then �x, y, z�w = �x, y, z, a�d ∈ S[4]. If z = τa(d), then, by +Proposition 4.1(iii), +�x, y, z�w = �x, y, a, d, a�w ∈ S[4]. +If y = τa(d), then, by Proposition 4.1(v), +�x, y, z�w = �x, a, d, a, z�w ≡(2) �x, z, w, z, a�d. +If x = a or z = a, then �x, y, z�w ∈ S[4]. If w = a, then �x, y, z�w ∈ S[4], +by Proposition 4.1(ii). We may thus assume that z = b and w = c. If x = b, +then we see that �x, y, z�w ∈ S[4]. If x = c, then +�x, y, z�w ≡(2) �c, b, c, b, a�d ∈ S[3], +by Proposition 3.10(iii). +If x = τa(d), then, assuming without loss that y = b, +�x, y, z�w = �a, d, a, b, z�w. +If z = a, then by Proposition 4.1(iii), �x, y, z�w ∈ S[4]. +Hence we may +assume z = c, and by Proposition 4.1(ii) we may assume that w = a. In +this case, Proposition 4.1(iv) shows that �x, y, z�w ∈ S[4]. +Case 2. Suppose that three of the four elements x, y, z, w are equal to τa(d). +If x = y = z = τa(d), then of course �x, y, z�w = �τa(d)�w = �a, d, a�w ∈ +S[2]. For the other cases, we simply observe that +�τa(d), x, τa(d)�τa(d) = �τa(d), x�τa(d) = �a, d, a, x, a�d ∈ S[4], +�τa(d), τa(d), x�τa(d) = �x, a�d ∈ S[2], +�x, τa(d), τa(d)�τa(d) = �x, a�d ∈ S[2]. +Case 3. Exactly two of the four elements x, y, z, w are equal to τa(d). + +PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE +11 +We have +�τa(d), τa(d), z�w = �z�w ∈ S[1], +�x, τa(d), τa(d)�w = �x�w ∈ S[1], and +�x, y, τa(d)�τa(d) = �x, y, a�d ∈ S[3]. +If x = w = τa(d) and y, z ∈ {a, b, c}, then, by (vi), +�τa(d), y, z�τa(d) = �a, d, a, y, z, a�d ≡(4) �a, a, z, y, a�d ∈ S[3]. +Next, if x = w = τa(d) and y, z ∈ {a, b, c}, then, by Lemma 3.11(ii) (with +τa(d) in place of a), �τa(d), y, τa(d)�w ∈ S[4]. +Finally, if y = w = τa(d) and x, z ∈ {a, b, c}, then, by Lemma 3.11(i) +�x, τa(d), z�τa(d) ∈ S[4]. +□ +The following corollary will play an important role in the proof of Propo- +sition 4.5. +Corollary 4.3. Let a, b, c, d ∈ S and let T = {a, τa(b), τa(c), d}. +Then +T[4] ⊆ S[6]. +Proof. Let S′ = {a, b, c, τa(d)} as in Lemma 4.2 and notice that T = τa(S′), +i.e., T is obtained from S′ by applying τa on each element. By Proposi- +tion 3.9, for all x1, . . . , xk, y ∈ S′ we have +�τa(x1), . . . , τa(xk)�τa(y) = �a, x1, . . . , xk, a�τa(y) = τa(�x1, . . . , xk�y), +so T[i] = τa(S′[i]) for all i. +By Lemma 4.2, we have S′[3] ⊆ S[4]. Now +S′[4] = �a�S′[3] ∪ �b�S′[3] ∪ �c�S′[3] ∪ �τa(d)�S′[3] +⊆ S[5] ∪ �a, d, a�S[4], +and hence +T[4] = �a�S′[4] ⊆ �a�S[5] ∪ �d, a�S[4] ⊆ S[6]. +□ +Proposition 4.4. Let a, b, c, d ∈ S. Then +�a, b, c, a, b, c�d ≡(4) �b, c, a, b, c, a�d ≡(4) �c, a, b, c, a, b�d. +Proof. Let S′ = {a, b, c, τa(d)}. By Lemma 4.2, we have S′[3] ⊆ S[4]. We +can thus apply Proposition 4.1(vii) with respect to S′ to get +�c, b, τa(d), c, b�a ≡(4) �b, c, a, b, c�τa(d), +hence +�c, b, a, d, a, c, b�a ≡(4) �b, c, a, b, c, a�d. +(4.2) +On the other hand, we apply �c, b, a, d� to the equivalence in Lemma 3.11(iii) +(with b and c interchanged) to get +�c, b, a, d, a, c, b�a +≡(4) δ�c, b, a, d, a�c − ǫa,b�c, b, a, d, a, c�b + �c, b, a, d, b, a�c + +12 +TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV +for some δ ∈ F. Now �c, b, a, d, a�c ∈ S[4] by Proposition 4.1(iii). Also, by +Proposition 4.1(v) and Proposition 3.10(iii), we have +�c, b, a, d, a, c�b ≡(3) �c, b, c, b, c, a�d ∈ S[4]. +Thus, by Proposition 4.1(vii), +�c, b, a, d, a, c, b�a ≡(4) �c, b, a, d, b, a�c ≡(4) �c, a, b, c, a, b�d. +(4.3) +Combining (4.2) and (4.3), we see that +�b, c, a, b, c, a�d ≡(4) �c, a, b, c, a, b�d. +It now suffices to cyclically permute a, b, c to also get the other equivalence. +□ +We now come to the final and most challenging rewriting rule, which +will effectively put a bound on the dimension of 4-generated primitive axial +algebras of Jordan type. +Proposition 4.5. Let a, b, c, d ∈ S. Then �d, a, b, c, a, b, c�d ∈ S[6]. +Proof. Let +T = {τd(a), τd(b), c, d}. +By Corollary 4.3, we have T[4] ⊆ S[6]. By Proposition 4.4 applied to T, this +implies that +�τd(a), τd(b), c, τd(a), τd(b), c�d ≡(6) �c, τd(a), τd(b), c, τd(a), τd(b)�d. +(4.4) +We will proceed in two steps: We first show that +�τd(a), τd(b), c, τd(a), τd(b), c�d ≡(6) �d, a, c, b, a, c, b�d, +(4.5) +and then we show that +�c, τd(a), τd(b), c, τd(a), τd(b)�d ∈ S[6]. +(4.6) +Interchanging the role of b and c, it will then follow from (4.4), (4.5) and (4.6) +that �d, a, b, c, a, b, c�d ∈ S[6]. +Step 1. Proof of (4.5). +By Lemma 3.11(i) applied on �d, c�d, we have +�τd(a), τd(b), c, τd(a), τd(b), c�d += �d, a, b, d, c, d, a, b, d, c�d += �d, a, b, d, c, d, a, b�c + ǫc,d�d, a, b, d, c, d, a, b�d +− ǫc,d�d, a, b, d, c, d, a, b, d�c. +Now let γ = −ǫc,τd(b); then by Proposition 4.1(vi) and Proposition 4.1(v), +we have +�d, a, b, d, c, d, a, b, d�c ≡(6) �d, a, b, d, d, b, a, d�c + γ�d, a, b, d, c, d, a�b +≡(0) γ�d, a, b, d, c, d, a�b +≡(4) γ�d, a, b, a, b, a, d�c ∈ S[6], + +PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE +13 +by Proposition 3.10(iii). +Also, by Proposition 4.1(iv), we have +�d, a, b, d, c, d, a, b�d +≡(6) �d, a, b, d, c, b, a�d − ǫb,d�d, a, b, d, c, d, a�b +≡(6) �d, a, b, d, c, b, d�a − ǫb,d�d, a, b, a, b, a, d�c +(by 4.1(i) and 4.1(v)) +≡(6) �d, a, d, b, a, d, b�c − ǫb,d�d, a, b, a, b, a, c�d +(by 4.1(vii) and 4.1(i)) +∈ S[6], +by Proposition 4.4 and Proposition 3.10(iii). +Finally, by Proposition 3.10(ii), +�d, a, b, d, c, d, a, b�c +≡(6) �d, a, b, c, d, c, a, b�c +≡(6) �d, a, b, c, d, b, a�c − ǫb,c�d, a, b, c, d, c, a�b +(by 4.1(iv)) +≡(6) �d, a, b, c, d, b, c�a − ǫb,c�d, a, b, a, b, a, c�d +(by 4.1(i) and 4.1(v)) +≡(5) �d, a, c, b, a, c, b�d +(by 4.1(vii) and 3.10(iii)). +This proves (4.5). +Step 2. Proof of (4.6). +By Lemma 3.11(iii), there exists δ ∈ F with +�c, τd(a), τd(b), c, τd(a), τd(b)�d += �c, d, a, b, d, c, d, a, b�d +≡(6) δ�c, d, a, b, d, c, d�a − ǫb,d�c, d, a, b, d, c, d, a�b ++ �c, d, a, b, d, c, b, d�a. +Now by Proposition 4.1(iii), �c, d, a, b, d, c, d�a ∈ S[6]. By Proposition 4.1(v) +and Proposition 3.10(iii), we have +�c, d, a, b, d, c, d, a�b ≡(5) �c, d, a, b, a, b, a, d�c ∈ S[6]. +Finally, by Proposition 4.1(vii) and Proposition 4.4, we also have +�c, d, a, b, d, c, b, d�a ≡(6) �c, d, a, d, b, a, d, b�c +≡(6) �c, d, d, b, a, d, b, a�c = �c, b, a, d, b, a�c ∈ S[6]. +This proves (4.6) and thus finishes the proof of this proposition. +□ +5. 4-generated primitive axial algebras of Jordan type +We are now ready to prove our main result. Although it requires some +care to write down the proof, the hard work has already been done in Propo- +sitions 4.1, 4.4 and 4.5. + +14 +TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV +Theorem 5.1. Assume that A is generated by a set S = {a, b, c, d} of 4 +axes. Then A = S[6] and A is at most 81-dimensional. +More precisely, let G be the group Sym(S) of all permutations of S. De- +fine2 +Γ0 = {a}G, +Γ1 = {�a�b}G, +Γ2 = {�a, b�c}G, +Γ3 = {�a, b, c�d}G, +Γ4 = {�a, b, a, c�d, �a, b, c, a�d}G, +Γ5 = {�a, b, c, a, b�d}G, +Γ6 = {�a, b, c, a, b, c�d}G. +Then for each i ∈ {1, . . . , 6}, we have S[i] = ⟨Γ0, . . . , Γi⟩. In particular, +A = ⟨Γ0, . . . , Γ6⟩. +Moreover, there is some redundancy in these spanning sets: The dimen- +sion of each of the S[i] is at most 4, 10, 22, 34, 61, 73 and 81, respectively. +Proof. For each i ≤ 6, let T[i] be the subspace of A spanned by Γ0, . . . , Γi. +Obviously, we have T[i] ≤ S[i] for each i. We will show recursively that +for each i ≤ 6, S[i] = T[i], and that S[7] = S[6]. We will, at the same +time, compute the maximal possible dimension of each T[i]. Notice that for +each i, the subspace S[i + 1] is spanned by S[i] and all elements obtained +by applying the four operations �a�, �b�, �c� and �d� on the elements of S[i]. +In order to go from S[i] = T[i] to the next step S[i + 1], it will suffice, by +G-symmetry, to apply these four operations on the given representative of +the set Γi. +i = 0. Obviously, T[0] = ⟨a, b, c, d⟩ = S[0], and dim S[0] ≤ 4. +i = 1. We have �a�a = a ∈ S[0], whereas applying any of the other three +operations �b�, �c�, �d� on the representative a ∈ Γ0 results in an +element of Γ1, so S[1] ≤ T[1]. +By Proposition 4.1(i), we have �b�a ≡(0) �a�b, so the 12 possible +elements of Γ1 come in pairs that are linearly dependent modulo S[0]. +Hence dim S[1] ≤ 4 + 12/2 = 10. +i = 2. We have �a, a�b = b ∈ S[0] and �b, a�b ∈ S[1] by Proposition 4.1(ii). +On the other hand, �c, a�b and �d, a�b belong to Γ2 and hence to +T[2]. Hence S[2] ≤ T[2]. +By Proposition 4.1(i), we have �a, b�c +≡(1) �a, c�b, so the 24 +possible elements of Γ2 come in pairs that are linearly dependent +modulo S[1]. Hence dim S[2] ≤ 10 + 24/2 = 22. +2There is some obvious abuse of notation here: a priori, the group G does not act on A, +so when we write an expression like {�a, b, c�d}G, we really mean {�aρ, bρ, cρ�(dρ) | ρ ∈ G}. + +PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE +15 +i = 3. We have �a, a, b�c = �b�c ∈ S[1], and we have �b, a, b�c ∈ S[2] by +Proposition 4.1(iii) and �c, a, b�c ∈ S[2] by Proposition 4.1(iv). On +the other hand, �d, a, b�c belongs to Γ3 and hence to T[3]. Hence +S[3] ≤ T[3]. +By Proposition 4.1(i), we have �a, b, c�d ≡(2) �a, b, d�c, so the 24 +possible elements of Γ3 come in pairs that are linearly dependent +modulo S[2]. Hence dim S[3] ≤ 22 + 24/2 = 34. +i = 4. We have �a, a, b, c�d = �b, c�d ∈ S[2]. On the other hand, �b, a, b, c�d +and �c, a, b, c�d belong to Γ4 and hence to T[4]. Finally, �d, a, b, c�d ≡(3) +�d, a, b, d�c ∈ Γ4, so �d, a, b, c�d belongs to ⟨S[3], Γ4⟩ ≤ T[4]. Hence +S[4] ≤ T[4]. +By Proposition 4.1(v), the 24 possible elements of {�a, b, a, c�d}G +come in 8-tuples that are pairwise linearly dependent modulo S[3]: +�a, b, a, c�d ≡(1) �c, d, c, a�b ≡(3) �c, d, c, b�a ≡(1) �b, a, b, c�d +≡(3) �b, a, b, d�c ≡(1) �d, c, d, b�a ≡(3) �d, c, d, a�b ≡(1) �a, b, a, d�c. +On the other hand, there are no such equivalences between the 24 +possible elements of {�a, b, c, a�d}G. Hence dim S[4] ≤ 34 + 24/8 + +24 = 61. +i = 5. First, because �a, b, a, c�d is 3-equivalent to an element beginning +with any of the generators a, b, c, d, we see that applying any of the +four operators �a�, �b�, �c�, �d� on this element will result in an +element already contained in S[3]. +Next, we apply these operators on �a, b, c, a�d. +Of course, we +again have �a, a, b, c, a�d ∈ S[3]. Next, by Proposition 3.10(ii) and +Proposition 4.1(iii), we have �b, a, b, c, a�d ≡(3) �a, b, a, c, a�d ∈ S[4], +and by Proposition 4.1(vi), we have �d, a, b, c, a�d ∈ S[4]. Finally, +�c, a, b, c, a�d ∈ Γ5. Hence S[5] ≤ T[5]. +By Proposition 4.1(vii), the 24 possible elements of Γ5 come in +pairs that are linearly dependent modulo S[4]. Hence dim S[5] ≤ +61 + 24/2 = 73. +i = 6. We have �a, a, b, c, a, b�d = �b, c, a, b�d ∈ S[4]. By Proposition 3.10(ii) +and Proposition 4.1(v), we have +�b, a, b, c, a, b�d ≡(4) �a, b, a, c, a, b�d ≡(3) �a, b, b, d, b, a�c ∈ S[4]. +Next, �c, a, b, c, a, b�d ∈ Γ6, and finally, by Proposition 4.1(vii), we +also have �d, a, b, c, a, b�d ≡(4) �d, b, a, d, b, a�c ∈ Γ6. Hence S[6] ≤ +T[6]. +By Proposition 4.4, the 24 possible elements of Γ6 come in triples +that are pairwise linearly dependent module S[5]. Hence dim S[6] ≤ +73 + 24/3 = 81. +i = 7. We have �a, a, b, c, a, b, c�d ∈ S[5], and by Proposition 4.4, it follows +that also �b, a, b, c, a, b, c�d and �c, a, b, c, a, b, c�d belong to S[5]. Fi- +nally, by Proposition 4.5, we also have �d, a, b, c, a, b, c�d ∈ S[6]. We +conclude that S[7] = S[6], and therefore A = S[6]. +□ + +16 +TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV +References +[Asc97] +M. Aschbacher. 3-transposition groups, volume 124 of Cambridge Tracts in +Mathematics. Cambridge University Press, Cambridge, 1997. +[Asc00] +M. Aschbacher. Finite group theory, volume 10 of Cambridge Studies in +Advanced Mathematics. Cambridge University Press, Cambridge, second +edition, 2000. +[CH95] +H. Cuypers and J. I. Hall. The 3-transposition groups with trivial center. +J. Algebra, 178(1):149–193, 1995. +[DMPSVC20] T. De Medts, S. F. Peacock, S. Shpectorov, and M. Van Couwenberghe. +Decomposition algebras and axial algebras. J. Algebra, 556:287–314, 2020. +[DMR17] +T. De Medts and F. Rehren. Jordan algebras and 3-transposition groups. +J. Algebra, 478:318–340, 2017. +[GS20] +I. Gorshkov and A. Staroletov. On primitive 3-generated axial algebras of +Jordan type. J. Algebra, 563:74–99, 2020. +[Hal22] +J. I. Hall. Generating finite 3-transposition groups. J. Algebra, 607:338–371, +2022. +[HRS15a] +J. I. Hall, F. Rehren, and S. Shpectorov. Primitive axial algebras of Jordan +type. J. Algebra, 437:79–115, 2015. +[HRS15b] +J. I. Hall, F. Rehren, and S. Shpectorov. Universal axial algebras and a +theorem of Sakuma. J. Algebra, 421:394–424, 2015. +[HS95] +J. I. Hall and L. H. Soicher. Presentations of some 3-transposition groups. +Comm. Algebra, 23(7):2517–2559, 1995. +[HSS18] +J. I. Hall, Y. Segev, and S. Shpectorov. On primitive axial algebras of +Jordan type. Bull. Inst. Math. Acad. Sin. (N.S.), 13(4):397–409, 2018. +[Jac68] +N. Jacobson. Structure and representations of Jordan algebras. American +Mathematical Society Colloquium Publications, Vol. XXXIX. American +Mathematical Society, Providence, R.I., 1968. +[KMS20] +S. M. S. Khasraw, J. McInroy, and S. Shpectorov. On the structure of axial +algebras. Trans. Amer. Math. Soc., 373(3):2135–2156, 2020. +[LB83] +J. Lacaze and L. B´en´eteau. The automorphism group of the smallest non- +affine Hall triple system. In Combinatorial mathematics (Marseille-Luminy, +1981), volume 75 of North-Holland Math. Stud., pages 387–391. North- +Holland, Amsterdam, 1983. +Tom De Medts, Department of Mathematics: Algebra and Geometry, Ghent +University, Krijgslaan 281 – S25, 9000 Gent, Belgium +Email address: tom.demedts@ugent.be +Louis Rowen, Department of Mathematics, Bar-Ilan University, Ramat Gan, +Israel +Email address: rowen@math.biu.ac.il +Yoav Segev, Department of Mathematics, Ben-Gurion University, Beer- +Sheva 84105, Israel +Email address: yoavs@math.bgu.ac.il + diff --git a/aNE0T4oBgHgl3EQfnQFP/content/tmp_files/load_file.txt b/aNE0T4oBgHgl3EQfnQFP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e8cd1de5ef972adff6cdc141dea6222fe178f8be --- /dev/null +++ b/aNE0T4oBgHgl3EQfnQFP/content/tmp_files/load_file.txt @@ -0,0 +1,721 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf,len=720 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='02509v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='RA] 6 Jan 2023 PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE TOM DE MEDTS LOUIS ROWEN YOAV SEGEV Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We show that primitive 4-generated axial algebras of Jor- dan type are at most 81-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Introduction Axial algebras were introduced in 2015 by Jonathan Hall, Felix Rehren and Sergey Shpectorov [HRS15b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' They are non-associative commutative algebras generated by axes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', idempotents for which the left multiplica- tion operator is semisimple and such that the resulting eigenspaces multiply according to a given fusion law (see §2 for precise definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In the easiest interesting case, these multiplication operators admit pre- cisely 3 eigenvalues 0, 1 and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' A typical example is provided by Jordan algebras, where each idempotent gives rise to a Peirce decomposition of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In this case, we have η = 1 2, and the fusion law is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' ∗ 1 0 η 1 {1} ∅ {η} 0 ∅ {0} {η} η {η} {η} {1, 0} Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The Jordan fusion law Φ(η) We call the axial algebras with a fusion law Φ(η) axial algebras of Jordan type η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Other than Jordan algebras themselves, there are other interesting examples of axial algebras of Jordan type (for arbitrary values of η ̸= 0, 1), namely the Matsuo algebras arising from 3-transposition groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In this case, the dimension of the algebra is equal to the size of the normal gen- erating set of 3-transpositions of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (See Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5 below for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=') The classification of 3-transposition groups has a long history (see [CH95, Hal22] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' It is a highly non-trivial fact that finitely generated 3-transposition groups are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In fact, this is a consequence Date: January 9, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 1 2 TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV of the classification of finite simple groups, and a direct proof of this fact would be very valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (See [CH95, Theorem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=') One possible approach for such a direct proof is precisely via the corre- sponding Matsuo algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' More generally, we ask the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (We refer to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3 below for the precise meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=') Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let A be a primitive axial algebra of Jordan type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Assume that A is generated by a finite set of axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Can we conclude that A is finite- dimensional?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Notice that, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8 below, a positive answer to this question would show, in particular, that finitely generated 3-transposition groups are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In fact, for η ̸= 1 2, it is equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' It is natural to try to answer this question for an increasing number of axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' For 2-generated primitive axial algebras of Jordan type, this is almost trivial: such algebras are at most 3-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (In fact, much more can be said: [HRS15a, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1] gives a complete classification of such algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=') For 3-generated algebras, this question was answered affirmatively in the recent paper [GS20]: such algebras are at most 9-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Our main result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Primitive 4-generated axial algebras of Jordan type η are at most 81-dimensional, for any η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Moreover, this result is best possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' To go from 3-generated to 4-generated primitive axial algebras of Jordan type is a large step that required substantial new ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In fact, in our new setup, it is almost a triviality to recover the earlier result from [GS20] that such 3-generated algebras are at most 9-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' One of the key ideas is that we will almost never use the actual multiplication in the algebra, but instead, we use sequences of Miyamoto involutions (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' These sequences will allow us to formulate many “rewriting rules” that we can use to systematically deal with larger and larger expressions, until we eventually “wrap up” so that we can reduce every possible expression of length larger than 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The precise meaning of this will be explained below and can be seen in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1, which is a more detailed version of our Main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' It is worth pointing out that going to the next step, primitive 5-generated axial algebras of Jordan type, is expected to be increasingly more difficult, because the upper bound of the dimension will be at least 312 = 531441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (In fact, this is our conjectured upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=') In addition, one of the examples (of dimension 306936) arises from the largest sporadic Fischer group Fi24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Primitive axial algebras of Jordan type Throughout the paper, F will be a commutative field with char F ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' All our algebras will be commutative but non-associative1 F-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 1As usual, non-associative means “not necessarily associative”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE 3 For the definition of fusion laws and axial algebras, we rely on [DMPSVC20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) A fusion law is a pair (X, ∗), where X is a set and ∗ is a map from X × X to 2X, where 2X denotes the power set of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' A fusion law (X, ∗) is called symmetric if x ∗ y = y ∗ x for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) The Jordan fusion law is the fusion law with X = {0, 1, η} (where η is just a symbol) and with ∗ given by Table 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let Φ = (X, ∗) be a fusion law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) A Φ-decomposition of an algebra A is a direct sum decomposition A = � x∈X Ax (as vector spaces) such that AxAy ⊆ Ax∗y for all x, y ∈ X, where AY := � y∈Y Ay for all Y ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) A Φ-decomposition algebra is a triple (A, I, Ω) where A is an F-algebra, I is an index set and Ω is a tuple of Φ-decompositions of A indexed by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In other words, for each i ∈ I, we have a corresponding Φ-de- composition A = � x∈X A(i) x of the algebra A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let Φ = (X, ∗) be a fusion law with 1 ∈ X ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) For each a ∈ A, we write ada for the left multiplication by a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', ada: A → A: x �→ ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) An element a ∈ A is called a Φ-axis if it is idempotent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', a2 = a) and the decomposition of A into the eigenspaces for ada is a Φ-decom- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iii) The algebra A is a Φ-axial algebra if it is generated by a set of Φ-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' This makes A into a Φ-decomposition algebra (with I identified with the given set of axes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iv) A Φ-axial algebra A is primitive if for each axis a of the generating Φ-axes of A, the 1-eigenspace A(a) 1 is 1-dimensional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', is equal to Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (v) An axial algebra of Jordan type η is a Φ-axial algebra for the fusion law Φ = Φ(η) as in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' As we mentioned in the introduction, the two main sources of examples of axial algebras of Jordan type are (1) Jordan algebras, and (2) Matsuo algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We give some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let J be a Jordan algebra over F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', J is a unital commu- tative non-associative algebra such that a2(ab) = a(a2b) for all a, b ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If e ∈ J is an idempotent, then it is an axis for the Jordan fusion law Φ(1 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' this is the famous Peirce decomposition for Jordan algebras (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', [Jac68, Chapter III]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In particular, if J is generated by idempotents, then it is an axial algebra of Jordan type 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let (G, D) be a 3-transposition group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', G is a group and D ⊆ G is a generating set of involutions, closed under conjugation in G, such that the product of any two elements in D has order at most 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let 4 TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV η ∈ F\\{0, 1} be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then the Matsuo algebra Mη(G, D) is the algebra with basis D, with multiplication given by de := \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 e if d = e 0 if o(de) = 2 η 2(d + e − f) if o(de) = 3, where f = de = ed in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By [HRS15a, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5], Mη(G, D) is a primitive axial algebra of Jordan type η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Axial algebras of Jordan type, and more generally any type of decom- position algebras where the fusion law admits a Z/2-grading, admit many involutory automorphisms, the so-called Miyamoto involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) A Z/2-grading of a fusion law (X, ∗) is a map θ: X → Z/2 such that x ∗ y ⊆ θ−1(θ(x) + θ(y)) for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' For instance, the Jordan fusion law from Table 1 is Z/2-graded with θ(0) = θ(1) = 0 and θ(η) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) If (A, I, Ω) is a Φ-decomposition algebra for a Z/2-graded fusion law (X, ∗), then for each i ∈ I, we define a Miyamoto involution τi : A → A: ax �→ (−1)θ(x)ax, when ax ∈ A(i) x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In other words, τi fixes the 0-graded elements and negates the 1-graded elements with respect to the i-th decomposition of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8 below is an important motivation for the main result of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let (G, D) be a 3-transposition group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The following are equivalent: (a) G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (b) D is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (c) Mη(G, D) is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Of course, (a) implies (b), and (b) and (c) are equivalent because Mη(G, D) has dimension |D|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In particular, the dimension of Mη(G, D) is independent of the choice of the base field F and of η ∈ F, so to show that (c) implies (a), we may assume that F is a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then A := Mη(G, D) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By [DMR17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 325] (which relies on [Asc97, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 92, Example (4)]), G/Z(G) is embedded in Aut(A), so G/Z(G) is a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By a theorem of Schur, [Asc00, (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='9), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 168], the derived subgroup G′ is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Since G/G′ is an abelian group generated by a finite number of involutions, it is finite, so we conclude that G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The following are equivalent: (a) Every finitely generated 3-transposition group (G, D) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (b) Every primitive axial algebra A of Jordan type η ̸= 1 2 generated by a finite set of axes X is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (a) ⇒ (b) Let A and X be as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' For x ∈ X, let τx be the Miyamoto involution associated with x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By [HRS15a, Theorem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 105], the group G = ⟨τx | x ∈ X⟩ is a 3-transposition group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By the assumption, G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By [HRS15a, Corollary (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 81], A is spanned by {xg | x ∈ X, g ∈ G}, so A is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (b) ⇒ (a) Let (G, D) be a finitely generated 3-transposition group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then G is generated by a finite number of elements from D, hence the algebra Mη(G, D) is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Thus, by the assumption, it is finite- dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='7 then tells us that G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ In order to get an idea about the complexity of the primitive 4-generated axial algebras of Jordan type, it is useful to look at the list of 4-generated 3-transposition groups first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In particular, this will provide us with an exam- ple of such an algebra of dimension 81, which is precisely the upper bound that we will obtain in our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let (G, D) be a 3-transposition group generated by 4 ele- ments from D (but not by less than 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then its central type is one of the following: (i) W(A4), the Weyl group of type A4 (with |D| = 10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) W(D4), the Weyl group of type D4 (with |D| = 12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iii) 33 : Sym(4) (with |D| = 18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iv) 21+6 : SU3(2)′ (with |D| = 36);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (v) Hall’s 3-transposition group [310]: 2 (with |D| = 81) or its affine quo- tient 33+3 : 2 (with |D| = 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The definition of central type, and the proof of this fact (together with the size of D in each case) can be found in [HS95, Proposition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2)], where the authors point out that this classification has been proven indepen- dently by Zara, Hall and Moori;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' the first written source seems to be Zara’s (unpublished) thesis from 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ The unique 3-transposition group in this list attaining the upper bound |D| = 81 is particularly interesting because it arises as a 3-transposition subgroup of the sporadic Fischer groups Fi23 and Fi24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We give an explicit construction of the resulting Matsuo algebra, based on [LB83, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In fact, we had implemented this example on a computer to experiment with identities, which is how some of our ideas arose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let D be the 4-dimensional vector space over the field F3 (so |D| = 81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We first set (x1, x2, x3, x4) • (y1, y2, y3, y4) := � x1 + y1, x2 + y2, x3 + y3, x4 + y4 + (x1y2 − x2y1)(x3 − y3) � for all xi, yi ∈ F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Next, we set d ∗ e := (d • e) • (d • e) 6 TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV for all d, e ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' For any η ∈ F \\ {0, 1}—recall that F is still our arbitrary base field of characteristic different from 2—we now define an F-algebra with basis D, and with multiplication given by de := � e if d = e η 2(d + e − d ∗ e) if d ̸= e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then by combining [LB83] with Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5, we see that this is precisely the Matsuo algebra corresponding to Hall’s 3-transposition group [310]: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Method From now on, we assume that A is a primitive axial algebra of Jordan type η generated by a finite set S of axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' For each i ≥ 0, we set S[i] := ⟨τa1τa2 · · · τaℓ(b) | ℓ ≤ i, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , aℓ, b ∈ S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In particular, S[0] = ⟨S⟩, and the S[i] form an ascending chain of subspaces of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Our goal is to show that S[n] = A for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The following proposition tells us that we can do this by showing that the ascending chain of the S[i] stabilizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Assume that S[n] = S[n+1] for some n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then A = S[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Following [HRS15a, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 81], we define the closure of the set S of axes to be the smallest set C of axes of A containing S such that for each a ∈ C, we have τa(C) ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In fact, C = {τa1τa2 · · · τaℓ(b) | ℓ ≥ 0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , aℓ, b ∈ S};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' see, for instance, [KMS20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' It now suffices to observe that if S[n] = S[n + 1], then S[n] = S[ℓ] for all ℓ ≥ n, hence S[n] = ⟨C⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By [HRS15a, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 81], however, A is spanned by C, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ From now on, when we refer to an arbitrary axis of A, we will always mean an element of the closure C of S (which is indeed always an axis for the same fusion law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The following two definitions will play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) We let �a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , aℓ� := τa1τa2 · · · τaℓ for all axes a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , aℓ ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) For all x, y ∈ A, we set x ≡(i) y ⇐⇒ x − y ∈ S[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Notice that x ≡(i) y implies x ≡(j) y for all j ≥ i, and also implies that �a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , aℓ�x ≡(i+ℓ) �a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , aℓ�y for all a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , aℓ ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE 7 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The notation �a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , aℓ� will also be used when the ai are axes that are not necessarily contained in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Some care is needed with the use of the equivalence relations ≡(i) in such a situation, as these relations are always meant with respect to the given generating set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By [HSS18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1], primitive axial algebras of Jordan type always admit a (necessarily unique) normalized symmetric Frobenius form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) A bilinear form (·, ·): A×A → F is called a (normal- ized) Frobenius form on A if (xy, z) = (x, yz) for all x, y, z ∈ A and, in addition, (a, a) = 1 for each axis a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) It will be useful to introduce the notation ǫx,y := 1 − 2 η(x, y) for all x, y ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a ∈ A be an axis and x ∈ A be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then τa(x) = x + 2 η(a, x)a − 2 ηax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' This is [HSS18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3] combined with the statement from [HSS18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1] that (a, x) = ϕa(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In [HSS18], their Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3 is used, in fact, in the proof of their Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1 (the existence of the Frobenius form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' On the other hand, if we already assume the existence of the Frobenius form to begin with, then there is an easy direct proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6 by simply decomposing x with respect to the eigenspaces for the axis a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6 has the following immediate but useful consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a, b ∈ A be axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then: (i) �a�b − �b�a = ǫa,b(b − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) If a ∈ S and x ∈ S[i], then (− 2 η)ax ≡(0) �a�x − x ≡(i) �a�x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6, we have τa(b) − τb(a) = � 1 − 2 η(a, b) � (b − a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) This follows immediately from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ We recall the following important fact, which we will be using over and over again, often without explicitly mentioning it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We have �a, b, a� = �τa(b)� for all axes a, b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' This follows from [HRS15a, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 103] and the fact that τa ∈ Aut(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ The following result is a first instance of how useful it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a, b ∈ S and x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then: (i) �a, b, a�x ≡(1) x − 2 ητa(b)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 8 TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV (ii) �a, b, a�x − �b, a, b�x ≡(1) ǫa,b(�b�x − �a�x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In particular, if x ∈ S[i] for some i ≥ 0, then �a, b, a�x ≡(i+1) �b, a, b�x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iii) �b, a, b, a�x ≡(2) �a, b�x + ǫa,b(x − �b, a�x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In particular, if x ∈ S[i] for some i ≥ 2, then �b, a, b, a�x ≡(i) �a, b�x − ǫa,b�b, a�x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) We apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='9 to x and use Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6 on the right-hand side to get �a, b, a�x = x + 2 η(τa(b), x)�a�b − 2 ητa(b)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1) Since �a�b ∈ S[1], the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) Interchanging a and b in (i) and subtracting gives, using Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8(i), �a, b, a�x − �b, a, b�x ≡(1) (− 2 η)ǫa,b(b − a)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8(ii), however, (− 2 η)(bx − ax) ≡(0) (�b�x − x) − (�a�x − x) = �b�x − �a�x, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iii) This follows immediately by applying τb on (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a, b, c ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then: (i) �a, b�a = ǫa,ba + b − ǫa,b�a�b ≡(0) −ǫa,b�a�b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) �a, b, a�c = αc − α�a�b + �c, a�b where α = ǫτa(b),c ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iii) �a, b, c�a ≡(0) δ�a�b − ǫa,c�a, b�c + �c, a�b for some δ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8(i), �a, b�a = �a�(�b�a − �a�b + �a�b) = ǫa,b�a�(a − b) + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) Let α = ǫτa(b),c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By substituting τa(b) for a and c for b in Corol- lary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8(i), we get �τa(b)�c − �c, a�b = α(c − �a�b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The result now follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iii) By Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8(i), we have �a, b, c�a = �a, b, a�c + �a, b� � �c�a − �a�c � = �a, b, a�c + ǫa,c�a, b�(a − c), so (iii) follows from (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Rewriting rules In this section, we will gradually build up “rewriting rules” that will allow us to simplify certain expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' As the length of the expressions increases, the proofs become more and more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a, b, c, d ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then: (i) �a�b ≡(0) �b�a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) �a, b�a ∈ S[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE 9 (iii) �a, b, a�c ≡(1) �c, a�b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iv) �a, b, c�a ≡(1) �c, b�a − ǫa,c�a, b�c (v) �a, b, a, c�d ≡(1) �c, d, c, a�b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (vi) �a, b, c, d, b�a ≡(2) �b, d, c, b�a − ǫa,τb(d)�a, b, c�d ≡(3) �b, d, c, b�a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (vii) �a, b, c, a, b�d ≡(3) �b, a, d, b, a�c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (i) This follows from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (ii) By (i), we have �a, b�a ≡(1) �a, a�b = b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (Of course, this also follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=') (iii) This follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (iv) This follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11(iii) and (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (v) We have �a, b, a, c�d − �c, d, c, a�b = �τa(b)�τc(d) − �τc(d)�τa(b), which is contained in ⟨τa(b) − τc(d)⟩ ≤ S[1] by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='8(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (vi) We have �a, b, c, d, b�a = �a, τb(c), τb(d)�a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Now let S′ = {a, τb(c), τb(d)} and apply (iv) with respect to this set S′ in place of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Notice that S′[1] ≤ S[2], because �τb(c)�a = �b, c, b�a ∈ S[2] (by (iii)), �τb(c)�τb(d) = �b, c, b, b�d = �b, c�d ∈ S[2], so we see that indeed �x�y ∈ S[2] for all x, y ∈ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence �a, τb(c), τb(d)�a ≡(2) �τb(d), τb(c)�a − ǫa,τb(d)�a, τb(c)�τb(d) = �b, d, c, b�a − ǫa,τb(d)�a, b, c�d so we conclude that indeed �a, b, c, d, b�a ≡(2) �b, d, c, b�a − ǫa,τb(d)�a, b, c�d ≡(3) �b, d, c, b�a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (vii) We start from �b, a, c, a, b�d = �τbτa(c)�d = �d�τbτa(c) + ǫd,τbτa(c)(d − τbτa(c)) = �d, b, a�c + ǫd,τbτa(c)(d − �b, a�c) ≡(0) �d, b, a�c − ǫd,τbτa(c)�b, a�c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In particular, �b, a, c, a, b�d ∈ S[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Moreover, applying �b, a� to this equivalence yields �b, a, b, a, c, a, b�d ≡(2) �b, a, d, b, a�c − ǫd,τbτa(c)�b, a, b, a�c ≡(2) �b, a, d, b, a�c, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1) where the last equivalence holds because by (iii) and (iv), we have �b, a, b, a�c ≡(2) �b, c, a�b ∈ S[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 10 TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV We now apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(iii) with x = �c, a, b�d ∈ S[3], which gives �b, a, b, a, c, a, b�d ≡(3) �a, b, c, a, b�d − ǫa,b�b, a, c, a, b�d ≡(3) �a, b, c, a, b�d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The claim follows by combining this with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ For our next rewriting rule in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4, we first need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a, b, c, d ∈ S and let S′ = {a, b, c, τa(d)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then S′[3] ⊆ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let �x, y, z�w be any element with x, y, z, w ∈ S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Of course, if none of these four elements is equal to τa(d), then �x, y, z�w ∈ S[3] ⊆ S[4], and if all four elements are equal to τa(d), then �x, y, z�w = τa(d) ∈ S[1] ⊆ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Suppose that only one of these four elements is equal to τa(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If w = τa(d), then �x, y, z�w = �x, y, z, a�d ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If z = τa(d), then, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iii), �x, y, z�w = �x, y, a, d, a�w ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If y = τa(d), then, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(v), �x, y, z�w = �x, a, d, a, z�w ≡(2) �x, z, w, z, a�d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If x = a or z = a, then �x, y, z�w ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If w = a, then �x, y, z�w ∈ S[4], by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We may thus assume that z = b and w = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If x = b, then we see that �x, y, z�w ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If x = c, then �x, y, z�w ≡(2) �c, b, c, b, a�d ∈ S[3], by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If x = τa(d), then, assuming without loss that y = b, �x, y, z�w = �a, d, a, b, z�w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If z = a, then by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iii), �x, y, z�w ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence we may assume z = c, and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(ii) we may assume that w = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In this case, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iv) shows that �x, y, z�w ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Suppose that three of the four elements x, y, z, w are equal to τa(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If x = y = z = τa(d), then of course �x, y, z�w = �τa(d)�w = �a, d, a�w ∈ S[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' For the other cases, we simply observe that �τa(d), x, τa(d)�τa(d) = �τa(d), x�τa(d) = �a, d, a, x, a�d ∈ S[4], �τa(d), τa(d), x�τa(d) = �x, a�d ∈ S[2], �x, τa(d), τa(d)�τa(d) = �x, a�d ∈ S[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Exactly two of the four elements x, y, z, w are equal to τa(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE 11 We have �τa(d), τa(d), z�w = �z�w ∈ S[1], �x, τa(d), τa(d)�w = �x�w ∈ S[1], and �x, y, τa(d)�τa(d) = �x, y, a�d ∈ S[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' If x = w = τa(d) and y, z ∈ {a, b, c}, then, by (vi), �τa(d), y, z�τa(d) = �a, d, a, y, z, a�d ≡(4) �a, a, z, y, a�d ∈ S[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Next, if x = w = τa(d) and y, z ∈ {a, b, c}, then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11(ii) (with τa(d) in place of a), �τa(d), y, τa(d)�w ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Finally, if y = w = τa(d) and x, z ∈ {a, b, c}, then, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11(i) �x, τa(d), z�τa(d) ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ The following corollary will play an important role in the proof of Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a, b, c, d ∈ S and let T = {a, τa(b), τa(c), d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then T[4] ⊆ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let S′ = {a, b, c, τa(d)} as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2 and notice that T = τa(S′), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', T is obtained from S′ by applying τa on each element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposi- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='9, for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , xk, y ∈ S′ we have �τa(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , τa(xk)�τa(y) = �a, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , xk, a�τa(y) = τa(�x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , xk�y), so T[i] = τa(S′[i]) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2, we have S′[3] ⊆ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Now S′[4] = �a�S′[3] ∪ �b�S′[3] ∪ �c�S′[3] ∪ �τa(d)�S′[3] ⊆ S[5] ∪ �a, d, a�S[4], and hence T[4] = �a�S′[4] ⊆ �a�S[5] ∪ �d, a�S[4] ⊆ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a, b, c, d ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then �a, b, c, a, b, c�d ≡(4) �b, c, a, b, c, a�d ≡(4) �c, a, b, c, a, b�d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let S′ = {a, b, c, τa(d)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2, we have S′[3] ⊆ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We can thus apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vii) with respect to S′ to get �c, b, τa(d), c, b�a ≡(4) �b, c, a, b, c�τa(d), hence �c, b, a, d, a, c, b�a ≡(4) �b, c, a, b, c, a�d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2) On the other hand, we apply �c, b, a, d� to the equivalence in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11(iii) (with b and c interchanged) to get �c, b, a, d, a, c, b�a ≡(4) δ�c, b, a, d, a�c − ǫa,b�c, b, a, d, a, c�b + �c, b, a, d, b, a�c 12 TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV for some δ ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Now �c, b, a, d, a�c ∈ S[4] by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Also, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(v) and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(iii), we have �c, b, a, d, a, c�b ≡(3) �c, b, c, b, c, a�d ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Thus, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vii), �c, b, a, d, a, c, b�a ≡(4) �c, b, a, d, b, a�c ≡(4) �c, a, b, c, a, b�d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3), we see that �b, c, a, b, c, a�d ≡(4) �c, a, b, c, a, b�d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' It now suffices to cyclically permute a, b, c to also get the other equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ We now come to the final and most challenging rewriting rule, which will effectively put a bound on the dimension of 4-generated primitive axial algebras of Jordan type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let a, b, c, d ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then �d, a, b, c, a, b, c�d ∈ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Let T = {τd(a), τd(b), c, d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='3, we have T[4] ⊆ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4 applied to T, this implies that �τd(a), τd(b), c, τd(a), τd(b), c�d ≡(6) �c, τd(a), τd(b), c, τd(a), τd(b)�d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4) We will proceed in two steps: We first show that �τd(a), τd(b), c, τd(a), τd(b), c�d ≡(6) �d, a, c, b, a, c, b�d, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5) and then we show that �c, τd(a), τd(b), c, τd(a), τd(b)�d ∈ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6) Interchanging the role of b and c, it will then follow from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6) that �d, a, b, c, a, b, c�d ∈ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11(i) applied on �d, c�d, we have �τd(a), τd(b), c, τd(a), τd(b), c�d = �d, a, b, d, c, d, a, b, d, c�d = �d, a, b, d, c, d, a, b�c + ǫc,d�d, a, b, d, c, d, a, b�d − ǫc,d�d, a, b, d, c, d, a, b, d�c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Now let γ = −ǫc,τd(b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' then by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vi) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(v), we have �d, a, b, d, c, d, a, b, d�c ≡(6) �d, a, b, d, d, b, a, d�c + γ�d, a, b, d, c, d, a�b ≡(0) γ�d, a, b, d, c, d, a�b ≡(4) γ�d, a, b, a, b, a, d�c ∈ S[6], PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE 13 by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Also, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iv), we have �d, a, b, d, c, d, a, b�d ≡(6) �d, a, b, d, c, b, a�d − ǫb,d�d, a, b, d, c, d, a�b ≡(6) �d, a, b, d, c, b, d�a − ǫb,d�d, a, b, a, b, a, d�c (by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(i) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(v)) ≡(6) �d, a, d, b, a, d, b�c − ǫb,d�d, a, b, a, b, a, c�d (by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vii) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(i)) ∈ S[6], by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Finally, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(ii), �d, a, b, d, c, d, a, b�c ≡(6) �d, a, b, c, d, c, a, b�c ≡(6) �d, a, b, c, d, b, a�c − ǫb,c�d, a, b, c, d, c, a�b (by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iv)) ≡(6) �d, a, b, c, d, b, c�a − ǫb,c�d, a, b, a, b, a, c�d (by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(i) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(v)) ≡(5) �d, a, c, b, a, c, b�d (by 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vii) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(iii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='11(iii), there exists δ ∈ F with �c, τd(a), τd(b), c, τd(a), τd(b)�d = �c, d, a, b, d, c, d, a, b�d ≡(6) δ�c, d, a, b, d, c, d�a − ǫb,d�c, d, a, b, d, c, d, a�b + �c, d, a, b, d, c, b, d�a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Now by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iii), �c, d, a, b, d, c, d�a ∈ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(v) and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(iii), we have �c, d, a, b, d, c, d, a�b ≡(5) �c, d, a, b, a, b, a, d�c ∈ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Finally, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vii) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4, we also have �c, d, a, b, d, c, b, d�a ≡(6) �c, d, a, d, b, a, d, b�c ≡(6) �c, d, d, b, a, d, b, a�c = �c, b, a, d, b, a�c ∈ S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='6) and thus finishes the proof of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 4-generated primitive axial algebras of Jordan type We are now ready to prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Although it requires some care to write down the proof, the hard work has already been done in Propo- sitions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 14 TOM DE MEDTS, LOUIS ROWEN, YOAV SEGEV Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Assume that A is generated by a set S = {a, b, c, d} of 4 axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then A = S[6] and A is at most 81-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' More precisely, let G be the group Sym(S) of all permutations of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' De- fine2 Γ0 = {a}G, Γ1 = {�a�b}G, Γ2 = {�a, b�c}G, Γ3 = {�a, b, c�d}G, Γ4 = {�a, b, a, c�d, �a, b, c, a�d}G, Γ5 = {�a, b, c, a, b�d}G, Γ6 = {�a, b, c, a, b, c�d}G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Then for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , 6}, we have S[i] = ⟨Γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , Γi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In particular, A = ⟨Γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , Γ6⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Moreover, there is some redundancy in these spanning sets: The dimen- sion of each of the S[i] is at most 4, 10, 22, 34, 61, 73 and 81, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' For each i ≤ 6, let T[i] be the subspace of A spanned by Γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' , Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Obviously, we have T[i] ≤ S[i] for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We will show recursively that for each i ≤ 6, S[i] = T[i], and that S[7] = S[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We will, at the same time, compute the maximal possible dimension of each T[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Notice that for each i, the subspace S[i + 1] is spanned by S[i] and all elements obtained by applying the four operations �a�, �b�, �c� and �d� on the elements of S[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In order to go from S[i] = T[i] to the next step S[i + 1], it will suffice, by G-symmetry, to apply these four operations on the given representative of the set Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Obviously, T[0] = ⟨a, b, c, d⟩ = S[0], and dim S[0] ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We have �a�a = a ∈ S[0], whereas applying any of the other three operations �b�, �c�, �d� on the representative a ∈ Γ0 results in an element of Γ1, so S[1] ≤ T[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(i), we have �b�a ≡(0) �a�b, so the 12 possible elements of Γ1 come in pairs that are linearly dependent modulo S[0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence dim S[1] ≤ 4 + 12/2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We have �a, a�b = b ∈ S[0] and �b, a�b ∈ S[1] by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' On the other hand, �c, a�b and �d, a�b belong to Γ2 and hence to T[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence S[2] ≤ T[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(i), we have �a, b�c ≡(1) �a, c�b, so the 24 possible elements of Γ2 come in pairs that are linearly dependent modulo S[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence dim S[2] ≤ 10 + 24/2 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' 2There is some obvious abuse of notation here: a priori, the group G does not act on A, so when we write an expression like {�a, b, c�d}G, we really mean {�aρ, bρ, cρ�(dρ) | ρ ∈ G}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' PRIMITIVE 4-GENERATED AXIAL ALGEBRAS OF JORDAN TYPE 15 i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We have �a, a, b�c = �b�c ∈ S[1], and we have �b, a, b�c ∈ S[2] by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iii) and �c, a, b�c ∈ S[2] by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' On the other hand, �d, a, b�c belongs to Γ3 and hence to T[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence S[3] ≤ T[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(i), we have �a, b, c�d ≡(2) �a, b, d�c, so the 24 possible elements of Γ3 come in pairs that are linearly dependent modulo S[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence dim S[3] ≤ 22 + 24/2 = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' i = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We have �a, a, b, c�d = �b, c�d ∈ S[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' On the other hand, �b, a, b, c�d and �c, a, b, c�d belong to Γ4 and hence to T[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Finally, �d, a, b, c�d ≡(3) �d, a, b, d�c ∈ Γ4, so �d, a, b, c�d belongs to ⟨S[3], Γ4⟩ ≤ T[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence S[4] ≤ T[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(v), the 24 possible elements of {�a, b, a, c�d}G come in 8-tuples that are pairwise linearly dependent modulo S[3]: �a, b, a, c�d ≡(1) �c, d, c, a�b ≡(3) �c, d, c, b�a ≡(1) �b, a, b, c�d ≡(3) �b, a, b, d�c ≡(1) �d, c, d, b�a ≡(3) �d, c, d, a�b ≡(1) �a, b, a, d�c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' On the other hand, there are no such equivalences between the 24 possible elements of {�a, b, c, a�d}G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence dim S[4] ≤ 34 + 24/8 + 24 = 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' i = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' First, because �a, b, a, c�d is 3-equivalent to an element beginning with any of the generators a, b, c, d, we see that applying any of the four operators �a�, �b�, �c�, �d� on this element will result in an element already contained in S[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Next, we apply these operators on �a, b, c, a�d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Of course, we again have �a, a, b, c, a�d ∈ S[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Next, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(ii) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(iii), we have �b, a, b, c, a�d ≡(3) �a, b, a, c, a�d ∈ S[4], and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vi), we have �d, a, b, c, a�d ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Finally, �c, a, b, c, a�d ∈ Γ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence S[5] ≤ T[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vii), the 24 possible elements of Γ5 come in pairs that are linearly dependent modulo S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence dim S[5] ≤ 61 + 24/2 = 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' i = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We have �a, a, b, c, a, b�d = �b, c, a, b�d ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='10(ii) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(v), we have �b, a, b, c, a, b�d ≡(4) �a, b, a, c, a, b�d ≡(3) �a, b, b, d, b, a�c ∈ S[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Next, �c, a, b, c, a, b�d ∈ Γ6, and finally, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='1(vii), we also have �d, a, b, c, a, b�d ≡(4) �d, b, a, d, b, a�c ∈ Γ6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence S[6] ≤ T[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='4, the 24 possible elements of Γ6 come in triples that are pairwise linearly dependent module S[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Hence dim S[6] ≤ 73 + 24/3 = 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' i = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' We have �a, a, b, c, a, b, c�d ∈ S[5], 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Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', 373(3):2135–2156, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' [LB83] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Lacaze and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' B´en´eteau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' The automorphism group of the smallest non- affine Hall triple system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' In Combinatorial mathematics (Marseille-Luminy, 1981), volume 75 of North-Holland Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=', pages 387–391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' North- Holland, Amsterdam, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content=' Tom De Medts, Department of Mathematics: Algebra and Geometry, Ghent University, Krijgslaan 281 – S25, 9000 Gent, Belgium Email address: tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='demedts@ugent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='be Louis Rowen, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel Email address: rowen@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='biu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='il Yoav Segev, Department of Mathematics, Ben-Gurion University, Beer- Sheva 84105, Israel Email address: yoavs@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='bgu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} +page_content='il' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE0T4oBgHgl3EQfnQFP/content/2301.02509v1.pdf'} diff --git a/aNE4T4oBgHgl3EQfOwxI/content/tmp_files/2301.04967v1.pdf.txt b/aNE4T4oBgHgl3EQfOwxI/content/tmp_files/2301.04967v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3aad0b812fde5532f40ac7bf32a636ce8f5abc00 --- /dev/null +++ b/aNE4T4oBgHgl3EQfOwxI/content/tmp_files/2301.04967v1.pdf.txt @@ -0,0 +1,1911 @@ +Shadows of Kerr-Vaidya-like black holes +Hai Siong Tan +University of Pennsylvania, Perelman School of Medicine, Department of Radiation Oncology, +Philadelphia, USA +Jan 2023 +Abstract +In this work, we study the shadow boundary curves of rotating time-dependent black hole solutions +which have well-defined Kerr and Vaidya limits. These solutions are constructed by applying the +Newman-Janis algorithm to a spherically symmetric seed metric conformal to the Vaidya solution +with a mass function that is linear in Eddington-Finkelstein coordinates. Equipped with a confor- +mal Killing vector field, this class of solution exhibits separability of null geodesics, thus allowing +one to develop an analytic formula for the boundary curve of its shadow. We find a simple power +law describing the dependence of the mean radius and asymmetry factor of the shadow on the +accretion rate. Applicability of our model to recent Event Horizon Telescope observations of M87∗ +and Sgr A∗ is also discussed. +1 +arXiv:2301.04967v1 [gr-qc] 12 Jan 2023 + +Contents +1 +Introduction +2 +2 +A family of rotating Vaidya-like black hole solutions +4 +2.1 +Conformal factors, coordinate charts and the Newman-Janis algorithm . . . . . . . . +4 +2.2 +Horizons in the solution parameter space . . . . . . . . . . . . . . . . . . . . . . . . . +6 +3 +Null geodesics, photon spheres and shadow formulas +8 +3.1 +On null geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +3.2 +Shadow formulas from photon region . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +4 +Portraits of the shadow +10 +4.1 +Scaling laws for variation of R and A with µ +. . . . . . . . . . . . . . . . . . . . . . +11 +4.2 +On the shadows of M87∗ and Sagittarius A∗ as observed by EHT . . . . . . . . . . . +14 +5 +Discussion +16 +A Global geometry and matching spacetimes via junction conditions +17 +B On the reference frame of the shadow observer +19 +B.1 +In the limit of a = 0 +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +B.2 +Observers in the {v, w, θ, φ} chart and aberration formulas . . . . . . . . . . . . . . . +19 +1 +Introduction +Recent Event Horizon Telescope (EHT) observations of horizon-scale shadow images of M87∗ [1] +and Sgr A∗ [2] have furnished not only a direct visual evidence of black holes, but have also led to +many new constraints on various potential deviations from General Relativity. The boundary curve +of the black hole shadow emerges from light rays that spiral asymptotically from the photon region +demarcating the boderline between light rays that will eventually be captured by the black hole and +those that escape to infinity [3].i The geometry of this boundary curve depends on the background +metric which could thus be probed by EHT observations [7, 8, 9] . +For example, the shadow +geometry of Sgr∗ has been used to exclude the central object being a Reissner-Nordstrom-type +naked singularity or a traversable Misner-Thorne wormhole [8]. +Surrounding the black hole shadow is an emission ring of which structure is sensitive to a rich +set of astrophysical phenomena, such as radiative transfer, that characterize the matter-energy +accretion process. Typically, general relativistic magnetohydrodynamic (GRMHD) simulations are +used to model the accretion flow processes [7, 8, 10, 11]. For the EHT experiments, they have +revealed the emission ring properties to be consistent with a number of accretion models built +iThis curve is termed as the ‘critical curve’ by Gralla et al. in [4] and ‘apparent boundary’ by Bardeen in [5]. See +for example the review of [6] for an extensive discussion of basic ideas and history. +2 + +upon the background of a Kerr black hole [7, 8]. The spacetime metric in these simulations is +assumed to be purely Kerr spacetime throughout, with the energy-momentum tensor capturing +the magnetic field and average plasma properties [10, 12]. In [13, 14, 15], it was noted that the +shadow size and shape is hardly influenced by the accretion details, and thus serves as a pristine +signature of spacetime geometry. An implicit assumption is that the backreaction of the GRMHD +energy-momentum tensor on the metric has a negligible influence on the shadow and could thus be +ignored in deriving its geometry. +In this paper, we study the shadow boundary curves of a class of rotating time-dependent +black hole solutions of which metric is a deformation of the Kerr solution described by a small +dimensionless parameter µ. In the limit of vanishing spin, our spacetime reduces to a well-known +model of spherically accreting black hole - the Vaidya spacetime with a mass function µv, with v +being an ingoing Eddington-Finkelstein coordinate and µ being the mass accretion rate constant +in natural units. The latter solution was studied most recently in [16] where the authors derived +and examined its shadow characteristics analytically. Most crucially, an analytic treatment of the +shadow was possible by virtue of the existence of a Carter constant leading to separability of its +null geodesic equations. This is related to a conformal Killing symmetry associated with the linear +mass function, and hence its choice, for it enables the authors of [16] to derive explicit formulas +for the radius of the photon sphere and the shadow angular diameter. One main motivation of our +work here is to seek a rotating generalization of the analytic treatment in [16]. This would serve +as a simple model of a backreacted Kerr-like geometry that is accreting mass, and for which an +analytic derivation of its shadow geometry is possible. For readers familiar with exact solutions in +GR, a natural candidate would be the Kerr-Vaidya solution [17] which can be obtained by replacing +the constant Kerr mass with a variable mass function in the original Kerr line element expressed in +Eddington-Finkelstein coordinates. Unfortunately, as we’ll elaborate later, this solution does not +offer any additional Carter constant that could lead to its null geodesic equations being separable. +We construct our solutions by applying the Newman-Janis algorithm [18] to a spherically sym- +metric seed metric conformal to the Vaidya solution with the mass function that is linear in +Eddington-Finkelstein coordinates. Fortunately, this solution-generating technique turns out to +preserve the conformal Killing vector field in the original Vaidya metric, leading to separability of +null geodesics, and ultimately allows us to develop an analytic formula for the boundary curve of +its shadow. The solution space is parametrized by {a, µ, Ms} where {a, Ms} are the spin and mass +parameters of Kerr spacetime in the vanishing µ limit. Like the Kerr solution, there are regions in +the moduli space which do not pertain to black holes. Motivated by phenomenological interests, we +focus on the regime of parameters where our solution has event horizons like those of Kerr, with the +conformal Killing horizon at a large distance away from the shadow observer and the outer horizon. +Thus, our solution serves as a simple model of an accreting Kerr-like geometry not globally but for +a finite spatial domain defined by the interior of the conformal Killing horizon. Generically, the +shadow geometry is sensitive to the choice of coordinates. We work in a chart which reduces to +the Kerr spacetime in Boyer-Lindquist coordinates in the limit µ = 0, and the Vaidya spacetime +in Eddington-Finkelstein-like coordinates in the limit a = 0. Correspondingly, we verified that our +shadow formulas reduce consistently to those of Kerr [19] and Vaidya [16] under these limits. +As reviewed in for example [6], analytic derivations in cases that allow them complement numer- +ical studies of shadow geometry in general. For example, for Schwarzschild spacetime, the angular +diameter of its shadow is ∼ 3 +√ +3Ms/Ro for a distant observer located at the radial coordinate Ro +[20]. This numerical value has turned out to be very useful as a guide in the analysis of shadow +size and shape in EHT’s recent observations [1, 2]. For our shadow analysis here, we find a simple +power law describing the dependence of the mean radius and asymmetry factor of the shadow on +3 + +the accretion rate. The latter describes the departure of the shadow from circularity and has been +constrained in M87∗ studies by EHT team [1]. When applied to the parameters of M87∗ and Sgr +A∗, our analysis of shadow geometry appears to indicate that the effect of µ is very small, and +thus provides support for the assumption of using the pure Kerr metric throughout in GRMHD +simulations. Our results, in addition, yield an empirical formula that parametrizes the variation of +mean radius and asymmetry factor with accretion rate explicitly, and can thus be used to anticipate +when backreaction of accretion on the metric may be significant. +Our paper is organized as follows. In Section 2, we present the construction of a class of Kerr- +Vaidya-like solutions and elaborate on some basic aspects of its geometry and moduli space, followed +by a derivation of some analytical formulas for shadow geometry in Section 3. In Section 4, we +present several visual plots of the shadow and examine how the mean radius and asymmetry factor +of the shadows vary with various parameters. We also include a brief discussion on recent EHT +observations of M87∗ and Sgr A∗ in relation to our model geometry. Finally, we end with some +concluding remarks in Section 5. Appendix A presents an extension of our solution obtained by +matching the spacetime at some cutoff distance to Kerr-like solutions that are asymptotically flat +via Darmois-Israel junction conditions. In Appendix B, for completeness, we develop an aberration +formula for observers in another reference frame which, in the zero spin limit, reduces to another +class of observers discussed previously in [16] for Vaidya spacetime. +2 +A family of rotating Vaidya-like black hole solutions +We begin with the Vaidya metric in the coordinatesii +ds2 += +− +� +1 − 2m(v) +w +� +dv2 + 2dvdw + w2 � +dθ2 + sin2 θdφ2� +, +(1) +with the domains v ∈ (0, ∞), w ∈ (0, ∞), θ ∈ (0, π), φ ∈ (0, 2π). We note that m(v) is a mass +function that can be used to model a time-dependent black hole of which exterior is described by +(1). The solution (1) solves the field equations in ordinary GR with the energy momentum tensor +T µν = m(v)KµKν, Kν∂ν = ∂w which is typically interpreted as that of a null dust moving in +the direction of decreasing w, with the black hole accreting (radiating) mass if m′(v) is positive +(negative). +2.1 +Conformal factors, coordinate charts and the Newman-Janis algorithm +In this work, we restrict ourselves to the case where m(v) = µv, where µ is a positive constant. In +this case, the geometry admits a conformal Killing vector field. To see this, we define ∂/∂T as the +conformal Killing vector and make a coordinate transformation as follows. +v = r0eT/r0, +w = reT/r0, +(2) +where r0 is a positive constant with dimension of length. This brings (1) to +ds2 = e2T/r0 +� +− +� +1 − 2µr0 +r +− 2r +r0 +� +dT 2 + 2dTdr + r2(dθ2 + sin2 θdφ2) +� +, +(3) +iiThe unusual choice of the symbol w to denote radial distance for this line element is solely due to shortage of +conventions for the many different radial coordinates that we’ll use throughout this paper. +4 + +with T ∈ (−∞, ∞), r ∈ (0, ∞). In this form, the metric is conformal to a manifestly static spacetime +which can be taken to generate a rotating solution via Newman-Janis algorithm. But first, we seek +a temporal coordinate such that constant time slices are 3-dimensional spatial manifolds. Defining +T = t + Υ(r), +Υ(r) = +� r +d ˜R +� +1 − 2µro +˜R +− 2 ˜R +r0 +�−1 +, +(4) +the line element then reads +ds2 = e +2(t+Υ(r)) +r0 +� +− +� +1 − 2M(r) +r +� +dt2 + +� +1 − 2M(r) +r +�−1 +dR2 + r2(dθ2 + sin2 θdφ2) +� +≡ Ω2(t, r)ds2 +static, +(5) +where t ∈ (−∞, ∞) and +M(r) ≡ µr0 + r2 +r0 +. +If we restrict ourselves to the spacetime patch where the conformal Killing vector field ∂ +∂t is timelike, +then letting 1 − 2M/r > 0 leads to the domain +r ∈ (Rh, Rc), +Rh,c = r0 +4 +� +1 ± +� +1 − 16µ +� +, +µ < 1/16, +where Rh is the black hole horizon and Rc denotes the conformal Killing horizon. The Schwarzschild +limit can be obtained as a double scaling limit as follows. +µ → 0, r0 → ∞, µro = Ms, +(6) +where Ms is a finite mass parameter equivalent to the ADM mass of the limiting Schwarzschild +black hole. We now apply the Newman-Janis algorithmiii to the metric ds2 +static which leads to a +metric endowed with angular momentum +ds2 +static → ds2 +rotating += +− +� +1 − 2M(r)r +Σ +� +dt2 − 4M(r)ar sin2 θ +Σ +dφdt ++ +� +r2 + a2 + 2M(r)a2r sin2 θ +Σ +� +sin2 θdφ2 + Σ +∆dr2 + Σdθ2, +(7) +where a is the spin parameter and +Σ = r2 + a2 cos2 θ, +∆ = r2 − 2M(r)r + a2. +We will also modify the exponential argument of the conformal factor Ω(t, r) in (5) as follows +t + Υ(r) → t + Υa(r), +Υa(r) ≡ +� r +dr +r2 + a2 +r2 − 2M(r)r + a2 . +(8) +The full metric then reads +ds2 = e +2(t+Υa(r)) +r0 +ds2 +rotating, +(9) +with ds2 +rotating, Υa(r) being defined in (7) and (8) respectively. In the scaling limit of (6), the line +element (9) reduces to Kerr spacetime in Boyer-Lindquist coordinates. +iiiTo be precise, this algorithm carries with it the assumption of asymptotic flatness in the generated metric which +doesn’t hold for our solution though. +5 + +Now, like the Kerr solution where M is instead just a constant, the metric (9) has singularities +at the roots of ∆ = 0. To extend the spacetime beyond these singularities, we can perform a +coordinate transformation +˜T = t + Υa(r), +˜φ = −φ − a +� r +dr +1 +r2 − 2M(r)r + a2 +(10) +which leads to +ds2 += +e +2µ +Ms ˜T +� +− +� +1 − +2M(r)r +r2 + a2 cos2 θ +� +(d ˜T + a sin2 θd˜φ)2 + 2(d ˜T + a sin2 θd˜φ)(dr + a sin2 θd˜φ) ++(r2 + a2 cos2 θ)dΩ2 +� +, +(11) +In the a = 0 limit, we recover Vaidya spacetime in the conformally static coordinates of (3), whereas +the µ = 0 limit (as in (6)) takes the metric to that of Kerr in ingoing Eddington-Finkelstein +coordinates. +We note that in (11), replacing M(r) → µ ˜T and removing the conformal factor +e +2µ +Ms ˜T yields the Kerr-Vaidya solution [17] which evidently isn’t equipped with the conformal Kiling +symmetry. This leads to non-separability of null geodesics which would not allow us to solve for +the shadow boundary curve analytically. +Our main interest in this class of time-dependent solutions lies in its property of being locally +deformable to the Kerr geometry in Boyer-Lindquist coordinates in the µ → 0, µr0 → Ms limit, +and, in the limit of a = 0, to the Vaidya solution in a chart where it’s conformally static. This gives +us a model of local geometry that approximates both spacetimes in a coordinate system suitable for +deriving the analytical form of the black hole shadow. For this specific purpose, we work in the +{t, r, θ, φ} chart (line element in (9)), where the spacetime is conformal to a Kerr-like solution in +Boyer-Lindquist coordinates. In Section 2.2, we explore the parameter space {a, µ, Ms} in greater +detail. +2.2 +Horizons in the solution parameter space +In (9), setting grr = 0 yields the following cubic equation in r. +− 2µ +Ms +r3 + r2 − 2Msr + a2 = 0. +(12) +In certain regimes of the parameter space of {µ, a}, one could find event horizons. Consider the +root space of the cubic equation (12) of which discriminant reads (henceforth, we define a → a/Ms +to be a dimensionless parameter) +D = 4 +� +1 − a2 − 16µ + 18a2µ − 27a4µ2� +. +The sign of D determines the number of real roots to grr = 0. As a quadratic equation in µ, we +can derive the curves along which D = 0, which read +µ± = −8 + 9a2 ± +� +4 − 3a2�3/2 +27a4 +. +(13) +They enclose the region which pertains to three distinct roots — two event horizons (outer and +inner) and the conformal Killing horizon, and they intersect at the point (see Fig. 1 ) +(ae, µe) = +� 2 +√ +3, 1 +12 +� +, +(14) +6 + +which represents a generalized extremal limit. At any constant µ ∈ (0, 1 +12), the upper bound on +a is given by taking µ = µ−(a) along which the inner and outer event horizons coincide. Along +µ = µ+(a), the conformal Killing horizon coincides with the outer event horizon. +μ+ +μ- +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +a +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +μ +Figure 1: Graph depicting the parameter space (µ, a) of our family of solutions. The curve µ+ begins +at (0, 1 +16) and contains solutions where the outer event horizon and conformal Killing horizon are +degenerate. The curve µ− contains all the extremal solutions with degenerate outer and inner event +horizons and with a finite conformal Killing horizon radius. Both curves merge at the extremal +point ( 2 +√ +3, 1 +12) at which there is only apparent horizon at r = 2Ms. Our family of solutions can +be seen as parametric deformations of the Vaidya solution (vertical axis) and the Kerr solution +(horizontal axis). +As depicted in Fig. 1, the region enclosed by the axes and the two curves has non-degenerate +outer, inner event horizons and conformal Killing horizon. The small µ ≪ 1 region of this enclosed +segment is of closer phenomenological interest to us. Let us consider a generic point in this region. +Ordering the roots of (12) as Ri < Re < Ra, we find that up to first few orders in µ, a : +Re += +Ms +�� +1 + +� +1 − a2 +� ++ 4 + 4 +√ +1 − a2 − 5a2 − 3a2√ +1 − a2 + a4 +2(1 − a2) +u + O(µ2) +� += +Ms [(2 + 8µ) − (2µ + 1/2)a + . . .] +(15) +Ri += +Ms +�� +1 − +� +1 − a2 +� ++ 4 − 4 +√ +1 − a2 − 5a2 + 3a2√ +1 − a2 + a4 +2(1 − a2) +µ + O(µ2) +� += +Ms +�a +2 + a2 +8 − µa3 +16 + a3 +16 + . . . +� +(16) +Ra += +Ms +2µ − 2Ms − 8µMs + 2aµMs + . . . +(17) +The radii Re, Ri are those of the outer and inner event horizons smoothly connected to their +corresponding expressions in the ordinary Kerr solution, whereas Ra is the radius of the conformal +Killing horizon associated with the conformal Killing vector ∂t (this symmetry is preserved by +the Newman-Janis algorithm we implemented). For a generic point within our domain of interest +(region enclosed by curves and axes in Fig. 1 ), our shadow observer is a timelike observer located +at some distance Re < r < Ra away from the outer event or conformal Killing horizon. +7 + +3 +Null geodesics, photon spheres and shadow formulas +3.1 +On null geodesics +Metrics of the form ds2 +rotating in (9) admit null geodesics which are separable. This condition was +shown for the Kerr solution in for example [6, 19], and for other well-motivated forms of mass +function M(r) in [21]. Such a property is preserved within its conformal class, and in particular +by our line element ds2 in (9). To appreciate this, we first recall that for a pair of metrics which +are conformally related, say gµν = Ω2(x)˜gµν, their geodesic equations are related by +d2xα +dη2 + Γα +βν +dxβ +dη +dxν +dη = 0 = d2xα +dη2 + ˜Γα +βν +dxβ +dη +dxν +dη − 2Ω−1 dΩ +dη +dxα +dη , +(18) +where ˜Γα +βν are the Christoffel symbols associated with the metric ˜g. The last term of (18) can be +rewritten as +d2xα +dλ2 + ˜Γα +βν +dxβ +dλ +dxν +dλ = 0, +dλ +dη = Ω2(xα(η)). +(19) +Thus, a suitable redefinition of the affine parameter leads to identical forms of the geodesic equations +for the pair of conformally related metrics. In particular, we expect separability of null geodesic +equations just like the Kerr solution or more generally the Kerr-like solutions studied in [21]. +Our solution has a conformal Killing vector field K ∼ ∂t which naturally gives a quantity +conserved along its null geodesics. In parallel with the notion of energy in the static case, we call +this conserved quantity ˜E = KµPµ. Also, the metric components are independent of φ, and we call +L = pφ the associated conserved quantity. These symmetries motivate the use of the Hamilton- +Jacobi formalism for geodesics where one first defines an auxiliary action S(κ, xµ) obeying +∂S +∂κ + 1 +2gµνpµpν = 0, +pµ = gµνpν = gµν ∂S +∂xν . +(20) +By virtue of the nature of conserved quantities, we adopt the following ansatz for the action S: +S = − ˜Et + Lφ + Sr(r) + Sθ(θ) + 1 +2µ2κ, +µ = −pαpα. +(21) +By construction, the 4-momenta pµ = dxµ +dη = gµν∂νS(x) satisfies the geodesic equation, with κ = 0 +for null geodesics. We find that the Hamilton-Jacobi equation (20) leads to +− ∆(r)(∂rSr)2 + [(r2 + a2)E − aL]2/∆(r) = (∂θSθ)2 + (L − aE sin2 θ)2/ sin2 θ = K, +(22) +where the constant K indicates separability. For deriving the shadow formulas, we need the explicit +expressions for the 4-momenta which we find to simplify as +dt +dη += +1 +Σ(r)e− 2(t+Υa(r)) +r0 +� +−a(aE sin2 θ − L) + (r2 + a2)P(r) +∆(r) +� +, +(23) +dr +dη += +± +1 +Σ(r)e− 2(t+Υa(r)) +r0 +� +R(r), +(24) +dθ +dη += +± +1 +Σ(r)e− 2(t+Υa(r)) +r0 +� +Ξ(θ), +(25) +dφ +dη += +1 +Σ(r)e− 2(t+Υa(r)) +r0 +� +− +� +aE − +L +sin2 θ +� ++ aP(r)/∆(r) +� +, +(26) +8 + +where +P(r) ≡ E(r2 + a2) − aL, R(r) ≡ P(r)2 − K∆(r), Ξ(θ) ≡ Q + cos2 θ +� +a2E2 − L2/ sin2 θ +� +, +with Q being conventionally called the Carter constant defined by +Q ≡ K − (L − aE)2. +(27) +At this point, we note that apart from the specific form of our mass function M(r) = Ms + r2/r0 +implicitly contained in ∆(r) = r2 − 2M(r)r + a2, the 4-momenta expressions in (23) - (26) are +identical to those found in [21] up to the conformal factor e− 2(t+Υa(r)) +r0 +. This is consistent with the +general relations governing conformally related metrics as described in eqns (18) and (19). +3.2 +Shadow formulas from photon region +For our analysis of the shadow, we define the constants of motion +η ≡ Q +E2 , +ξ ≡ L +E . +(28) +Any null geodesics with r = Rp for some constant Rp leads to the condition +R(Rp) = P(Rp)2 − K∆(Rp) = 0. +Further, if the orbit is unstable then setting d2r/dη2 = 0 yields +R′(rp) = 0. +After some algebra, we find that these couple of equations lead to the constants of motion +a L +E /M 2 +s += +4(1 + µR2 +p)R2 +p − (3µR2 +p + Rp + 1)(R2 +p + a2) +−3µR2p − 1 + Rp +, +(29) +K +E2 /M 2 +s += +((R2 +p + a2) − aL/E)2 +R2p − 2(1 + µR2p)Rp + a2 . +(30) +These constants of motion also characterize non-spherical geodesics which asymptotically approach +those confined to spheres defined by R(Rp) = R′(Rp) = 0. In eqns. (29) and (30), and henceforth, +for notational simplicity, we define Rp, a in units of Ms. Now consider an observer at the position +(Ro, θinc) and described by an orthonormal tetrad as follows. +e0 = e− µT +Ms (r2 + a2)∂t + a∂φ +√ +Σ∆ +, e1 = e− µT +Ms +� +1 +Σ∂θ, +e2 = −e− µT +Ms ∂φ + a sin2 θ∂t +√ +Σ sin θ +, e3 = −e− µT +Ms +� +∆ +Σ ∂r. +(31) +As explained in for example [6, 19], this choice leads to e0 being the 4-velocity of the shadow +observer with e0 ± e3 being tangential to the principal null congruences. (The various expressions +in (31) differ from those in [6, 19] by the conformal factor which we need to take into account for +an orthonormalized set of basis vectors.) The tangent vector of each light ray reaching the observer +reads +d +dη = pµ∂µ = ˙r∂r + ˙θ∂θ + ˙φ∂φ + ˙t∂t = α(−e0 + sin θ cos φe1 + sin θ sin φe2 + cos θe3), +(32) +9 + +where α = gµνpµeν +0. Equating the coefficients after evaluating both sides of (32) at (Ro, θinc) yields +sin Φ = LE(Rp) − a sin2 θinc +� +KE(Rp) sin θinc +, sin Θ = +� +∆(Ro)KE(Rp) +R2o − aLE(Rp) + a2 , +(33) +where +LE ≡ +L +M2s E , +KE ≡ +K +M2s E2 , +and we have switched to a different notation for the celestial coordinates Φ, Θ for clarity having +derived their eventual expressions. In particular, we note that θinc measures the angle between the +black hole’s spin axis as defined by θ = 0(the zero locus of gφφ) and the observer, with e3 being +parallel to the line of sight connecting the observer to the origin of the Boyer-Lindquist-like chart +in (9). +The photon region consists of spherical orbits with radii bounded in the domain +Rp ∈ (Rp,min, Rp,max), +(34) +where Rp,min, Rp,max are defined by setting sin Φ = +1, −1 respectively, i.e. +aLE(Rp,min) += +a2 sin2 θinc + a +� +KE(Rp,min) sin(θinc), +(35) +aLE(Rp,max) += +−a2 sin2 θinc − a +� +KE(Rp,max) sin(θinc). +(36) +In the vanishing a limit, the width of the photon region (Rp,min, Rp,max) goes to zero, and collapses +to a single value of Rp which is the photon sphere radius of Vaidya spacetime. In this limit, from +(35), +lim +a→0 aLE → R2 +p +3 + µR2 +p − Rp +Rp − 1 − 3µR2p += 0 ⇒ Rp = 1 +2µ +� +1 − +� +1 − 12µ +� +, +where we have restricted the root to be smaller than the conformal Killing horizon radius. This is +indeed the expression obtained in [16]. Further taking the µ = 0 limit yields Rp = 3 which is the +radius of Schwarzschild photon sphere. In the a = 0 limit, our expression for sin Θ in (33) reduces +to eqn. (41) of [16] which is the sine of the angular radius of the Vaidya solution’s shadow. +4 +Portraits of the shadow +In this Section, we discuss geometrical properties of the shadow in more details. We first recall +that the coordinate system describing the shadow observer has a conformal Killing horizon Ra +obtained as the largest root of grr = 0 in the line element (9) which we reproduce explicitly below +for convenience. +ds2 += +e +2(t+Υa(r)) +r0 +� +− +� +1 − 2M(r)r +Σ +� +dt2 − 4M(r)ar sin2 θ +Σ +dφdt ++ +� +r2 + a2 + 2M(r)a2r sin2 θ +Σ +� +sin2 θdφ2 + Σ +∆dr2 + Σdθ2 +� +, +(37) +where M(r) = Ms + r2 +r0 and Υa(r) ≡ +� r dr +r2+a2 +r2−2M(r)r+a2 . For our shadow observer located at some +radial distance Ro, we would also like gtt < 0 for all values of θ, leading to the condition +Rh < Ro < Rc < Ra, +Rh,c = r0 +4 +� +1 ∓ +� +1 − 16µ +� +, +(38) +10 + +where Rh, Rc are the event and Killing horizons of the Vaidya solution in the a = 0 limit. This also +implies that for some fixed Ro, we have an upper bound on +µ < µb ≡ Ms +Ro − 2Ms +2R2o +, +(39) +since we wish to have Ro < Rc. For the visual representation of the shadow, we follow the convention +used by Johannsen and Psaltis in [13]. This is essentially the orthonormal tetrad we used in deriving +the shadow formula, with the x, y coordinates being +x += +Ro sin (Θ(Rp, Ro)) sin (Φ(Rp, θinc)) , +(40) +y += +±Ro sin (Θ(Rp, Ro)) cos(Φ(Rp, θinc)). +(41) +These coordinates parametrize the observer’s plane upon which the shadow is projected.iv From +(33), noting that LE and KE are odd and even in the spin parameter a respectively, one can +straightforwardly identify the discrete symmetry +a → −a, θinc → −θinc, +which implies in particular that a < 0 shadows can be obtained from their a > 0 counterparts by +a reflection in the y-axis (for all our shadow plots, we take a > 0). +In quantifying the shape of the shadow, we follow the work of Johannsen and Psaltis in [13] who +introduced the asymmetry parameter A to describe departure from circularity. +A = 2 +� +� +� +� +� 2π +0 +dα (R − R)2 +� 2π +0 +dα +, tan α = y +x, R ≡ +� +(x − D)2 + y2, D ≡ |xmax + xmin| +2 +, +(42) +and R = +� 2π dα R/2π is the averaged radius projected upon the observer’s plane. These quantities +were mentioned in the EHT paper [1] for M87∗, and we checked that our shadow geometries for +the background Kerr solution (with µ = 0) yield comparable features obtained previously in the +work of Johannsen and Psaltis [13]. In Figure 2, we picked a few values of a, θinc for a black hole +located at some Ro to demonstrate how the shadow curve changes with µ. +4.1 +Scaling laws for variation of R and A with µ +The parameter space for the shadow geometry is spanned by {a, θinc, Ro, µ}. Computing the +shadow’s mean radius and asymmetry factor for a range of parameters, we find a simple em- +pirical scaling law that describes the variation of R, A with the accretion rate parameter µ and +other parameters as follows. +R = Ro (a, θinc, Ro) +� +1 − +µ +µb(Ro), A = Ao (a, θinc, Ro) +� +1 − +µ +µb(Ro), +(43) +where µb(Ro) is the upper bound (39) on the accretion rate allowed by our model for some fixed +observer distance Ro, obtained by setting the conformal Killing horizon to be the observer distance. +ivAnother set of projection coordinates used for plotting the black hole shadow is the (α, β) parameters of Bardeen +which would not be entirely suitable in our case since our solution is not asymptotically flat. See for example the +review of [6] which discusses the relations between Bardeen’s impact parameters and others such as stereographic +coordinates, etc. +11 + +������� +-4 +-2 +2 +4 +-4 +-2 +2 +4 +Kerr,a=0.1,θinc=17∘ +Kerr-Vaidya,μ=1×10-12 +Kerr-Vaidya,μ=5×10-12 +(a) +������� +-6 +-4 +-2 +2 +4 +-4 +-2 +2 +4 +Kerr,a=0.94,θinc=17∘ +Kerr-Vaidya,μ=1×10-12 +Kerr-Vaidya,μ=5×10-12 +(b) +������� +-6 +-4 +-2 +2 +4 +-4 +-2 +2 +4 +Kerr,a=0.5,θinc=90∘ +Kerr-Vaidya,μ=1×10-12 +Kerr-Vaidya,μ=5×10-12 +(c) +������� +-8 +-6 +-4 +-2 +2 +-4 +-2 +2 +4 +Kerr,a=0.94,θinc=90∘ +Kerr-Vaidya,μ=1×10-12 +Kerr-Vaidya,μ=5×10-12 +(d) +Figure 2: A visual representation of the shadow projected upon the observer plane at Ro/Ms = +5.4 × 1010 with θinc = 17◦ for (a) and (b), θinc = 90◦ in (c) and (d). +The upper bound on +µb ∼ 9.26 × 10−12. Horizontal axes are scaled in units of Ms. +The dependence on µ appears as a separate factor independent of the other shadow parameters, +with the functions Ro (a, θinc, Ro) , Ao (a, θinc, Ro) describing the radius and asymmetry factor at +µ = 0. The form of (43) implies that for µ ≪ 1, R0 ≫ Ms, the fractional decrease in the mean +radius and the asymmetry factor that is induced by a non-zero µ scales approximately as +δR +R = δA +A ≈ −µ Ro +Ms +. +(44) +In Figure 3, we plot these functions for a few values of spin at a fixed R0 and θinc. These plots +are expectedly similar to the corresponding ones presented in [13] and [14]. At higher spin values, +the asymmetry factor and mean radius exhibit a greater range of values over the θinc domain. +At any θinc, increasing a increases Ao but decreases Ro. The form of (43) also implies that the +asymmetry factor expressed in units of the mean radius is independent of µ, with +A +R = Ao +Ro +(a, θinc, Ro) . +(45) +12 + +������� +20 +40 +60 +80 +θinc/deg +0.05 +0.10 +0.15 +0.20 +0.25 +�/Ms +a=0.5 +a=0.6 +a=0.7 +a=0.8 +a=0.9 +(a) +������� +0 +20 +40 +60 +80 +θinc/deg +4.90 +4.95 +5.00 +5.05 +5.10 +5.15 +5.20 +R /Ms +(b) +Figure 3: Graphs depicting how R, A vary with angle θinc, at µ = 0. +We plot the functions +Ro (a, θinc, Ro) , Ao (a, θinc, Ro) at five values of a at Ro/Ms = 5.4 × 1010 and θinc = 17◦. +In Figures 4, 5 and 6, we plot the empirical fitting curves (described by (43) ) that depict how +R, A vary with the accretion parameter µ and each of the parameters {a, θinc, Ro} separately. +������� +2.×10-12 +4.×10-12 +6.×10-12 +8.×10-12 +μ +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +�/Ms +a=0.5 +a=0.75 +a=0.9 +(a) +������� +2.×10-12 +4.×10-12 +6.×10-12 +8.×10-12 +μ +1 +2 +3 +4 +5 +R/Ms +(b) +Figure 4: Graphs depicting how R, A vary with µ for a = 0.5, 0.75, 0.9, with Ro/Ms = 5.4 × +1010, θinc = 17◦. +������� +2.×10-12 +4.×10-12 +6.×10-12 +8.×10-12 +μ +0.05 +0.10 +0.15 +0.20 +0.25 +�/Ms +θinc=17∘ +θinc=45∘ +θinc=90∘ +(a) +������� +2.×10-12 +4.×10-12 +6.×10-12 +8.×10-12 +μ +1 +2 +3 +4 +5 +R/Ms +(b) +Figure 5: Graphs depicting how R, A vary with µ for θinc = 17◦, 45◦, 90◦, with Ro/Ms = 5.4 × +1010, a = 0.9. +13 + +������� +0 +5.×10-12 +1.×10-11 +1.5×10-11 +2.×10-11 +2.5×10-11 +μ +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +�/Ms +Ro/Ms=2.0×1010 +Ro/Ms=4.2×1010 +Ro/Ms=5.4×1010 +(a) +������� +0 +5.×10-12 +1.×10-11 +1.5×10-11 +2.×10-11 +2.5×10-11 +μ +0 +1 +2 +3 +4 +5 +R/Ms +(b) +Figure 6: Graphs depicting how R, A vary with Ro for R0/Ms = 2.0 × 1010, 4.2 × 1010, 5.4 × 1010 +with θinc = 17◦ and a = 0.9. +4.2 +On the shadows of M87∗ and Sagittarius A∗ as observed by EHT +The M87∗ black hole was found to be about 16.8 Mpc away, with a mass Ms ≃ 6.5 × 109M⊙[1]. +The ensemble of accretion models used by the EHT team [1, 7] involved mass rates that ranged +from about 2×10−7 to 4×10−4 times the Eddington rate ˙MEdd. In their work, ˙MEdd ∼ 137M⊙/yr, +and we find that this translates into +µM87 = ˙M × GM⊙ +Yr × c2 ∼ ˙M × 1.56 × 10−13 ∈ +� +4 × 10−18, 9 × 10−15� +, +where +˙M is mass rate in units of M⊙/yr. This is smaller than the upper bound µb ∼ 9.2 × 10−12 +for Ro ∼ 16.8 Mpc. Equivalently, for this range of µ, the conformal Killing horizon size falls within +Rc/Ms ∼ +� +5.9 × 1013, 1.16 × 1017� +, +which lies beyond Ro/Ms ∼ 5.4 × 1010. Thus, the observed distance to M87∗ and estimates of +mass accretion are well within the domains of validity of our simple model geometry. Now it was +estimated in [22] that the angle of inclination is around 17◦. Corresponding to this value, in Figure +7, we plot the variation of the shape parameters R, A and their ratio with the spin parameter a. +The range of values of R translates into the shadow angular diameter ∼ (36.9µas, 39.6µas) which +is comparable to the measured emission ring diameter in the EHT experiment [1]. The maximum +A/R ratio is about 0.01 which is within limits of the upper bound of 10% indicated in [1]. The +highest µ ∼ 8.5 × 10−15 in the ensemble of models considered in [1] translates only to a fractional +shift of 0.05 % in R, A. +Sgr A∗ has been observed to located near the dynamical center of our galaxy at a distance +Ro ∼ 8 kpc away, with a dense concentration of mass Ms ∼ 4 × 106M⊙. In contrast to M87∗ +where its prominent jet provides robust constraints on source orientation with respect to the line +of sight, fixing it to be ∼ 17◦, there is no such constraint unfortunately on Sgr A∗ [2]. However, +GRMHD models appeared to have favored θinc < 50◦, with accretion rate of order-of-magnitude +10−9 − 10−8M⊙ yr−1. These models were equipped with spin parameter values of a = 0.5, 0.94 [2]. +As mentioned in [2], in the earlier works of Quataert [23] and Baganoff [24], the captured accretion +rate was estimated to be 10−6−10−5M⊙ yr−1 from Chandra observations of thermal bremsstrahlung +emission at the vicinity of the gas capture radius. Most recently, in [11], a promising model was +identified in which θinc ≤ 30◦, and accretion models of ˙M ∼ 5.2−9.5×10−9M⊙/yr were examined. +14 + +������� +0.2 +0.4 +0.6 +0.8 +1.0 +a +4.9 +5.0 +5.1 +5.2 +R/Ms +0.2 +0.4 +0.6 +0.8 +1.0 +a +0.01 +0.02 +0.03 +0.04 +0.05 +�/Ms +0.2 +0.4 +0.6 +0.8 +1.0 +a +0.002 +0.004 +0.006 +0.008 +0.010 +�/R +Figure 7: Graphs showing how the mean radius R, asymmetry factor A and their ratio vary with +a, with Ro/Ms = 5.4 × 1010 (pertaining to EHT observation of M87∗), θinc = 17◦. +Even for the accretion rate +˙M = 10−5M⊙ yr−1 this translates into merely +µsgr ∼ 1.6 × 10−18. +Like the case of M87∗, this turns out to be smaller than the upper bound µb ∼ 1.2 × 10−11 for +Ro ∼ 8 kpc. Equivalently, taking this value of µsgr, the conformal Killing horizon size is +Rc/Ms ∼ 3.2 × 1017, +which lies beyond Ro/Ms ∼ 4.2 × 1010. Thus again, both observer distance and (estimated) mass +accretion rates are well within the domains of validity of our simple model geometry. +In Figure 8, we plot the variation of the shape parameters R, A and their ratio with the spin +parameter a, at a few representative values of θinc. Over the domain of θinc ∈ (10◦, 50◦), the range +of shadow angular diameters is ∼ (47.6µas, 51.2µas) which is comparable to the shadow diameter +estimate 48.7±7.0µas in the EHT experiment [2]. The asymmetry factor-to-mean radius ratio A/R +ratio increases with θinc, and can be as high as ∼ 10% for θinc = 50◦. It would be interesting to study +this geometrical signature for the EHT’s Sgr A∗ shadow image. In [8], the EHT team mentioned in +passing that the sparse interferometric coverage of 2017 observations led to significant uncertainties +in circularity measurements which were thus not quantified yet, but future EHT observations with +additional telescopes may place constraints on the circularity. Finally, let us mention that the +effect of µ on R, A is even smaller for Sgr A∗, inducing only a fractional change of 10−6 in these +quantities. +������� +● +● +● +● +● +● +● +● +● +● +● +● +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +0.2 +0.4 +0.6 +0.8 +1.0 +a +4.9 +5.0 +5.1 +5.2 +R/Ms +● +● +● +● +● +● +● +● +● +●● ● +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ +▲ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +● +θinc=10∘ +▲ +θinc=30∘ +■ +θinc=50∘ +0.2 +0.4 +0.6 +0.8 +1.0 +a +0.1 +0.2 +0.3 +0.4 +�/Ms +● +● +● +● +● +● +● +● +● +●● ● +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ ▲ +▲ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +0.2 +0.4 +0.6 +0.8 +1.0 +a +0.02 +0.04 +0.06 +0.08 +�/R +Figure 8: Graphs showing how the mean radius R, asymmetry factor A and their ratio vary with +a, with Ro/Ms = 4.2 × 1010 (pertaining to EHT observation of Sgr A∗), θinc = 10◦, 30◦, 50◦. +15 + +5 +Discussion +We have presented a study of the shadow geometry for a class of spacetime metric that is Kerr- +Vaidya-like in nature. The family of time-dependent black hole solutions we constructed in this +work has well-defined Kerr and Vaidya limits. Agnostic to the source and the underlying theory, +we had conceived of its form starting from the Vaidya solution with a mass function that is lin- +ear in Eddington-Finkelstein coordinates since this particular class of solutions furnishes a model +for accretion and is equipped with a conformal Killing isometry that leads to separability of null +geodesics. After expressing it as being conformal to a solution that is Schwarzschild-like with a +radial coordinate-dependent mass function, the Newman-Janis algorithm was applied to obtain a +Kerr-like solution that reduces to the Kerr solution in the limit of vanishing accretion parameter +µ. In real-life applications, the dimensionless accretion rate µ is expected to be very small, for +instance, for the recent M87∗ and Sgr A∗ observations, the highest model estimates of µ are of +the order 10−15, 10−18 respectively. Thus, our model geometry can be considered as a small µ- +deformation of the Kerr solution that preserves separability of null geodesics, or from a different +perpsective, a rotating generalization of the Vaidya solution. For a finite spatial domain, it could +act as a simple model of a Kerr-like geometry that takes into account the backreaction of accre- +tion. The existence of a conformal Killing vector field allows us to solve for the shadow geometry +straightforwardly yet also brings with it the subtlety of a horizon that should be located beyond +the shadow observer. Equivalently, at any fixed observer distance, there exists an upper bound to +µ (eqn.(39)) for applicability of our model. +In our study of the variation of the mean radius R and asymmetry factor A with regards to +various parameters — {µ, a, θinc, Ro}, we found a simple empirical scaling law of the form in (43). +In particular, this implies that at small µ ≪ 1 and large observer distance Ro ≫ Ms, the fractional +change induced by turning on the accretion rate parameter µ reads simply as +δR +R = δA +A ≈ −µ Ro +Ms +. +(46) +To our knowledge, we have not encountered any previous descriptive relations between shadow geo- +metrical features and accretion rate, black hole and observer parameters of the form similar to (43) +or (46)v.GRMHD simulations (e.g. [10, 12, 26]) typically assume the validity of the background +metric being purely Kerr in some suitable coordinate system, with the complicated astrophysics +of accretion contained within the choice of the energy-momentum tensor. Even for GRMHD sim- +ulations involving spacetime metrics motivated by beyond-GR theories (e.g. [8]), the accretion +parameter is rarely involved in describing the background spacetime. In [13, 14] and most recently +in [15], it was found that the accretion details do not appear to influence the shadow geometry +which is sensitive only to the background metric. The assumption here is that backreactions of +accretion on the metric are insignificant. Indeed, in applying our model to EHT observations of +M87∗ and Sgr A∗, we found that the most generous estimates of µ in [1] and [2] yield fractional +changes of R and A of the order of 10−4 and 10−6 respectively, consistent with such an assumption. +In addition, our model yields an explicit relation in (43) that describes how, at least in principle, +the shadow geometry changes with accretion rate. From (46), it may seem that for situations where +the (fractional) experimental uncertainties δRe/Re, δAe/Ae are of the order µRo/Ms, then metric +vIn [25], it was found that increasing the flux of infalling gas into a Schwarzschild black hole (and hence the accretion +rate) by increasing an axion-plasmon coupling parameter decreases the size of the shadow which qualitatively agrees +with the variation of R with µ of our model. It would be interesting to study if the results in [25] could be cast in a +similar form as (43) or (46). +16 + +backreactions due to accretion may be important in analyzing geometrical details of the black hole +shadow. +A limitation of our model geometry is that it is only asymptotically Ricci-flat and not Minkowskian. +As a model of an effective Kerr-like geometry with accretion backreaction, it is thus valid only for +a finite spatial domain. Imposing a stricter condition for gtt < 0 in (37), in our exposition of the +black hole shadow analysis, we have taken the observer location Ro to lie within the interior of +the sphere defined by the conformal Killing horizon of the limiting Vaidya spacetime, i.e. R < Rc +in the coordinate system of (37). +(For M87∗ and Sgr A∗, Rc/Ro ∼ 103, 107 respectively.) +It +would be interesting to seek a refinement of our model geometry in one that allows for separa- +bility of null geodesics while being asymptotically flat. Towards this ideal goal, in Appendix A, +we sketched a globally well-defined solution obtained by matching the spacetime at some cutoff +distance to asymptotically flat Kerr-like solutions using Darmois-Israel junction conditions, leaving +more realistic constructions for future work. +Acknowledgments +I am grateful to Jan de Boer, Chong-Sun Chu, Ori Ganor, Petr Horava, Daniel Robbins and Neal +Snyderman for sharing with me their insights on various aspects of gravitational physics and their +moral support over the years. +A +Global geometry and matching spacetimes via junction condi- +tions +For the Vaidya spacetime, the conformally static chart in (3) has a coordinate singularity at the +conformal Killing horizon. It can be continued beyond that via (2). Similarly, from (11), we can +perform the coordinate transformation (2) +v = r0e +˜T/r0, +w = re +˜T/r0, +(47) +after which the line element takes the form +ds2 += +−F +�w +v +� � +dv + av sin2 θ +r0 +d˜φ +�2 ++ 2 +� +dv + av sin2 θ +r0 +d˜φ +� � +dw + av sin2 θ +r0 +d˜φ +� +−2aw sin2 θ +r0 +dvd˜φ + +� +w2 + v2a2 cos2 θ +r2 +0 +� +dΩ2, +(48) +where +F +�w +v +� += −1 + 2 +� +µ + ( w +v )2� w +v +� w +v +�2 + a2 cos2 θ +r2 +0 +− 2w +v . +The Vaidya metric in the chart (1) is obtained in the a = 0 limit, whereas the double scaling +limit of (6) brings it to Kerr in Eddington-Finkelstein chart. The metric in the chart {v, w, θ, ˜φ} +is convenient for examining the asymptotic infinity of the spacetime. Taking the limit of infinite w +yields the following asymptotic form +lim +w→∞ ds2 ∼ −dv2 + 2dvdw + w2dΩ2 + 2aµ +Ms +sin2 θ +� +wdvd˜φ + vdwd˜φ +� +. +(49) +17 + +At radial infinity, while the spacetime is Ricci-flat, there are still non-zero Ricci and Einstein +tensor components which read Rθθ = −Gθθ = − a2 +r2 +0 sin2 θ, R˜φ˜φ = 3G˜φ˜φ = −3a2 +r2 +0 sin4 θ, with other +components being zero. In the following, we briefly discuss how one could impose suitable initial +and final boundary conditions to the mass-accreting geometry by suitably matching our solution +to Kerr solutions representing the start or end-point of the mass accretion process, yielding a more +realistic global geometry via the use of Darmois-Israel junction conditions. +For the Vaidya spacetime, we can cut-and-paste the spacetime geometry to that of Schwarzschild +spacetime at some advanced v where the accretion ends, and to an initial geometry such as +Minkowski or another Schwarzschild solution at some earlier v. For example, in (1), instead of +a function monotonically increasing in v, we can refine the mass function to be +m(ν) = +� +� +� +� +� +0, +ν = 0 +µν, +0 < ν < νf +µνf, +ν ≥ νf, +(50) +and formally, we obtain a global geometry constructed by matching Vaidya to Minkowski at ν = 0 +and Schwarzschild solution with ADM mass µνf at ν = νf, yielding a more realistic model. +We adopt a similar construction to extend our geometry suitably to past and future infinities in +this vein. Let Gi, Gf be the initial and final geometries that we match our Kerr-Vaidya-like solution +at times ˜T = { ˜Ti, ˜Tf} respectively, working in the chart of { ˜T, r, θ, ˜φ} (11). We denote the matched +induced metric on Gi, Gf by Σi, Σf, and that of our Kerr-Vaidya-like solutions by G evaluated on +{ ˜Ti, ˜Tf}. The first Darmois-Israel junction condition is the continuity of the induced metric at +these two junctions. +Σi = G| ˜T= ˜Ti Σf = G| ˜T= ˜Tf . +(51) +For simplicity, we consider glueing geometries of which induced metric at the matching surface +can be easily expressed in the coordinate systems we discussed earlier. For asymptotic flatness in +G, Gi, Gf, consider modifying the line element in (11) as follows. Defining the bracketed expression +in (11) to be ds2 +K, we now replace it with +ds2 +K = +� +� +� +� +� +� +� +� +� +� +� +� +� +− +� +1 − +2M(r)r +r2+a2 cos2 θ +� +(d ˜T + a sin2 θd˜φ)2 + 2(d ˜T + a sin2 θd˜φ)(dr + a sin2 θd˜φ) ++(r2 + a2 cos2 θ)dΩ2, +r < Rm +− +� +1 − +2MKr +r2+a2 cos2 θ +� +(d ˜T + a sin2 θd˜φ)2 + 2(d ˜T + a sin2 θd˜φ)(dr + a sin2 θd˜φ) ++(r2 + a2 cos2 θ)dΩ2, +r > Rm +(52) +where r = Rm is a matching surface beyond which ds2 +K is the Kerr line element with mass MK, +MK = Ms +� +1 + µR2 +m +M2s +� +. +We set the cutoff radius Rm to be at some radial distance beyond the observer of the shadow.vi +Such a spacetime is parametrized not only by {a, µ, Ms} but also by the cutoff parameter Rm. The +full metric still takes the form +G(a, µ, MK, Rm) : ds2 = e +2µ +Ms ˜T ds2 +K(a, µ, MK, Rm), +(53) +viOne natural choice would be to take Rm = Rc - the conformal Killing horizon of the limiting Vaidya spacetime. +18 + +but with ds2 +K defined as in (52). Next, at ˜T = { ˜Ti, ˜Tf}, we cut-and-paste G to Gi, Gf respectively +which are defined by +Gi : ds2 = ds2 +K(ai, µ, Mi +K, Ri +m), +Gf : ds2 = ds2 +K(af, µ, Mf +K, Rf +m). +(54) +The initial and final matching surfaces are spacelike surfaces, and the various parameters are related +as +e +˜ +Ti +r0 Rm = Ri +m, e +˜ +Ti +r0 MK = Mi +K, e +˜ +Ti +r0 a = ai, +(55) +e +˜ +Tf +r0 Rm = Rf +m, e +˜ +Tf +r0 MK = Mf +K, e +˜ +Tf +r0 a = af. +(56) +The initial geometry can be taken to be arbitrarily close to Minkowski spacetime with ˜Ti → −∞ +with (Rm, MK, a) being some finite set of parameters. +The eventual geometry from the mass +accretion process of duration ∆ ˜T = ˜Tf − ˜Ti is that of the Kerr solution (for R > Rf +m) with mass +and spin parameters being +Mf +K = eµ ∆ ˜ +T +Ms Mi +K, af = eµ ∆ ˜ +T +Ms ai. +(57) +From (57), one can easily see that our model geometry manifestly represents an accretion process +where the fractional increase in both the mass and spin parameters occur at a rate of µ. +B +On the reference frame of the shadow observer +B.1 +In the limit of a = 0 +In the limit of a = 0, the tetrad basis (31) defining the reference frame of our shadow observer +reduces to the following. +e0 = e−µ ˜T/Ms +� +1 − 2M +r +∂t, e1 = e−µ ˜T/Ms +r +∂θ, e2 = −e−µ ˜T/Ms +r sin θ ∂φ, e3 = −e−µ ˜T/Ms +�� +1 − 2M +r +� +∂r +(58) +In the chart { ˜T, r, θ, ˜φ}, we replace ∂t → ∂ ˜T , ∂r → ∂ ˜T +∂r +∂ +∂ ˜T + ∂ +∂r which leads to +e0 = e−µ ˜T/Ms +√f +∂ ˜T , e1 = e−µ ˜T/Ms +r +∂θ, e2 = −e−µ ˜T/Ms +r sin θ ∂φ, e3 = −e−µ ˜T/Ms +√f +� +∂ ˜T + f∂r +� +(59) +where f = 1− 2M(r) +r +. This is the tetrad basis used in [16] for Vaidya spacetime’s shadow calculation, +relevant for an observer with 4-velocity e0, at constant θ, φ, r. +B.2 +Observers in the {v, w, θ, φ} chart and aberration formulas +The coordinate system {v, w, θ, φ} avoids the horizons as coordinate singularities. +In [16], the +shadow observed by an observer with 4-velocity ∼ +∂ +∂v was derived using an aberration formula +being applied to the shadow angle formula. +In general, for a pair of reference frames (S, S′), aberration formulas relating the coordinates of +their celestial spheres can be derived from the expression (32) after we express the 4-velocity e′ +0 of +the S′ reference frame as a linear combination of the original tetrad basis components, writing +e′ +0 = e0 + V kek +√ +1 − v2 , +(60) +19 + +where ⃗V is the relative 3-velocity of S′ observer. +Consider an observer of which e′ +0 is a linear +combination of e0 and another basis vector e3. Its tetrad basis components read +e′ +0 = +1 +√ +1 − V 2 (e0 + V e3), e′ +3 = +1 +√ +1 − V 2 (e3 + V e0), e′ +1,2 = e1,2. +(61) +Taking the inner product between e′ +0, e′ +3 and the tangent vector expression in (32) in both unprimed +and primed coordinates, we obtain the aberration formulas +cos θ′ = V + cos θ +1 + V cos θ, +φ′ = φ, +V = −e′ +0 · e3 +e′ +0 · e0 +. +(62) +In [16], the observer with e′ +0 ∼ +∂ +∂v was also considered. After a coordinate transformation from +{ ˜T, r, θ, ˜φ} used for the tetrad basis in (59), one can show that +e′ +0 ∼ ∂ +∂v = +1 +� +1 − 2Ms +r +e− T +r0 +� +∂ ˜T − r +r0 +∂r +� += +1 +√ +1 − V 2 (e0 + V e3), V = +r2 +r2 + rr0 +� +1 − 2M(r) +r +� (63) +where e0, e1, e2, e3 are as defined in (59). In [16], the same expression for the relative velocity V +was obtained with an aberration relation tan2 θ′ +2 = 1−V +1+V tan2 θ +2 that we verified to be identical to +(62). +Let us now consider an appropriate S′ observer for our Kerr-Vaidya-like geometry. We note +that for the S observer, its 4-velocity e0 is a linear combination of ∂ ˜T and ∂˜φ, or in the Boyer- +Lindquist-like chart, a linear combination of ∂t and ∂φ. The angular component is such that e0 ±e3 +are tangential to the principal null congruences of the metric. Relating between v and t, we choose +the following S′ observer with +e′ +0 ∼ ∂ +∂v + +r0a +v(r2 + a2) +∂ +∂ ˜φ += +r0 +v +� +1 + +r(r2 + a2) +r0(r2 − 2Mr + a2) +� +∂t − r +v∂r − +r0a +v(r2 − 2Mr + a2)∂φ ++ +r0a +v(r2 + a2) +∂ +∂ ˜φ +. +(64) +This choice of e′ +0 is also uniquely the one that allows us to write +e′ +0 ∼ e0 + we3, +(65) +for some relative 3-velocity w. Thus the aberration formulas in (62) apply similarly. Recall that in +(31), the relevant basis tetrad components read +e0 = e−µ ˜T/Ms (r2 + a2)∂t + a∂φ +√ +Σ∆ +, e3 = −e−µ ˜T/Ms +� +∆ +Σ ∂r, +(66) +which allows us to read off the 3-velocity as +w = +r2 + a2 +r2 + a2 + r0 +r (r2 − 2Mr + a2). +(67) +In the limit of vanishing a, we recover the 3-velocity v for the observer in Vaidya spacetime with +e0 ∼ +∂ +∂ ˜T as derived in [16]. In the limit µ → 0, up to leading order, we have +w ≈ µ +� +r2 + a2 +Ms +r (r2 − 2Msr + a2) +� ++ O(µ2). +(68) +Thus, this reference frame may be relevant for theoretical situations where the observer’s velocity +is proportional to the strength of the accretion rate, although arguably not so for realistic EHT +observations where the accretion is hardly expected to backreact on the metric significantly to +affect the 3-velocity of the shadow observer in such a manner. +20 + +References +[1] K. Akiyama et al. [Event Horizon Telescope], “First M87 Event Horizon Telescope Re- +sults. I. 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Magnetic Field Structure near The Event Horizon,” Astrophys. J. Lett. 910, no.1, L13 +(2021) doi:10.3847/2041-8213/abe4de [arXiv:2105.01173 [astro-ph.HE]]. +22 + diff --git a/aNE4T4oBgHgl3EQfOwxI/content/tmp_files/load_file.txt b/aNE4T4oBgHgl3EQfOwxI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fe40522628db2a3d3fcc1f51ca706b9e933344c8 --- /dev/null +++ b/aNE4T4oBgHgl3EQfOwxI/content/tmp_files/load_file.txt @@ -0,0 +1,977 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf,len=976 +page_content='Shadows of Kerr-Vaidya-like black holes Hai Siong Tan University of Pennsylvania, Perelman School of Medicine, Department of Radiation Oncology, Philadelphia, USA Jan 2023 Abstract In this work, we study the shadow boundary curves of rotating time-dependent black hole solutions which have well-defined Kerr and Vaidya limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' These solutions are constructed by applying the Newman-Janis algorithm to a spherically symmetric seed metric conformal to the Vaidya solution with a mass function that is linear in Eddington-Finkelstein coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Equipped with a confor- mal Killing vector field, this class of solution exhibits separability of null geodesics, thus allowing one to develop an analytic formula for the boundary curve of its shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We find a simple power law describing the dependence of the mean radius and asymmetry factor of the shadow on the accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Applicability of our model to recent Event Horizon Telescope observations of M87∗ and Sgr A∗ is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='04967v1 [gr-qc] 12 Jan 2023 Contents 1 Introduction 2 2 A family of rotating Vaidya-like black hole solutions 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 Conformal factors, coordinate charts and the Newman-Janis algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 On the shadows of M87∗ and Sagittarius A∗ as observed by EHT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 14 5 Discussion 16 A Global geometry and matching spacetimes via junction conditions 17 B On the reference frame of the shadow observer 19 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 In the limit of a = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 19 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 Observers in the {v, w, θ, φ} chart and aberration formulas .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 19 1 Introduction Recent Event Horizon Telescope (EHT) observations of horizon-scale shadow images of M87∗ [1] and Sgr A∗ [2] have furnished not only a direct visual evidence of black holes, but have also led to many new constraints on various potential deviations from General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The boundary curve of the black hole shadow emerges from light rays that spiral asymptotically from the photon region demarcating the boderline between light rays that will eventually be captured by the black hole and those that escape to infinity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='i The geometry of this boundary curve depends on the background metric which could thus be probed by EHT observations [7, 8, 9] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For example, the shadow geometry of Sgr∗ has been used to exclude the central object being a Reissner-Nordstrom-type naked singularity or a traversable Misner-Thorne wormhole [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Surrounding the black hole shadow is an emission ring of which structure is sensitive to a rich set of astrophysical phenomena, such as radiative transfer, that characterize the matter-energy accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Typically, general relativistic magnetohydrodynamic (GRMHD) simulations are used to model the accretion flow processes [7, 8, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For the EHT experiments, they have revealed the emission ring properties to be consistent with a number of accretion models built iThis curve is termed as the ‘critical curve’ by Gralla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' in [4] and ‘apparent boundary’ by Bardeen in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' See for example the review of [6] for an extensive discussion of basic ideas and history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 2 upon the background of a Kerr black hole [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The spacetime metric in these simulations is assumed to be purely Kerr spacetime throughout, with the energy-momentum tensor capturing the magnetic field and average plasma properties [10, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In [13, 14, 15], it was noted that the shadow size and shape is hardly influenced by the accretion details, and thus serves as a pristine signature of spacetime geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' An implicit assumption is that the backreaction of the GRMHD energy-momentum tensor on the metric has a negligible influence on the shadow and could thus be ignored in deriving its geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In this paper, we study the shadow boundary curves of a class of rotating time-dependent black hole solutions of which metric is a deformation of the Kerr solution described by a small dimensionless parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In the limit of vanishing spin, our spacetime reduces to a well-known model of spherically accreting black hole - the Vaidya spacetime with a mass function µv, with v being an ingoing Eddington-Finkelstein coordinate and µ being the mass accretion rate constant in natural units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The latter solution was studied most recently in [16] where the authors derived and examined its shadow characteristics analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Most crucially, an analytic treatment of the shadow was possible by virtue of the existence of a Carter constant leading to separability of its null geodesic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This is related to a conformal Killing symmetry associated with the linear mass function, and hence its choice, for it enables the authors of [16] to derive explicit formulas for the radius of the photon sphere and the shadow angular diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' One main motivation of our work here is to seek a rotating generalization of the analytic treatment in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This would serve as a simple model of a backreacted Kerr-like geometry that is accreting mass, and for which an analytic derivation of its shadow geometry is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For readers familiar with exact solutions in GR, a natural candidate would be the Kerr-Vaidya solution [17] which can be obtained by replacing the constant Kerr mass with a variable mass function in the original Kerr line element expressed in Eddington-Finkelstein coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Unfortunately, as we’ll elaborate later, this solution does not offer any additional Carter constant that could lead to its null geodesic equations being separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We construct our solutions by applying the Newman-Janis algorithm [18] to a spherically sym- metric seed metric conformal to the Vaidya solution with the mass function that is linear in Eddington-Finkelstein coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Fortunately, this solution-generating technique turns out to preserve the conformal Killing vector field in the original Vaidya metric, leading to separability of null geodesics, and ultimately allows us to develop an analytic formula for the boundary curve of its shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The solution space is parametrized by {a, µ, Ms} where {a, Ms} are the spin and mass parameters of Kerr spacetime in the vanishing µ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Like the Kerr solution, there are regions in the moduli space which do not pertain to black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Motivated by phenomenological interests, we focus on the regime of parameters where our solution has event horizons like those of Kerr, with the conformal Killing horizon at a large distance away from the shadow observer and the outer horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Thus, our solution serves as a simple model of an accreting Kerr-like geometry not globally but for a finite spatial domain defined by the interior of the conformal Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Generically, the shadow geometry is sensitive to the choice of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We work in a chart which reduces to the Kerr spacetime in Boyer-Lindquist coordinates in the limit µ = 0, and the Vaidya spacetime in Eddington-Finkelstein-like coordinates in the limit a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Correspondingly, we verified that our shadow formulas reduce consistently to those of Kerr [19] and Vaidya [16] under these limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' As reviewed in for example [6], analytic derivations in cases that allow them complement numer- ical studies of shadow geometry in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For example, for Schwarzschild spacetime, the angular diameter of its shadow is ∼ 3 √ 3Ms/Ro for a distant observer located at the radial coordinate Ro [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This numerical value has turned out to be very useful as a guide in the analysis of shadow size and shape in EHT’s recent observations [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For our shadow analysis here, we find a simple power law describing the dependence of the mean radius and asymmetry factor of the shadow on 3 the accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The latter describes the departure of the shadow from circularity and has been constrained in M87∗ studies by EHT team [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' When applied to the parameters of M87∗ and Sgr A∗, our analysis of shadow geometry appears to indicate that the effect of µ is very small, and thus provides support for the assumption of using the pure Kerr metric throughout in GRMHD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Our results, in addition, yield an empirical formula that parametrizes the variation of mean radius and asymmetry factor with accretion rate explicitly, and can thus be used to anticipate when backreaction of accretion on the metric may be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In Section 2, we present the construction of a class of Kerr- Vaidya-like solutions and elaborate on some basic aspects of its geometry and moduli space, followed by a derivation of some analytical formulas for shadow geometry in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In Section 4, we present several visual plots of the shadow and examine how the mean radius and asymmetry factor of the shadows vary with various parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We also include a brief discussion on recent EHT observations of M87∗ and Sgr A∗ in relation to our model geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Finally, we end with some concluding remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Appendix A presents an extension of our solution obtained by matching the spacetime at some cutoff distance to Kerr-like solutions that are asymptotically flat via Darmois-Israel junction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In Appendix B, for completeness, we develop an aberration formula for observers in another reference frame which, in the zero spin limit, reduces to another class of observers discussed previously in [16] for Vaidya spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 2 A family of rotating Vaidya-like black hole solutions We begin with the Vaidya metric in the coordinatesii ds2 = − � 1 − 2m(v) w � dv2 + 2dvdw + w2 � dθ2 + sin2 θdφ2� , (1) with the domains v ∈ (0, ∞), w ∈ (0, ∞), θ ∈ (0, π), φ ∈ (0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We note that m(v) is a mass function that can be used to model a time-dependent black hole of which exterior is described by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The solution (1) solves the field equations in ordinary GR with the energy momentum tensor T µν = m(v)KµKν, Kν∂ν = ∂w which is typically interpreted as that of a null dust moving in the direction of decreasing w, with the black hole accreting (radiating) mass if m′(v) is positive (negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 Conformal factors, coordinate charts and the Newman-Janis algorithm In this work, we restrict ourselves to the case where m(v) = µv, where µ is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In this case, the geometry admits a conformal Killing vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' To see this, we define ∂/∂T as the conformal Killing vector and make a coordinate transformation as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' v = r0eT/r0, w = reT/r0, (2) where r0 is a positive constant with dimension of length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This brings (1) to ds2 = e2T/r0 � − � 1 − 2µr0 r − 2r r0 � dT 2 + 2dTdr + r2(dθ2 + sin2 θdφ2) � , (3) iiThe unusual choice of the symbol w to denote radial distance for this line element is solely due to shortage of conventions for the many different radial coordinates that we’ll use throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 4 with T ∈ (−∞, ∞), r ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In this form, the metric is conformal to a manifestly static spacetime which can be taken to generate a rotating solution via Newman-Janis algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' But first, we seek a temporal coordinate such that constant time slices are 3-dimensional spatial manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Defining T = t + Υ(r), Υ(r) = � r d ˜R � 1 − 2µro ˜R − 2 ˜R r0 �−1 , (4) the line element then reads ds2 = e 2(t+Υ(r)) r0 � − � 1 − 2M(r) r � dt2 + � 1 − 2M(r) r �−1 dR2 + r2(dθ2 + sin2 θdφ2) � ≡ Ω2(t, r)ds2 static, (5) where t ∈ (−∞, ∞) and M(r) ≡ µr0 + r2 r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' If we restrict ourselves to the spacetime patch where the conformal Killing vector field ∂ ∂t is timelike, then letting 1 − 2M/r > 0 leads to the domain r ∈ (Rh, Rc), Rh,c = r0 4 � 1 ± � 1 − 16µ � , µ < 1/16, where Rh is the black hole horizon and Rc denotes the conformal Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The Schwarzschild limit can be obtained as a double scaling limit as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' µ → 0, r0 → ∞, µro = Ms, (6) where Ms is a finite mass parameter equivalent to the ADM mass of the limiting Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We now apply the Newman-Janis algorithmiii to the metric ds2 static which leads to a metric endowed with angular momentum ds2 static → ds2 rotating = − � 1 − 2M(r)r Σ � dt2 − 4M(r)ar sin2 θ Σ dφdt + � r2 + a2 + 2M(r)a2r sin2 θ Σ � sin2 θdφ2 + Σ ∆dr2 + Σdθ2, (7) where a is the spin parameter and Σ = r2 + a2 cos2 θ, ∆ = r2 − 2M(r)r + a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We will also modify the exponential argument of the conformal factor Ω(t, r) in (5) as follows t + Υ(r) → t + Υa(r), Υa(r) ≡ � r dr r2 + a2 r2 − 2M(r)r + a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (8) The full metric then reads ds2 = e 2(t+Υa(r)) r0 ds2 rotating, (9) with ds2 rotating, Υa(r) being defined in (7) and (8) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In the scaling limit of (6), the line element (9) reduces to Kerr spacetime in Boyer-Lindquist coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' iiiTo be precise, this algorithm carries with it the assumption of asymptotic flatness in the generated metric which doesn’t hold for our solution though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 5 Now, like the Kerr solution where M is instead just a constant, the metric (9) has singularities at the roots of ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' To extend the spacetime beyond these singularities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' we can perform a coordinate transformation ˜T = t + Υa(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' ˜φ = −φ − a � r dr 1 r2 − 2M(r)r + a2 (10) which leads to ds2 = e 2µ Ms ˜T � − � 1 − 2M(r)r r2 + a2 cos2 θ � (d ˜T + a sin2 θd˜φ)2 + 2(d ˜T + a sin2 θd˜φ)(dr + a sin2 θd˜φ) +(r2 + a2 cos2 θ)dΩ2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (11) In the a = 0 limit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' we recover Vaidya spacetime in the conformally static coordinates of (3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' whereas the µ = 0 limit (as in (6)) takes the metric to that of Kerr in ingoing Eddington-Finkelstein coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We note that in (11), replacing M(r) → µ ˜T and removing the conformal factor e 2µ Ms ˜T yields the Kerr-Vaidya solution [17] which evidently isn’t equipped with the conformal Kiling symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This leads to non-separability of null geodesics which would not allow us to solve for the shadow boundary curve analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Our main interest in this class of time-dependent solutions lies in its property of being locally deformable to the Kerr geometry in Boyer-Lindquist coordinates in the µ → 0, µr0 → Ms limit, and, in the limit of a = 0, to the Vaidya solution in a chart where it’s conformally static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This gives us a model of local geometry that approximates both spacetimes in a coordinate system suitable for deriving the analytical form of the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For this specific purpose, we work in the {t, r, θ, φ} chart (line element in (9)), where the spacetime is conformal to a Kerr-like solution in Boyer-Lindquist coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2, we explore the parameter space {a, µ, Ms} in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 Horizons in the solution parameter space In (9), setting grr = 0 yields the following cubic equation in r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' − 2µ Ms r3 + r2 − 2Msr + a2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (12) In certain regimes of the parameter space of {µ, a}, one could find event horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Consider the root space of the cubic equation (12) of which discriminant reads (henceforth, we define a → a/Ms to be a dimensionless parameter) D = 4 � 1 − a2 − 16µ + 18a2µ − 27a4µ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The sign of D determines the number of real roots to grr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' As a quadratic equation in µ, we can derive the curves along which D = 0, which read µ± = −8 + 9a2 ± � 4 − 3a2�3/2 27a4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (13) They enclose the region which pertains to three distinct roots — two event horizons (outer and inner) and the conformal Killing horizon, and they intersect at the point (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 1 ) (ae, µe) = � 2 √ 3, 1 12 � , (14) 6 which represents a generalized extremal limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' At any constant µ ∈ (0, 1 12), the upper bound on a is given by taking µ = µ−(a) along which the inner and outer event horizons coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Along µ = µ+(a), the conformal Killing horizon coincides with the outer event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' μ+ μ- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='10 μ Figure 1: Graph depicting the parameter space (µ, a) of our family of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The curve µ+ begins at (0, 1 16) and contains solutions where the outer event horizon and conformal Killing horizon are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The curve µ− contains all the extremal solutions with degenerate outer and inner event horizons and with a finite conformal Killing horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Both curves merge at the extremal point ( 2 √ 3, 1 12) at which there is only apparent horizon at r = 2Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Our family of solutions can be seen as parametric deformations of the Vaidya solution (vertical axis) and the Kerr solution (horizontal axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 1, the region enclosed by the axes and the two curves has non-degenerate outer, inner event horizons and conformal Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The small µ ≪ 1 region of this enclosed segment is of closer phenomenological interest to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Let us consider a generic point in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Ordering the roots of (12) as Ri < Re < Ra, we find that up to first few orders in µ, a : Re = Ms �� 1 + � 1 − a2 � + 4 + 4 √ 1 − a2 − 5a2 − 3a2√ 1 − a2 + a4 2(1 − a2) u + O(µ2) � = Ms [(2 + 8µ) − (2µ + 1/2)a + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='] (15) Ri = Ms �� 1 − � 1 − a2 � + 4 − 4 √ 1 − a2 − 5a2 + 3a2√ 1 − a2 + a4 2(1 − a2) µ + O(µ2) � = Ms �a 2 + a2 8 − µa3 16 + a3 16 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' � (16) Ra = Ms 2µ − 2Ms − 8µMs + 2aµMs + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (17) The radii Re, Ri are those of the outer and inner event horizons smoothly connected to their corresponding expressions in the ordinary Kerr solution, whereas Ra is the radius of the conformal Killing horizon associated with the conformal Killing vector ∂t (this symmetry is preserved by the Newman-Janis algorithm we implemented).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For a generic point within our domain of interest (region enclosed by curves and axes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 1 ), our shadow observer is a timelike observer located at some distance Re < r < Ra away from the outer event or conformal Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 7 3 Null geodesics, photon spheres and shadow formulas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 On null geodesics Metrics of the form ds2 rotating in (9) admit null geodesics which are separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This condition was shown for the Kerr solution in for example [6, 19], and for other well-motivated forms of mass function M(r) in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Such a property is preserved within its conformal class, and in particular by our line element ds2 in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' To appreciate this, we first recall that for a pair of metrics which are conformally related, say gµν = Ω2(x)˜gµν, their geodesic equations are related by d2xα dη2 + Γα βν dxβ dη dxν dη = 0 = d2xα dη2 + ˜Γα βν dxβ dη dxν dη − 2Ω−1 dΩ dη dxα dη , (18) where ˜Γα βν are the Christoffel symbols associated with the metric ˜g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The last term of (18) can be rewritten as d2xα dλ2 + ˜Γα βν dxβ dλ dxν dλ = 0, dλ dη = Ω2(xα(η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (19) Thus, a suitable redefinition of the affine parameter leads to identical forms of the geodesic equations for the pair of conformally related metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In particular, we expect separability of null geodesic equations just like the Kerr solution or more generally the Kerr-like solutions studied in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Our solution has a conformal Killing vector field K ∼ ∂t which naturally gives a quantity conserved along its null geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In parallel with the notion of energy in the static case, we call this conserved quantity ˜E = KµPµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Also, the metric components are independent of φ, and we call L = pφ the associated conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' These symmetries motivate the use of the Hamilton- Jacobi formalism for geodesics where one first defines an auxiliary action S(κ, xµ) obeying ∂S ∂κ + 1 2gµνpµpν = 0, pµ = gµνpν = gµν ∂S ∂xν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (20) By virtue of the nature of conserved quantities, we adopt the following ansatz for the action S: S = − ˜Et + Lφ + Sr(r) + Sθ(θ) + 1 2µ2κ, µ = −pαpα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (21) By construction, the 4-momenta pµ = dxµ dη = gµν∂νS(x) satisfies the geodesic equation, with κ = 0 for null geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We find that the Hamilton-Jacobi equation (20) leads to − ∆(r)(∂rSr)2 + [(r2 + a2)E − aL]2/∆(r) = (∂θSθ)2 + (L − aE sin2 θ)2/ sin2 θ = K, (22) where the constant K indicates separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For deriving the shadow formulas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' we need the explicit expressions for the 4-momenta which we find to simplify as dt dη = 1 Σ(r)e− 2(t+Υa(r)) r0 � −a(aE sin2 θ − L) + (r2 + a2)P(r) ∆(r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (23) dr dη = ± 1 Σ(r)e− 2(t+Υa(r)) r0 � R(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (24) dθ dη = ± 1 Σ(r)e− 2(t+Υa(r)) r0 � Ξ(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (25) dφ dη = 1 Σ(r)e− 2(t+Υa(r)) r0 � − � aE − L sin2 θ � + aP(r)/∆(r) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (26) 8 where P(r) ≡ E(r2 + a2) − aL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' R(r) ≡ P(r)2 − K∆(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Ξ(θ) ≡ Q + cos2 θ � a2E2 − L2/ sin2 θ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' with Q being conventionally called the Carter constant defined by Q ≡ K − (L − aE)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (27) At this point, we note that apart from the specific form of our mass function M(r) = Ms + r2/r0 implicitly contained in ∆(r) = r2 − 2M(r)r + a2, the 4-momenta expressions in (23) - (26) are identical to those found in [21] up to the conformal factor e− 2(t+Υa(r)) r0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This is consistent with the general relations governing conformally related metrics as described in eqns (18) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 Shadow formulas from photon region For our analysis of the shadow, we define the constants of motion η ≡ Q E2 , ξ ≡ L E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (28) Any null geodesics with r = Rp for some constant Rp leads to the condition R(Rp) = P(Rp)2 − K∆(Rp) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Further, if the orbit is unstable then setting d2r/dη2 = 0 yields R′(rp) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' After some algebra, we find that these couple of equations lead to the constants of motion a L E /M 2 s = 4(1 + µR2 p)R2 p − (3µR2 p + Rp + 1)(R2 p + a2) −3µR2p − 1 + Rp , (29) K E2 /M 2 s = ((R2 p + a2) − aL/E)2 R2p − 2(1 + µR2p)Rp + a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (30) These constants of motion also characterize non-spherical geodesics which asymptotically approach those confined to spheres defined by R(Rp) = R′(Rp) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (29) and (30), and henceforth, for notational simplicity, we define Rp, a in units of Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Now consider an observer at the position (Ro, θinc) and described by an orthonormal tetrad as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' e0 = e− µT Ms (r2 + a2)∂t + a∂φ √ Σ∆ , e1 = e− µT Ms � 1 Σ∂θ, e2 = −e− µT Ms ∂φ + a sin2 θ∂t √ Σ sin θ , e3 = −e− µT Ms � ∆ Σ ∂r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (31) As explained in for example [6, 19], this choice leads to e0 being the 4-velocity of the shadow observer with e0 ± e3 being tangential to the principal null congruences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (The various expressions in (31) differ from those in [6, 19] by the conformal factor which we need to take into account for an orthonormalized set of basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=') The tangent vector of each light ray reaching the observer reads d dη = pµ∂µ = ˙r∂r + ˙θ∂θ + ˙φ∂φ + ˙t∂t = α(−e0 + sin θ cos φe1 + sin θ sin φe2 + cos θe3), (32) 9 where α = gµνpµeν 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Equating the coefficients after evaluating both sides of (32) at (Ro, θinc) yields sin Φ = LE(Rp) − a sin2 θinc � KE(Rp) sin θinc , sin Θ = � ∆(Ro)KE(Rp) R2o − aLE(Rp) + a2 , (33) where LE ≡ L M2s E , KE ≡ K M2s E2 , and we have switched to a different notation for the celestial coordinates Φ, Θ for clarity having derived their eventual expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In particular, we note that θinc measures the angle between the black hole’s spin axis as defined by θ = 0(the zero locus of gφφ) and the observer, with e3 being parallel to the line of sight connecting the observer to the origin of the Boyer-Lindquist-like chart in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The photon region consists of spherical orbits with radii bounded in the domain Rp ∈ (Rp,min, Rp,max), (34) where Rp,min, Rp,max are defined by setting sin Φ = +1, −1 respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' aLE(Rp,min) = a2 sin2 θinc + a � KE(Rp,min) sin(θinc), (35) aLE(Rp,max) = −a2 sin2 θinc − a � KE(Rp,max) sin(θinc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (36) In the vanishing a limit, the width of the photon region (Rp,min, Rp,max) goes to zero, and collapses to a single value of Rp which is the photon sphere radius of Vaidya spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In this limit, from (35), lim a→0 aLE → R2 p 3 + µR2 p − Rp Rp − 1 − 3µR2p = 0 ⇒ Rp = 1 2µ � 1 − � 1 − 12µ � , where we have restricted the root to be smaller than the conformal Killing horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This is indeed the expression obtained in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Further taking the µ = 0 limit yields Rp = 3 which is the radius of Schwarzschild photon sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In the a = 0 limit, our expression for sin Θ in (33) reduces to eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (41) of [16] which is the sine of the angular radius of the Vaidya solution’s shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 4 Portraits of the shadow In this Section, we discuss geometrical properties of the shadow in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We first recall that the coordinate system describing the shadow observer has a conformal Killing horizon Ra obtained as the largest root of grr = 0 in the line element (9) which we reproduce explicitly below for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' ds2 = e 2(t+Υa(r)) r0 � − � 1 − 2M(r)r Σ � dt2 − 4M(r)ar sin2 θ Σ dφdt + � r2 + a2 + 2M(r)a2r sin2 θ Σ � sin2 θdφ2 + Σ ∆dr2 + Σdθ2 � , (37) where M(r) = Ms + r2 r0 and Υa(r) ≡ � r dr r2+a2 r2−2M(r)r+a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For our shadow observer located at some radial distance Ro, we would also like gtt < 0 for all values of θ, leading to the condition Rh < Ro < Rc < Ra, Rh,c = r0 4 � 1 ∓ � 1 − 16µ � , (38) 10 where Rh, Rc are the event and Killing horizons of the Vaidya solution in the a = 0 limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This also implies that for some fixed Ro, we have an upper bound on µ < µb ≡ Ms Ro − 2Ms 2R2o , (39) since we wish to have Ro < Rc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For the visual representation of the shadow, we follow the convention used by Johannsen and Psaltis in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This is essentially the orthonormal tetrad we used in deriving the shadow formula, with the x, y coordinates being x = Ro sin (Θ(Rp, Ro)) sin (Φ(Rp, θinc)) , (40) y = ±Ro sin (Θ(Rp, Ro)) cos(Φ(Rp, θinc)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (41) These coordinates parametrize the observer’s plane upon which the shadow is projected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='iv From (33), noting that LE and KE are odd and even in the spin parameter a respectively, one can straightforwardly identify the discrete symmetry a → −a, θinc → −θinc, which implies in particular that a < 0 shadows can be obtained from their a > 0 counterparts by a reflection in the y-axis (for all our shadow plots, we take a > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In quantifying the shape of the shadow, we follow the work of Johannsen and Psaltis in [13] who introduced the asymmetry parameter A to describe departure from circularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' A = 2 � � � � � 2π 0 dα (R − R)2 � 2π 0 dα , tan α = y x, R ≡ � (x − D)2 + y2, D ≡ |xmax + xmin| 2 , (42) and R = � 2π dα R/2π is the averaged radius projected upon the observer’s plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' These quantities were mentioned in the EHT paper [1] for M87∗, and we checked that our shadow geometries for the background Kerr solution (with µ = 0) yield comparable features obtained previously in the work of Johannsen and Psaltis [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In Figure 2, we picked a few values of a, θinc for a black hole located at some Ro to demonstrate how the shadow curve changes with µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 Scaling laws for variation of R and A with µ The parameter space for the shadow geometry is spanned by {a, θinc, Ro, µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Computing the shadow’s mean radius and asymmetry factor for a range of parameters, we find a simple em- pirical scaling law that describes the variation of R, A with the accretion rate parameter µ and other parameters as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' R = Ro (a, θinc, Ro) � 1 − µ µb(Ro), A = Ao (a, θinc, Ro) � 1 − µ µb(Ro), (43) where µb(Ro) is the upper bound (39) on the accretion rate allowed by our model for some fixed observer distance Ro, obtained by setting the conformal Killing horizon to be the observer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' ivAnother set of projection coordinates used for plotting the black hole shadow is the (α, β) parameters of Bardeen which would not be entirely suitable in our case since our solution is not asymptotically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' See for example the review of [6] which discusses the relations between Bardeen’s impact parameters and others such as stereographic coordinates, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 11 ������� 4 2 2 4 4 2 2 4 Kerr,a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1,θinc=17∘ Kerr-Vaidya,μ=1×10-12 Kerr-Vaidya,μ=5×10-12 (a) ������� 6 4 2 2 4 4 2 2 4 Kerr,a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='94,θinc=17∘ Kerr-Vaidya,μ=1×10-12 Kerr-Vaidya,μ=5×10-12 (b) ������� 6 4 2 2 4 4 2 2 4 Kerr,a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5,θinc=90∘ Kerr-Vaidya,μ=1×10-12 Kerr-Vaidya,μ=5×10-12 (c) ������� 8 6 4 2 2 4 2 2 4 Kerr,a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='94,θinc=90∘ Kerr-Vaidya,μ=1×10-12 Kerr-Vaidya,μ=5×10-12 (d) Figure 2: A visual representation of the shadow projected upon the observer plane at Ro/Ms = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 × 1010 with θinc = 17◦ for (a) and (b), θinc = 90◦ in (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The upper bound on µb ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='26 × 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Horizontal axes are scaled in units of Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The dependence on µ appears as a separate factor independent of the other shadow parameters, with the functions Ro (a, θinc, Ro) , Ao (a, θinc, Ro) describing the radius and asymmetry factor at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The form of (43) implies that for µ ≪ 1, R0 ≫ Ms, the fractional decrease in the mean radius and the asymmetry factor that is induced by a non-zero µ scales approximately as δR R = δA A ≈ −µ Ro Ms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (44) In Figure 3, we plot these functions for a few values of spin at a fixed R0 and θinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' These plots are expectedly similar to the corresponding ones presented in [13] and [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' At higher spin values, the asymmetry factor and mean radius exhibit a greater range of values over the θinc domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' At any θinc, increasing a increases Ao but decreases Ro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The form of (43) also implies that the asymmetry factor expressed in units of the mean radius is independent of µ, with A R = Ao Ro (a, θinc, Ro) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (45) 12 ������� 20 40 60 80 θinc/deg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='25 �/Ms a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='7 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9 (a) ������� 0 20 40 60 80 θinc/deg 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='90 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='95 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='20 R /Ms (b) Figure 3: Graphs depicting how R, A vary with angle θinc, at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We plot the functions Ro (a, θinc, Ro) , Ao (a, θinc, Ro) at five values of a at Ro/Ms = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 × 1010 and θinc = 17◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In Figures 4, 5 and 6, we plot the empirical fitting curves (described by (43) ) that depict how R, A vary with the accretion parameter µ and each of the parameters {a, θinc, Ro} separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' ������� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='030 �/Ms a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='75 a=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9 (a) ������� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 μ 1 2 3 4 5 R/Ms (b) Figure 4: Graphs depicting how R, A vary with µ for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9, with Ro/Ms = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 × 1010, θinc = 17◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' ������� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='25 �/Ms θinc=17∘ θinc=45∘ θinc=90∘ (a) ������� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 μ 1 2 3 4 5 R/Ms (b) Figure 5: Graphs depicting how R, A vary with µ for θinc = 17◦, 45◦, 90◦, with Ro/Ms = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 × 1010, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 13 ������� 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5×10-11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5×10-11 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='030 �/Ms Ro/Ms=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0×1010 Ro/Ms=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2×1010 Ro/Ms=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4×1010 (a) ������� 0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5×10-11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='×10-11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5×10-11 μ 0 1 2 3 4 5 R/Ms (b) Figure 6: Graphs depicting how R, A vary with Ro for R0/Ms = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 × 1010, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 × 1010, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 × 1010 with θinc = 17◦ and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 On the shadows of M87∗ and Sagittarius A∗ as observed by EHT The M87∗ black hole was found to be about 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 Mpc away, with a mass Ms ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5 × 109M⊙[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The ensemble of accretion models used by the EHT team [1, 7] involved mass rates that ranged from about 2×10−7 to 4×10−4 times the Eddington rate ˙MEdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In their work, ˙MEdd ∼ 137M⊙/yr, and we find that this translates into µM87 = ˙M × GM⊙ Yr × c2 ∼ ˙M × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='56 × 10−13 ∈ � 4 × 10−18, 9 × 10−15� , where ˙M is mass rate in units of M⊙/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This is smaller than the upper bound µb ∼ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 × 10−12 for Ro ∼ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 Mpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Equivalently, for this range of µ, the conformal Killing horizon size falls within Rc/Ms ∼ � 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9 × 1013, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='16 × 1017� , which lies beyond Ro/Ms ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 × 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Thus, the observed distance to M87∗ and estimates of mass accretion are well within the domains of validity of our simple model geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Now it was estimated in [22] that the angle of inclination is around 17◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Corresponding to this value, in Figure 7, we plot the variation of the shape parameters R, A and their ratio with the spin parameter a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The range of values of R translates into the shadow angular diameter ∼ (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9µas, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6µas) which is comparable to the measured emission ring diameter in the EHT experiment [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The maximum A/R ratio is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='01 which is within limits of the upper bound of 10% indicated in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The highest µ ∼ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5 × 10−15 in the ensemble of models considered in [1] translates only to a fractional shift of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='05 % in R, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Sgr A∗ has been observed to located near the dynamical center of our galaxy at a distance Ro ∼ 8 kpc away, with a dense concentration of mass Ms ∼ 4 × 106M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In contrast to M87∗ where its prominent jet provides robust constraints on source orientation with respect to the line of sight, fixing it to be ∼ 17◦, there is no such constraint unfortunately on Sgr A∗ [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' However, GRMHD models appeared to have favored θinc < 50◦, with accretion rate of order-of-magnitude 10−9 − 10−8M⊙ yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' These models were equipped with spin parameter values of a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='94 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' As mentioned in [2], in the earlier works of Quataert [23] and Baganoff [24], the captured accretion rate was estimated to be 10−6−10−5M⊙ yr−1 from Chandra observations of thermal bremsstrahlung emission at the vicinity of the gas capture radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Most recently, in [11], a promising model was identified in which θinc ≤ 30◦, and accretion models of ˙M ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='5×10−9M⊙/yr were examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 14 ������� 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 R/Ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='05 �/Ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='010 �/R Figure 7: Graphs showing how the mean radius R, asymmetry factor A and their ratio vary with a, with Ro/Ms = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 × 1010 (pertaining to EHT observation of M87∗), θinc = 17◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Even for the accretion rate ˙M = 10−5M⊙ yr−1 this translates into merely µsgr ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 × 10−18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Like the case of M87∗, this turns out to be smaller than the upper bound µb ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 × 10−11 for Ro ∼ 8 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Equivalently, taking this value of µsgr, the conformal Killing horizon size is Rc/Ms ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 × 1017, which lies beyond Ro/Ms ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 × 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Thus again, both observer distance and (estimated) mass accretion rates are well within the domains of validity of our simple model geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In Figure 8, we plot the variation of the shape parameters R, A and their ratio with the spin parameter a, at a few representative values of θinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Over the domain of θinc ∈ (10◦, 50◦), the range of shadow angular diameters is ∼ (47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6µas, 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2µas) which is comparable to the shadow diameter estimate 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='7±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0µas in the EHT experiment [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The asymmetry factor-to-mean radius ratio A/R ratio increases with θinc, and can be as high as ∼ 10% for θinc = 50◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' It would be interesting to study this geometrical signature for the EHT’s Sgr A∗ shadow image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In [8], the EHT team mentioned in passing that the sparse interferometric coverage of 2017 observations led to significant uncertainties in circularity measurements which were thus not quantified yet, but future EHT observations with additional telescopes may place constraints on the circularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Finally, let us mention that the effect of µ on R, A is even smaller for Sgr A∗, inducing only a fractional change of 10−6 in these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' ������� ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 R/Ms ●● ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ θinc=10∘ ▲ θinc=30∘ ■ θinc=50∘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 �/Ms ●● ● ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='0 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='08 �/R Figure 8: Graphs showing how the mean radius R, asymmetry factor A and their ratio vary with a, with Ro/Ms = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 × 1010 (pertaining to EHT observation of Sgr A∗), θinc = 10◦, 30◦, 50◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 15 5 Discussion We have presented a study of the shadow geometry for a class of spacetime metric that is Kerr- Vaidya-like in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The family of time-dependent black hole solutions we constructed in this work has well-defined Kerr and Vaidya limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Agnostic to the source and the underlying theory, we had conceived of its form starting from the Vaidya solution with a mass function that is lin- ear in Eddington-Finkelstein coordinates since this particular class of solutions furnishes a model for accretion and is equipped with a conformal Killing isometry that leads to separability of null geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' After expressing it as being conformal to a solution that is Schwarzschild-like with a radial coordinate-dependent mass function, the Newman-Janis algorithm was applied to obtain a Kerr-like solution that reduces to the Kerr solution in the limit of vanishing accretion parameter µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In real-life applications, the dimensionless accretion rate µ is expected to be very small, for instance, for the recent M87∗ and Sgr A∗ observations, the highest model estimates of µ are of the order 10−15, 10−18 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Thus, our model geometry can be considered as a small µ- deformation of the Kerr solution that preserves separability of null geodesics, or from a different perpsective, a rotating generalization of the Vaidya solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For a finite spatial domain, it could act as a simple model of a Kerr-like geometry that takes into account the backreaction of accre- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The existence of a conformal Killing vector field allows us to solve for the shadow geometry straightforwardly yet also brings with it the subtlety of a horizon that should be located beyond the shadow observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Equivalently, at any fixed observer distance, there exists an upper bound to µ (eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (39)) for applicability of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In our study of the variation of the mean radius R and asymmetry factor A with regards to various parameters — {µ, a, θinc, Ro}, we found a simple empirical scaling law of the form in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In particular, this implies that at small µ ≪ 1 and large observer distance Ro ≫ Ms, the fractional change induced by turning on the accretion rate parameter µ reads simply as δR R = δA A ≈ −µ Ro Ms .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (46) To our knowledge, we have not encountered any previous descriptive relations between shadow geo- metrical features and accretion rate, black hole and observer parameters of the form similar to (43) or (46)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='GRMHD simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' [10, 12, 26]) typically assume the validity of the background metric being purely Kerr in some suitable coordinate system, with the complicated astrophysics of accretion contained within the choice of the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Even for GRMHD sim- ulations involving spacetime metrics motivated by beyond-GR theories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' [8]), the accretion parameter is rarely involved in describing the background spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In [13, 14] and most recently in [15], it was found that the accretion details do not appear to influence the shadow geometry which is sensitive only to the background metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The assumption here is that backreactions of accretion on the metric are insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Indeed, in applying our model to EHT observations of M87∗ and Sgr A∗, we found that the most generous estimates of µ in [1] and [2] yield fractional changes of R and A of the order of 10−4 and 10−6 respectively, consistent with such an assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In addition, our model yields an explicit relation in (43) that describes how, at least in principle, the shadow geometry changes with accretion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' From (46), it may seem that for situations where the (fractional) experimental uncertainties δRe/Re, δAe/Ae are of the order µRo/Ms, then metric vIn [25], it was found that increasing the flux of infalling gas into a Schwarzschild black hole (and hence the accretion rate) by increasing an axion-plasmon coupling parameter decreases the size of the shadow which qualitatively agrees with the variation of R with µ of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' It would be interesting to study if the results in [25] could be cast in a similar form as (43) or (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 16 backreactions due to accretion may be important in analyzing geometrical details of the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' A limitation of our model geometry is that it is only asymptotically Ricci-flat and not Minkowskian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' As a model of an effective Kerr-like geometry with accretion backreaction, it is thus valid only for a finite spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Imposing a stricter condition for gtt < 0 in (37), in our exposition of the black hole shadow analysis, we have taken the observer location Ro to lie within the interior of the sphere defined by the conformal Killing horizon of the limiting Vaidya spacetime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' R < Rc in the coordinate system of (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (For M87∗ and Sgr A∗, Rc/Ro ∼ 103, 107 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=') It would be interesting to seek a refinement of our model geometry in one that allows for separa- bility of null geodesics while being asymptotically flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Towards this ideal goal, in Appendix A, we sketched a globally well-defined solution obtained by matching the spacetime at some cutoff distance to asymptotically flat Kerr-like solutions using Darmois-Israel junction conditions, leaving more realistic constructions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Acknowledgments I am grateful to Jan de Boer, Chong-Sun Chu, Ori Ganor, Petr Horava, Daniel Robbins and Neal Snyderman for sharing with me their insights on various aspects of gravitational physics and their moral support over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' A Global geometry and matching spacetimes via junction condi- tions For the Vaidya spacetime, the conformally static chart in (3) has a coordinate singularity at the conformal Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' It can be continued beyond that via (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Similarly, from (11), we can perform the coordinate transformation (2) v = r0e ˜T/r0, w = re ˜T/r0, (47) after which the line element takes the form ds2 = −F �w v � � dv + av sin2 θ r0 d˜φ �2 + 2 � dv + av sin2 θ r0 d˜φ � � dw + av sin2 θ r0 d˜φ � −2aw sin2 θ r0 dvd˜φ + � w2 + v2a2 cos2 θ r2 0 � dΩ2, (48) where F �w v � = −1 + 2 � µ + ( w v )2� w v � w v �2 + a2 cos2 θ r2 0 − 2w v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The Vaidya metric in the chart (1) is obtained in the a = 0 limit, whereas the double scaling limit of (6) brings it to Kerr in Eddington-Finkelstein chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The metric in the chart {v, w, θ, ˜φ} is convenient for examining the asymptotic infinity of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Taking the limit of infinite w yields the following asymptotic form lim w→∞ ds2 ∼ −dv2 + 2dvdw + w2dΩ2 + 2aµ Ms sin2 θ � wdvd˜φ + vdwd˜φ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (49) 17 At radial infinity, while the spacetime is Ricci-flat, there are still non-zero Ricci and Einstein tensor components which read Rθθ = −Gθθ = − a2 r2 0 sin2 θ, R˜φ˜φ = 3G˜φ˜φ = −3a2 r2 0 sin4 θ, with other components being zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In the following, we briefly discuss how one could impose suitable initial and final boundary conditions to the mass-accreting geometry by suitably matching our solution to Kerr solutions representing the start or end-point of the mass accretion process, yielding a more realistic global geometry via the use of Darmois-Israel junction conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For the Vaidya spacetime, we can cut-and-paste the spacetime geometry to that of Schwarzschild spacetime at some advanced v where the accretion ends, and to an initial geometry such as Minkowski or another Schwarzschild solution at some earlier v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For example, in (1), instead of a function monotonically increasing in v, we can refine the mass function to be m(ν) = � � � � � 0, ν = 0 µν, 0 < ν < νf µνf, ν ≥ νf, (50) and formally, we obtain a global geometry constructed by matching Vaidya to Minkowski at ν = 0 and Schwarzschild solution with ADM mass µνf at ν = νf, yielding a more realistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We adopt a similar construction to extend our geometry suitably to past and future infinities in this vein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Let Gi, Gf be the initial and final geometries that we match our Kerr-Vaidya-like solution at times ˜T = { ˜Ti, ˜Tf} respectively, working in the chart of { ˜T, r, θ, ˜φ} (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We denote the matched induced metric on Gi, Gf by Σi, Σf, and that of our Kerr-Vaidya-like solutions by G evaluated on { ˜Ti, ˜Tf}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The first Darmois-Israel junction condition is the continuity of the induced metric at these two junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Σi = G| ˜T= ˜Ti Σf = G| ˜T= ˜Tf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (51) For simplicity, we consider glueing geometries of which induced metric at the matching surface can be easily expressed in the coordinate systems we discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' For asymptotic flatness in G, Gi, Gf, consider modifying the line element in (11) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Defining the bracketed expression in (11) to be ds2 K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' we now replace it with ds2 K = � � � � � � � � � � � � � − � 1 − 2M(r)r r2+a2 cos2 θ � (d ˜T + a sin2 θd˜φ)2 + 2(d ˜T + a sin2 θd˜φ)(dr + a sin2 θd˜φ) +(r2 + a2 cos2 θ)dΩ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' r < Rm − � 1 − 2MKr r2+a2 cos2 θ � (d ˜T + a sin2 θd˜φ)2 + 2(d ˜T + a sin2 θd˜φ)(dr + a sin2 θd˜φ) +(r2 + a2 cos2 θ)dΩ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' r > Rm (52) where r = Rm is a matching surface beyond which ds2 K is the Kerr line element with mass MK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' MK = Ms � 1 + µR2 m M2s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We set the cutoff radius Rm to be at some radial distance beyond the observer of the shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='vi Such a spacetime is parametrized not only by {a, µ, Ms} but also by the cutoff parameter Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The full metric still takes the form G(a, µ, MK, Rm) : ds2 = e 2µ Ms ˜T ds2 K(a, µ, MK, Rm), (53) viOne natural choice would be to take Rm = Rc - the conformal Killing horizon of the limiting Vaidya spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 18 but with ds2 K defined as in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Next, at ˜T = { ˜Ti, ˜Tf}, we cut-and-paste G to Gi, Gf respectively which are defined by Gi : ds2 = ds2 K(ai, µ, Mi K, Ri m), Gf : ds2 = ds2 K(af, µ, Mf K, Rf m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (54) The initial and final matching surfaces are spacelike surfaces, and the various parameters are related as e ˜ Ti r0 Rm = Ri m, e ˜ Ti r0 MK = Mi K, e ˜ Ti r0 a = ai, (55) e ˜ Tf r0 Rm = Rf m, e ˜ Tf r0 MK = Mf K, e ˜ Tf r0 a = af.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (56) The initial geometry can be taken to be arbitrarily close to Minkowski spacetime with ˜Ti → −∞ with (Rm, MK, a) being some finite set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The eventual geometry from the mass accretion process of duration ∆ ˜T = ˜Tf − ˜Ti is that of the Kerr solution (for R > Rf m) with mass and spin parameters being Mf K = eµ ∆ ˜ T Ms Mi K, af = eµ ∆ ˜ T Ms ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (57) From (57), one can easily see that our model geometry manifestly represents an accretion process where the fractional increase in both the mass and spin parameters occur at a rate of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' B On the reference frame of the shadow observer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1 In the limit of a = 0 In the limit of a = 0, the tetrad basis (31) defining the reference frame of our shadow observer reduces to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' e0 = e−µ ˜T/Ms � 1 − 2M r ∂t, e1 = e−µ ˜T/Ms r ∂θ, e2 = −e−µ ˜T/Ms r sin θ ∂φ, e3 = −e−µ ˜T/Ms �� 1 − 2M r � ∂r (58) In the chart { ˜T, r, θ, ˜φ}, we replace ∂t → ∂ ˜T , ∂r → ∂ ˜T ∂r ∂ ∂ ˜T + ∂ ∂r which leads to e0 = e−µ ˜T/Ms √f ∂ ˜T , e1 = e−µ ˜T/Ms r ∂θ, e2 = −e−µ ˜T/Ms r sin θ ∂φ, e3 = −e−µ ˜T/Ms √f � ∂ ˜T + f∂r � (59) where f = 1− 2M(r) r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' This is the tetrad basis used in [16] for Vaidya spacetime’s shadow calculation, relevant for an observer with 4-velocity e0, at constant θ, φ, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='2 Observers in the {v, w, θ, φ} chart and aberration formulas The coordinate system {v, w, θ, φ} avoids the horizons as coordinate singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In [16], the shadow observed by an observer with 4-velocity ∼ ∂ ∂v was derived using an aberration formula being applied to the shadow angle formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In general, for a pair of reference frames (S, S′), aberration formulas relating the coordinates of their celestial spheres can be derived from the expression (32) after we express the 4-velocity e′ 0 of the S′ reference frame as a linear combination of the original tetrad basis components, writing e′ 0 = e0 + V kek √ 1 − v2 , (60) 19 where ⃗V is the relative 3-velocity of S′ observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Consider an observer of which e′ 0 is a linear combination of e0 and another basis vector e3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Its tetrad basis components read e′ 0 = 1 √ 1 − V 2 (e0 + V e3), e′ 3 = 1 √ 1 − V 2 (e3 + V e0), e′ 1,2 = e1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (61) Taking the inner product between e′ 0, e′ 3 and the tangent vector expression in (32) in both unprimed and primed coordinates, we obtain the aberration formulas cos θ′ = V + cos θ 1 + V cos θ, φ′ = φ, V = −e′ 0 · e3 e′ 0 · e0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (62) In [16], the observer with e′ 0 ∼ ∂ ∂v was also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' After a coordinate transformation from { ˜T, r, θ, ˜φ} used for the tetrad basis in (59), one can show that e′ 0 ∼ ∂ ∂v = 1 � 1 − 2Ms r e− T r0 � ∂ ˜T − r r0 ∂r � = 1 √ 1 − V 2 (e0 + V e3), V = r2 r2 + rr0 � 1 − 2M(r) r � (63) where e0, e1, e2, e3 are as defined in (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In [16], the same expression for the relative velocity V was obtained with an aberration relation tan2 θ′ 2 = 1−V 1+V tan2 θ 2 that we verified to be identical to (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Let us now consider an appropriate S′ observer for our Kerr-Vaidya-like geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' We note that for the S observer, its 4-velocity e0 is a linear combination of ∂ ˜T and ∂˜φ, or in the Boyer- Lindquist-like chart, a linear combination of ∂t and ∂φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' The angular component is such that e0 ±e3 are tangential to the principal null congruences of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Relating between v and t, we choose the following S′ observer with e′ 0 ∼ ∂ ∂v + r0a v(r2 + a2) ∂ ∂ ˜φ = r0 v � 1 + r(r2 + a2) r0(r2 − 2Mr + a2) � ∂t − r v∂r − r0a v(r2 − 2Mr + a2)∂φ + r0a v(r2 + a2) ∂ ∂ ˜φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (64) This choice of e′ 0 is also uniquely the one that allows us to write e′ 0 ∼ e0 + we3, (65) for some relative 3-velocity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Thus the aberration formulas in (62) apply similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Recall that in (31), the relevant basis tetrad components read e0 = e−µ ˜T/Ms (r2 + a2)∂t + a∂φ √ Σ∆ , e3 = −e−µ ˜T/Ms � ∆ Σ ∂r, (66) which allows us to read off the 3-velocity as w = r2 + a2 r2 + a2 + r0 r (r2 − 2Mr + a2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (67) In the limit of vanishing a, we recover the 3-velocity v for the observer in Vaidya spacetime with e0 ∼ ∂ ∂ ˜T as derived in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' In the limit µ → 0, up to leading order, we have w ≈ µ � r2 + a2 Ms r (r2 − 2Msr + a2) � + O(µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' (68) Thus, this reference frame may be relevant for theoretical situations where the observer’s velocity is proportional to the strength of the accretion rate, although arguably not so for realistic EHT observations where the accretion is hardly expected to backreact on the metric significantly to affect the 3-velocity of the shadow observer in such a manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 20 References [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Akiyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' [Event Horizon Telescope], “First M87 Event Horizon Telescope Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Magnetic Field Structure near The Event Horizon,” Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 910, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='1, L13 (2021) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='3847/2041-8213/abe4de [arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='01173 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content='HE]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} +page_content=' 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE4T4oBgHgl3EQfOwxI/content/2301.04967v1.pdf'} diff --git 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100644 index 0000000000000000000000000000000000000000..f7c3f027034b1bbb1af14103b56db24d36495c7d --- /dev/null +++ b/c9AzT4oBgHgl3EQfnv2o/content/tmp_files/2301.01586v1.pdf.txt @@ -0,0 +1,746 @@ +1 +Post-Quantum Key Agreement Protocol based on +Non-Square Integer Matrices + +HUGO DANIEL SCOLNIK 1, 2, 3, 4 +hugo@dc.uba.ar, hscolnik@gmail.com + +JUAN PEDRO HECHT 3 +phecht@dc.uba.ar, qubit101@gmail.com + +1Instituto de Ciencias de la Computación, Universidad de Buenos Aires and CONICET, Buenos Aires, Argentina. +2Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, +Buenos Aires, Argentina. +3Maestria en Seguridad Informática – Facultades de Ciencias Económicas, Ciencias Exactas y Naturales, +Ingeniería, Universidad de Buenos Aires, Buenos Aires, Argentina. +4Corresponding author + +Abstract. We present in this paper an algorithm for exchanging session keys, coupled with an hashing encryption +module. We show schemes designed for their potential invulnerability to classical and quantum attacks. In turn, +if the parameters included were appropriate, brute-force attacks exceed the (five) security levels used in the NIST +competition of new post-quantum standards. The original idea consists of products of rectangular matrices in Zp +as public values and whose factorization is provably an NP-complete problem. We present running times as a +function of the explored parameters and their link with operational safety. To our knowledge there are no classical +and quantum attacks of polynomial complexity available at hand, remaining only the systematic exploration of +the private-key space. + +Keywords: cryptography, integer matrices, modular arithmetic, key exchange, discrete post quantum algorithm. + +Contribution of the authors: both authors contributed equally to this paper. + +Statements and declarations: the authors declare no competing financial interests. No funding was received for +conducting this study. +MSC classification: 11T71, 14G50, 94A60, 81P94 + +1. Introduction +As it is very well known generating secure key exchange algorithms is a priority for +implementing asymmetric protocols [17]. The idea of public key cryptography goes back +to the work of James Ellis [8] and the seminal work of Diffie-Hellman [7] which was the +first practical solution universally used in SSL, TLS, SSH, IPsec, PKI, Signal, etc. +On the other hand, the imminent appearance of quantum computers able to implement +Shor's and Grover's algorithms [2] which seriously affect the currently used cryptographic +methods, led to the current research efforts in Post Quantum Cryptography (PQC). +This paper was inspired by E. Stickels’s proposals [27] which were cryptanalyzed by V. +Shpilrain [21] and C. Mullan [20]. More recently, S. Kanwal and R. Ali [12] published an +interesting protocol but it was also cryptanalyzed by J. Liu et al. [15]. A natural alternative +was to use rank-deficient matrices but this has been cryptanalyzed by F.Virdia using Jordan +canonical forms [29]. + + + + +It is worthwhile to point out that in the NIST competition for standardization of post- +quantum protocols [22], there is none based on the use of non-commutative algebraic +systems [21], those dedicated to key exchange protocols (KEP) and their canonical +asymmetric cryptosystems, derived using a simple hashing scheme. + +This paper aims to provide alternative solutions in this regard. Daniel Brown [5] presented +an attack on the early versions of our algorithm [26, v:20221116:142416], which relies on +the computation of the characteristic polynomial of the public elements. This approach +seems impractical when applied to real-world parameters but led to an updated version +using ��� field operations [26, v: 20221123:145424]. As this latter kind of attack would +be susceptible to a potential Menezes-Wu attack [18] (a fact pointed out by F. Virdia [29]), +we reconsider here our first methodology as a usable and secure key-agreement protocol. + +2. The notation used in this work +p: prime integer, Zp: set of non-negative residuals mod p, products in Zp (represented by +dots), ||: concatenation, Det[A]: the determinant of matrix A, AT: transpose of matrix A, +A(i, j): matrix component of the i-th row and j-th column, ∈����: random uniform selection +in a closed interval, ⨁ : bitwise XOR. + +3. Paper organization +First, we present an overall description of the proposed algorithm and the corresponding +protocol, the proof that Alice and Bob will derive a common key, security considerations, +and finally some experimental results and a discussion. + +4. Overall description +The algorithm starts by choosing a prime p shared by Alice and Bob who generate two +rectangular matrices each, the first one with more rows than columns and the second one +with inverse dimensions, and t is the number of iterations. For each iteration and every +entry, a random integer s ∈���� [(p-1)/2, p -1] is chosen as the module, employing the +algorithm given in [6]. + +Following this scheme, Alice calculates two matrices A1k and B1k in each cycle (k=1,…,t) +and computes Uk utilizing the matrix product +Uk = A1k . B1k (mod p) (k=1,2,3,…,t) +The vector U=(U1, U2, U3, …, Ut) is sent to Bob. Analogously Bob computes +Vk = A2k . B2k (mod p) (k=1,2,3,…,t) +and the vector V=(V1, V2, V3,…, Vt) is sent to Alice. +We prove that: +A-KEYk = Det[A1kT . Vk. B1kT] (mod p) +and +B-KEYk = Det[A2kT .Uk. B2kT] (mod p) +are equal in each k-cycle. +Finally, Alice computes the hashing of A-CONCAT = A-KEY1 || A-KEY2 ||… || A-KEYt, +and Bob the hashing of B-CONCAT = B-KEY1 || B-KEY2 ||… || B-KEYt which are equal, +and hence this is the shared key. + +A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices + + +3 +We must observe this protocol is highly parameterizable since we can change the +dimensions of the matrices, the number of cycles, the primes, etc. The numerical results +(see below) show a very complex shared key can be obtained in a fraction of a second +using a standard processor. + + +5. key exchange algorithm +ALGORITHM 1: PQC multiKEP +COMMENTS +The key Exchange Algorithm (KEP) uses several cycles as defined below. +INPUT: see the initial configuration. +OUTPUT: shared session key of 512-bits. +INITIAL CONFIGURATION (PUBLIC VALUES): +p: a shared prime number that can be obtained randomly. +rows[X|, columns[X]: dimensions of the matrices X:{A, B}, where +rowsA=columnsB, columnsA=rowsB and rowsA > columnsA. +rowsA is a value whose maximum is a predefined rowmax value. Our +proposal is rowmax=100, rowsA ∈���� [5, rowmax] and columnsA + ∈���� [4, rowsA-1]. +t: number of iterations +H( ): hashing SHA3-512. + +ALICE +1. for k=1 to t +2. +for i=1 to rowsA +3. +for j=1 to columnsA +4. +A1k (i,j) ∈���� [(p-1)/2, p -1] +5. +next j +6. +next i +7. +for i=1 to rowsB +8. +for j=1 to columnsB +9. + B1k (i,j) ∈���� [(p-1)/2, p -1] +10. +next j +11. +next i +12. +Uk = A1k . B1k (mod p) +13. next k +14. Send the vector U = (U1, … , Ut) to Bob + + + + + + + + + +BOB +15. for k=1 to t +16. +for i=1 to rowsA +17. +for j=1 to columnsA +18. + A2k (i,j) ∈���� [(p-1)/2, p -1] +19. +next j +20. +next i +21. +for i=1 to rowsB +22. +for j=1 to columnsB +23. + B2 k (i,j) ∈���� [(p-1)/2, p -1] +24. +next j +25. +next i +26. +Vk = A2k . B2k (mod p) +27. next k +28. Send the vector V = (V1, … ,Vt) to Alice +SESSION KEY OBTAINED BY ALICE +29. for k=1 to t +30. A-KEYk = Det[A1kT . Vk. B1kT] (mod p) +31. next k +32. A-CONCAT = A-KEY1 || A-KEY2 ||… || A-KEYt +33. KEYalice =H( A-CONCAT ) + +SESSION KEY OBTAINED BY BOB +34. for k=1 to t +35. B-KEYk = Det[A2kT .Uk. B2kT] (mod p) +36. next k +37. B-CONCAT = B-KEY1 || B-KEY2 ||… || B-KEYt +38. KEYbob =H( B-CONCAT ) + +6. keys equality proof +Lemma 1: +The keys given by Algorithm 1 are equal, that is KEYalice = KEYbob + +Proof: it is very simple taking into account the elementary properties det(X)=det(��), +det(XY)=det(X)det(Y), (��)� = ���� where X, Y are square matrices of the same +dimension We have to prove that for every k (all operations (mod p) ) + +Det[A1kT . Vk . B1kT)= Det[A2kT .Uk . B2kT]. Since Vk = A2k . B2k and Uk = A1k. B1k the +keys can be written as follows: KEYalice= Det[A1kT .Vk. B1kT) = Det[A1kT . A2k. B2k . B1kT) += Det[(A2kT.A1k)T. (B1k.B2k)T] and KEYbob= Det[A2kT.Uk . B2kT]= Det[A2kT. A1k . B1k . +B2kT] ∎ + +A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices + + +5 +7. Derived cipher algorithm +ALGORITHM 2: PQC multiKEP + simple hashing cipher +Observation: Vector U was received by Bob +Insert here algorithm 1 (up to line 27.) +BOB CIPHERS A MESSAGE TO ALICE +1. msg = 512-bit message from Bob +2. for k=1 to t +3. B-KEYk = Det[A2kT . Uk . B2kT] (mod p) +4. next k +5. B-CONCAT = B-KEY1 || B-KEY2 ||… || B-KEYt +HASHING CIPHER (C, D): +6. C = V (* see Algorithm 1 *) +7. D = H[ B-CONCAT ] ⨁ msg +8. Send (C, D) to Alice +ALICE RECOVERS THE MESSAGE FROM BOB +9. for k=1 to t +10. A-KEYk = Det[A1kT . Vk . B1kT] (mod p) +11. next k +12. A-CONCAT = A-KEY1 || A-KEY2 ||… || A-KEYt +13. msg =H[ A-CONCAT ] ⨁ D +The above algorithms can be enhanced by combining modular operations in Zp with +polynomial multiplications in a finite field like ��� as in [26], see Discussion below [13, +4]. A convenient choice would be m=8 and any of the 30 primitive polynomials of that +order [14]. + +8. KEP and HASHING cipher protocols +It is necessary to define a protocol allowing for an interchange of information between +Alice and Bob asynchronously to achieve the following objectives: + + +deferred communications + +check the integrity of the exchanged information + +mutual authentication to avoid attacks from active adversaries (e.g. man-in-the- +middle) + +block replay attacks + +availability of the exchanged information + +perfectly defined formats + +The following protocol aims at fulfilling these requirements. + + + + +PROTOCOL: KEP AND CIPHER PUBLIC DATA EXCHANGES +INPUT: any kind of data to be exchanged between entities. +OUTPUT: encapsulated message (msg). +INITIAL CONFIGURATION: +msg: any kind of information to be exchanged between entities. +Universal-Keyed Message Authentication Code (UMAC): here proposed to +assure strong symmetric authentication [3, 11]. +ID: any elsewhere predefined and sender-receiver shared identification Tag. +K: sender-receiver shared key. +Tag: a smart label that can store any sort of information from identification +numbers as a brief description for each entity. Here the tag is Tag = HMAC- +SHA3-512 (HM ∥ Nonce). See Fig 1 and more in [3] +HM : NHK(msg1) ∥ NHK(msg2) ∥ · · · ∥ NHK(msgr) ∥ Len, see NH definition +in [3]. +Nonce: pseudorandom and unique number that changes with each generated +tag. +Timestamp: formatted date and time. + +MESSAGE AUTHENTICATION +1. Acquire K and msg +2. Define a fixed-length Nonce +3. Generate UMAC/SHA3-512/ID/Data +4. Encapsulate the concatenation of: [ msg || UMAC/SHA3-512 (HM || +Nonce) || Timestamp] into a file +5. Send the file and nonce to the receiver + +MESSAGE VALIDATION +1. Acquire at any time the sent file +2. Recover msg and UMAC/SHA3-512 (HM || Nonce) +3. Verify integrity and sender’s identity using K, Nonce, and msg +4. Accept or Dismiss msg according to the verification result + +9. Security against brute force attacks +Lemma 2: + +The complexity of attacking Algorithm 1 by brute force is given by (ndim . q)2. t, +where p is the shared prime , q= +��� +� and ndim=rowsA x columnsA. + + + + + + + + +A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices + + +7 +Proof: + +To obtain the session key the attacker has to factorize matrices Uk = A1k . B1k (mod +p), k=1,…,t where the maximum value that can appear in every entry (i, j) is p-1, +independently of ndim (see Algorithm 1). Thus, defining q= +��� +� ; for each entry of each +matrix, it is necessary to try q values, that is (ndim.q)2 possibilities. The attack is +successful only if the factorization can be obtained for each k=1,…,t, something that +can be easily avoided by choosing appropriate parameters ∎ + +Example: +If p = 2147483647, ndim=100 x 90, t =10, then the complexity is ~ 274 + +10. Security against analytical attacks + +It must be stated that Uk, Vk are always singular matrices, therefore blocking inversions +and subsequent linear attacks. This is a consequence of the fact that the rank of the product +of two (rectangular) matrices must be less or equal to the minimum rank of each one, +therefore the rank of the product Uk (or Vk) of dimension rows x rows (rows > cols) is less +than rows, so its determinant is zero. + +The core of the function defining the session keys (lines 30 and 35 of Algorithm 1) +comprises the product of four rectangular matrices, a public component (Uk,Vk) flanked +by two private matrices (ANk, BNk, N={1,2}). +The determinant is distributive concerning this product, and rearranging the product to be +the multiplication of two public elements (Uk . Vk). As mentioned before, it does not work +over real-life parameters. This fact forces the factorization of the (Uk,Vk) components, +which is computationally hard. + +The factorizations Uk = A1k . B1k (mod p), k=1,…,t can be solved in Uk ∈ Rn×n , A1k ∈ +Rn×m , B1k ∈ Rm×n (real realm), using the SVD (Singular Value Decomposition) if there are +no restrictions regarding the nonnegativity of the factors, but if they are imposed as in our +proposal, then Vavasis and references therein [28] proved that the problem is NP-hard in +the continuous case. For the Boolean case, N. Gillis [10] wrote: “…for exact factorizations, +the rank-one problem is trivial. For higher ranks, Boolean factorization is equivalent to +finding a rectangle covering of the matrix U. This is equivalent to the so-called biclique +problem (given a bipartite graph defined by U, find the smallest number of complete +bipartite subgraphs that cover the graph) which is NP-complete as proved in [24]. (sic)”. + +The Boolean NP-complete complexity was formally proven by Miettinen and Neumann +[19]. + +11. Semantic security of algorithm 2 + +Here we provide an informal view of this aspect, based on concepts mostly derived from +Bellare’s work [1]. In a nutshell, semantical security measures the resistance of any +encryption algorithm to attacks using chosen plaintext or ciphertext selected by the +attacker, who has access to the encryption and decryption modules working as oracles; +without knowledge of the key selected for enciphering [1]. The semantic security term is + + + +strongly related to other definitions: the one-way functions and the non-malleability of +ciphertexts. +Indistinguishability under chosen plaintext attack (represented as IND-CPA) is equivalent +to the property of semantic security and is considered a basic requirement for +most provably secure public-key cryptosystems [1]. One-way refers to bidirectional +functions that have a probabilistic-polynomial time algorithm that converts domains into +codomains, but no such algorithm is known that inverts the procedure. Non-malleability +refers to the resistance to modify slightly the ciphertext to obtain meaningful recovered +plaintext [1]. The next concept to define is the indistinguishability of different ciphertexts +of two similar but different plaintexts, an attacker could not assign a ciphertext of one of +them to any one of the plaintexts. This feature is generally presented as a game between a +challenger (the algorithm defender) and an adversary (the algorithm attacker) [1]. The +challenger generates a key pair PK, SK (public key and secret key respectively), based on +any security parameter k (which can be the key size in bits), and publishes PK to the +adversary. The challenger retains SK. Here we describe the adaptative version of the game. +The adversary may perform any number of encryptions, decryptions, or any other +operations. (The adversary is a probabilistic polynomial Turing Machine) [1]. Eventually, +the adversary submits two distinct chosen plaintexts m0 and m1 to the challenger (of the +same length). The challenger selects a bit b ∈ {0, 1} uniformly at random, and sends the +challenge ciphertext C = E (PK, mb) back to the adversary. The adversary is free to perform +any number of additional computations or decryptions (except C, this step is the adaptative +phase of the attack). Finally, its outputs in polynomial time a guess for the value of b [1]. +The adversary wins the game if it guesses the bit b, and winning means the algorithm is +not indistinguishable and secure, else the algorithm reaches the strongest available security +level: IND-CCA2. (Indistinguishable chosen ciphertext adaptative attack). Formally, a +cryptosystem is indistinguishable under an adaptative chosen ciphertext attack if no +adversary can win the above game with probability p greater than 1/2+ ∈k, where ∈k ≤ 1/πK +(πK arbitrary polynomial function) and ∈k is defined as a negligible function in the security +parameter k [1]. For Algorithm 2 we prove the IND-CPA security level and explain how +it could be easily adapted to reach the IND-CCA2 security level. The use of the UMAC +function [3] in our Protocol fills this need in such a way that the practical implementation +of Algorithms 1 and 2 culminates with the desired provable-security level. + +12. NIST PQC security level of algorithm 2 + +NIST bases its classification on the range of security strengths offered by the existing NIST +standards in symmetric cryptography, which NIST expects to offer significant resistance +to quantum cryptanalysis. In particular, NIST defines a separate category for each of the +following security requirements [23]. As previously described in the brute-force attack +section, reasonable parameters exceed largely Level-1 PQC security. + +13. A toy numerical example + Shared parameters: prime p = 5303, rowA=3, columnsA=2, t=2 + + + + + + + + + + +A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices + + +9 +ALICE + +Iteration 1 + +Alice A11 = + 1123 341 + 14 238 + 1041 13 + +Alice B11= + 1525 1019 1561 + 1561 716 862 + +Alice U1 = + 1707 4410 5290 + 446 4372 4284 + 1009 4184 2883 +Iteration 2 + +Alice A12 = + 665 1338 + 622 38 + 505 1617 + +Alice B12= + 925 1412 598 + 364 463 409 + +Alice U2 = + 4436 4695 978 + 549 4954 379 + 416 3406 3500 +BOB + +Iteration 1 + +Bob A21 = + 802 2435 + 1206 3408 + 707 3723 + +Bob B21= + 1174 2805 1242 + 3110 814 550 + +Bob V1 = + 3083 5209 2014 + 3429 159 4847 + 4831 2322 3791 +Iteration 2 + +Bob A22 = + 656 13 + 1900 107 + 611 1537 + +Bob B22 = + 2192 1270 845 + 820 1022 2194 + + Bob V2 = + 893 3229 4815 + 4837 3429 117 + 1182 2858 1374 + +Alice computes A-KEYk = Det[A1kT .Vk. B1kT] (mod p) for k=1, 2 +A-KEY1 = 3207 +A-KEY2 = 2121 +Alice’s key = +0c3322f92446b51e3372d2a7bd2b81265bb96f32fa38562e4c02414e3c73d85ca4b358363b8792461d4033c1d76 +23589c0f6c07ab01e33b6a7294019e125c779 +Bob computes B-KEYk = Det[A2kT .Uk . B2kT] (mod p) for k=1, 2 +B-KEY1 = 3207 +B-KEY2 = 2121 +Bob’s key = +0c3322f92446b51e3372d2a7bd2b81265bb96f32fa38562e4c02414e3c73d85ca4b358363b8792461d4033c1d76 +23589c0f6c07ab01e33b6a7294019e125c779 + +Example of the cipher algorithm +Bob ciphers a message to Alice +B-KEY = 32072121 +PLAINTEXT msg (formatted as a 64-byte string with spaces appended on the right) = "This is a secret +communication.” +CIPHERTEXT C = + 3083 5209 2014 + 893 3229 4815 + 3429 159 4847 +4837 3429 117 + 4831 2322 3791 +1182 2858 1374 +CIPHERTEXT D = +(88,91,75,138,4,47,198,62,82,82,161,194,222,89,228,82,123,218,0,95,151,77,56,71,47,99,53,39,83,29,246,124, +132,147,120,22,27,167,178,102,61,96,19,225,247,66,21,169,224,214,224,90,144,62,19,150,135,9,96,57,193,5, +231,89) + + + +Alice recovers the message +A-KEY = 32072121 +RECOVERED msg = “This is a secret communication.” + +14. Numerical experiments +Programmed in RUST +OS: Windows 10 Pro (64 bits) +Processor: Intel(R) Core(TM) i7-3770K CPU @ 3.50GHz +RAM: 8,00 GB +The CPU times reported in the following Table 1 are the mean values of 10 runs for each combination of the +variables, prime p = 231-1= 2147483647 (~31 bits) + + + +rowsA +columnsA +Cycles +Complexity + +CPU time in +milliseconds +5 +4 +10 +~272 +0.94 +5 +4 +20 +~273 +1.15 +5 +4 +100 + ~275 +2.69 +6 +5 +10 + ~273 +0.99 +6 +5 +20 + ~274 +1.39 +6 +5 +100 +~276 +3.95 +20 +19 +10 +~280 +8.92 +20 +19 +20 +~281 +13.45 +20 +19 +100 +~284 +74.47 +100 +99 +10 +~289 +755.38 +100 +99 +20 +~291 +1503.54 +100 +99 +100 +~293 +7488.44 +Table1. Algorithm 1 complexity and throughput time as a function + of the parameters. + +Using 128 bits integers in RUST with prime p = 18446744073709551113 (~64 bits): + +rowsA +columnsA +Cycles +Complexity + +CPU time in +milliseconds +5 +4 +10 +~2138 +1.29 +6 +5 +10 +~2139 +0.98 +20 +19 +10 +~2146 +26.84 +100 +99 +10 +~2156 +3203.48 +Table 2. Algorithm 1 complexity and throughput time as a function + of the parameters. +15. Discussion + +Algorithm 1 has been implemented in different computer languages and shows that +extremely high complexity can be achieved in fractions of a second on a standard I7 +processor. The fact that by modifying the input variables (number of rows, columns, +primes, iterations) practically any security level can be easily obtained without resorting +to multiple precision, leads to very fast implementations. Depending upon the computer +architecture and software implementation, larger primes can be used for reaching higher +complexity levels. + + +A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices + + +3 +An enhancement is to use ��� polynomial fields [26], recurring to very fast field +multiplications based on hard-coded discrete logarithm tables of any field generator base, +on some steps. In such a way the lack of uniformity in the operations constitutes an +additional barrier to possible algebraic attacks. This proposal could be very easily +implemented without downgrading performance [4]. + +It is particularly important to use the UMAC function in the Protocol because it is similar +to Merkle's trees for PQC digital signatures [2] and also plays the role of achieving +maximal semantic security [1] and simultaneously strengthens its post-quantum character. +Note: as Black et al. [3] state "the security of UMAC is rigorously proven, in the sense of +giving exact and quantitatively strong results which demonstrate an inability to forge +UMAC-authenticated messages assuming an inability to break the underlying +cryptographic primitive. (sic)” + +16. Conclusions +The algorithm presented in this paper is such that very high complexity can be reached +using small primes, normal precision, and small rectangular matrices, leading to very fast +computer implementations. + +Acknowledgments: to D. R. L. Brown, D. B. Szyld, L. Liberti, N. Gillis, F. Virdia, J. Di +Mauro, I. Córdoba, S. Barzola, G. Cucatti and Y. Alis for many interesting discussions, +theoretical insights and computer implementations. + +References +1. +Bellare, M., Desai, A., Pointcheval, D., Rogaway, P.: Relations among notions of security for public-key +encryption schemes. In Annual International Cryptology Conference, Springer, Berlin, Heidelberg, pp. 26-45 +(1998) +2. +Bernstein D., Lange T.: Post-Quantum Cryptography, Nature, 149,188-194 (2017) +3. +Black, J., Halevi, S., Krawczyk, H., Krovetz, T., & Rogaway, P.: UMAC: Fast and secure message +authentication, Annual International Cryptology Conference, pp. 216-233, Springer, Berlin, Heidelberg (1999) +4. +Daemen, J., Rijmen, V.: AES Proposal: Rijndael, AES algorithm submission, September 3 (1999) +5. +Brown, D .R. L.: private communication (2022) +6. +Di Mauro, J., Salazar, E. & Scolnik, H.D.: Design and implementation of a novel cryptographically secure +pseudorandom number generator. J Cryptogr Eng, 12, 255–265 https://doi.org/10.1007/s13389-022-00297-8 +(2022) +7. +Diffie, W., Hellman, M.: New directions in cryptography”, IEEE Transactions on Information Theory, 22, 6, +644-654 (1976) +8. +Ellis, J. H.: The possibility of non-secret digital encryption, CESG Research Report (1970) +9. +Fujisaki, E., Okamoto, T.: Secure Integration of Asymmetric and Symmetric Encryption Schemes, in M. +Wiener (Ed.): CRYPTO’99, LNCS 1666, Springer-Verlag (1999) +10. Gillis N., Private communication (2022) +11. Internet Engineering Task Force (IETF), UMAC RFC 4418, https://datatracker.ietf.org/doc/rfc4418/ (2006). +Accessed 15 September 2022 +12. Kanwal, S., Ali, R.: A cryptosystem with noncommutative platform groups. Neural Computing and +Applications, 29: 11, 1273-1278 (2018). +13. Lee, G.T.: Abstract Algebra, Springer Undergraduate Mathematics Series, https://doi.org/10.1007/978-3-319- +77649-1_3 (2018) +14. Lidl, R., Niederreiter, H.: Introduction to Finite Fields and their Applications, Cambridge University Press, +Cambridge (1997) +15. Liu, J., Jia, J., Zhang, H., Yu, R., Yu, Y., Wu, W.: Cryptanalysis of a cryptosystem with non-commutative +platform groups. China Communications, 15(2), 67-73 (2018) +16. Maurer, U., Renner, R., Holenstein, C.: Indifferentiability, Impossibility Results on Reductions, and +Applications to the Random Oracle Methodology, M. Naor (Ed.): TCC 2004, LNCS 2951, pp. 21–39, Springer- +Verlag, (2004) + + + +17. Menezes, A. J., Van Oorschot, P., Vanstone, S.: Handbook of applied cryptography. The CRC Press series on +discrete mathematics and its applications, CRC-Press (1997) +18. Menezes, A. J., Wu, Y. H. : The discrete logarithm problem in GL (n, q). Ars Combinatoria, 47, 23-32 (1997) +19. Miettinen, P., Neumann, S.: Recent Developments in Boolean Matrix Factorization, Ninth International Joint +Conference on Artificial Intelligence (IJCAI-20), a slightly extended version of the survey is available at +preprint https://doi.org/10.48550/arXiv.2012.03127 (2020) +20. Mullan, C.: Some Results in Group-Based Cryptography, Thesis submitted to the University of London for the +Degree of Doctor of Philosophy (2020) +21. Myasnikov, A., Shpilrain, V., Ushakov A.: Non-commutative Cryptography and Complexity of Group- +theoretic Problems, Mathematical Surveys and Monographs, AMS Volume 177 (2011) +22. NIST Computer Security Resource Center: Competition for standardization of post-quantum protocols (PQC), +https://csrc.nist.gov/projects/post-quantum-cryptography (2017). Accessed 15 September 2022 +23. NIST Computer Security Resource Center: Post-Quantum Security, https://csrc.nist.gov/projects/post- +quantum-cryptography/post-quantum-cryptography-standardization/evaluation-criteria/security-(evaluation- +criteria) (2022). Accessed 15 September 2022 +24. Peeters R.: The maximum edge biclique problem is NP-complete, Discrete Applied Mathematics, Volume 131, +Issue 3, pp 651-654 (2003) +25. Rotman, J. J.: Advanced Modern Algebra, vol. 114, American Mathematical Soc. (2010) +26. Scolnik, H. D., Hecht, J. P.: A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based +on Rectangular Matrices, https://eprint.iacr.org/2022/1370 (2022) +27. Stickel, E.: A new method for exchanging secret keys. Proceedings of the Third International Conference on +Information Technology and Applications (ICITA05), Contemporary Mathematics, IEEE Computer Society, +2, 426–430 (2005) +28. Vavasis, S.: On the complexity of nonnegative matrix factorization, SIAM J. Optim., 20, pp. 1364–1377 +(2010), +29. Virdia, F.: private communication (2022) + + + diff --git a/c9AzT4oBgHgl3EQfnv2o/content/tmp_files/load_file.txt b/c9AzT4oBgHgl3EQfnv2o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..042c190891d8f79fcaf2875ed6beb296fd300396 --- /dev/null +++ b/c9AzT4oBgHgl3EQfnv2o/content/tmp_files/load_file.txt @@ -0,0 +1,553 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf,len=552 +page_content='1 Post-Quantum Key Agreement Protocol based on Non-Square Integer Matrices HUGO DANIEL SCOLNIK 1, 2, 3, 4 hugo@dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='uba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='ar, hscolnik@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='com JUAN PEDRO HECHT 3 phecht@dc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='uba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='ar, qubit101@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='com 1Instituto de Ciencias de la Computación, Universidad de Buenos Aires and CONICET, Buenos Aires, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 2Departamento de Computación, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 3Maestria en Seguridad Informática – Facultades de Ciencias Económicas, Ciencias Exactas y Naturales, Ingeniería, Universidad de Buenos Aires, Buenos Aires, Argentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 4Corresponding author Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' We present in this paper an algorithm for exchanging session keys, coupled with an hashing encryption module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' We show schemes designed for their potential invulnerability to classical and quantum attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' In turn, if the parameters included were appropriate, brute-force attacks exceed the (five) security levels used in the NIST competition of new post-quantum standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The original idea consists of products of rectangular matrices in Zp as public values and whose factorization is provably an NP-complete problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' We present running times as a function of the explored parameters and their link with operational safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' To our knowledge there are no classical and quantum attacks of polynomial complexity available at hand, remaining only the systematic exploration of the private-key space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Keywords: cryptography, integer matrices, modular arithmetic, key exchange, discrete post quantum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Contribution of the authors: both authors contributed equally to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Statements and declarations: the authors declare no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' No funding was received for conducting this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' MSC classification: 11T71, 14G50, 94A60, 81P94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Introduction As it is very well known generating secure key exchange algorithms is a priority for implementing asymmetric protocols [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The idea of public key cryptography goes back to the work of James Ellis [8] and the seminal work of Diffie-Hellman [7] which was the first practical solution universally used in SSL, TLS, SSH, IPsec, PKI, Signal, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=" On the other hand, the imminent appearance of quantum computers able to implement Shor's and Grover's algorithms [2] which seriously affect the currently used cryptographic methods, led to the current research efforts in Post Quantum Cryptography (PQC)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' This paper was inspired by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Stickels’s proposals [27] which were cryptanalyzed by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Shpilrain [21] and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Mullan [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' More recently, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Kanwal and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Ali [12] published an interesting protocol but it was also cryptanalyzed by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A natural alternative was to use rank-deficient matrices but this has been cryptanalyzed by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Virdia using Jordan canonical forms [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' It is worthwhile to point out that in the NIST competition for standardization of post- quantum protocols [22], there is none based on the use of non-commutative algebraic systems [21], those dedicated to key exchange protocols (KEP) and their canonical asymmetric cryptosystems, derived using a simple hashing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' This paper aims to provide alternative solutions in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Daniel Brown [5] presented an attack on the early versions of our algorithm [26, v:20221116:142416], which relies on the computation of the characteristic polynomial of the public elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' This approach seems impractical when applied to real-world parameters but led to an updated version using ��� field operations [26, v: 20221123:145424].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' As this latter kind of attack would be susceptible to a potential Menezes-Wu attack [18] (a fact pointed out by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Virdia [29]), we reconsider here our first methodology as a usable and secure key-agreement protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The notation used in this work p: prime integer, Zp: set of non-negative residuals mod p, products in Zp (represented by dots), ||: concatenation, Det[A]: the determinant of matrix A, AT: transpose of matrix A, A(i, j): matrix component of the i-th row and j-th column, ∈����: random uniform selection in a closed interval, ⨁ : bitwise XOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Paper organization First, we present an overall description of the proposed algorithm and the corresponding protocol, the proof that Alice and Bob will derive a common key, security considerations, and finally some experimental results and a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Overall description The algorithm starts by choosing a prime p shared by Alice and Bob who generate two rectangular matrices each, the first one with more rows than columns and the second one with inverse dimensions, and t is the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' For each iteration and every entry, a random integer s ∈���� [(p-1)/2, p -1] is chosen as the module, employing the algorithm given in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Following this scheme, Alice calculates two matrices A1k and B1k in each cycle (k=1,…,t) and computes Uk utilizing the matrix product Uk = A1k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1k (mod p) (k=1,2,3,…,t) The vector U=(U1, U2, U3, …, Ut) is sent to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Analogously Bob computes Vk = A2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2k (mod p) (k=1,2,3,…,t) and the vector V=(V1, V2, V3,…, Vt) is sent to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' We prove that: A-KEYk = Det[A1kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1kT] (mod p) and B-KEYk = Det[A2kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2kT] (mod p) are equal in each k-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Finally, Alice computes the hashing of A-CONCAT = A-KEY1 || A-KEY2 ||… || A-KEYt, and Bob the hashing of B-CONCAT = B-KEY1 || B-KEY2 ||… || B-KEYt which are equal, and hence this is the shared key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices 3 We must observe this protocol is highly parameterizable since we can change the dimensions of the matrices, the number of cycles, the primes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The numerical results (see below) show a very complex shared key can be obtained in a fraction of a second using a standard processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' key exchange algorithm ALGORITHM 1: PQC multiKEP COMMENTS The key Exchange Algorithm (KEP) uses several cycles as defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' INPUT: see the initial configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' OUTPUT: shared session key of 512-bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' INITIAL CONFIGURATION (PUBLIC VALUES): p: a shared prime number that can be obtained randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' rows[X|, columns[X]: dimensions of the matrices X:{A, B}, where rowsA=columnsB, columnsA=rowsB and rowsA > columnsA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' rowsA is a value whose maximum is a predefined rowmax value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Our proposal is rowmax=100, rowsA ∈���� [5, rowmax] and columnsA ∈���� [4, rowsA-1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' t: number of iterations H( ): hashing SHA3-512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' ALICE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for k=1 to t 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for i=1 to rowsA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for j=1 to columnsA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A1k (i,j) ∈���� [(p-1)/2, p -1] 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next j 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next i 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for i=1 to rowsB 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for j=1 to columnsB 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1k (i,j) ∈���� [(p-1)/2, p -1] 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next j 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next i 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Uk = A1k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1k (mod p) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next k 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Send the vector U = (U1, … , Ut) to Bob BOB 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for k=1 to t 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for i=1 to rowsA 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for j=1 to columnsA 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A2k (i,j) ∈���� [(p-1)/2, p -1] 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next j 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next i 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for i=1 to rowsB 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for j=1 to columnsB 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2 k (i,j) ∈���� [(p-1)/2, p -1] 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next j 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next i 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Vk = A2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2k (mod p) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next k 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Send the vector V = (V1, … ,Vt) to Alice SESSION KEY OBTAINED BY ALICE 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for k=1 to t 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A-KEYk = Det[A1kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1kT] (mod p) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next k 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A-CONCAT = A-KEY1 || A-KEY2 ||… || A-KEYt 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' KEYalice =H( A-CONCAT ) SESSION KEY OBTAINED BY BOB 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for k=1 to t 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B-KEYk = Det[A2kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2kT] (mod p) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next k 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B-CONCAT = B-KEY1 || B-KEY2 ||… || B-KEYt 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' KEYbob =H( B-CONCAT ) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' keys equality proof Lemma 1: The keys given by Algorithm 1 are equal, that is KEYalice = KEYbob Proof: it is very simple taking into account the elementary properties det(X)=det(��), det(XY)=det(X)det(Y), (��)� = ���� where X, Y are square matrices of the same dimension We have to prove that for every k (all operations (mod p) ) Det[A1kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Vk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1kT)= Det[A2kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2kT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Since Vk = A2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2k and Uk = A1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1k the keys can be written as follows: KEYalice= Det[A1kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1kT) = Det[A1kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1kT) = Det[(A2kT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='A1k)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' (B1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='B2k)T] and KEYbob= Det[A2kT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2kT]= Det[A2kT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A1k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2kT] ∎ A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Derived cipher algorithm ALGORITHM 2: PQC multiKEP + simple hashing cipher Observation: Vector U was received by Bob Insert here algorithm 1 (up to line 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=') BOB CIPHERS A MESSAGE TO ALICE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' msg = 512-bit message from Bob 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for k=1 to t 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B-KEYk = Det[A2kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2kT] (mod p) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next k 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B-CONCAT = B-KEY1 || B-KEY2 ||… || B-KEYt HASHING CIPHER (C, D): 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' C = V (* see Algorithm 1 *) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' D = H[ B-CONCAT ] ⨁ msg 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Send (C, D) to Alice ALICE RECOVERS THE MESSAGE FROM BOB 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for k=1 to t 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A-KEYk = Det[A1kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Vk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1kT] (mod p) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' next k 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A-CONCAT = A-KEY1 || A-KEY2 ||… || A-KEYt 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' msg =H[ A-CONCAT ] ⨁ D The above algorithms can be enhanced by combining modular operations in Zp with polynomial multiplications in a finite field like ��� as in [26], see Discussion below [13, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A convenient choice would be m=8 and any of the 30 primitive polynomials of that order [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' KEP and HASHING cipher protocols It is necessary to define a protocol allowing for an interchange of information between Alice and Bob asynchronously to achieve the following objectives: deferred communications check the integrity of the exchanged information mutual authentication to avoid attacks from active adversaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' man in the middle) block replay attacks availability of the exchanged information perfectly defined formats The following protocol aims at fulfilling these requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' PROTOCOL: KEP AND CIPHER PUBLIC DATA EXCHANGES INPUT: any kind of data to be exchanged between entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' OUTPUT: encapsulated message (msg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' INITIAL CONFIGURATION: msg: any kind of information to be exchanged between entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Universal-Keyed Message Authentication Code (UMAC): here proposed to assure strong symmetric authentication [3, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' ID: any elsewhere predefined and sender-receiver shared identification Tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' K: sender-receiver shared key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Tag: a smart label that can store any sort of information from identification numbers as a brief description for each entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Here the tag is Tag = HMAC- SHA3-512 (HM ∥ Nonce).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' See Fig 1 and more in [3] HM : NHK(msg1) ∥ NHK(msg2) ∥ · · · ∥ NHK(msgr) ∥ Len, see NH definition in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Nonce: pseudorandom and unique number that changes with each generated tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Timestamp: formatted date and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' MESSAGE AUTHENTICATION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Acquire K and msg 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Define a fixed-length Nonce 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Generate UMAC/SHA3-512/ID/Data 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Encapsulate the concatenation of: [ msg || UMAC/SHA3-512 (HM || Nonce) || Timestamp] into a file 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Send the file and nonce to the receiver MESSAGE VALIDATION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Acquire at any time the sent file 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Recover msg and UMAC/SHA3-512 (HM || Nonce) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Verify integrity and sender’s identity using K, Nonce, and msg 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Accept or Dismiss msg according to the verification result 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Security against brute force attacks Lemma 2: The complexity of attacking Algorithm 1 by brute force is given by (ndim .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' q)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' t, where p is the shared prime , q= ��� � and ndim=rowsA x columnsA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices 7 Proof: To obtain the session key the attacker has to factorize matrices Uk = A1k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1k (mod p), k=1,…,t where the maximum value that can appear in every entry (i, j) is p-1, independently of ndim (see Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Thus, defining q= ��� � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' for each entry of each matrix, it is necessary to try q values, that is (ndim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='q)2 possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The attack is successful only if the factorization can be obtained for each k=1,…,t, something that can be easily avoided by choosing appropriate parameters ∎ Example: If p = 2147483647, ndim=100 x 90, t =10, then the complexity is ~ 274 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Security against analytical attacks It must be stated that Uk, Vk are always singular matrices, therefore blocking inversions and subsequent linear attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' This is a consequence of the fact that the rank of the product of two (rectangular) matrices must be less or equal to the minimum rank of each one, therefore the rank of the product Uk (or Vk) of dimension rows x rows (rows > cols) is less than rows, so its determinant is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The core of the function defining the session keys (lines 30 and 35 of Algorithm 1) comprises the product of four rectangular matrices, a public component (Uk,Vk) flanked by two private matrices (ANk, BNk, N={1,2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The determinant is distributive concerning this product, and rearranging the product to be the multiplication of two public elements (Uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Vk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' As mentioned before, it does not work over real-life parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' This fact forces the factorization of the (Uk,Vk) components, which is computationally hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The factorizations Uk = A1k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1k (mod p), k=1,…,t can be solved in Uk ∈ Rn×n , A1k ∈ Rn×m , B1k ∈ Rm×n (real realm), using the SVD (Singular Value Decomposition) if there are no restrictions regarding the nonnegativity of the factors, but if they are imposed as in our proposal, then Vavasis and references therein [28] proved that the problem is NP-hard in the continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' For the Boolean case, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Gillis [10] wrote: “…for exact factorizations, the rank-one problem is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' For higher ranks, Boolean factorization is equivalent to finding a rectangle covering of the matrix U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' This is equivalent to the so-called biclique problem (given a bipartite graph defined by U, find the smallest number of complete bipartite subgraphs that cover the graph) which is NP-complete as proved in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' (sic)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The Boolean NP-complete complexity was formally proven by Miettinen and Neumann [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Semantic security of algorithm 2 Here we provide an informal view of this aspect, based on concepts mostly derived from Bellare’s work [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' In a nutshell, semantical security measures the resistance of any encryption algorithm to attacks using chosen plaintext or ciphertext selected by the attacker, who has access to the encryption and decryption modules working as oracles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' without knowledge of the key selected for enciphering [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The semantic security term is strongly related to other definitions: the one-way functions and the non-malleability of ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Indistinguishability under chosen plaintext attack (represented as IND-CPA) is equivalent to the property of semantic security and is considered a basic requirement for most provably secure public-key cryptosystems [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' One-way refers to bidirectional functions that have a probabilistic-polynomial time algorithm that converts domains into codomains, but no such algorithm is known that inverts the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Non-malleability refers to the resistance to modify slightly the ciphertext to obtain meaningful recovered plaintext [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The next concept to define is the indistinguishability of different ciphertexts of two similar but different plaintexts, an attacker could not assign a ciphertext of one of them to any one of the plaintexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' This feature is generally presented as a game between a challenger (the algorithm defender) and an adversary (the algorithm attacker) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The challenger generates a key pair PK, SK (public key and secret key respectively), based on any security parameter k (which can be the key size in bits), and publishes PK to the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The challenger retains SK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Here we describe the adaptative version of the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The adversary may perform any number of encryptions, decryptions, or any other operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' (The adversary is a probabilistic polynomial Turing Machine) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Eventually, the adversary submits two distinct chosen plaintexts m0 and m1 to the challenger (of the same length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The challenger selects a bit b ∈ {0, 1} uniformly at random, and sends the challenge ciphertext C = E (PK, mb) back to the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The adversary is free to perform any number of additional computations or decryptions (except C, this step is the adaptative phase of the attack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Finally, its outputs in polynomial time a guess for the value of b [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The adversary wins the game if it guesses the bit b, and winning means the algorithm is not indistinguishable and secure, else the algorithm reaches the strongest available security level: IND-CCA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' (Indistinguishable chosen ciphertext adaptative attack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Formally, a cryptosystem is indistinguishable under an adaptative chosen ciphertext attack if no adversary can win the above game with probability p greater than 1/2+ ∈k, where ∈k ≤ 1/πK (πK arbitrary polynomial function) and ∈k is defined as a negligible function in the security parameter k [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' For Algorithm 2 we prove the IND-CPA security level and explain how it could be easily adapted to reach the IND-CCA2 security level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The use of the UMAC function [3] in our Protocol fills this need in such a way that the practical implementation of Algorithms 1 and 2 culminates with the desired provable-security level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' NIST PQC security level of algorithm 2 NIST bases its classification on the range of security strengths offered by the existing NIST standards in symmetric cryptography, which NIST expects to offer significant resistance to quantum cryptanalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' In particular, NIST defines a separate category for each of the following security requirements [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' As previously described in the brute-force attack section, reasonable parameters exceed largely Level-1 PQC security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A toy numerical example Shared parameters: prime p = 5303,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' rowA=3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' columnsA=2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='t=2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='ALICE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Iteration 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Alice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='A11 = 1123 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='341 14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='238 1041 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Alice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='B11= 1525 1019 1561 1561 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='716 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='862 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Alice ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='U1 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='1707 4410 5290 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='446 4372 4284 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='1009 4184 2883 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Iteration 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Alice A12 = 665 1338 622 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='38 505 1617 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Alice B12= 925 1412 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='598 364 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='463 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='409 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Alice U2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='4436 4695 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='978 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='549 4954 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='416 3406 3500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='BOB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Iteration 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Bob A21 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='802 2435 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='1206 3408 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='707 3723 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Bob B21= 1174 2805 1242 3110 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='814 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='550 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Bob V1 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='3083 5209 2014 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='3429 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='159 4847 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='4831 2322 3791 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Iteration 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Bob A22 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='656 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='1900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='107 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='611 1537 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Bob B22 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='2192 1270 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='845 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='820 1022 2194 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Bob V2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='893 3229 4815 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='4837 3429 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='117 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='1182 2858 1374 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Alice computes A-KEYk = Det[A1kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B1kT] (mod p) for k=1, 2 A-KEY1 = 3207 A-KEY2 = 2121 Alice’s key = 0c3322f92446b51e3372d2a7bd2b81265bb96f32fa38562e4c02414e3c73d85ca4b358363b8792461d4033c1d76 23589c0f6c07ab01e33b6a7294019e125c779 Bob computes B-KEYk = Det[A2kT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='Uk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B2kT] (mod p) for k=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 2 B-KEY1 = 3207 B-KEY2 = 2121 Bob’s key = 0c3322f92446b51e3372d2a7bd2b81265bb96f32fa38562e4c02414e3c73d85ca4b358363b8792461d4033c1d76 23589c0f6c07ab01e33b6a7294019e125c779 Example of the cipher algorithm Bob ciphers a message to Alice B-KEY = 32072121 PLAINTEXT msg (formatted as a 64-byte string with spaces appended on the right) = "This is a secret communication.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' CIPHERTEXT C = 3083 5209 2014 893 3229 4815 3429 159 4847 4837 3429 117 4831 2322 3791 1182 2858 1374 CIPHERTEXT D = (88,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='91,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='75,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='138,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='47,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='62,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='82,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='82,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='161,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='194,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='222,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='89,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='228,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='82,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='123,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='95,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='151,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='77,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='56,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='71,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='47,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='99,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='53,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='83,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='29,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='246,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='124,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 132,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='147,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='120,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='167,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='178,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='102,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='96,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='225,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='247,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='66,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='169,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='224,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='214,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='224,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='90,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='144,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='62,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='135,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='96,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='57,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='193,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 231,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='89) Alice recovers the message A-KEY = 32072121 RECOVERED msg = “This is a secret communication.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Numerical experiments Programmed in RUST OS: Windows 10 Pro (64 bits) Processor: Intel(R) Core(TM) i7-3770K CPU @ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='50GHz RAM: 8,00 GB The CPU times reported in the following Table 1 are the mean values of 10 runs for each combination of the variables, prime p = 231-1= 2147483647 (~31 bits) rowsA columnsA Cycles Complexity CPU time in milliseconds 5 4 10 ~272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='94 5 4 20 ~273 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='15 5 4 100 ~275 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='69 6 5 10 ~273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='99 6 5 20 ~274 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='39 6 5 100 ~276 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='95 20 19 10 ~280 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='92 20 19 20 ~281 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='45 20 19 100 ~284 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='47 100 99 10 ~289 755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='38 100 99 20 ~291 1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='54 100 99 100 ~293 7488.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='44 Table1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Algorithm 1 complexity and throughput time as a function of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Using 128 bits integers in RUST with prime p = 18446744073709551113 (~64 bits): rowsA columnsA Cycles Complexity CPU time in milliseconds 5 4 10 ~2138 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='29 6 5 10 ~2139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='98 20 19 10 ~2146 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='84 100 99 10 ~2156 3203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content='48 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Algorithm 1 complexity and throughput time as a function of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Discussion Algorithm 1 has been implemented in different computer languages and shows that extremely high complexity can be achieved in fractions of a second on a standard I7 processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' The fact that by modifying the input variables (number of rows, columns, primes, iterations) practically any security level can be easily obtained without resorting to multiple precision, leads to very fast implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Depending upon the computer architecture and software implementation, larger primes can be used for reaching higher complexity levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' A New Post-Quantum Key Agreement Protocol and Derived Cryptosystem Based on Rectangular Matrices 3 An enhancement is to use ��� polynomial fields [26], recurring to very fast field multiplications based on hard-coded discrete logarithm tables of any field generator base, on some steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' In such a way the lack of uniformity in the operations constitutes an additional barrier to possible algebraic attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' This proposal could be very easily implemented without downgrading performance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=" It is particularly important to use the UMAC function in the Protocol because it is similar to Merkle's trees for PQC digital signatures [2] and also plays the role of achieving maximal semantic security [1] and simultaneously strengthens its post-quantum character." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Note: as Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' [3] state "the security of UMAC is rigorously proven, in the sense of giving exact and quantitatively strong results which demonstrate an inability to forge UMAC-authenticated messages assuming an inability to break the underlying cryptographic primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' (sic)” 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Conclusions The algorithm presented in this paper is such that very high complexity can be reached using small primes, normal precision, and small rectangular matrices, leading to very fast computer implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Acknowledgments: to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Brown, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Szyld, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Liberti, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Gillis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Virdia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9AzT4oBgHgl3EQfnv2o/content/2301.01586v1.pdf'} +page_content=' Di Mauro, I.' 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100644 index 0000000000000000000000000000000000000000..aa9efa70f47e1a1792e146b089fb8df7f2980003 --- /dev/null +++ b/c9FRT4oBgHgl3EQfTTda/content/tmp_files/2301.13532v1.pdf.txt @@ -0,0 +1,1799 @@ +Population-wise Labeling of Sulcal Graphs using +Multi-graph Matching +R.Yadav⋆‡†, F.X. Dup´e†, S. Takerkart⋆, G. Auzias⋆ +⋆Institut de Neurosciences de la Timone UMR 7289, Aix-Marseille Universit´e, CNRS +‡Aix Marseille Universit´e, Institut Marseille Imaging, Marseille, France +†Laboratoire d’Informatique et Syst`emes UMR 7020, Aix-Marseille Universit´e, CNRS +February 1, 2023 +Abstract +Population-wise matching of the cortical fold is necessary to identify biomarkers +of neurological or psychiatric disorders. The difficulty comes from the massive inter- +individual variations in the morphology and spatial organization of the folds. This +task is challenging at both methodological and conceptual levels. In the widely used +registration-based techniques, these variations are considered as noise and the match- +ing of folds is only implicit. Alternative approaches are based on the extraction and +explicit identification of the cortical folds. In particular, representing cortical folding +patterns as graphs of sulcal basins – termed sulcal graphs – enables to formalize the +task as a graph-matching problem. In this paper, we propose to address the problem +of sulcal graph matching directly at the population level using multi-graph matching +techniques. First, we motivate the relevance of multi-graph matching framework in +this context. We then introduce a procedure to generate populations of artificial sulcal +graphs, which allows us benchmarking several state of the art multi-graph matching +methods. Our results on both artificial and real data demonstrate the effectiveness of +multi-graph matching techniques to obtain a population-wise consistent labeling of +cortical folds at the sulcal basins level. +Keywords: brain, sulcal graphs, multi-graph matching, sulcal pits, MRI. +1 +Introduction +1.1 +Quantitative comparison across brains is a crucial but open question +Comparing features extracted from brain MRI across individuals is necessary for estimat- +ing population statistics and ultimately discover markers of diseases. However, this task +presents several challenges at both the methodological and conceptual levels. Indeed, the +features extracted from two different individual brains are defined in two different mathe- +matical spaces. Comparing such features thus requires to address the methodological prob- +lem of transferring them into a common space. The task of transferring information from +one brain to another or to a common space consists in defining spatial correspondences +i +arXiv:2301.13532v1 [stat.ML] 31 Jan 2023 + +across these objects by compensating for their variations in their respective geometry. The +challenge lies in the massive inter-individual variations of the morphology of the brain +and in particular the geometry of the cortical surface, which make the identification of +such spatial correspondences an ill-posed problem. As a consequence, any solution to this +problem inevitably requires to introduce additional constraints based on assumptions on +the biological validity of the resulting spatial correspondences, which constitutes a chal- +lenge at the conceptual level. Indeed, the assumptions and constraints introduced in the +definition of the spatial correspondences actually influence the derived statistics measured +on the population of interest, and could thus be considered as a source of bias in the anal- +ysis Van Essen, Glasser, Dierker, Harwell, and Coalson (2012). +One widely used approach to tackle this problem – termed here as the registration- +based approach – consists in defining a mapping between each individual brain and an +atlas serving as the common space by estimating a spatial transformation. As pointed +above, the process of building the atlas and defining the associated projection operator +which minimizes the error induced by the transformation remains an open research ques- +tion. As a consequence, several registration techniques and atlases co-exist in the field, +and tools to enable comparison across atlases are then required (Devlin & Poldrack, 2007; +Van Essen & Dierker, 2007). The variety of atlases, projection mechanisms and descriptors +illustrate the ongoing exploration of putative biologically relevant features used to define +these correspondences across individuals. One of the most widely used registration-based +approach (Fischl, Sereno, Tootell, & Dale, 1999) defines a mapping between cortical sur- +faces by imposing the alignment of a combination of curvature and convexity features es- +timated from a 2D mesh representing the geometry of the cortex. The cortical surface of a +given subject is projected onto the atlas by matching its curvature and convexity, under the +assumption that aligning these features induces biologically relevant anatomo-functional +correspondences. In this process, as in any registration-based approach, variations across +individuals are considered as noise or confounding perturbations to be minimized, includ- +ing variations in the topology and number of folds (sulci). More generally, the registration- +based approach might be seen as an over simplification of the problem since potentially +relevant geometrical information is not taken into account. +Alternative approaches consist in characterizing the geometry and organization of the +cortical folds in each individual and then compare these features across the population. +1.2 +Characterizing cortical folding patterns using graphs +Several approaches have been proposed to characterize cortical folding patterns, such as +gyrification index, fractal dimension and curvature (Armstrong, Schleicher, Omran, Cur- +tis, & Zilles, 1995; Cachia et al., 2008; Im et al., 2006). Although these measures capture +relevant morphological features, they do not explicitly reflect the topology, i.e the spatial +relationships between sulci. Mangin et al. (2004) introduced an analysis framework based +on the automatic extraction and labeling of the sulci allowing to characterize their shape, +size and pattern in terms of e.g. sulcus area, depth and length. This representation of the +ii + +cortical geometry has been used for instance to characterize populations of healthy sub- +jects (Duchesnay et al., 2007), to quantify potential deviations from normal populations in +various conditions such as schizophrenia (Cachia et al., 2008) and autism spectrum disor- +der (Auzias et al., 2014), or to estimate the heritability of the folding patterns (Pizzagalli +et al., 2020). Pursuing on this line of research, the sulcal pits were introduced as a concept +allowing to decompose the sulci into smaller pieces and thus access finer scale geometrical +information. As described in details in Auzias, Brun, Deruelle, and Coulon (2015); Im et +al. (2010), each fold is divided into sulcal basins that are defined as concavities in the white +matter surface bounded by convex ridges, and the deepest point in each basin defines the +associated sulcal pit. More recently, Im et al. (2011); Takerkart, Auzias, Brun, and Coulon +(2017) represented the geometrical relationships between sulcal basins as a sulcal graph. A +sulcal graph is constructed by considering each sulcal basin (or associated pit) as a node, +while the edges connect only adjacent basins and thus represent their spatial organization. +Various geometrical features of a sulcal basin can then be attributed to graph nodes (such +as the depth of the pit, its 3d coordinates...), while the spatial organization of the basins is +encoded in the topology of the graph. Figure 1 illustrates this decomposition of the cortical +folds into sulcal basins allowing to represent this complex geometry as a sulcal graph. +Figure 1: Example of sulcal graphs from three individual brains, superimposed with the +underlying decomposition of the cortical surface in sulcal basins. Sulcal basins are shown +in different colours, and their corresponding node in the graph are represented as spherical +dots in the lower panel. The color of each node in the graph illustrates the value of a given +attribute such as for instance the area or depth of corresponding sulcal basin. +These sulcal graphs constitute particularly relevant representations because: 1) varia- +tions across individuals are preserved and are manifested as changes in both the topology +of the graph and the value of the attributes attached to the nodes and edges; 2) the de- +sign of tools for the quantitative characterization of these variations can benefit from the +extensive body of methods from the graph processing literature. +1.3 +Problem statement and contributions +In the present work, we focus on the task of matching together a set of sulcal graphs in or- +der to define biologically relevant correspondences across a population of subjects, under the +specific constraint of explicitly taking into account the variations in folding patterns. Before +iii + +moving to the formalization, we more precisely situate this problem with respect to the +conceptual question of defining correspondences across individuals, and with respect to +the methodological problem of graph matching. +1.3.1 +Unsupervised comparison and matching of sulcal graphs +Comparing brains using sulcal graphs is highly relevant because all the geometrical infor- +mation about the macroscopic cortical folding can be encoded into such graphs. However, +several challenges need to be addressed in this context: 1) the large inter-individual varia- +tions in brain anatomy induce complex variations across sulcal graphs, including in their +topology; 2) sulcal graphs can be contaminated by noise resulting from the imperfect seg- +mentation of the individual cortical surface and corresponding sulcal basins; 3) there is +no consensus on a nomenclature or atlas at the scale of sulcal basins covering the whole +brain, that is a prerequisite to tackle the matching problem as a supervised learning task. +Indeed, few studies investigated the matching of cortical folds across individuals as a su- +pervised task (Behnke et al., 2003; Borne, Rivi`ere, Mancip, & Mangin, 2020; Rivi`ere et al., +2002). All these works focused at the scale of sulci, i.e. considering large folds consist- +ing of several of our sulcal basins. To our knowledge, only Lyu et al. (2021) attempted +to tackle this problem at finer scale, probably because of the massive amount of efforts +needed to gather sufficient amount of manually labeled data Voorhies, Miller, Yao, Bunge, +and Weiner (2021). Indeed, ambiguities due to variations across individuals in the folding +patterns become overwhelming at finer scale than sulci. This is illustrated by the tedious +works advancing the definition of a fined-grained nomenclature of folds Sprung-Much +and Petrides (2020) and their relationship with underlying function Willbrand et al. (2022). +The lack of widely accepted fined-grained nomenclature is also blatant in the related field +of brain parcellation: more than 20 different fine-grained atlases co-exist (Eickhoff, Yeo, +& Genon, 2018), and even the most advanced multi-modal atlas (Glasser et al., 2016) was +validated only on a small portion of the cortex. +Matching sulcal graphs across individuals is thus a very challenging problem. Instead +of relying on the few existing labeled data-sets that clearly deserve further validation, we +decided to approach this question as an unsupervised learning task. +We now describe the few studies that have attempted to tackle the question of unsu- +pervised labeling of sulcal graphs. The first approach was proposed by Im et al. (2010) +and consisted in computing a map of the spatial density of sulcal pits across a population +of subjects. This density map was computed by accumulating the pits from the different +individuals in each vertex of an average surface after aligning the folds using a registra- +tion technique. A watershed algorithm was then applied to this density map in order to +separate the main clusters of sulcal pits, empirically defined as the regions of high density. +An arbitrary label was then associated to each cluster, hereby defining an ad-hoc labeling +of the pits across individuals, depending on the cluster to which they contributed in the +density map. This procedure implicitly defines a matching of sulcal pits and correspond- +ing basins across individuals. Exemplar applications of this method can be found in e.g. +iv + +Auzias et al. (2015); Im et al. (2010); Le Guen et al. (2017), with illustrations of density maps +and induced labeling for various populations. We refer in the following to this category of +methods as Auzias et al. since we used the open source implementation from this paper. +The main limitation of this approach is that the labeling is driven only by the coordinates +of the sulcal pit. +Kaltenmark et al. (2020) introduced an alternative procedure for labeling the sulcal +basins, hereby considering the geometry of the basin surrounding each sulcal pit in addi- +tion to its spatial location. We refer to this method as Kaltenmark et al. in the following. +The authors of (Kaltenmark et al., 2020) also raised the question of the consistency of the +labeling, a notion that we will develop further below. In this method, an explicit constraint +is imposed to restrict the labeling to only one node per subject for each label. In addition, +the nodes for which the labeling is ambiguous – i.e. for which several labels are equally +plausible – remain unlabelled, which is often denoted as partial matching in the literature on +graph processing. Importantly, the spatial relationships between adjacent sulcal basins and +pits are never taken into account in any of these methods, since the different pits/basins +from each subject are considered independently. In contrast, in the present work our aim +is to exploit the spatial organization of the adjacent basins stored in the sulcal graph rep- +resentation. +Few publications investigated the potential of graph matching in the context of sulcal +graphs. In Im et al. (2011), the spectral graph matching technique (Leordeanu & Hebert, +2005) was applied to a set of 48 monozygotic twins, comparing a pair at a time. This study +showed that the similarity of the sulcal graphs across pairs of twins are higher than for +unrelated pairs, demonstrating the genetic influence on sulcal patterns, and the relevance +of graph matching techniques in this context. This approach was used in follow-up papers +from the same group, e.g for comparing brain lobes in Morton et al. (2019) or for matching +individuals onto an atlas in Im et al. (2017). +In the work by Meng et al. (2018), a population of 677 neonates was analyzed based +on a sulcal graph comparison method similar to the one of Im et al. (2011). The authors +proposed to use different features of the sulcal pits such as the pit position, the pit depth, +the basin area, the basin boundary and the pit local connectivity to construct different +similarity matrices, one per feature, and merge them into a single one using a matrix fusion +technique (B. Wang et al., 2014). A clustering algorithm was then applied to the fused +similarity matrix to identify sub-populations of sulcal graphs, associated to specific folding +patterns in the central, cingulate and superior temporal regions. +Critically, all these previous studies relied only on pairwise graph matching techniques. +Comparing pairs of graphs independently, in the presence of noise and large inter-individual +variations, is clearly sub-optimal. +1.3.2 +Multi-graph matching: a relevant framework for population studies +Given the large variations across subjects and imperfect sulcal basins extraction, examining +jointly a group of sulcal graphs is key to reveal meaningful information not accessible by +v + +considering only pairs of subjects. This is the translation to sulcal graphs of the basic +idea behind general population studies, that allowed researchers to uncover some of the +mechanisms underlying the anatomo-functional organization of the brain. We follow this +principle by investigating for the first time the potential of multi-graph matching techniques +in the context of sulcal graphs. By considering several brains together, the geometrical +information that is shared by the majority of individuals should help to regularize the +matching problem and allow to identify putative noisy graph nodes in a more robust way +than with pairwise matching. The multi-graph matching framework has the potential to +uncover population-wise invariant patterns in sulcal graphs without imposing a priori, +potentially biasing, assumptions. +1.3.3 +Contributions +In our previous work (Buskulic, Dup´e, Takerkart, & Auzias, 2021), we introduced a frame- +work to generate a set of synthetic sulcal graphs representative of a population, and used it +to benchmark state of the art pairwise matching techniques in the context of sulcal graphs. +In Yadav, Dup´e, Takerkart, and Auzias (2022), we provided a proof of concept of the rele- +vance of multi-graph matching techniques in this context. In the present study, we extend +these preliminary studies in several directions. +First, we introduce an improved simulation framework to generate populations of arti- +ficial sulcal graphs and demonstrate their biological plausibility through a quantitative +comparison with real data. Secondly, we benchmark a selection of recently published +multi-graph matching techniques against the best pairwise technique for this task (iden- +tified in from Buskulic et al. (2021)), and report variations in performances that would +clearly impact potential real-world applications, e.g in a clinical context. Finally, we com- +pare qualitatively and quantitatively the different graph matching techniques, as well as +the previously published approaches Auzias et al. and Kaltenmark et al, on a real data- +set of 137 subjects. In addition, our experiments demonstrate the feasibility of comparing +a large population of sulcal graphs based on multi-graph matching techniques, in fully +acceptable computing times. All the source code and data will be shared openly upon +publication at https://www.github.com/gauzias/sulcal graphs matching. +2 +Formal problem and state of the art +In this section, we define formally the problem of matching sulcal graphs, as well as the +multi-graph framework. We then give an overview of the different methods proposed in +the literature and provide a more detailed description of the multi-graph matching meth- +ods included in our experiments. +2.1 +Undirected attributed Sulcal graphs +We consider a population of N sulcal graphs, noted G1 ... GN, representing the cortical +folding pattern of an hemisphere from N different individuals. The sulcal graph from a +vi + +given subject q is an undirected attributed graph formally defined as a quadruplet Gq = +(Vq, Eq, AV +q , AE +q ), where Vq = {v1, v2, . . . , vnq} are the nodes in the graph and |Gq| = nq is +the number of nodes. Eq ⊆ Vq × Vq defines the set of eq edges. AV +q = {aV +v1, aV +v2, . . . , aV +vnq } is +the set of attributes associated to each node in Vq, and AE +q = {aE +e1, aE +e2, . . . , aE +eeq } is the set +of attributes associated with each edge in Eq. Note that the number of nodes nq and edges +eq and corresponding attributes varies across graphs. As illustrated on Fig.2, the sulcal +graph from each subject is then mapped onto the same common spherical domain using +the surface inflation and registration tools from freesurfer v.5.1.0 (https://surfer.nmr +.mgh.harvard.edu/, see Fischl et al. (1999) for details). The matching is computed in +this common spherical domain. In this work, we consider as attributes of the nodes the +3D coordinates of the sulcal pits on the sphere. Regarding the attributes of the edges, we +compute the length of the edge on the sphere as an approximation of the geodesic distance +between neighboring pits. +Figure 2: The sulcal graph from each subject is transferred onto a common sphere using +the inflation and spherical registration tools from freesurfer. The sulcal graphs from every +subjects can then be mapped onto either the common sphere or onto an average surface for +visualization. Note that the spatial dispersion of the nodes of the graphs on the common +spaces is heterogeneous, with dense clusters in cortical regions where the variations across +individuals are lower. +2.2 +Generalities and overview of pairwise graph matching methods +Pairwise graph matching refers to the problem of finding correspondences between the +nodes of two graphs G1 and G2. This problem is usually divided into two categories: exact +and partial matching. Exact matching methods consider graph matching to be a special +case of the graph isomorphism problem. It aims at finding the bijection between two +graphs, which implies that both the nodes and edges of the different graphs are strictly +matched. This requirement is too strict for most real-world tasks and in particular in our +context where the number of nodes and edges varies across graphs. Therefore, we focus on +the partial matching problem. This problem can be formulated as a Quadratic Assignment +Problem (QAP) (Loiola, Silva, & Galati, 2007). Although different forms of QAP exist, +the vast majority of the literature has focused on Lawler’s QAP (Lawler, 1963). Given +two graphs G1 and G2 with number of nodes |G1| = n1 and |G2| = n2 respectively, the +Lawler’s QAP consists in searching for the assignment matrix X12 ∈ {0, 1}n1×n2 such that +X12[i, j] = 1 indicates that υi ∈ V1 corresponds to υj ∈ V2 and X12[i, j] = 0 otherwise, +vii + +Common space +sphere +average surfaceresulting from the following optimization problem: +max J(X12) = vec(X12)⊤Φ12 vec(X12) , +(1) +subject to X121n2 = 1n1, X⊤ +121n1 ≤ 1n2, X12 ∈ {0, 1}n1×n2 , +where vec(X12) denotes the column wise vectorization of X12; 1n1 and 1n2 denote the +column vectors of all ones of size n1 and n2; and Φ12 ∈ [0, 1]n1n2×n1n2 is the affinity matrix +that is given as an input. The diagonal entries of Φ12 encode the similarity across nodes +whereas non-diagonal entries encode the similarity across edges between the two graphs. +The computation of the affinity matrix is context-dependent, and we detail the approach +used in the present work in section 4.1. +The computation and storage in memory of the very large matrix Φ12 impedes the +scalability of the matching problem based on this formulation. A solution to tackle this +limitation is to reformulate the matching as a Koopmans-Beckmann’s problem (F. Zhou & +De la Torre, 2015) that is a special case of Lawler’s QAP: +max J(X12) = tr(Ψ⊤ +12X12) + tr(A1X12A2X⊤ +12) , +(2) +subject to X121n2 = 1n1, X⊤ +121n1 ≤ 1n2, X12 ∈ {0, 1}n1×n2 , +where Ψ12 ∈ [0, 1]n1×n2 denotes the affinity matrix across nodes, and A1 ∈ Rn1×n1 and +A2 ∈ Rn2×n2 are the weighted adjacency matrices of two graphs respectively such that +A[i, j] = wij if edge (vi, vj) exists with weight wij and A[i, j] = 0 otherwise. Koopmans- +Beckmann’s formulation is a special case of Lawler’s where the edges can only be weighted +by a scalar value (i.e. cannot support a vector of attributes on edges). Under this constraint, +we can decompose the large matrix Φ12 into three smaller matrices Ψ12, A1 and A2, which +provides better scalability than Lawler’s QAP. +These two formulations are combinatorial QAPs and are known to be NP-hard prob- +lems. Most methods therefore relax the hard constraints given in Eq.(1) and (2) and provide +approximate solutions. Various approaches have been proposed to relax these problems, +leading to a variety of graph matching methods. Discussing these methods is beyond the +scope of this work but we refer interested readers to the review Yan, Yin, et al. (2016). +Going back to our specific context, we reported in Buskulic et al. (2021) a benchmark +of the pairwise methods SMAC (Spectral Matching with Affine Constraints) (Cour, Srini- +vasan, & Shi, 2007), IPFP (Integer Projected Fixed Point algorithm) (Leordeanu, Hebert, & +Sukthankar, 2009), RRWM (Reweighted Random Walks for graph Matching) (Hutchison et +al., 2010), and KerGM (Kernelized Graph Matching) (Zhang, Xiang, Wu, Xue, & Nehorai, +2019). We observed that KerGM clearly outperforms the others in our context. Conceptu- +ally, KerGM well suits sulcal graphs as it relies on Frank-Wolfe optimization that allows +to follow an optimisation path that respects the constraint on each step. This induces a +robustness to the presence of noise in graphs that is crucial in our context. In the present +work, KerGM is included in our benchmark as a representative of pairwise approaches. It +viii + +is also used to define the initialization of all the multi-graph methods that are introduced +in next section. +2.3 +The multi-graph matching problem +We now focus on the problem of jointly matching a population of N graphs {G1, . . . , GN}, +starting from pairwise assignment matrices Xij between graphs Gi and Gj (computed with +KerGM in this work). The key concept behind multi-graph matching is the cycle consis- +tency. This concept states that a matching between two graphs Gi and Gj should be the +same if we go through an intermediate graph Gk to create a new mapping. Formally, a +perfectly consistent, bijective mapping (every node is matched to one and only one other +node) would satisfy : +Xik = XijXjk , +(3) +for any i, j and k with i ̸= j ̸= k. A common way to estimate consistency at the population +level is to compute the full bulk assignment matrix X ∈ {0, 1}m×m with m = �N +q=1 |Gq|, +that is obtained by assembling all individual pairwise matrices: +X = +� +������ +X11 +X12 +· · · +X1N +X21 +X22 +· · · +X2N +... +... +... +... +XN1 +XN2 +· · · +XNN +� +������ +Intuitively, enforcing the consistency constraint will induce a reduction of the rank of this +bulk matrix. Multi-graph matching techniques can be divided into three categories as +follows. +The first category of approaches explicitly aim at minimizing the rank of the bulk +matrix using various approaches (Chen, Guibas, & Huang, 2014; Hu, Huang, Thibert, & +Guibas, 2018; Pachauri, Kondor, & Singh, 2013; Q. Wang, Zhou, & Daniilidis, 2018). For +instance, (Bernard, Thunberg, Swoboda, & Theobalt, 2019) solves a global optimization +problem by using a projected power iterative method, and we detailed further (X. Zhou, +Zhu, & Daniilidis, 2015). +The second category of techniques does not explicitly minimize the rank of the bulk +matrix but rely on other types of formalization aiming at increasing the consistency across +all graphs (Yan, Cho, Zha, Yang, & Chu, 2016; Yan, Cho, et al., 2016; Yan et al., 2014, 2013; +Yan, Wang, Zha, Yang, & Chu, 2015). +Finally, the third category corresponds to deep learning approaches that show promis- +ing performances in supervised tasks compared to previous methods, but are not suited +for unsupervised tasks (Rol´ınek et al., 2020; R. Wang, Yan, & Yang, 2019, 2020a, 2020b, 2021; +Yu, Wang, Yan, & Li, 2019, 2021; Zanfir & Sminchisescu, 2018). +Some other interesting methods exploit the concept of consistency in order to solve +the problem of jointly matching multiple images (Faktor & Irani, 2013; Rubinstein, Joulin, +ix + +Kopf, & Liu, 2013; Tron, Zhou, Esteves, & Daniilidis, 2017; T. Zhou, Jae Lee, Yu, & Efros, +2015). However, these do not take into account the connectivity of the graphs. +2.4 +Selection of the methods included in our benchmark +We used the following criteria to select the methods included in our benchmark: (i) Avail- +ability of code. We included only methods for which the authors have made their code +openly available in order to avoid reimplementation issues and to ensure the full repro- +ducibility of our results. (ii) Methods exploiting graph topology. We selected the methods that +take into account the topology of the graph, which is crucial to exploit the spatial adja- +cency information encoded in the sulcal graphs. (iii) Scalability. Since we are interested in +performing population studies over large sets of individuals, we excluded methods that +do not provide acceptable scalability. (iv) Unsupervised methods. Finally, as motivated in +the introduction, we focus on unsupervised methods in the present study. +The method that satisfy these selection criteria are mALS (X. Zhou et al., 2015), mSync +(Pachauri et al., 2013) and CAO (Yan, Cho, et al., 2016). We provide a detailed description +of each of these methods below. In our experiments, these multigraph graph-matching +techniques will be compared with the pariwise approach KerGM, and with the two meth- +ods from the literature specifically designed for labeling sulcal graphs already described +in Sec. 1.3.1: Auzias et al. (Auzias et al., 2015) and Kaltenmark et al. (Kaltenmark et al., +2020). +2.5 +Description of the selected multi-graph matching methods +As described in section 2.3, the general objective of multi-graph matching methods is to +match the nodes across several graphs together by enforcing consistency. +The authors of CAO (Yan, Cho, et al., 2016) propose to maximize the affinity infor- +mation and impose consistency at the same time instead of considering them separately. +They assume that enforcing consistency acts as a regularizer in the affinity objective func- +tion, particularly when the matching is ambiguous due to noise. The approach is based +on the search of an intermediate graph Gq that allows to optimize the affinity score while +progressively inducing consistency. They introduce the unitary consistency across a set of +N pairwise matching solutions X for a graph Gq as: +Cu(Gq, X) = 1 − +�N−1 +i=1 +�N +j=i+1 +��Xij − XiqXqj +�� +F /2 +nqN(N − 1)/2 +, +(4) +where∥.∥F is the Frobenius norm. The authors propose several approaches to balance be- +tween consistency and affinity, leading to different variants of CAO. In particular, their +best algorithm is able to elicit outlier nodes during the optimization, which is highly rel- +evant in our context. However, the use of affinity information along with consistency +and outlier elicitation increase the computational complexity of the method to O(N4). As +a consequence, only the least resource-demanding algorithm CAOcst did scale with the +x + +memory requirements imposed by the size of our graphs and number of subjects in our +populations. We thus refer to that particular version in the rest of this article. This version +of CAO enforces consistency through Eq.4, but ignores the affinity information. +The approach mSync (Pachauri et al., 2013) consists in estimating a mapping of each +Xij to a common universe of assignment matrices, of size d: +max +{Ui,Uj}∈P +N +� +i=1 +N +� +j=1 +⟨UiUj, Xij⟩ , +(5) +with P = {U ∈ {0, 1}nq×d | U1d = 1nq}. +(6) +Since solving eq.5 is intractable in most applications, the authors relax the problem into +a generalized Rayleigh problem. They further propose to use a reference graph in order to +estimate the mapping to the universe. In the implementation provided by the authors, the +first graph in the collection G1 is selected as the reference graph. +In mALS X. Zhou et al. (2015), the authors formalize the multi-graph matching as the +following low rank matrix recovery problem: +f(X) = − +N +� +i=1 +N +� +j=1 +⟨Ψij, Xij⟩ + α⟨1, X⟩ + λ∥X∥∗ , += −⟨K − α1, X⟩ + λ∥X∥∗ , +(7) +where, ⟨., .⟩ is the inner product, α controls the weight on sparsity, and K = {Ψij}N +i,j=1 is +the set of affinity matrices given as input. The cycle consistency is induced by the nuclear +norm ∥X∥∗ that controls for the rank of X while ⟨1, X⟩ favors bijective matchings across +graphs. Importantly, X is treated as a real matrix such that X ∈ [0, 1] The matrix is binarized +at the end of the optimization process using a threshold value t that is set by default as +to t = 0.5. In, addition, the authors leverage the work by Hastie, Mazumder, Lee, and +Zadeh (2015) and Cabral, De la Torre, Costeira, and Bernardino (2013) for decomposing X +which allows to solve the problem in a lower dimension space using the ADMM method +(Eckstein & Bertsekas, 1992). +3 +Generation of a population of synthetic sulcal graphs +A primary objective of our work is to investigate and evaluate different multi-graph match- +ing techniques in the context of sulcal graphs. However, as mentioned in the introduction, +there is no ground truth matching available for such graphs. We tackle this problem by +designing a procedure allowing to generate a population of artificial sulcal graphs with +correspondences defined by construction. Such populations of artificial graphs will con- +stitute a ground truth against which the different matching methods can then be bench- +marked. Generating artificial sulcal graphs for the purpose of a benchmark study induces +the two following constraints: 1) The artificial graphs should be biologically plausible, i.e. +xi + +they should respect as much as possible the intrinsic properties of a population of real +sulcal graphs. 2) The generation of the artificial graphs should be as simple and straight- +forward as possible in order to facilitate the comparison of the performances obtained in +the benchmark study and the interpretation of the differences, i.e. the generation proce- +dure should rely only on a limited number of parameters, and potential biases should be +avoided. As detailed below, these two contradictory constraints are balanced in the design +of our generation procedure. +The procedure is summarized in Algo. 1 and consists in two main steps. First, we +generate a set of points on the common spherical domain, that will serve as reference nodes. +Then, we impose several types of perturbations to this set of reference nodes in order +to generate a corresponding population of artificial sulcal graphs, while preserving the +correspondences across graphs, i.e. the ground-truth matching. Such procedure provides +the ground truth matching across the population, while controlling for the nature and +amount of variations across artificial sulcal graphs (corresponding to different subjects in +real data). +Algorithm 1 Procedure to generate a population of artificial sulcal grahs +Require: N, nref, κ, µpert, σpert, p +Step1: create reference nodes +▷ See Sec.3.1 +for j = 1..10000 do +Sample nref points on the sphere +Compute the minimum geodesic distance +end for +Choose the set of points with the largest min distance. +Step 2: generate a population of sulcal graphs +▷ See Sec.3.2 +for i = 1..N do +Perturb location of the reference nodes +▷ See Sec.3.2.1 +Add outliers and suppress some nodes +▷ See Sec.3.2.2 +Compute the edges of the graph +▷ See Sec.3.2.3 +end for +3.1 +Generation of a set of reference nodes +The first step consists in generating a set of reference nodes on the spherical domain while +controlling for two specific distinct parameters : the number of nodes noted nref, that is +typically set to match the average number of nodes across a real population, and the mini- +mum distance between the nodes. Indeed, the nodes of the real sulcal graphs cannot be closer +to each other than a minimum distance since they correspond to depth maxima that are +not located in the immediate proximity of the boundary of sulcal basins (see Auzias et al. +(2015) for further description of the extraction of sulcal pits and basins). As a consequence +the spatial distribution of the nodes on the sphere cannot be fully random. In order to +generate this set of nref points on a sphere with pseudo-random spatial distribution, we +adopted a simple brute force approach: we sample a set of nref points over the surface of +the sphere 10000 times; and we select the set that has the largest minimum geodesic dis- +xii + +tance between neighbouring points. As we will show in sec.4.3.1, 10000 times is sufficient +to get a set of reference nodes with a minimum distance between points that is realistic. +Technically, the uniform sampling of points on the sphere is achieved by generating ran- +dom rotations of the unit vector as described in Blaser, Fryzlewicz, Blaser, and Fryzlewicz +(2016); Lef`evre et al. (2018). +At this stage, we have defined on purpose a set of reference nodes that matches a real +population in terms of size and of minimal distance between nodes. The next step consists +in perturbing the reference nodes in order to generate the population of synthetic sulcal +graphs. +3.2 +Generation of an individual sulcal graphs +We now add perturbations of different natures to this set of reference nodes in order +to obtain a population of artificial sulcal graphs, that corresponds to different subjects. +These perturbations aim at mimicking the inter-individual variations that are observed in +a healthy population, by affecting the features of the nodes and edges, but also the topol- +ogy of the graphs. In order to generate a population of N artificial sulcal graphs, these +operations are repeated N times independently. +3.2.1 +Perturbation of the location of the reference nodes +The first step consists in adding random noise to the coordinates of the reference nodes on +the sphere, in order to model the inter-individual variability that exists in the location of +the sulcal pits. We used the von Mises-Fisher (vMF) distribution that is an approximation +of Gaussian distribution on a sphere (Von Mises, 1964). The two parameters of the vMF +distribution µ and κ can be seen as the equivalent of the mean and of the inverse of the +standard deviation (κ ∝ 1/σ) for a Gaussian distribution. Therefore, we iterate across the +reference nodes, and for each reference node, we produce a noisy one by sampling from +the distribution vMF(µ, κ), where µ is the coordinates of this reference node. We control +for the amount of noise on the coordinates of the perturbed nodes through the value of +the parameter κ, that is common to all nodes from the reference set. Smaller values for +κ will induce larger variations across the artificial sulcal graphs within the population. +Importantly, note that since we perturb each node of the reference set independently, we +keep the correspondence between each noisy node and its reference node, which will allow +defining our ground truth matching at the population level. +3.2.2 +Addition of outliers and suppression of nodes +Next, we simulate the inter-individual variations in the number of nodes across the sulcal +graphs, which is of crucial importance for generating biologically plausible artificial pop- +ulations. The aim is to model both false positive and false negative matchings, i.e. respec- +tively nodes that are present in the reference set but not in a given graph, and nodes that +are present in the graph but not in the reference set. This is achieved by randomly adding +xiii + +a certain number no of nodes on top of the perturbed nodes – hereafter called outlier nodes, +and by deleting ns nodes amongst the perturbed nodes – hereafter called suppressed nodes. +In order to randomly draw no and ns, we use the β-binomial distribution B(ν, α, β), which +is a distribution of non-negative integers. The parameter ν denotes the size of the support +of the distribution, i.e the maximal value that can be sampled. The parameters α and β can +be set so that B(ν, α, β) approximates a Gaussian distribution. We describe the setting of +these parameters and precise their link with µ and σ of a Gaussian in Appendix A. Since +we want the average number of nodes across the population of perturbed graphs µsimu +to match the number of nodes in the reference set nref, we set µo = µs = µpert and also +σo = σs = σpert. This formulation allows us to control the standard deviation of the num- +ber of nodes across the population of artificial graphs with the two parameters µpert and +σpert. +3.2.3 +Construction of the edges +The last step consists in constructing each artificial sulcal graph with the sets of perturbed +nodes as follows. We first compute the three-dimensional convex hull of each set of per- +turbed nodes located on the sphere. This yields a triangulation where only neighboring +nodes on the sphere are connected, which is a simple way to simulate the region adjacency +graph that is constructed from the sulcal basins in the real data. However, the average +node degree in such triangulations is higher than for real sulcal graphs. Therefore, we +finally delete a small percentage p of the edges in these triangulations, in order to obtain +artificial graphs which match the average degree of real sulcal graphs. +Note that since the construction of the edges occurs after the previous perturbation +steps (perturbations of the location, addition of outlier nodes and suppression of nodes), +the resulting artificial sulcal graphs can show variations in their topology across individu- +als of a population, as we observe in real data, making them biologically-plausible in that +respect. +4 +Experiments and results +4.1 +Computation of the affinity matrices +As described in Sec.2.2, we initialize all the multigraph matching methods using the pair- +wise results obtain from KerGM, which relies on the formalization of Eq. 2. We thus need +to compute the affinity matrices Ψij, Ai, Aj that store the similarity between nodes and +edges across every pairs of graphs in the population. +In the present work, we compute these affinity matrices using Gaussian kernels applied +to the attributes. For two nodes v ∈ G and v′ ∈ G ′ the affinity value is computed using the +kernel defined as exp (−γV ���aV +v − aV +v′ +��� +2 +2) and for two edges e ∈ G and e′ ∈ G ′ the kernel +is defined as exp (−γE���aE +e − aE +e′ +��� +2 +2). To estimate appropriate values for γV and γE we +use a heuristic proposed in Takerkart et al. (2017) that consists in using a cross-validation +xiv + +scheme to compute the inverse of the median of the distribution across all possible pairs +of nodes/edges, independently for each attribute (3D coordinates on the sphere for the +nodes and the geodesic distance for the edges). +4.2 +Dummy nodes +Most graph matching methods assume a constant number of nodes across the graphs to +be matched, which is not the case in our case (both synthetic and real graphs). We use +the classical approach from the graph matching literature which consists in adding dummy +nodes to smaller graphs so that all the graphs get the same number of nodes as the largest +graph in the population. For each of these dummy nodes, we assign to 0 the correspond- +ing values in the node and edge affinity matrices. This makes the optimization problem +defined in Eq.2 independent from dummy nodes. +4.3 +Benchmark on synthetic sulcal graphs +4.3.1 +Description of synthetic data sets +We first tuned empirically the parameters to the values µpert = 12, σpert = 4 and p = 10% to +obtain variations in our synthetic graph populations that are in line with what is observed +in real data. The distribution for number of nodes in the real data population is 88.27±4.72 +likewise in our simulated population for a randomly chosen κ value the distribution for +number of nodes is 88.15 ± 4.45 for the selected value of µpert and σpert and is consistent +across all κ values across all trials. We further provide in Appendix B additional materials +showing the matching distributions between our simulated graphs and real data. +Furthermore, we varied the value of κ ∈ [100, 200, 400, 1000], which controls the amount +of variations across synthetic graphs within a population. Note that κ controls the spread +of nodes coordinates around the reference nodes, which in turn induces variations in the +topology and attributes of synthetic graphs. +For each value of κ, we generate 10 populations of N = 137 synthetic graphs (which +corresponds to the number of subjects in our real population; see below) and report the +average and standard deviation of the metrics described below. As illustrated on Fig.3, +our populations of synthetic graphs show variations that are qualitatively very close to +those observed across real graphs. +4.3.2 +Evaluation metrics for synthetic data sets +In order to evaluate the different matching methods on simulated graphs, we use the clas- +sical precision, recall and F1-score: +Precision = +True Positives +True Positives + False Positives ∈ [0, 1] +(8) +xv + +Figure 3: a) Real sulcal graphs from three randomly chosen individuals, and projected +on the average surface. b) Simulated graphs randomly chosen for κ = 1000, showing +the ground-truth correspondence across graphs in color. Nodes in black represent the +outlier nodes that have no correspondence. c) Illustration of the impact of κ on the spatial +dispersion of nodes: the nodes of six simulated graphs are shown on the average surface +for κ = 1000 (left) and , κ = 200 (right). The spread across the nodes for each cluster varies +according to κ, while outlier nodes in black have random locations. +Recall = +True Positives +True Positives + False Negatives ∈ [0, 1] +(9) +F1 = 2(precision × recall) +precision + recall ∈ [0, 1] +(10) +Thus, Precision is a ratio between the True positives(number of correct matches predicted +by the algorithms) and all the positives(number of matches by the algorithms). Whereas, Recall +is a ratio between True positives and True positives along with False negatives(number +of correct matches not predicted by the algorithms). Finally, the F1 score provides a balance +between Precision and Recall. A F1-score of 1 reflects the ability of the algorithm to obtain a +perfect matching of inlier nodes and accurate identification of outlier nodes. These metrics +are relevant in our context to detect matching with outliers alongside the incorrect matches. +xvi + +a) +b) +c)4.3.3 +Results on synthetic data sets +We report of Fig 4 the mean and standard deviation of Precision, Recall and F1-score, com- +puted across the 10 synthetic populations for each value of κ. +First, we find that two multi-graph matching methods, mALS and mSync, vastly and +consistently outperform KerGM, which has been identified as the best pairwise matching +method for this task in Buskulic et al. (2021). This confirms our main hypothesis: consid- +ering the matching problem on the whole population using multi-graph matching allows +an important gain in performance compared to only considering pairs of graphs. +Then, we observe a gradual decline in the performances of all methods as the noise +increases (decrease of κ), as expected. The performances of the multigraph approaches +mALS and mSync resist much more to this increase in variability than the pairwise ap- +proach. The performances of mSync are limited more specifically by the lower precision +at any level of noise. This suggests that the difference in performances between the two +methods are mainly due to the hard constraint on the consistency in mSync that seems too +restrictive. On the other hand, the recall indicates that mSync is more robust to increasing +noise than mALS, with very close value when κ = 100. However, mALS performs better +for lower noise values. Overall, mALS shows the best F1-score for every κ values, thanks +to a very high precision combined with very good recall. Indeed, the F1-score for mALS is +above 0.7 even for κ = 200 which corresponds to a configuration where the noise is quite +strong. +Finally, the performances of CAO are very low, even lower than the pairwise technique +KerGM. Such poor performances are likely a consequence of the optimization that consid- +ers only the consistency but ignores the affinity of nodes. As already mentioned in Sec.2.5, +the other versions of CAO proposed in Yan, Cho, et al. (2016) could show much higher +performances but did not scale with the size of our data. +Figure 4: F1-score, Precision and recall for κ ∈ [1000, 400, 200, 100]. For each method, we +plot the average across the 10 simulated populations as a line and the standard deviation +as the shaded region of the same color. +4.4 +Application to real data +4.4.1 +Preprocessing of real data +For the evaluation on real data, we use the sulcal graphs from 137 young healthy adults +taken from the publicly available database OASIS (Marcus et al., 2007). The preprocessing +xvii + +1.0 +1.0 +1.0 +0.9 +0.9 +0.9 +0.8 +0.8 +0.8 +0.7 +0.7 +0. +2 0.6 +2 0.4 +E0 +0.3 +0 +0.2 +KerGM + 0.2 +KerGM + 0.2 +KerGM +msync +msync +mALS +msync + 0.1 + 0.1 +mALS +0.1 +mALS +0.0 +CAO +1000 +400 +200 +100 +0.0 1 +1000 +400 +200 +100 +0.0 +kappa +1000 +400 +200 +100 +kappa +kappaof these data (brain tissues segmentation, mesh extraction and sulcal graphs construction) +has been detailed in Auzias et al. (2015); Takerkart et al. (2017). Across this population, the +number of nodes is 88 ± 4, with a maximum size of 101 nodes/pits. Dummy nodes are +thus added to all other graphs to get a constant size of 101, as explained above. +4.4.2 +Evaluation metrics used with real data +In absence of ground truth matching for real data, we cannot compute the same scores as +for the simulation experiments. We therefore combine a set of quantitative metrics with +some qualitative assessments, which we describe below. +Consistency +According to Yan, Cho, et al. (2016), we compute the node consistency as follows: Given +Gk ∈ {Gq}N +q=1 and the bulk matrix X, for node vk ∈ Gk, with index i(vk) ∈ {1, . . . , |Gk|}, its +consistency is defined by: +C(vk, X) = 1 − +�N−1 +i=1 +�N +j=i+1 ||Y(vk, :)||F /2 +N(N − 1)/2 +, ∈ (0, 1], +(11) +where || · ||F is the Frobenius norm, Y = Xkj − XkiXij and Y(vk, :) is the i(vk)-th row of matrix +Y. Note that it is different from Eq. 4 which estimates the consistency at the graph level. +This consistency measure is computed for each node of each graph, including dummy +nodes. A value of 1 corresponds to the ideal case where each graph only contains nodes +that have been matched in a consistent manner. This consistency measure cannot distin- +guish the matches of real nodes to dummy nodes from valid matches across real nodes. +For methods imposing an explicit constraint on the consistency, a value of 1 is expected +(and not informative), but for the other methods this measure is relevant and allows to +assess the spatial pattern of the consistency across clusters. +Qualitative and quantitative assessment of the labeling induced by the matching +In terms of potential applications of the graph matching to sulcal graphs, a major out- +come is the labeling of graph nodes that is induced. As already mentioned in the introduc- +tion, the assessment of the quality of the labeling and thus of the biological relevance of +the matching across individuals is an ill-posed problem. The first problem is to retrieve a +labeling from the assignment matrix resulting from the matching. In the case of a perfectly +consistent matching where each node of each graph would be matched to one and only one +node from every other graph in the population, the labeling would be trivial and would +consist in simply associating a label to each row or column of the assignment matrix. This +situation is however impossible since the number of nodes varies across individuals within +our population of interest. Therefore, in the present work we take the largest graph as a +reference, and we associate an arbitrary label to each of its nodes and then propagate these +labels to every other graphs based on the assignment matrix resulting from each method. +Once the labeling of the nodes is retrieved, the nodes that share the same label across +subjects are grouped together into what we will designate as clusters, that are different +depending on the matching method. We then compute the coordinates of the centroid of +xviii + +each cluster, which enables to evaluate qualitatively the spatial distribution of the different +clusters across the cortical surface. +This qualitative assessment is complemented with a quantitative measure of the com- +pactness of the clusters. For this, we compute the silhouette coefficient of each node from +each graph. As proposed in Rousseeuw (1987), the silhouette of a node corresponds to the +ratio between the average Euclidean distance to the other nodes in the cluster and its dis- +tance to other nearby clusters. Since the distances are computed on the spherical domain, +the use of Euclidean distance is sub-optimal but the errors induced are very low and inde- +pendent from the matching method. The silhouette coefficient of a cluster is then obtained +by averaging the silhouette values from corresponding nodes. +4.4.3 +Results on real data +We first report in Table 1 the quantitative measures that allow us to compare the different +techniques at the whole brain level: the number of clusters (thus of labels) obtained with +each method, the silhouette measure averaged across all nodes and graphs, the percent- +age of nodes remaining unlabeled, the consistency measure averaged across all nodes and +graphs, and the computing time. +Table 1: Quantitative measures computed at the whole brain scale. +Method +Num. +silhouette +Perc. +consistency +cpu time +clusters +unmatched +(min) +mALS +82 +0.55 ± 0.22 +28.4 +0.91 ± 0.08 +783 +Kaltenmark et al +94 +0.44 ± 0.23 +17.0 +1.0 ± 0.0 +∼ 180 +Auzias et al +104 +0.49 ± 0.18 +0 +0.82 ± 0.15 +∼ 30 +mSync +101 +0.08 ± 0.49 +0 +1.0 ± 0.0 +31 +CAO +101 +−0.12 ± 0.45 +0 +1.0 ± 0.0 +3255 +KerGM +101 +−0.04 ± 0.34 +0 +0.30 ± 0.17 +1362 +The number of clusters and percentage of unmatched nodes indicate that the two meth- +ods that allow partial matching mALS and Kaltenmark et al. result in a lower number +of clusters, suggesting that the ambiguous nodes remain unlabeled instead of enforcing +their matching into potentially unreliable clusters. The three methods mSync, CAO and +KerGM enforce the matching of every nodes, and result in a number of clusters equal +to the size of the largest graph in the set, i.e. 101. The method Auzias et al. results in +more clusters than the size of the largest graph, suggesting that some clusters correspond +to highly variable nodes that cannot be matched consistently across individuals. This is +confirmed by the consistency measure which is lower than for mALS. The consistency +of Auzias et al. is still much higher than the value of 0.30 obtained with the pairwise +technique KerGM. Note that the methods mSync and CAO explicitly enforce a perfect +consistency, but this is possible only when considering the dummy nodes as pointed in +section 4.4.2. Also note that the method Kaltenmark et al. also gets a perfect consistency. +This is a consequence of the explicit constraint imposed in this technique by allowing one +and only one node per subject to be matched for any given cluster. +xix + +The silhouette measures illustrate that a high consistency can be associated with a low +compactness of the clusters as e.g. for CAO and mSync that get values close to the one +of the pairwise technique KerGM. The methods Auzias et al. and Kaltenmark et al. get +much higher silhouette values which is expected since these techniques enforce the match- +ing of nodes based essentially on their spatial proximity on the surface. The silhouette +value of mALS is higher than these two techniques. Overall, mALS results in high sil- +houette and consistency values, at the cost of a high number of unmatched nodes (28.4%) +compared to Kaltenmark et al. and Auzias et al., indicating that this method was much +more conservative in the matching, leaving more ambiguous nodes unmatched. +We then illustrate the matching across nodes from the different graphs (subjects), ob- +tained for each method on Fig. 5. The number and location of the different centroids (larger +circles) is informative of the spatial distribution of the clusters of nodes across the cortical +surface, for each method. On the first column (mALS and Kaltenmark et al.) some nodes +remain unlabelled and are represented in black. The clusters seem more compact than for +the methods of the second column (Auzias et al. and mSync) that do not allow any node to +remain unlabeled. On the third column (CAO and kerGM) the matching looks noisy, with +clusters overlapping between eachother in almost every cortical location, which illustrates +the poor anatomical relevance of the matching. +Figure 5: Labeling and corresponding cluster centroids (larger circles) for each method. +Dots in black in the first column (for mALS and Kaltenmark et al.) correspond to un- +matched nodes. See text for further description. +For further evaluation of the performances of the different techniques, we show on +Fig. 6 the silhouette values of every nodes across all graphs as well as the centroids of +each cluster as a larger circle. The high silhouette values of the centroids for the methods +mALS, Auzias et al. and Kaltenmark et al. are visible with mostly red and orange cen- +troids. In contrast, we observe more centroids in green and blue for CAO and KerGM. +Together with Table 1, this figure illustrates the poor performances of pairwise matching +xx + +mALS +Auzias et al. +CAO +msync +kerGM +Kaltenmarket al.approach with high spatial dispersion of nodes corresponding to each cluster for KerGM, +associated to very low silhouette coefficients. The method mSync results in higher silhou- +ette coefficients for some nodes, but lower value for others (nodes and centroids in blue +on Fig. 6), indicating that the matching was enforced also for ambiguous nodes located in +highly variable regions. This is a consequence of the hard consistency constraint in mSync +imposing a matching that is consistent across all graphs by construction, even in highly +variable regions. For Auzias et al., we observe that the clusters are organized around re- +gions of high nodes density, but the nodes located relatively far from the centroids have a +lower silhouette value (nodes in green on Fig. 6). These observations are consistent with +the algorithm that is based on a watershed applied to the sulcal pits density map as de- +scribed in Sec.1.3.1. For both mSync and Auzias et al., we observe some clusters with low +silhouette value located close to each other, suggesting that the number of clusters is too +high. +The techniques mALS and Kaltenmark et al. result in much higher silhouette values, +which is expected since they do not force the matching of highly variable nodes that are +left unlabeled. The unlabeled nodes have a very low silhouette value (in violet on Fig. 6), +but since they do not belong to any cluster, this does not reduce the silhouette values of +clusters. Note that even for these methods, the clusters get closer with lower silhouette +values in highly variable regions such as the anterior frontal and occipital lobes. +Across the different methods, we observe that the clusters showing a higher silhou- +ette value relative to other clusters are located systematically in the same regions that are +known to be less variable across individuals, such as the central sulcus, and the insula. For +these clusters, the silhouette values are close across methods, confirming the lower ambi- +guity in the matching in these regions. In highly variable regions, the different methods +produce different matchings. For instance in the occipital lobe, the clusters produced by +Kaltenmark et al. show lower silhouette values compared to mALS, but we observe the +opposite effect in the anterior frontal lobe. +On Fig. 7, we show the consistency for every nodes and centroids, for the three methods +that do not explicitly enforce a perfect consistency. Clearly, the pairwise technique KerGM +results in inconsistent matching for every clusters, including the regions where the vari- +ations across individuals are known to be low (no centroid in green, even in the central +sulcus and the insula). For mALS and Auzias et al., we can observe the spatial variations +of the consistency across cortical regions. Again, higher consistency is obtained in less +variable regions (central sulcus, insula) for both techniques, and relatively lower values +are visible in the frontal and occipital regions. The consistency is higher for mALS than +Auzias et al. for every clusters. Note that the spatial pattern of the consistency measure for +mALS is anatomically relevant, with a consistent matching in the insula, the central and +pre-central regions, and less consistent in the peri-sylvian regions. At a more local scale, +we observe a cluster in the superior temporal sulcus that is more consistent than those lo- +cated anteriorly or posteriorly, which is in line with previous studies describing variations +and stabilities across individuals in this region Leroy et al. (2015). +xxi + +Figure 6: For each method, we show the silhouette coefficient of each node from every +graphs, as well as corresponding centroids as larger circles. Each centroid (larger circles) +is colored according to the average of the silhouette coefficient of corresponding nodes. +Figure 7: Node consistency computed for each node of each graph with respect to the +remaining graphs, and then averaged across graphs. We adapted the colorbar to visualize +the differences between the three methods, with the pariwise technique KerGM showing +much lower values. +5 +Discussion +In this work, we explored the potential of graph matching methods applied to a popula- +tion of sulcal graphs to uncover correspondences across individuals driven by the local +patterns of folds. Indeed, these graph matching methods take into account the charac- +teristics of individual sulcal basins as well as their topological organization to construct +the correspondences. Our results on both simulations and real data support the biological +relevance of the correspondences across individual resulting from multi-graph matching +techniques. +xxii + +mALS +Auzias et al. +CAO +Kaltenmarket al +kerGM +ms +ync0.7 +mALS +Auzias et al +kerGM5.1 +Relevance of simulated graphs relative to real data for evaluating matching +techniques +To overcome the lack of ground truth for real data, we proposed a procedure allowing to +generate artificial graphs that approximate the features of real sulcal graphs while control- +ling the variations across graphs. This simulation procedure enabled to benchmark various +pairwise and multi-graph matching techniques. The evaluation of the performances of the +different methods and their robustness to controlled variations in the simulated graphs +was informative for probing their effectiveness in this context. The performances of the +pairwise approach KerGM were limited even when the level of perturbations was mini- +mal. Note that we reported in Buskulic et al. (2021) that alternative pairwise techniques +perform even worse on this task. Amongst the different multi-graph matching techniques +that were tested, mALS showed better performances than the others in all conditions, and +a good robustness to increasing noise levels. These observations were confirmed by our +application on real data. Overall, our set of experiments confirmed the intuition that multi- +graph matching techniques are highly relevant in our context, while pairwise techniques +show limited performances and might thus be restricted to initialization purpose. +Of note, our aim was not to push the biological plausibility of our simulated graphs. +Keeping the simulations simple enables straightforward interpretation of the variations in +the performances across the different approaches. This trade-off is visible in the procedure +in particular when we sample the reference nodes uniformly on the sphere. Indeed, our +simulation procedure cannot produce realistic non-uniform spatial distribution of nodes +across the population. While this could be achieved by adapting the sampling of the refer- +ence points, this would induce variations in the performances of the matching techniques +depending on the location on the sphere, which in turn would make the comparison across +methods much more difficult. +Beyond the present work, our procedure for simulating sulcal graphs could be instru- +mental to assess future improvements in graph matching techniques. +5.2 +Considerations relative to deep-learning approaches and potential method- +ological improvements +As already mentioned in section 2.3, many other graph matching techniques can be found +in the literature but were not included in the present work. More specifically, deep learning +approaches outperform traditional approaches in supervised learning task (LeCun, Ben- +gio, & Hinton, 2015). Recent works such as e.g. (Scarselli, Gori, Tsoi, Hagenbuchner, & +Monfardini, 2009; Xu, Hu, Leskovec, & Jegelka, 2019) showed that the structural informa- +tion can be learnt by a Graph Neural Network(GNN), providing that manually labelled +ground-truth data is available. +In addition, the rise of semi-supervised learning approaches represents an opportu- +nity in the context of graphs with partial matching ground-truth. Such approaches are +worth considering in our context, since we observed marked variations across cortical re- +xxiii + +gions in the ambiguity of the matching. The work by Fey, Lenssen, Morris, Masci, and +Kriege (2020) considers a semi-supervised framework for handling the matching problem +where the ground-truth correspondence are only given for a small subset of nodes. In +addition, their approach imposes an explicit inductive bias to find correspondences across +graphs, based on neighbourhood consensus that does not allow adjacent nodes from being +mapped to different regions in other graphs. This is appealing in the case of sulcal graph +matching where we would like to enforce the matching of nodes located in some specific +regions more than in others. Such a framework could benefit from the recent work Lyu et +al. (2021) on context-aware data augmentation, which could be instrumental to overcome +the bottleneck of the lack of ground-truth labeling data. +Another avenue for potential gains in performance consists in improving the defini- +tion and integration of the attributes on nodes and edges. Many other geometrical fea- +tures could be considered to enrich the attributes on nodes, such as e.g. shape index and +curvedness Awate, Yushkevich, Song, Licht, and Gee (2010), or the local gyrification index +Rabiei, Richard, Coulon, and Lef`evre (2017). On the other hand, the literature on learning +edge representations is very scarce Hsu, Shen, and Cremers (2022), and the attributes on +edges are most often reduced to a scalar value (i.e. a simple weight). In particular, the +methods included in the present work cannot handle vectors of attributes on edges. Some +recent deep learning methods such as (R. Wang et al., 2021) can exploit such vectors of at- +tributes, but their scalability is limited by the size of the affinity matrices. We proposed in +Dup´e, Yadav, Auzias, and Takerkart (2022) to overcome this limitation by leveraging the +recent matrix factorization method from Zhang et al. (2019). We will further investigate +the potential of these methods in our future studies. +5.3 +Data-driven nomenclature of sulcal basins +The present work extends previously reported experimental results illustrating the major +impact of the labeling strategy on the induced correspondences and data-driven nomencla- +ture. In Kaltenmark et al. (2020), the number of clusters obtained at the group level varied +from 90 to 114 for the right hemisphere using either the approach proposed in Kaltenmark +et al. (2020) or Auzias et al. (2015) respectively, on the same population of subjects. We +provide a much more detailed comparison. We show on Fig. 8 the superimposition of the +centroids from different methods on the same average surface. This visualization shows +that the location of some of the centroids are very consistent across methods (indicated by +arrows), corresponding to cortical regions where variations across individuals are known +to be low, such as the central sulcus, the insula, the inferior precentral or superior tem- +poral sulcus. Other clusters differ across methods. The clusters indicated by squares are +those resulting from Auzias et al. and mSync (resp.) that do not match clusters from +Kaltenmark et al. (see also Table 1). These are located either in highly variable regions +such as the frontal lobe, or on the top of gyri such as the superior temporal gyrus and +the inferior frontal gyrus. While mALS and Kaltenmark et al. result in centroids that are +highly similar (crosses and rings are often superimposed on the panel on the left), the two +xxiv + +methods do not result in the same matching in highly variable regions such as the inferior +frontal and parietal regions (indicated by diamonds). In conjunction with our results on +synthetic (Sec.4.3.3) and real data (Sec.4.4.3), these observations confirm that the concep- +tual differences between the approaches yield different matchings and thus different cor- +respondences across individuals. Indeed, graph-matching techniques such as mALS are +able to take into account the topological information encoded in the graphs, i.e the spatial +organization of neighbouring folds, while Kaltenmark et al. relies only on the geometry +of sulcal basins, considering different folds separately. +Figure 8: Superimposition of the centroids from mALS, Auzias et al., and mSync shown +as crosses with those of Kaltenmark et al. shown as green rings. mSync is representative +of the methods kerGM and CAO that also result in 101 clusters. The arrows point to +centroids that are robust across methods. The squares indicate centroids corresponding to +small clusters located on gyri. Diamonds indicate centroids that differ between mALS and +Kaltenmark et al.. +The next step will be to assess the biological relevance of the induced correspondences +across subjects by visualizing the matching on the cortical surface of the individuals. Given +the variations across methods in the location of clusters observed on Fig.8, we expect to +observe important differences between the techniques at the individual level, especially +in highly variable regions such as the parietal lobe. More specifically, our expectation is +that graph matching techniques should allow solving potential anatomical ambiguities in +a much more relevant way than Auzias et al. and Kaltenmark et al., by exploiting the +topological information of the neighbouring folding pattern. +6 +Conclusion +In this study, we explored the potential of several graph matching methods chosen from +the literature to define population-wise correspondences across individual cortical geome- +tries. In the absence of a ground-truth labeling for real data, we first proposed a procedure +to generate simulated sulcal graphs that follow the intrinsic structure and properties of +real sulcal graph. We then compared the approaches on our simulated sulcal graphs with +ground-truth correspondences defined by construction. +We also evaluated the methods on real data. We computed the silhouette value of +each node of the graph that measures the degree of compactness of each cluster, giving +xxv + +mALS +Auzias et alus insights on the matching across graphs produced by the different methods. We also +computed a consistency measure that gave us an insight on the variability across the pop- +ulation for each cluster. The results obtained on real data were compared with two other +methods from the literature. +Overall, our experiments on both artificial and real data showed the high relevance +of multi-graph methods for sulcal graph matching. We observed that mALS and mSync +outperform CAO and the pairwise approach KerGM. While mALS proved to be very +robust to noise compared to other methods, the much lower complexity of mSync makes +it also a relevant candidate for further studies and extensions to larger populations. +7 +Funding sources +The data used in the preparation of this article were obtained from OASIS: Cross-Sectional: +Principal Investigators: D. Marcus, R, Buckner, J, Csernansky J. Morris; P50 AG05681, P01 +AG03991, P01 AG026276, R01 AG021910, P20 MH071616, U24 RR021382. The project lead- +ing to this publication has received funding from Excellence Initiative of Aix-Marseille +University - A*MIDEX, a french ”Investissements d’Avenir” programme (AMX-19-IET- +002). +The research leading to these results has also been supported by the ANR Sul- +calGRIDS Project, Grant ANR-19-CE45-0014 and the ERA-NET NEURON MULTI-FACT +Project, Grant ANR-21-NEU2-0005 funded by the French National Research Agency. +8 +Declaration of competing interest +The authors declare that they have no known competing financial interests or personal +relationships that could have appeared to influence the work reported in this paper. +9 +Author contributions +We report individual contributions to the paper using the relevant CRediT roles: +R.Yadav:Conceptualization; Data curation; Formal analysis; Investigation; Methodol- +ogy; Software; Visualization; Writing - original draft; Writing - review & editing. +F.-X.Dup´e: Conceptualization; Formal analysis; Funding acquisition; Investigation; +Methodology; Supervision; Writing - original draft; Writing - review & editing. +S.Takerkart: Conceptualization; Data curation; Formal analysis; Investigation; Method- +ology; Software; Supervision; Writing - original draft; Writing - review & editing. +G.Auzias: Conceptualization; Data curation; Formal analysis; Funding acquisition; In- +vestigation; Methodology; Project administration; Resources; Software; Supervision; Vali- +dation; Visualization; Writing - original draft; Writing - review & editing. +xxvi + +References +Armstrong, E., Schleicher, A., Omran, H., Curtis, M., & Zilles, K. 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In Proceedings of the ieee international conference on computer vision (pp. +4032–4040). +Appendices +A +Additional description of the β-binomial distribution +In the following section we provide additional description of the formulation of the β- +binomial distribution. +The β-binomial distribution is parameterized by ν, α and β. The parameter ν defines +the size of support (in our case, maximum number of no/ns). The setting of ν can impact +the skewness of the distribution but the shape will be Gaussian as long as the value is +sufficiently large. We show in figure A.1 the β-binomial distributions for different values +of ν, with α = 7.15 and β = 28.62. In this work, we set ν = 30 which is sufficient to get a +distribution close to Gaussian for all combinations of α and β parameters that are relevant +in our context. +Although α and β are not trivial to calibrate, they can be related to µ and σ of a Gaussian +xxxiii + +Figure A.1: Effect on β-binomial mass function for different values of ν fixing α = 7.15 and +β = 28.62 +distribution using the following formulae: +ρ = ν − µ +µ +, +α = +(1 + ρ)2σ − n2ρ +(νρ(1 + ρ) − σ(1 + ρ)3 , +β = ν − µ +µ +α +(A.1) +which allows us to control for µ and σ, i.e. the amount of nodes to suppress(ns) and +outliers to add(no). Figure A.1 illustrates how we can obtain distributions close to two +Gaussian with identical µ but different σ value, by controlling α and β. +Figure A.1: β-binomial distributions for identical mean: µ, µ1 = 12 but different standard +deviations: σ = 3, σ1 = 5. The dotted lines signifies the mean of the distribution where as +the shaded area is the standard deviation across 5 trials. +B +Supplementary data showing the fit between simulated and +real graphs +As stated in section in 4.3.1 we empirically set the simulation parameters µpert, σpert and +p = 10% such that our simulated graphs follow the intrinsic properties of real graphs. With +the choice of µpert, σpert we estimate the corresponding α and β of β-binomial distribution +xxxiv + +0.14 +V= 30 +α = 4.29, β = 6.43 / μ = 12, o = 5 +αl = 45.99, β1 = 68.99 / μl = 12, 01 = 3 +0.12 +0.10 += +P(P +0.06 +0.04- +0.02 +0.00 +5 +10 +15 +20 +25 +X0.8 +V = 15 +V = 30 +09 = ↑ +0.08 +0.08 +0.7 +0.6 +0.06 +nsity +0.0 +0.2 +0.02 +0.1 +in +15 +20 +5 +20 +X +Xas described in Appendix A. This allows us to generate graphs with similar mean and +standard deviation of number of nodes as in the real data. This control on the number +of nodes is independent from the other types of perturbations we induce. In particular, +we show on Figure B.1 the match between simulated and real graphs for various values +of the parameter controlling for the perturbation of the coordinates of the nodes, κ. This +figure shows the density distribution for the number of nodes in the simulated graphs for +different values of κ, compared to number of nodes in the real population. The largely +overlapping distributions confirm the match of the number of nodes, for any value of κ. +Figure B.1: Distribution for number of node in the simulated population corresponding to +different κ value with the distribution for number of nodes in the real sulcal graphs. +In addition, we also compare the distributions of the geodesic length of the edges which +serves as the feature on the edges. Figure B.2 shows the distributions of mean geodesic dis- +tance across a populations of real data and simulated graphs for different values of κ. The +shaded area surrounding each curve shows the standard deviation across a population of +137 graphs in both real and simulated population. As stated in section 3.1 the distance be- +tween the nodes in the real graphs are larger than a minimum distance, which is illustrated +by the flat portion of the blue curve for low geodesic distances. Our simulations do not +reproduce this feature, as expected from the uniform sampling of the location of outliers +nodes that can get close to previous nodes (figure 3.b,c). Note that the fit is good for larger +geodesic distance values. +Finally, we show on Figure B.3 the distribution of the degree of nodes for simulated +and real graphs. The degree corresponds to the number of neighbors of each node, and is +thus indicative of the local topology of the graphs. This figure confirms the good match +between simulated and real data, independently of κ that controls the perturbation level. +Overall, all our measures confirmed a good match between simulated and real graphs, +for any value of κ. +xxxv + +0.10 +Graphs +original +kappa_100 +0.08 +kappa_200 +kappa_400 +kappa_1000 +Density +0.06 +0.04 +0.02 +0.00 +70 +75 +80 +85 +90 +95 +100 +105 +110 +Number of nodesFigure B.2: Distribution for geodesic distance in the simulated and real population of 137 +graphs. The shaded region corresponds to standard deviations across graphs in the popu- +lation. +Figure B.3: Degree distribution in the simulated and real population of 137 graphs. The +shaded region corresponds to standard deviations across graphs in the population. +xxxvi + +0.035 +0.035 +Simulations +0.03 +0.03 +Real data +0.025 +0.025 +Proportion +Kappa = 100 +Proportion +0.02 +0.02 +Kappa = 200 +0.015 +0.015 +0.01 +0.01 +0.005 +0.005 +0 +20 +40 +60 +80 +100 +120 +0 +20 +40 +60 +80 +100 +120 +Geodesic distance +Geodesic distance +0.035 +0.035 +0.03 +0.03 +0.025 +0.025 +Kappa = 400 +roportion +Kappa = 1000 +0.02 +0.02 +0.015 +0.015 +0.01 +0.01 +0.005 +0.005 +0 +20 +40 +60 +80 +100 +120 +0 +20 +40 +60 +80 +100 +120 +Geodesic distance +Geodesic distance0.35 +0.35 +Simulations +0.3 +0.3 +Real data +0.25 +0.25 +0.2 +Proportion +0.2 +Kappa = 100 +Kappa = 200 +0.15 +0.15 +0.1 +0.1 +0.05 +0.05 +0 +0 +2 +6 +8 +10 +12 +14 +0 +2 +8 +10 +12 +14 +Degree +Dearee +0.35 +0.35 +0.3 +0.3 +0.25 +0.25 +Kappa = 400 +Proportion +Kappa = 1000 +0.2 +0.2 +0.15 +2 0.15 +0.1 +0.1 +0.05 +0.05 +- +0 +2 +4 +6 +8 +10 +12 +14 +0 +2 +4 +6 +8 +10 +12 +14 +Degree +Degree \ No newline at end of file diff --git a/c9FRT4oBgHgl3EQfTTda/content/tmp_files/load_file.txt b/c9FRT4oBgHgl3EQfTTda/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6efd300ad0a919d3c6e7b37646732a9af70848f7 --- /dev/null +++ b/c9FRT4oBgHgl3EQfTTda/content/tmp_files/load_file.txt @@ -0,0 +1,1903 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf,len=1902 +page_content='Population-wise Labeling of Sulcal Graphs using Multi-graph Matching R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='Yadav⋆‡†, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Dup´e†, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Takerkart⋆, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Auzias⋆ ⋆Institut de Neurosciences de la Timone UMR 7289, Aix-Marseille Universit´e, CNRS ‡Aix Marseille Universit´e, Institut Marseille Imaging, Marseille, France †Laboratoire d’Informatique et Syst`emes UMR 7020, Aix-Marseille Universit´e, CNRS February 1, 2023 Abstract Population-wise matching of the cortical fold is necessary to identify biomarkers of neurological or psychiatric disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The difficulty comes from the massive inter- individual variations in the morphology and spatial organization of the folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This task is challenging at both methodological and conceptual levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In the widely used registration-based techniques, these variations are considered as noise and the match- ing of folds is only implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Alternative approaches are based on the extraction and explicit identification of the cortical folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In particular, representing cortical folding patterns as graphs of sulcal basins – termed sulcal graphs – enables to formalize the task as a graph-matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In this paper, we propose to address the problem of sulcal graph matching directly at the population level using multi-graph matching techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' First, we motivate the relevance of multi-graph matching framework in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We then introduce a procedure to generate populations of artificial sulcal graphs, which allows us benchmarking several state of the art multi-graph matching methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Our results on both artificial and real data demonstrate the effectiveness of multi-graph matching techniques to obtain a population-wise consistent labeling of cortical folds at the sulcal basins level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Keywords: brain, sulcal graphs, multi-graph matching, sulcal pits, MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Quantitative comparison across brains is a crucial but open question Comparing features extracted from brain MRI across individuals is necessary for estimat- ing population statistics and ultimately discover markers of diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' However, this task presents several challenges at both the methodological and conceptual levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, the features extracted from two different individual brains are defined in two different mathe- matical spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Comparing such features thus requires to address the methodological prob- lem of transferring them into a common space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The task of transferring information from one brain to another or to a common space consists in defining spatial correspondences i arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='13532v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='ML] 31 Jan 2023 across these objects by compensating for their variations in their respective geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The challenge lies in the massive inter-individual variations of the morphology of the brain and in particular the geometry of the cortical surface, which make the identification of such spatial correspondences an ill-posed problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As a consequence, any solution to this problem inevitably requires to introduce additional constraints based on assumptions on the biological validity of the resulting spatial correspondences, which constitutes a chal- lenge at the conceptual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, the assumptions and constraints introduced in the definition of the spatial correspondences actually influence the derived statistics measured on the population of interest, and could thus be considered as a source of bias in the anal- ysis Van Essen, Glasser, Dierker, Harwell, and Coalson (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' One widely used approach to tackle this problem – termed here as the registration- based approach – consists in defining a mapping between each individual brain and an atlas serving as the common space by estimating a spatial transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As pointed above, the process of building the atlas and defining the associated projection operator which minimizes the error induced by the transformation remains an open research ques- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As a consequence, several registration techniques and atlases co-exist in the field, and tools to enable comparison across atlases are then required (Devlin & Poldrack, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Van Essen & Dierker, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The variety of atlases, projection mechanisms and descriptors illustrate the ongoing exploration of putative biologically relevant features used to define these correspondences across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' One of the most widely used registration-based approach (Fischl, Sereno, Tootell, & Dale, 1999) defines a mapping between cortical sur- faces by imposing the alignment of a combination of curvature and convexity features es- timated from a 2D mesh representing the geometry of the cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The cortical surface of a given subject is projected onto the atlas by matching its curvature and convexity, under the assumption that aligning these features induces biologically relevant anatomo-functional correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In this process, as in any registration-based approach, variations across individuals are considered as noise or confounding perturbations to be minimized, includ- ing variations in the topology and number of folds (sulci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' More generally, the registration- based approach might be seen as an over simplification of the problem since potentially relevant geometrical information is not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Alternative approaches consist in characterizing the geometry and organization of the cortical folds in each individual and then compare these features across the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Characterizing cortical folding patterns using graphs Several approaches have been proposed to characterize cortical folding patterns, such as gyrification index, fractal dimension and curvature (Armstrong, Schleicher, Omran, Cur- tis, & Zilles, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Cachia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Although these measures capture relevant morphological features, they do not explicitly reflect the topology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e the spatial relationships between sulci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Mangin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2004) introduced an analysis framework based on the automatic extraction and labeling of the sulci allowing to characterize their shape, size and pattern in terms of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' sulcus area, depth and length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This representation of the ii cortical geometry has been used for instance to characterize populations of healthy sub- jects (Duchesnay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2007), to quantify potential deviations from normal populations in various conditions such as schizophrenia (Cachia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2008) and autism spectrum disor- der (Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2014), or to estimate the heritability of the folding patterns (Pizzagalli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Pursuing on this line of research, the sulcal pits were introduced as a concept allowing to decompose the sulci into smaller pieces and thus access finer scale geometrical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As described in details in Auzias, Brun, Deruelle, and Coulon (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2010), each fold is divided into sulcal basins that are defined as concavities in the white matter surface bounded by convex ridges, and the deepest point in each basin defines the associated sulcal pit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' More recently, Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Takerkart, Auzias, Brun, and Coulon (2017) represented the geometrical relationships between sulcal basins as a sulcal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' A sulcal graph is constructed by considering each sulcal basin (or associated pit) as a node, while the edges connect only adjacent basins and thus represent their spatial organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Various geometrical features of a sulcal basin can then be attributed to graph nodes (such as the depth of the pit, its 3d coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='), while the spatial organization of the basins is encoded in the topology of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure 1 illustrates this decomposition of the cortical folds into sulcal basins allowing to represent this complex geometry as a sulcal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure 1: Example of sulcal graphs from three individual brains, superimposed with the underlying decomposition of the cortical surface in sulcal basins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Sulcal basins are shown in different colours, and their corresponding node in the graph are represented as spherical dots in the lower panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The color of each node in the graph illustrates the value of a given attribute such as for instance the area or depth of corresponding sulcal basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' These sulcal graphs constitute particularly relevant representations because: 1) varia- tions across individuals are preserved and are manifested as changes in both the topology of the graph and the value of the attributes attached to the nodes and edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2) the de- sign of tools for the quantitative characterization of these variations can benefit from the extensive body of methods from the graph processing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 Problem statement and contributions In the present work, we focus on the task of matching together a set of sulcal graphs in or- der to define biologically relevant correspondences across a population of subjects, under the specific constraint of explicitly taking into account the variations in folding patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Before iii moving to the formalization, we more precisely situate this problem with respect to the conceptual question of defining correspondences across individuals, and with respect to the methodological problem of graph matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Unsupervised comparison and matching of sulcal graphs Comparing brains using sulcal graphs is highly relevant because all the geometrical infor- mation about the macroscopic cortical folding can be encoded into such graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' However, several challenges need to be addressed in this context: 1) the large inter-individual varia- tions in brain anatomy induce complex variations across sulcal graphs, including in their topology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2) sulcal graphs can be contaminated by noise resulting from the imperfect seg- mentation of the individual cortical surface and corresponding sulcal basins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 3) there is no consensus on a nomenclature or atlas at the scale of sulcal basins covering the whole brain, that is a prerequisite to tackle the matching problem as a supervised learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, few studies investigated the matching of cortical folds across individuals as a su- pervised task (Behnke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Borne, Rivi`ere, Mancip, & Mangin, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Rivi`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' All these works focused at the scale of sulci, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' considering large folds consist- ing of several of our sulcal basins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' To our knowledge, only Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2021) attempted to tackle this problem at finer scale, probably because of the massive amount of efforts needed to gather sufficient amount of manually labeled data Voorhies, Miller, Yao, Bunge, and Weiner (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, ambiguities due to variations across individuals in the folding patterns become overwhelming at finer scale than sulci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This is illustrated by the tedious works advancing the definition of a fined-grained nomenclature of folds Sprung-Much and Petrides (2020) and their relationship with underlying function Willbrand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The lack of widely accepted fined-grained nomenclature is also blatant in the related field of brain parcellation: more than 20 different fine-grained atlases co-exist (Eickhoff, Yeo, & Genon, 2018), and even the most advanced multi-modal atlas (Glasser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2016) was validated only on a small portion of the cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Matching sulcal graphs across individuals is thus a very challenging problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Instead of relying on the few existing labeled data-sets that clearly deserve further validation, we decided to approach this question as an unsupervised learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We now describe the few studies that have attempted to tackle the question of unsu- pervised labeling of sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The first approach was proposed by Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2010) and consisted in computing a map of the spatial density of sulcal pits across a population of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This density map was computed by accumulating the pits from the different individuals in each vertex of an average surface after aligning the folds using a registra- tion technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' A watershed algorithm was then applied to this density map in order to separate the main clusters of sulcal pits, empirically defined as the regions of high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' An arbitrary label was then associated to each cluster, hereby defining an ad-hoc labeling of the pits across individuals, depending on the cluster to which they contributed in the density map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This procedure implicitly defines a matching of sulcal pits and correspond- ing basins across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Exemplar applications of this method can be found in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' iv Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Le Guen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2017), with illustrations of density maps and induced labeling for various populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We refer in the following to this category of methods as Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' since we used the open source implementation from this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The main limitation of this approach is that the labeling is driven only by the coordinates of the sulcal pit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2020) introduced an alternative procedure for labeling the sulcal basins, hereby considering the geometry of the basin surrounding each sulcal pit in addi- tion to its spatial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We refer to this method as Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The authors of (Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2020) also raised the question of the consistency of the labeling, a notion that we will develop further below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In this method, an explicit constraint is imposed to restrict the labeling to only one node per subject for each label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In addition, the nodes for which the labeling is ambiguous – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' for which several labels are equally plausible – remain unlabelled, which is often denoted as partial matching in the literature on graph processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Importantly, the spatial relationships between adjacent sulcal basins and pits are never taken into account in any of these methods, since the different pits/basins from each subject are considered independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In contrast, in the present work our aim is to exploit the spatial organization of the adjacent basins stored in the sulcal graph rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Few publications investigated the potential of graph matching in the context of sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2011), the spectral graph matching technique (Leordeanu & Hebert, 2005) was applied to a set of 48 monozygotic twins, comparing a pair at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This study showed that the similarity of the sulcal graphs across pairs of twins are higher than for unrelated pairs, demonstrating the genetic influence on sulcal patterns, and the relevance of graph matching techniques in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This approach was used in follow-up papers from the same group, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='g for comparing brain lobes in Morton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2019) or for matching individuals onto an atlas in Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In the work by Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2018), a population of 677 neonates was analyzed based on a sulcal graph comparison method similar to the one of Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The authors proposed to use different features of the sulcal pits such as the pit position, the pit depth, the basin area, the basin boundary and the pit local connectivity to construct different similarity matrices, one per feature, and merge them into a single one using a matrix fusion technique (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' A clustering algorithm was then applied to the fused similarity matrix to identify sub-populations of sulcal graphs, associated to specific folding patterns in the central, cingulate and superior temporal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Critically, all these previous studies relied only on pairwise graph matching techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Comparing pairs of graphs independently, in the presence of noise and large inter-individual variations, is clearly sub-optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Multi-graph matching: a relevant framework for population studies Given the large variations across subjects and imperfect sulcal basins extraction, examining jointly a group of sulcal graphs is key to reveal meaningful information not accessible by v considering only pairs of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This is the translation to sulcal graphs of the basic idea behind general population studies, that allowed researchers to uncover some of the mechanisms underlying the anatomo-functional organization of the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We follow this principle by investigating for the first time the potential of multi-graph matching techniques in the context of sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' By considering several brains together, the geometrical information that is shared by the majority of individuals should help to regularize the matching problem and allow to identify putative noisy graph nodes in a more robust way than with pairwise matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The multi-graph matching framework has the potential to uncover population-wise invariant patterns in sulcal graphs without imposing a priori, potentially biasing, assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 Contributions In our previous work (Buskulic, Dup´e, Takerkart, & Auzias, 2021), we introduced a frame- work to generate a set of synthetic sulcal graphs representative of a population, and used it to benchmark state of the art pairwise matching techniques in the context of sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In Yadav, Dup´e, Takerkart, and Auzias (2022), we provided a proof of concept of the rele- vance of multi-graph matching techniques in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In the present study, we extend these preliminary studies in several directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' First, we introduce an improved simulation framework to generate populations of arti- ficial sulcal graphs and demonstrate their biological plausibility through a quantitative comparison with real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Secondly, we benchmark a selection of recently published multi-graph matching techniques against the best pairwise technique for this task (iden- tified in from Buskulic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2021)), and report variations in performances that would clearly impact potential real-world applications, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='g in a clinical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Finally, we com- pare qualitatively and quantitatively the different graph matching techniques, as well as the previously published approaches Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and Kaltenmark et al, on a real data- set of 137 subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In addition, our experiments demonstrate the feasibility of comparing a large population of sulcal graphs based on multi-graph matching techniques, in fully acceptable computing times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' All the source code and data will be shared openly upon publication at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='com/gauzias/sulcal graphs matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2 Formal problem and state of the art In this section, we define formally the problem of matching sulcal graphs, as well as the multi-graph framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We then give an overview of the different methods proposed in the literature and provide a more detailed description of the multi-graph matching meth- ods included in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Undirected attributed Sulcal graphs We consider a population of N sulcal graphs, noted G1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' GN, representing the cortical folding pattern of an hemisphere from N different individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The sulcal graph from a vi given subject q is an undirected attributed graph formally defined as a quadruplet Gq = (Vq, Eq, AV q , AE q ), where Vq = {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' , vnq} are the nodes in the graph and |Gq| = nq is the number of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Eq ⊆ Vq × Vq defines the set of eq edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' AV q = {aV v1, aV v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' , aV vnq } is the set of attributes associated to each node in Vq, and AE q = {aE e1, aE e2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' , aE eeq } is the set of attributes associated with each edge in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that the number of nodes nq and edges eq and corresponding attributes varies across graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As illustrated on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2, the sulcal graph from each subject is then mapped onto the same common spherical domain using the surface inflation and registration tools from freesurfer v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 (https://surfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='nmr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='mgh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='edu/, see Fischl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (1999) for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The matching is computed in this common spherical domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In this work, we consider as attributes of the nodes the 3D coordinates of the sulcal pits on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Regarding the attributes of the edges, we compute the length of the edge on the sphere as an approximation of the geodesic distance between neighboring pits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure 2: The sulcal graph from each subject is transferred onto a common sphere using the inflation and spherical registration tools from freesurfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The sulcal graphs from every subjects can then be mapped onto either the common sphere or onto an average surface for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that the spatial dispersion of the nodes of the graphs on the common spaces is heterogeneous, with dense clusters in cortical regions where the variations across individuals are lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Generalities and overview of pairwise graph matching methods Pairwise graph matching refers to the problem of finding correspondences between the nodes of two graphs G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This problem is usually divided into two categories: exact and partial matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Exact matching methods consider graph matching to be a special case of the graph isomorphism problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' It aims at finding the bijection between two graphs, which implies that both the nodes and edges of the different graphs are strictly matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This requirement is too strict for most real-world tasks and in particular in our context where the number of nodes and edges varies across graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Therefore, we focus on the partial matching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This problem can be formulated as a Quadratic Assignment Problem (QAP) (Loiola, Silva, & Galati, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Although different forms of QAP exist, the vast majority of the literature has focused on Lawler’s QAP (Lawler, 1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Given two graphs G1 and G2 with number of nodes |G1| = n1 and |G2| = n2 respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' the Lawler’s QAP consists in searching for the assignment matrix X12 ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1}n1×n2 such that X12[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' j] = 1 indicates that υi ∈ V1 corresponds to υj ∈ V2 and X12[i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' j] = 0 otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' vii Common space sphere average surfaceresulting from the following optimization problem: max J(X12) = vec(X12)⊤Φ12 vec(X12) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (1) subject to X121n2 = 1n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' X⊤ 121n1 ≤ 1n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' X12 ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1}n1×n2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' where vec(X12) denotes the column wise vectorization of X12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1n1 and 1n2 denote the column vectors of all ones of size n1 and n2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and Φ12 ∈ [0, 1]n1n2×n1n2 is the affinity matrix that is given as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The diagonal entries of Φ12 encode the similarity across nodes whereas non-diagonal entries encode the similarity across edges between the two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The computation of the affinity matrix is context-dependent, and we detail the approach used in the present work in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The computation and storage in memory of the very large matrix Φ12 impedes the scalability of the matching problem based on this formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' A solution to tackle this limitation is to reformulate the matching as a Koopmans-Beckmann’s problem (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Zhou & De la Torre, 2015) that is a special case of Lawler’s QAP: max J(X12) = tr(Ψ⊤ 12X12) + tr(A1X12A2X⊤ 12) , (2) subject to X121n2 = 1n1, X⊤ 121n1 ≤ 1n2, X12 ∈ {0, 1}n1×n2 , where Ψ12 ∈ [0, 1]n1×n2 denotes the affinity matrix across nodes, and A1 ∈ Rn1×n1 and A2 ∈ Rn2×n2 are the weighted adjacency matrices of two graphs respectively such that A[i, j] = wij if edge (vi, vj) exists with weight wij and A[i, j] = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Koopmans- Beckmann’s formulation is a special case of Lawler’s where the edges can only be weighted by a scalar value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' cannot support a vector of attributes on edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Under this constraint, we can decompose the large matrix Φ12 into three smaller matrices Ψ12, A1 and A2, which provides better scalability than Lawler’s QAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' These two formulations are combinatorial QAPs and are known to be NP-hard prob- lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Most methods therefore relax the hard constraints given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (1) and (2) and provide approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Various approaches have been proposed to relax these problems, leading to a variety of graph matching methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Discussing these methods is beyond the scope of this work but we refer interested readers to the review Yan, Yin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Going back to our specific context, we reported in Buskulic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2021) a benchmark of the pairwise methods SMAC (Spectral Matching with Affine Constraints) (Cour, Srini- vasan, & Shi, 2007), IPFP (Integer Projected Fixed Point algorithm) (Leordeanu, Hebert, & Sukthankar, 2009), RRWM (Reweighted Random Walks for graph Matching) (Hutchison et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2010), and KerGM (Kernelized Graph Matching) (Zhang, Xiang, Wu, Xue, & Nehorai, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We observed that KerGM clearly outperforms the others in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Conceptu- ally, KerGM well suits sulcal graphs as it relies on Frank-Wolfe optimization that allows to follow an optimisation path that respects the constraint on each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This induces a robustness to the presence of noise in graphs that is crucial in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In the present work, KerGM is included in our benchmark as a representative of pairwise approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' It viii is also used to define the initialization of all the multi-graph methods that are introduced in next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 The multi-graph matching problem We now focus on the problem of jointly matching a population of N graphs {G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' , GN}, starting from pairwise assignment matrices Xij between graphs Gi and Gj (computed with KerGM in this work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The key concept behind multi-graph matching is the cycle consis- tency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This concept states that a matching between two graphs Gi and Gj should be the same if we go through an intermediate graph Gk to create a new mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Formally, a perfectly consistent, bijective mapping (every node is matched to one and only one other node) would satisfy : Xik = XijXjk , (3) for any i, j and k with i ̸= j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' A common way to estimate consistency at the population level is to compute the full bulk assignment matrix X ∈ {0, 1}m×m with m = �N q=1 |Gq|, that is obtained by assembling all individual pairwise matrices: X = � ������ X11 X12 · · X1N X21 X22 · · X2N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' XN1 XN2 · · XNN � ������ Intuitively, enforcing the consistency constraint will induce a reduction of the rank of this bulk matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Multi-graph matching techniques can be divided into three categories as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The first category of approaches explicitly aim at minimizing the rank of the bulk matrix using various approaches (Chen, Guibas, & Huang, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Hu, Huang, Thibert, & Guibas, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Pachauri, Kondor, & Singh, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Wang, Zhou, & Daniilidis, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For instance, (Bernard, Thunberg, Swoboda, & Theobalt, 2019) solves a global optimization problem by using a projected power iterative method, and we detailed further (X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Zhou, Zhu, & Daniilidis, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The second category of techniques does not explicitly minimize the rank of the bulk matrix but rely on other types of formalization aiming at increasing the consistency across all graphs (Yan, Cho, Zha, Yang, & Chu, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Yan, Cho, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2014, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Yan, Wang, Zha, Yang, & Chu, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Finally, the third category corresponds to deep learning approaches that show promis- ing performances in supervised tasks compared to previous methods, but are not suited for unsupervised tasks (Rol´ınek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Wang, Yan, & Yang, 2019, 2020a, 2020b, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Yu, Wang, Yan, & Li, 2019, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Zanfir & Sminchisescu, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Some other interesting methods exploit the concept of consistency in order to solve the problem of jointly matching multiple images (Faktor & Irani, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Rubinstein, Joulin, ix Kopf, & Liu, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Tron, Zhou, Esteves, & Daniilidis, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Zhou, Jae Lee, Yu, & Efros, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' However, these do not take into account the connectivity of the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4 Selection of the methods included in our benchmark We used the following criteria to select the methods included in our benchmark: (i) Avail- ability of code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We included only methods for which the authors have made their code openly available in order to avoid reimplementation issues and to ensure the full repro- ducibility of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (ii) Methods exploiting graph topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We selected the methods that take into account the topology of the graph, which is crucial to exploit the spatial adja- cency information encoded in the sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (iii) Scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Since we are interested in performing population studies over large sets of individuals, we excluded methods that do not provide acceptable scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (iv) Unsupervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Finally, as motivated in the introduction, we focus on unsupervised methods in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The method that satisfy these selection criteria are mALS (X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2015), mSync (Pachauri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2013) and CAO (Yan, Cho, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We provide a detailed description of each of these methods below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In our experiments, these multigraph graph-matching techniques will be compared with the pariwise approach KerGM, and with the two meth- ods from the literature specifically designed for labeling sulcal graphs already described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1: Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2015) and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='5 Description of the selected multi-graph matching methods As described in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3, the general objective of multi-graph matching methods is to match the nodes across several graphs together by enforcing consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The authors of CAO (Yan, Cho, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2016) propose to maximize the affinity infor- mation and impose consistency at the same time instead of considering them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' They assume that enforcing consistency acts as a regularizer in the affinity objective func- tion, particularly when the matching is ambiguous due to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The approach is based on the search of an intermediate graph Gq that allows to optimize the affinity score while progressively inducing consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' They introduce the unitary consistency across a set of N pairwise matching solutions X for a graph Gq as: Cu(Gq, X) = 1 − �N−1 i=1 �N j=i+1 ��Xij − XiqXqj �� F /2 nqN(N − 1)/2 , (4) where∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='∥F is the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The authors propose several approaches to balance be- tween consistency and affinity, leading to different variants of CAO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In particular, their best algorithm is able to elicit outlier nodes during the optimization, which is highly rel- evant in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' However, the use of affinity information along with consistency and outlier elicitation increase the computational complexity of the method to O(N4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As a consequence, only the least resource-demanding algorithm CAOcst did scale with the x memory requirements imposed by the size of our graphs and number of subjects in our populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We thus refer to that particular version in the rest of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This version of CAO enforces consistency through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4, but ignores the affinity information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The approach mSync (Pachauri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2013) consists in estimating a mapping of each Xij to a common universe of assignment matrices, of size d: max {Ui,Uj}∈P N � i=1 N � j=1 ⟨UiUj, Xij⟩ , (5) with P = {U ∈ {0, 1}nq×d | U1d = 1nq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (6) Since solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='5 is intractable in most applications, the authors relax the problem into a generalized Rayleigh problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' They further propose to use a reference graph in order to estimate the mapping to the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In the implementation provided by the authors, the first graph in the collection G1 is selected as the reference graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In mALS X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2015), the authors formalize the multi-graph matching as the following low rank matrix recovery problem: f(X) = − N � i=1 N � j=1 ⟨Ψij, Xij⟩ + α⟨1, X⟩ + λ∥X∥∗ , = −⟨K − α1, X⟩ + λ∥X∥∗ , (7) where, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='⟩ is the inner product, α controls the weight on sparsity, and K = {Ψij}N i,j=1 is the set of affinity matrices given as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The cycle consistency is induced by the nuclear norm ∥X∥∗ that controls for the rank of X while ⟨1, X⟩ favors bijective matchings across graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Importantly, X is treated as a real matrix such that X ∈ [0, 1] The matrix is binarized at the end of the optimization process using a threshold value t that is set by default as to t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In, addition, the authors leverage the work by Hastie, Mazumder, Lee, and Zadeh (2015) and Cabral, De la Torre, Costeira, and Bernardino (2013) for decomposing X which allows to solve the problem in a lower dimension space using the ADMM method (Eckstein & Bertsekas, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 3 Generation of a population of synthetic sulcal graphs A primary objective of our work is to investigate and evaluate different multi-graph match- ing techniques in the context of sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' However, as mentioned in the introduction, there is no ground truth matching available for such graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We tackle this problem by designing a procedure allowing to generate a population of artificial sulcal graphs with correspondences defined by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Such populations of artificial graphs will con- stitute a ground truth against which the different matching methods can then be bench- marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Generating artificial sulcal graphs for the purpose of a benchmark study induces the two following constraints: 1) The artificial graphs should be biologically plausible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' xi they should respect as much as possible the intrinsic properties of a population of real sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2) The generation of the artificial graphs should be as simple and straight- forward as possible in order to facilitate the comparison of the performances obtained in the benchmark study and the interpretation of the differences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' the generation proce- dure should rely only on a limited number of parameters, and potential biases should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As detailed below, these two contradictory constraints are balanced in the design of our generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The procedure is summarized in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1 and consists in two main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' First, we generate a set of points on the common spherical domain, that will serve as reference nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Then, we impose several types of perturbations to this set of reference nodes in order to generate a corresponding population of artificial sulcal graphs, while preserving the correspondences across graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' the ground-truth matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Such procedure provides the ground truth matching across the population, while controlling for the nature and amount of variations across artificial sulcal graphs (corresponding to different subjects in real data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Algorithm 1 Procedure to generate a population of artificial sulcal grahs Require: N, nref, κ, µpert, σpert, p Step1: create reference nodes ▷ See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 for j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='.10000 do Sample nref points on the sphere Compute the minimum geodesic distance end for Choose the set of points with the largest min distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Step 2: generate a population of sulcal graphs ▷ See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='.N do Perturb location of the reference nodes ▷ See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Add outliers and suppress some nodes ▷ See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Compute the edges of the graph ▷ See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 end for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Generation of a set of reference nodes The first step consists in generating a set of reference nodes on the spherical domain while controlling for two specific distinct parameters : the number of nodes noted nref, that is typically set to match the average number of nodes across a real population, and the mini- mum distance between the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, the nodes of the real sulcal graphs cannot be closer to each other than a minimum distance since they correspond to depth maxima that are not located in the immediate proximity of the boundary of sulcal basins (see Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2015) for further description of the extraction of sulcal pits and basins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As a consequence the spatial distribution of the nodes on the sphere cannot be fully random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In order to generate this set of nref points on a sphere with pseudo-random spatial distribution, we adopted a simple brute force approach: we sample a set of nref points over the surface of the sphere 10000 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and we select the set that has the largest minimum geodesic dis- xii tance between neighbouring points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As we will show in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1, 10000 times is sufficient to get a set of reference nodes with a minimum distance between points that is realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Technically, the uniform sampling of points on the sphere is achieved by generating ran- dom rotations of the unit vector as described in Blaser, Fryzlewicz, Blaser, and Fryzlewicz (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Lef`evre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' At this stage, we have defined on purpose a set of reference nodes that matches a real population in terms of size and of minimal distance between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The next step consists in perturbing the reference nodes in order to generate the population of synthetic sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Generation of an individual sulcal graphs We now add perturbations of different natures to this set of reference nodes in order to obtain a population of artificial sulcal graphs, that corresponds to different subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' These perturbations aim at mimicking the inter-individual variations that are observed in a healthy population, by affecting the features of the nodes and edges, but also the topol- ogy of the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In order to generate a population of N artificial sulcal graphs, these operations are repeated N times independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Perturbation of the location of the reference nodes The first step consists in adding random noise to the coordinates of the reference nodes on the sphere, in order to model the inter-individual variability that exists in the location of the sulcal pits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We used the von Mises-Fisher (vMF) distribution that is an approximation of Gaussian distribution on a sphere (Von Mises, 1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The two parameters of the vMF distribution µ and κ can be seen as the equivalent of the mean and of the inverse of the standard deviation (κ ∝ 1/σ) for a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Therefore, we iterate across the reference nodes, and for each reference node, we produce a noisy one by sampling from the distribution vMF(µ, κ), where µ is the coordinates of this reference node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We control for the amount of noise on the coordinates of the perturbed nodes through the value of the parameter κ, that is common to all nodes from the reference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Smaller values for κ will induce larger variations across the artificial sulcal graphs within the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Importantly, note that since we perturb each node of the reference set independently, we keep the correspondence between each noisy node and its reference node, which will allow defining our ground truth matching at the population level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Addition of outliers and suppression of nodes Next, we simulate the inter-individual variations in the number of nodes across the sulcal graphs, which is of crucial importance for generating biologically plausible artificial pop- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The aim is to model both false positive and false negative matchings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' respec- tively nodes that are present in the reference set but not in a given graph, and nodes that are present in the graph but not in the reference set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This is achieved by randomly adding xiii a certain number no of nodes on top of the perturbed nodes – hereafter called outlier nodes, and by deleting ns nodes amongst the perturbed nodes – hereafter called suppressed nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In order to randomly draw no and ns, we use the β-binomial distribution B(ν, α, β), which is a distribution of non-negative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The parameter ν denotes the size of the support of the distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e the maximal value that can be sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The parameters α and β can be set so that B(ν, α, β) approximates a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We describe the setting of these parameters and precise their link with µ and σ of a Gaussian in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Since we want the average number of nodes across the population of perturbed graphs µsimu to match the number of nodes in the reference set nref, we set µo = µs = µpert and also σo = σs = σpert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This formulation allows us to control the standard deviation of the num- ber of nodes across the population of artificial graphs with the two parameters µpert and σpert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 Construction of the edges The last step consists in constructing each artificial sulcal graph with the sets of perturbed nodes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We first compute the three-dimensional convex hull of each set of per- turbed nodes located on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This yields a triangulation where only neighboring nodes on the sphere are connected, which is a simple way to simulate the region adjacency graph that is constructed from the sulcal basins in the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' However, the average node degree in such triangulations is higher than for real sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Therefore, we finally delete a small percentage p of the edges in these triangulations, in order to obtain artificial graphs which match the average degree of real sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that since the construction of the edges occurs after the previous perturbation steps (perturbations of the location, addition of outlier nodes and suppression of nodes), the resulting artificial sulcal graphs can show variations in their topology across individu- als of a population, as we observe in real data, making them biologically-plausible in that respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4 Experiments and results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Computation of the affinity matrices As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2, we initialize all the multigraph matching methods using the pair- wise results obtain from KerGM, which relies on the formalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We thus need to compute the affinity matrices Ψij, Ai, Aj that store the similarity between nodes and edges across every pairs of graphs in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In the present work, we compute these affinity matrices using Gaussian kernels applied to the attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For two nodes v ∈ G and v′ ∈ G ′ the affinity value is computed using the kernel defined as exp (−γV ���aV v − aV v′ ��� 2 2) and for two edges e ∈ G and e′ ∈ G ′ the kernel is defined as exp (−γE���aE e − aE e′ ��� 2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' To estimate appropriate values for γV and γE we use a heuristic proposed in Takerkart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2017) that consists in using a cross-validation xiv scheme to compute the inverse of the median of the distribution across all possible pairs of nodes/edges, independently for each attribute (3D coordinates on the sphere for the nodes and the geodesic distance for the edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Dummy nodes Most graph matching methods assume a constant number of nodes across the graphs to be matched, which is not the case in our case (both synthetic and real graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We use the classical approach from the graph matching literature which consists in adding dummy nodes to smaller graphs so that all the graphs get the same number of nodes as the largest graph in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For each of these dummy nodes, we assign to 0 the correspond- ing values in the node and edge affinity matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This makes the optimization problem defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 independent from dummy nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 Benchmark on synthetic sulcal graphs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Description of synthetic data sets We first tuned empirically the parameters to the values µpert = 12, σpert = 4 and p = 10% to obtain variations in our synthetic graph populations that are in line with what is observed in real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The distribution for number of nodes in the real data population is 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='27±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='72 likewise in our simulated population for a randomly chosen κ value the distribution for number of nodes is 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='15 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='45 for the selected value of µpert and σpert and is consistent across all κ values across all trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We further provide in Appendix B additional materials showing the matching distributions between our simulated graphs and real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Furthermore, we varied the value of κ ∈ [100, 200, 400, 1000], which controls the amount of variations across synthetic graphs within a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that κ controls the spread of nodes coordinates around the reference nodes, which in turn induces variations in the topology and attributes of synthetic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For each value of κ, we generate 10 populations of N = 137 synthetic graphs (which corresponds to the number of subjects in our real population;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' see below) and report the average and standard deviation of the metrics described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As illustrated on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3, our populations of synthetic graphs show variations that are qualitatively very close to those observed across real graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Evaluation metrics for synthetic data sets In order to evaluate the different matching methods on simulated graphs, we use the clas- sical precision, recall and F1-score: Precision = True Positives True Positives + False Positives ∈ [0, 1] (8) xv Figure 3: a) Real sulcal graphs from three randomly chosen individuals, and projected on the average surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' b) Simulated graphs randomly chosen for κ = 1000, showing the ground-truth correspondence across graphs in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Nodes in black represent the outlier nodes that have no correspondence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' c) Illustration of the impact of κ on the spatial dispersion of nodes: the nodes of six simulated graphs are shown on the average surface for κ = 1000 (left) and , κ = 200 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The spread across the nodes for each cluster varies according to κ, while outlier nodes in black have random locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Recall = True Positives True Positives + False Negatives ∈ [0, 1] (9) F1 = 2(precision × recall) precision + recall ∈ [0, 1] (10) Thus, Precision is a ratio between the True positives(number of correct matches predicted by the algorithms) and all the positives(number of matches by the algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Whereas, Recall is a ratio between True positives and True positives along with False negatives(number of correct matches not predicted by the algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Finally, the F1 score provides a balance between Precision and Recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' A F1-score of 1 reflects the ability of the algorithm to obtain a perfect matching of inlier nodes and accurate identification of outlier nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' These metrics are relevant in our context to detect matching with outliers alongside the incorrect matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' xvi a) b) c)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 Results on synthetic data sets We report of Fig 4 the mean and standard deviation of Precision, Recall and F1-score, com- puted across the 10 synthetic populations for each value of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' First, we find that two multi-graph matching methods, mALS and mSync, vastly and consistently outperform KerGM, which has been identified as the best pairwise matching method for this task in Buskulic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This confirms our main hypothesis: consid- ering the matching problem on the whole population using multi-graph matching allows an important gain in performance compared to only considering pairs of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Then, we observe a gradual decline in the performances of all methods as the noise increases (decrease of κ), as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The performances of the multigraph approaches mALS and mSync resist much more to this increase in variability than the pairwise ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The performances of mSync are limited more specifically by the lower precision at any level of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This suggests that the difference in performances between the two methods are mainly due to the hard constraint on the consistency in mSync that seems too restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' On the other hand, the recall indicates that mSync is more robust to increasing noise than mALS, with very close value when κ = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' However, mALS performs better for lower noise values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Overall, mALS shows the best F1-score for every κ values, thanks to a very high precision combined with very good recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, the F1-score for mALS is above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='7 even for κ = 200 which corresponds to a configuration where the noise is quite strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Finally, the performances of CAO are very low, even lower than the pairwise technique KerGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Such poor performances are likely a consequence of the optimization that consid- ers only the consistency but ignores the affinity of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As already mentioned in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='5, the other versions of CAO proposed in Yan, Cho, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2016) could show much higher performances but did not scale with the size of our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure 4: F1-score, Precision and recall for κ ∈ [1000, 400, 200, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For each method, we plot the average across the 10 simulated populations as a line and the standard deviation as the shaded region of the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4 Application to real data 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Preprocessing of real data For the evaluation on real data, we use the sulcal graphs from 137 young healthy adults taken from the publicly available database OASIS (Marcus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The preprocessing xvii 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='6 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4 E0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 KerGM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 KerGM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 KerGM msync msync mALS msync 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 mALS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 mALS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 CAO 1000 400 200 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 1 1000 400 200 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 kappa 1000 400 200 100 kappa kappaof these data (brain tissues segmentation, mesh extraction and sulcal graphs construction) has been detailed in Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Takerkart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Across this population, the number of nodes is 88 ± 4, with a maximum size of 101 nodes/pits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Dummy nodes are thus added to all other graphs to get a constant size of 101, as explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Evaluation metrics used with real data In absence of ground truth matching for real data, we cannot compute the same scores as for the simulation experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We therefore combine a set of quantitative metrics with some qualitative assessments, which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Consistency According to Yan, Cho, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2016), we compute the node consistency as follows: Given Gk ∈ {Gq}N q=1 and the bulk matrix X, for node vk ∈ Gk, with index i(vk) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' , |Gk|}, its consistency is defined by: C(vk, X) = 1 − �N−1 i=1 �N j=i+1 ||Y(vk, :)||F /2 N(N − 1)/2 , ∈ (0, 1], (11) where || · ||F is the Frobenius norm, Y = Xkj − XkiXij and Y(vk, :) is the i(vk)-th row of matrix Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that it is different from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4 which estimates the consistency at the graph level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This consistency measure is computed for each node of each graph, including dummy nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' A value of 1 corresponds to the ideal case where each graph only contains nodes that have been matched in a consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This consistency measure cannot distin- guish the matches of real nodes to dummy nodes from valid matches across real nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For methods imposing an explicit constraint on the consistency, a value of 1 is expected (and not informative), but for the other methods this measure is relevant and allows to assess the spatial pattern of the consistency across clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Qualitative and quantitative assessment of the labeling induced by the matching In terms of potential applications of the graph matching to sulcal graphs, a major out- come is the labeling of graph nodes that is induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As already mentioned in the introduc- tion, the assessment of the quality of the labeling and thus of the biological relevance of the matching across individuals is an ill-posed problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The first problem is to retrieve a labeling from the assignment matrix resulting from the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In the case of a perfectly consistent matching where each node of each graph would be matched to one and only one node from every other graph in the population, the labeling would be trivial and would consist in simply associating a label to each row or column of the assignment matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This situation is however impossible since the number of nodes varies across individuals within our population of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Therefore, in the present work we take the largest graph as a reference, and we associate an arbitrary label to each of its nodes and then propagate these labels to every other graphs based on the assignment matrix resulting from each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Once the labeling of the nodes is retrieved, the nodes that share the same label across subjects are grouped together into what we will designate as clusters, that are different depending on the matching method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We then compute the coordinates of the centroid of xviii each cluster, which enables to evaluate qualitatively the spatial distribution of the different clusters across the cortical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This qualitative assessment is complemented with a quantitative measure of the com- pactness of the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For this, we compute the silhouette coefficient of each node from each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As proposed in Rousseeuw (1987), the silhouette of a node corresponds to the ratio between the average Euclidean distance to the other nodes in the cluster and its dis- tance to other nearby clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Since the distances are computed on the spherical domain, the use of Euclidean distance is sub-optimal but the errors induced are very low and inde- pendent from the matching method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The silhouette coefficient of a cluster is then obtained by averaging the silhouette values from corresponding nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 Results on real data We first report in Table 1 the quantitative measures that allow us to compare the different techniques at the whole brain level: the number of clusters (thus of labels) obtained with each method, the silhouette measure averaged across all nodes and graphs, the percent- age of nodes remaining unlabeled, the consistency measure averaged across all nodes and graphs, and the computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Table 1: Quantitative measures computed at the whole brain scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Method Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' silhouette Perc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' consistency cpu time clusters unmatched (min) mALS 82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='22 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='08 783 Kaltenmark et al 94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='23 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 ∼ 180 Auzias et al 104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='18 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='15 ∼ 30 mSync 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='49 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 31 CAO 101 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='45 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 3255 KerGM 101 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='34 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='17 1362 The number of clusters and percentage of unmatched nodes indicate that the two meth- ods that allow partial matching mALS and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' result in a lower number of clusters, suggesting that the ambiguous nodes remain unlabeled instead of enforcing their matching into potentially unreliable clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The three methods mSync, CAO and KerGM enforce the matching of every nodes, and result in a number of clusters equal to the size of the largest graph in the set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The method Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' results in more clusters than the size of the largest graph, suggesting that some clusters correspond to highly variable nodes that cannot be matched consistently across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This is confirmed by the consistency measure which is lower than for mALS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The consistency of Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' is still much higher than the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='30 obtained with the pairwise technique KerGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that the methods mSync and CAO explicitly enforce a perfect consistency, but this is possible only when considering the dummy nodes as pointed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Also note that the method Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' also gets a perfect consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This is a consequence of the explicit constraint imposed in this technique by allowing one and only one node per subject to be matched for any given cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' xix The silhouette measures illustrate that a high consistency can be associated with a low compactness of the clusters as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' for CAO and mSync that get values close to the one of the pairwise technique KerGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The methods Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' get much higher silhouette values which is expected since these techniques enforce the match- ing of nodes based essentially on their spatial proximity on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The silhouette value of mALS is higher than these two techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Overall, mALS results in high sil- houette and consistency values, at the cost of a high number of unmatched nodes (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4%) compared to Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', indicating that this method was much more conservative in the matching, leaving more ambiguous nodes unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We then illustrate the matching across nodes from the different graphs (subjects), ob- tained for each method on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The number and location of the different centroids (larger circles) is informative of the spatial distribution of the clusters of nodes across the cortical surface, for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' On the first column (mALS and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=') some nodes remain unlabelled and are represented in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The clusters seem more compact than for the methods of the second column (Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and mSync) that do not allow any node to remain unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' On the third column (CAO and kerGM) the matching looks noisy, with clusters overlapping between eachother in almost every cortical location, which illustrates the poor anatomical relevance of the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure 5: Labeling and corresponding cluster centroids (larger circles) for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Dots in black in the first column (for mALS and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=') correspond to un- matched nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' See text for further description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For further evaluation of the performances of the different techniques, we show on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 6 the silhouette values of every nodes across all graphs as well as the centroids of each cluster as a larger circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The high silhouette values of the centroids for the methods mALS, Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' are visible with mostly red and orange cen- troids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In contrast, we observe more centroids in green and blue for CAO and KerGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Together with Table 1, this figure illustrates the poor performances of pairwise matching xx mALS Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' CAO msync kerGM Kaltenmarket al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='approach with high spatial dispersion of nodes corresponding to each cluster for KerGM, associated to very low silhouette coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The method mSync results in higher silhou- ette coefficients for some nodes, but lower value for others (nodes and centroids in blue on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 6), indicating that the matching was enforced also for ambiguous nodes located in highly variable regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This is a consequence of the hard consistency constraint in mSync imposing a matching that is consistent across all graphs by construction, even in highly variable regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', we observe that the clusters are organized around re- gions of high nodes density, but the nodes located relatively far from the centroids have a lower silhouette value (nodes in green on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' These observations are consistent with the algorithm that is based on a watershed applied to the sulcal pits density map as de- scribed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For both mSync and Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', we observe some clusters with low silhouette value located close to each other, suggesting that the number of clusters is too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The techniques mALS and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' result in much higher silhouette values, which is expected since they do not force the matching of highly variable nodes that are left unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The unlabeled nodes have a very low silhouette value (in violet on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 6), but since they do not belong to any cluster, this does not reduce the silhouette values of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that even for these methods, the clusters get closer with lower silhouette values in highly variable regions such as the anterior frontal and occipital lobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Across the different methods, we observe that the clusters showing a higher silhou- ette value relative to other clusters are located systematically in the same regions that are known to be less variable across individuals, such as the central sulcus, and the insula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For these clusters, the silhouette values are close across methods, confirming the lower ambi- guity in the matching in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In highly variable regions, the different methods produce different matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For instance in the occipital lobe, the clusters produced by Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' show lower silhouette values compared to mALS, but we observe the opposite effect in the anterior frontal lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' On Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 7, we show the consistency for every nodes and centroids, for the three methods that do not explicitly enforce a perfect consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Clearly, the pairwise technique KerGM results in inconsistent matching for every clusters, including the regions where the vari- ations across individuals are known to be low (no centroid in green, even in the central sulcus and the insula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' For mALS and Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', we can observe the spatial variations of the consistency across cortical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Again, higher consistency is obtained in less variable regions (central sulcus, insula) for both techniques, and relatively lower values are visible in the frontal and occipital regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The consistency is higher for mALS than Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' for every clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that the spatial pattern of the consistency measure for mALS is anatomically relevant, with a consistent matching in the insula, the central and pre-central regions, and less consistent in the peri-sylvian regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' At a more local scale, we observe a cluster in the superior temporal sulcus that is more consistent than those lo- cated anteriorly or posteriorly, which is in line with previous studies describing variations and stabilities across individuals in this region Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' xxi Figure 6: For each method, we show the silhouette coefficient of each node from every graphs, as well as corresponding centroids as larger circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Each centroid (larger circles) is colored according to the average of the silhouette coefficient of corresponding nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure 7: Node consistency computed for each node of each graph with respect to the remaining graphs, and then averaged across graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We adapted the colorbar to visualize the differences between the three methods, with the pariwise technique KerGM showing much lower values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 5 Discussion In this work, we explored the potential of graph matching methods applied to a popula- tion of sulcal graphs to uncover correspondences across individuals driven by the local patterns of folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, these graph matching methods take into account the charac- teristics of individual sulcal basins as well as their topological organization to construct the correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Our results on both simulations and real data support the biological relevance of the correspondences across individual resulting from multi-graph matching techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' xxii mALS Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' CAO Kaltenmarket al kerGM ms ync0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='7 mALS Auzias et al kerGM5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 Relevance of simulated graphs relative to real data for evaluating matching techniques To overcome the lack of ground truth for real data, we proposed a procedure allowing to generate artificial graphs that approximate the features of real sulcal graphs while control- ling the variations across graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This simulation procedure enabled to benchmark various pairwise and multi-graph matching techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The evaluation of the performances of the different methods and their robustness to controlled variations in the simulated graphs was informative for probing their effectiveness in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The performances of the pairwise approach KerGM were limited even when the level of perturbations was mini- mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that we reported in Buskulic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2021) that alternative pairwise techniques perform even worse on this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Amongst the different multi-graph matching techniques that were tested, mALS showed better performances than the others in all conditions, and a good robustness to increasing noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' These observations were confirmed by our application on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Overall, our set of experiments confirmed the intuition that multi- graph matching techniques are highly relevant in our context, while pairwise techniques show limited performances and might thus be restricted to initialization purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Of note, our aim was not to push the biological plausibility of our simulated graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Keeping the simulations simple enables straightforward interpretation of the variations in the performances across the different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This trade-off is visible in the procedure in particular when we sample the reference nodes uniformly on the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, our simulation procedure cannot produce realistic non-uniform spatial distribution of nodes across the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' While this could be achieved by adapting the sampling of the refer- ence points, this would induce variations in the performances of the matching techniques depending on the location on the sphere, which in turn would make the comparison across methods much more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Beyond the present work, our procedure for simulating sulcal graphs could be instru- mental to assess future improvements in graph matching techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 Considerations relative to deep-learning approaches and potential method- ological improvements As already mentioned in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3, many other graph matching techniques can be found in the literature but were not included in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' More specifically, deep learning approaches outperform traditional approaches in supervised learning task (LeCun, Ben- gio, & Hinton, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Recent works such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (Scarselli, Gori, Tsoi, Hagenbuchner, & Monfardini, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Xu, Hu, Leskovec, & Jegelka, 2019) showed that the structural informa- tion can be learnt by a Graph Neural Network(GNN), providing that manually labelled ground-truth data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In addition, the rise of semi-supervised learning approaches represents an opportu- nity in the context of graphs with partial matching ground-truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Such approaches are worth considering in our context, since we observed marked variations across cortical re- xxiii gions in the ambiguity of the matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The work by Fey, Lenssen, Morris, Masci, and Kriege (2020) considers a semi-supervised framework for handling the matching problem where the ground-truth correspondence are only given for a small subset of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In addition, their approach imposes an explicit inductive bias to find correspondences across graphs, based on neighbourhood consensus that does not allow adjacent nodes from being mapped to different regions in other graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This is appealing in the case of sulcal graph matching where we would like to enforce the matching of nodes located in some specific regions more than in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Such a framework could benefit from the recent work Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2021) on context-aware data augmentation, which could be instrumental to overcome the bottleneck of the lack of ground-truth labeling data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Another avenue for potential gains in performance consists in improving the defini- tion and integration of the attributes on nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Many other geometrical fea- tures could be considered to enrich the attributes on nodes, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' shape index and curvedness Awate, Yushkevich, Song, Licht, and Gee (2010), or the local gyrification index Rabiei, Richard, Coulon, and Lef`evre (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' On the other hand, the literature on learning edge representations is very scarce Hsu, Shen, and Cremers (2022), and the attributes on edges are most often reduced to a scalar value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' a simple weight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In particular, the methods included in the present work cannot handle vectors of attributes on edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Some recent deep learning methods such as (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', 2021) can exploit such vectors of at- tributes, but their scalability is limited by the size of the affinity matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We proposed in Dup´e, Yadav, Auzias, and Takerkart (2022) to overcome this limitation by leveraging the recent matrix factorization method from Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We will further investigate the potential of these methods in our future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 Data-driven nomenclature of sulcal basins The present work extends previously reported experimental results illustrating the major impact of the labeling strategy on the induced correspondences and data-driven nomencla- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2020), the number of clusters obtained at the group level varied from 90 to 114 for the right hemisphere using either the approach proposed in Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2020) or Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2015) respectively, on the same population of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We provide a much more detailed comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We show on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 8 the superimposition of the centroids from different methods on the same average surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This visualization shows that the location of some of the centroids are very consistent across methods (indicated by arrows), corresponding to cortical regions where variations across individuals are known to be low, such as the central sulcus, the insula, the inferior precentral or superior tem- poral sulcus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Other clusters differ across methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The clusters indicated by squares are those resulting from Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and mSync (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=') that do not match clusters from Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (see also Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' These are located either in highly variable regions such as the frontal lobe, or on the top of gyri such as the superior temporal gyrus and the inferior frontal gyrus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' While mALS and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' result in centroids that are highly similar (crosses and rings are often superimposed on the panel on the left), the two xxiv methods do not result in the same matching in highly variable regions such as the inferior frontal and parietal regions (indicated by diamonds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In conjunction with our results on synthetic (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3) and real data (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3), these observations confirm that the concep- tual differences between the approaches yield different matchings and thus different cor- respondences across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Indeed, graph-matching techniques such as mALS are able to take into account the topological information encoded in the graphs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e the spatial organization of neighbouring folds, while Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' relies only on the geometry of sulcal basins, considering different folds separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure 8: Superimposition of the centroids from mALS, Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', and mSync shown as crosses with those of Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' shown as green rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' mSync is representative of the methods kerGM and CAO that also result in 101 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The arrows point to centroids that are robust across methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The squares indicate centroids corresponding to small clusters located on gyri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Diamonds indicate centroids that differ between mALS and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='. The next step will be to assess the biological relevance of the induced correspondences across subjects by visualizing the matching on the cortical surface of the individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Given the variations across methods in the location of clusters observed on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='8, we expect to observe important differences between the techniques at the individual level, especially in highly variable regions such as the parietal lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' More specifically, our expectation is that graph matching techniques should allow solving potential anatomical ambiguities in a much more relevant way than Auzias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' and Kaltenmark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', by exploiting the topological information of the neighbouring folding pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 6 Conclusion In this study, we explored the potential of several graph matching methods chosen from the literature to define population-wise correspondences across individual cortical geome- tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In the absence of a ground-truth labeling for real data, we first proposed a procedure to generate simulated sulcal graphs that follow the intrinsic structure and properties of real sulcal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We then compared the approaches on our simulated sulcal graphs with ground-truth correspondences defined by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We also evaluated the methods on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We computed the silhouette value of each node of the graph that measures the degree of compactness of each cluster, giving xxv mALS Auzias et alus insights on the matching across graphs produced by the different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We also computed a consistency measure that gave us an insight on the variability across the pop- ulation for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The results obtained on real data were compared with two other methods from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Overall, our experiments on both artificial and real data showed the high relevance of multi-graph methods for sulcal graph matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We observed that mALS and mSync outperform CAO and the pairwise approach KerGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' While mALS proved to be very robust to noise compared to other methods, the much lower complexity of mSync makes it also a relevant candidate for further studies and extensions to larger populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 7 Funding sources The data used in the preparation of this article were obtained from OASIS: Cross-Sectional: Principal Investigators: D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Marcus, R, Buckner, J, Csernansky J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Morris;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' P50 AG05681, P01 AG03991, P01 AG026276, R01 AG021910, P20 MH071616, U24 RR021382.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The project lead- ing to this publication has received funding from Excellence Initiative of Aix-Marseille University - A*MIDEX, a french ”Investissements d’Avenir” programme (AMX-19-IET- 002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The research leading to these results has also been supported by the ANR Sul- calGRIDS Project, Grant ANR-19-CE45-0014 and the ERA-NET NEURON MULTI-FACT Project, Grant ANR-21-NEU2-0005 funded by the French National Research Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 8 Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 9 Author contributions We report individual contributions to the paper using the relevant CRediT roles: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='Yadav:Conceptualization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Data curation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Formal analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Investigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Methodol- ogy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Software;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Visualization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Writing - original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='Dup´e: Conceptualization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Formal analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Funding acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Investigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Methodology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Writing - original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='Takerkart: Conceptualization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Data curation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Formal analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Investigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Method- ology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Software;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Writing - original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='Auzias: Conceptualization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Data curation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Formal analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Funding acquisition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In- vestigation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Methodology;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Project administration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Resources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Software;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Supervision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Vali- dation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Visualization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Writing - original draft;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Writing - review & editing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' xxvi References Armstrong, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', Schleicher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', Omran, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', Curtis, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', Jae Lee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', & Efros, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Flowweb: Joint image set alignment by weaving consistent, pixel-wise correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In Proceedings of the ieee conference on computer vision and pattern recognition (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 1191–1200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', Zhu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=', & Daniilidis, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Multi-image matching via fast alternating minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In Proceedings of the ieee international conference on computer vision (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' 4032–4040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Appendices A Additional description of the β-binomial distribution In the following section we provide additional description of the formulation of the β- binomial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The β-binomial distribution is parameterized by ν, α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The parameter ν defines the size of support (in our case, maximum number of no/ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The setting of ν can impact the skewness of the distribution but the shape will be Gaussian as long as the value is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' We show in figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 the β-binomial distributions for different values of ν, with α = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='15 and β = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In this work, we set ν = 30 which is sufficient to get a distribution close to Gaussian for all combinations of α and β parameters that are relevant in our context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Although α and β are not trivial to calibrate, they can be related to µ and σ of a Gaussian xxxiii Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1: Effect on β-binomial mass function for different values of ν fixing α = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='15 and β = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='62 distribution using the following formulae: ρ = ν − µ µ , α = (1 + ρ)2σ − n2ρ (νρ(1 + ρ) − σ(1 + ρ)3 , β = ν − µ µ α (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1) which allows us to control for µ and σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' the amount of nodes to suppress(ns) and outliers to add(no).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 illustrates how we can obtain distributions close to two Gaussian with identical µ but different σ value, by controlling α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1: β-binomial distributions for identical mean: µ, µ1 = 12 but different standard deviations: σ = 3, σ1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The dotted lines signifies the mean of the distribution where as the shaded area is the standard deviation across 5 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' B Supplementary data showing the fit between simulated and real graphs As stated in section in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 we empirically set the simulation parameters µpert, σpert and p = 10% such that our simulated graphs follow the intrinsic properties of real graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' With the choice of µpert, σpert we estimate the corresponding α and β of β-binomial distribution xxxiv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='14 V= 30 α = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='29, β = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='43 / μ = 12, o = 5 αl = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='99, β1 = 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='99 / μl = 12, 01 = 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='10 = P(P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='04- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='00 5 10 15 20 25 X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='8 V = 15 V = 30 09 = ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='06 nsity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 in 15 20 5 20 X Xas described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This allows us to generate graphs with similar mean and standard deviation of number of nodes as in the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This control on the number of nodes is independent from the other types of perturbations we induce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In particular, we show on Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 the match between simulated and real graphs for various values of the parameter controlling for the perturbation of the coordinates of the nodes, κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This figure shows the density distribution for the number of nodes in the simulated graphs for different values of κ, compared to number of nodes in the real population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The largely overlapping distributions confirm the match of the number of nodes, for any value of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1: Distribution for number of node in the simulated population corresponding to different κ value with the distribution for number of nodes in the real sulcal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' In addition, we also compare the distributions of the geodesic length of the edges which serves as the feature on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2 shows the distributions of mean geodesic dis- tance across a populations of real data and simulated graphs for different values of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The shaded area surrounding each curve shows the standard deviation across a population of 137 graphs in both real and simulated population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' As stated in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='1 the distance be- tween the nodes in the real graphs are larger than a minimum distance, which is illustrated by the flat portion of the blue curve for low geodesic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Our simulations do not reproduce this feature, as expected from the uniform sampling of the location of outliers nodes that can get close to previous nodes (figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Note that the fit is good for larger geodesic distance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Finally, we show on Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3 the distribution of the degree of nodes for simulated and real graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The degree corresponds to the number of neighbors of each node, and is thus indicative of the local topology of the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' This figure confirms the good match between simulated and real data, independently of κ that controls the perturbation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Overall, all our measures confirmed a good match between simulated and real graphs, for any value of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' xxxv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='10 Graphs original kappa_100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='08 kappa_200 kappa_400 kappa_1000 Density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='00 70 75 80 85 90 95 100 105 110 Number of nodesFigure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='2: Distribution for geodesic distance in the simulated and real population of 137 graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The shaded region corresponds to standard deviations across graphs in the popu- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='3: Degree distribution in the simulated and real population of 137 graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' The shaded region corresponds to standard deviations across graphs in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content=' xxxvi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='035 Simulations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='03 Real data 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='025 Proportion Kappa = 100 Proportion 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='02 Kappa = 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FRT4oBgHgl3EQfTTda/content/2301.13532v1.pdf'} +page_content='005 0 20 40 60 80 100 120 0 20 40 60 80 100 120 Geodesic distance Geodesic distance 0.' metadata={'source': 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Adinkras are graphical gadgets introduced by physicists to study +supersymmetry, which can be thought of as the Cayley graphs for supersym- +metry algebras. Improving the result of Iga et al., we determine the critical +group of an Adinkra given the 2-rank of the Laplacian of the underlying Cay- +ley graph. As a corollary, we show that the critical group is independent of +the signature of the Adinkra. The proof uses the monodromy pairing on these +critical groups. +1. Introduction +Computing the cokernel, or equivalently, the Smith Normal Form (SNF), of +an integer matrix is a recurrent topic in algebraic combinatorics [20]. +While +the definition is elementary, elaborated machinery has been used to approach +the task, and the answer to many seemingly simple matrices remain unknown. +A prominent notion within the topic is the critical group K(G) of a graph G, +which is (the torsion part of) the cokernel of the graph Laplacian L(G) [7, 18]. +A more specific direction is to compute the critical groups of graphs coming +from algebra, where ideas from representation theory, Gr¨obner theory, and p- +adic number theory have been applied [4, 6, 12, 14]. +In this paper, we study the critical groups of Adinkras, which are decorated +(colored, dashed) graphs introduced by physicists to encode special supersymme- +try algebras [13]. Roughly speaking, the vertices are particles, and the edges of +different colors correspond to the actions of different basis elements of the alge- +bra acting on these particles, in which an edge is dashed if the action flips the +sign of the particle. Therefore, Adinkras can be thought of as a signed analogue +of Cayley graphs for supersymmetry algebras. In [17], the authors initiated the +problem, and proved the following using a “deform to Z[x]-modules” argument: +Theorem 1.1. [17] Let A be an N-colored Adinkra on v vertices. Then the odd +component of K(A) is isomorphic to that of (Z/(N2 − N)Z)v/2. +However, the 2-Sylow subgroup of K(A) remains open, which seems tricky for +its connection with another problem. The underlying graph of an Adinkra is a +Cayley graph G of Fn +2, and the number of even invariant factors of K(A) equals +the corank of L(G) over F2, but this quantity is poorly-understood itself: even +the simplest case of hypercube was a conjecture of Reiner before being solved by +1 + +2 +CHI HO YUEN +Bai [1], and no good general description is known despite some recent progress +[14]. Nevertheless, we are able to bypass the apparent first hurdle and prove: +Theorem 1.2. Let A be an N-colored Adinkra on v vertices. Let the 2-rank of +the Laplacian of the underlying Cayley graph be v/2 − m; we have m ≥ 0 by +[14]. Then K(A) ∼= (Z/2Z)m ⊕ (Z/ N2−N +2 +Z)m ⊕ (Z/(N2 − N)Z)v/2−m. +Our proof differs from the main strategy in [17], and uses monodromy pairing +[19]. While the pairing is defined for any critical group, the regularity of Adinkras +yields a simple description of it [15]. We find a large “orthonormal” subset of +K(A), which implies that K(A) contains a large subgroup whose structure is +simple. Then we argue that the remaining part only depends on the 2-rank. To +the best of our knowledge, this is the first time the monodromy pairing is being +used to study the structure of specific critical groups. +As a corollary, we answer [17, Conjecture 1] in the affirmative: +Corollary 1.3. K(A) is independent of the signature of the Adinkra. +It is worth noting that while only the signed (dashed) graph structure of A was +used to define K(A), the existence of a compatible edge coloring is essential. It is +another curious instance in mathematics that admitting extra structure imposes +constraints on seemingly irrelevant invariants of the object. +2. Preliminaries +Unless otherwise specified, all (signed) graphs are finite, simple, and connected. +2.1. Adinkras. +Definition 2.1. A signed graph Gσ is a graph G together with an assignment +σ : E(G) → {±} (a signature of G). Switching a vertex flips the signs of the +edges incident to it. +Definition 2.2. For N ≥ 2, an N-colored Adinkra A is a signed graph with each +edge colored by one of N colors1, satisfying the following conditions: +(1) the graph is bipartite; +(2) every vertex is incident to exactly one edge of each color; +(3) for every pair of distinct colors, the graph restricted to edges of these +colors is a disjoint union of 4-cycles; +(4) each bi-colored 4-cycle contains an odd number of negative edges. +Example 2.3. In Figure 1, on the left is a 3-colored Adinkra supported on Q3, +and on the right is a 4-colored Adinkra supported on K4,4. Negative edges are +represented by dashed edges. +1Technically, the data here only defines a Cliffordinkra, as we need to further choose an acyclic +orientation of the underlying graph to define an Adinkra. However, there are no compatibility +conditions between the choice and the rest of the data, so we omit the orientation here. + +CRITICAL GROUPS OF ADINKRAS +3 +B +F +C +G +A +E +D +H +E +F +G +H +A +B +C +D +Figure 1. Two examples of Adinkras. +We have the following theorem concerning the underlying graph of an Adinkra. +Theorem 2.4. [8] A graph admits an Adinkra structure if and only if it is the +quotient of a hypercube Qt by a doubly even code (a subspace C of Ft +2 where the +size of the support of each element is divisible by 4), which is necessarily a Cayley +graph of Fn +2, where n = t − dim C. +2.2. Critical Groups and Their Monodromy Pairings. +Definition 2.5. The Laplacian of Gσ is L(Gσ) = D−Aσ, where D is the diagonal +matrix whose entries are the vertex degrees, and Aσ is the signed adjacency +matrix in which a positive (respectively, negative) edge xy is represented by +Axy = Ayx = 1 (respectively, −1). The Laplacian of an ordinary graph G can be +viewed/defined as L(Gσ) with σ ≡ +. +The critical group K(Gσ) is the torsion part of the cokernel of L(Gσ) over Z. In +the case of Adinkras (indeed, any unbalanced signed graphs), the cokernel itself +is finite, whereas it is always of rank 1 for ordinary graphs. +As two simple observations: (1) switching vertices perserves K(Gσ), and whether +the signed graph is an Adinkra; and (2) the rank of L(Gσ) over F2 (2-rank) is in- +dependent of σ, in particular, it is equal to that of L(G), and the number of even +invariant factors of K(Gσ) equals v minus (2-rank of L + rank of coker L(Gσ)). +The critical group of an Adinkra (or more generally, the torsion of the cokernel +of any symmetric integer matrix) is equipped with a canonical pairing ⟨·, ·⟩ taking +values in Q/Z, known as the monodromy pairing. It is related to several other +pairings in arithmetic geometry and discrete potential theory [3, 5]. +Definition 2.6. Let x, y ∈ Zv be two vectors representing two elements of K(A). +Choose a positive integer m such that L(A)f = mx for some f ∈ Zv. Then the +monodromy pairing between [x], [y] is ⟨[x], [y]⟩ := f T y +m ∈ Q/Z. +Proposition 2.7. [5, Lemma 1.1] The pairing is well-defined, bilinear, and sym- +metric. + +4 +CHI HO YUEN +3. Proof of the Main Theorem +Index the rows and columns of L(A) by V (A) ∼= Fn +2, and denote by {eu : u ∈ +V (A)} the standard basis of ZV (A). +We first collect some results on the Laplacians and critical groups of Adinkras +from [17] that can be obtained in a more elementary manner. +Theorem 3.1. Let A be an N-colored Adinkra on v vertices. Then L(A) has +exactly two distinct eigenvalues N ± +√ +N of equal multiplicities v/2, and |K(A)| = +det(L(A)) = (N2 − N)v/2. +The invariant factors f1 | f2 | . . . | fv of L(A) satisfy the relation fifv−i+1 = +N2 − N, ∀i. Moreover, for i > v/2, (N − 1) | fi. +The key (and neat) observation is that the signed boundaries of a family of +monochromatic edges are “orthonormal” with respect to ⟨·, ·⟩. +Proposition 3.2. Let A be an Adinkra of N ≥ 3 colors and let uv be an edge +of A of sign ǫ. Then ⟨eu − ǫev, eu − ǫev⟩ = +2 +N ̸= 0 ∈ Q/Z. Let xy be another +edge of the same color and of sign ǫ′. Then ⟨eu − ǫev, ex − ǫ′ey⟩ = 0. +Proof. Without loss of generality, we may assume ǫ = ǫ′ = + by switching. By +the first half of Theorem 3.1, L(A) has two distinct eigenvalues, so it satisfies the +condition in [15, Equation (3.3)], and we can solve m(eu − ev) = LAf by +(3.1) +(N2−N)(eu−ev) = LA[(N−1)eu−(N−1)ev+ +� +w∈N(u)\v +σ(uw)ew− +� +w∈N(v)\u +σ(vw)ew], +here N(x) is the neighborhood of the vertex x and σ(e) is the sign of the edge e. +By Definition 2.6 and (3.1), ⟨eu − ev, eu − ev⟩ = 2(N−1) +N2−N = 2 +N , which is non-zero +in Q/Z as N ≥ 3. +For the second statement, {x, y} and {u, v} are necessarily disjoint by (2) of +Definition 2.2. If x is adjacent to u along a positive edge of color c, (1) of Defini- +tion 2.2 guarantees that x is not adjacent to v, and (3) and (4) of Definition 2.2 +ensure y is adjacent to v along a negative edge of the same color but not to u. +Now (3.1) implies ⟨eu − ev, ex − ey⟩ = +1−1 +N2−N = 0; the cases when x is adjacent to +v and/or the edge is negative are essentially the same. The case when x is not +adjacent to u nor v is easier as y is not adjacent to u, v either, and the pairing is +simply +0−0 +N2−N = 0. +□ +Next, we state the result from [15] that explains how an orthonormal subset +implies a “rectangular” subgroup. +Theorem 3.3 ([15, Theorem 4.5]). Let G be a finite abelian group equipped with +a monodromy pairing ⟨·, ·⟩. Suppose there exist g1, . . . , gl ∈ G whose pairwise + +CRITICAL GROUPS OF ADINKRAS +5 +pairings are zero, and for every i, ⟨gi, gi⟩ = µ +η for relatively prime µ, η ∈ Z>0. +Then G contains a subgroup isomorphic to (Z/ηZ)l. +Proof of Theorem 1.2. The only Adinkra with parameter N = 2 is the 4-cycle +with 1 (or 3) negative edges, in which the theorem can be easily verified. +For N ≥ 3, fix a color of the Adinkra. By switching if necessary, we may +assume the v/2 edges of that color are all positive. +Applying the calculation +in Proposition 3.2 to Theorem 3.3 with η = N/ gcd(2, N), we know that K(A) +contains a subgroup isomorphic to (Z/ +N +gcd(2,N)Z)v/2. +Hence, by an elementary fact on invariant factors and subgroups (for reference, +see [15, Lemma 4.4]), for every i > v/2, +N +gcd(2,N) | fi. Combining that (N − 1) | +fi, ∀i > v/2, we have +N2−N +gcd(2,N) | fi, which in turn forces each fi with i ≤ v/2 to be +either 1 or 2 by the second half of Theorem 3.1. The number of even invariant +factors (necessarily 2) in the first half is m, so the number of invariant factors in +the second half that are equal to N2−N +2 +is also m, and the remaining non-trivial +invariant factors must be N2 − N, the claimed structure of K(A) follows. +□ +4. Non-generic Adinkras +Corollary 1.3 is straightforward from the main theorem. +Proof of Corollary 1.3. From the aforementioned observation, the signature does +not affect the 2-rank of the Laplacian, which determines the critical group. +□ +We recall some background of the corollary: while Theorem 2.4 classifies which +graphs admit an Adinkra structure, for a given such graph, there can be multiple +(even up to natural notions of isomorphism) Adinkra structures. In particular, +while K(A) is invariant under vertex switchings and color-preserving graph au- +tomorphisms, there can exist different Adinkra signatures on a graph G = Qt/C +that are not equivalent by these two operations, hence the original conjecture is +not a vacuous question. +Indeed, G admits inequivalent signatures if and only if C contains the all one +codeword 1 ∈ Ft +2 [11], and in the language of Cayley graphs, if and only if the +sum of generators is zero. These Cayley graphs are non-generic in the sense of +[14], and they are precisely the Cayley graphs on Fn +2 whose 2-rank drops below +2n−1, i.e., m > 0 in the main theorem. Therefore, the classes of Adinkras (or the +underlying graphs thereof) that behave non-trivially in terms of signatures and +critical groups turn out to be the same. +We use this opportunity to mention one more possible instance that the very +class of Adinkras is special. As referred to in the introduction, Theorem 1.1 was +proven by deforming the critical group into a Z[x]-module. This could be done by +considering the following matrix ˆL(A) over Z[x]: fix an arbitrary color c, replace +the diagonal entries of L(A) by x + (N − 1), and replace the off-diagonal entries + +6 +CHI HO YUEN +±1 corresponding to edges of color c by ±x. Since Z[x] is not a PID, it is not +obvious that the SNF of ˆL(A) exists, and it was conjectured in [17] that the SNF +exists if and only if K(A) ∼= (Z/(N2 − N)Z)v/2, which we now know the latter is +true if and only if A is generic. It was only stated in [17] as a fact without proof +that the forward direction is true, so we fill in the argument below: +Proposition 4.1. When A is non-generic, the SNF of ˆL(A) does not exist. +Proof. By [17, Corollary 29], the determinant of ˆL(A) is equal to +(2(N − 1)x + (N − 1)(N − 2))v/2. +Since A is non-generic, N must be an even number: e.g., if gcd(2, N) = 1, then +fi = (N − 1)N for all i > v/2 from the proof of the main theorem. +Suppose the SNF of ˆL(A) exists, and the invariant factors are ˆf1 | . . . | ˆfv. Then +whenever 2 or x + N−2 +2 +divides ˆfi for some i ≤ v/2, the same can be said for ˆfj +with j ≥ i, a contradiction to the fact that �v +i=1 ˆfi = ±2v/2(N −1)v/2(x+ N−2 +2 )v/2, +and that Z[x] is a UFD with 2, x + N−2 +2 +not dividing N − 1. Hence, ˆfi | (N − 1) +for i ≤ v/2. +The SNF of L(A) can be obtained from the SNF of ˆL(A) by setting x = 1 (see, +for example, [17, Lemma 27]), so the first half of the invariant factors of L(A) +must be all odd, a contradiction. +□ +5. Concluding Remarks +In some sense, Theorem 1.2 is the best possible result concerning K(A) unless +one is able to make progress on the 2-rank of Cayley graphs, which is a non- +trivial problem arguably orthogonal to the combinatorics of Adinkras2. However, +as Adinkras are related to multiple mathematical topics [16, 21], putting our +result in the context of those topics would be fruitful. +When studying the critical groups of Cayley graphs or many other graphs +from algebra, the results and/or their proofs are often directly related to the +algebraic origin of those graphs. On the contrary, the works on critical groups +of Adinkras so far mostly use the combinatorial axioms to develop alternative +algebraic setups for the problem. So it is interesting to interpret the result here +directly using supersymmetry algebras. For example, do supersymmetry algebras +corresponding to non-generic Adinkras also special in some way? +On the geometric side, every Adinkra can be canonically embedded to a Rie- +mann surface in the sense of Grothendieck’s dessins d’enfants, and some proper- +ties of those Riemann surfaces are related to the properties of Adinkras in a deep +manner [9, 10]. Meanwhile, the theory of critical groups is a discrete/tropical +2On the optimistic side, the author does not rule out the possibility of using Adinkras to +approach problems in Cayley graphs. + +CRITICAL GROUPS OF ADINKRAS +7 +analogue of the theory of divisors and Jacobians of algebraic curves [2]. Compar- +ing the two worlds via the embedding is another direction worth looking into. For +example, elements eu − ev’s considered in the proof of Proposition 3.2 are now +divisors on the Riemann surface, does the monodromy pairing on K(A) relate to +any notion there? +Finally, one can also ask if there are other families of graphs or signed graphs +whose critical groups can be approached in a similar fashion. More generally, can +the structure of every critical group be certified by demonstrating an “orthogonal +basis” with respect to ⟨·, ·⟩? +If not, how much information can the method +provide? +Acknowledgements +The author was supported by the Trond Mohn Foundation project “Algebraic +and Topological Cycles in Complex and Tropical Geometries” at the University +of Oslo. He also thanks Kevin Iga for reading an early draft. +References +[1] Hua Bai. On the critical group of the n-cube. Linear Algebra Appl., 369:251–261, 2003. +[2] Matthew Baker. Specialization of linear systems from curves to graphs. Algebra Number +Theory, 2(6):613–653, 2008. With an appendix by Brian Conrad. +[3] Matthew Baker and Farbod Shokrieh. Chip-firing games, potential theory on graphs, and +spanning trees. J. Combin. Theory Ser. A, 120(1):164–182, 2013. +[4] Georgia Benkart, Caroline Klivans, and Victor Reiner. Chip firing on Dynkin diagrams +and McKay quivers. Math. Z., 290(1-2):615–648, 2018. +[5] Siegfried Bosch and Dino Lorenzini. Grothendieck’s pairing on component groups of Jaco- +bians. Invent. Math., 148(2):353–396, 2002. +[6] David B. Chandler, Peter Sin, and Qing Xiang. The Smith and critical groups of Paley +graphs. J. Algebraic Combin., 41(4):1013–1022, 2015. +[7] Scott Corry and David Perkinson. Divisors and sandpiles. American Mathematical Society, +Providence, RI, 2018. +[8] C. F. Doran, M. G. Faux, S. J. Gates Jr., T. H¨ubsch, K. M. Iga, G. D. Landweber, and +R. L. Miller. Codes and Supersymmetry in One Dimension. Advances in Theoretical and +Mathematical Physics, 15(6):1909–1970, 2011. +[9] Charles Doran, Kevin Iga, Jordan Kostiuk, Greg Landweber, and Stefan M´endez-Diez. Ge- +ometrization of N-extended 1-dimensional supersymmetry algebras, I. Adv. Theor. Math. +Phys., 19(5):1043–1113, 2015. +[10] Charles Doran, Kevin Iga, Jordan Kostiuk, and Stefan M´endez-Diez. Geometrization of N- +extended 1-dimensional supersymmetry algebras, II. Adv. Theor. Math. Phys., 22(3):565– +613, 2018. +[11] Charles F. Doran, Kevin M. Iga, and Gregory D. Landweber. An application of cubical +cohomology to Adinkras and supersymmetry representations. Ann. Inst. Henri Poincar´e +D, 4(3):387–415, 2017. +[12] Joshua E. Ducey and Deelan M. Jalil. Integer invariants of abelian Cayley graphs. Linear +Algebra Appl., 445:316–325, 2014. + +8 +CHI HO YUEN +[13] Michael Faux and S. James Gates, Jr. Adinkras: A graphical technology for supersymmet- +ric representation theory. Phys. Rev. D (3), 71:065002, 2005. +[14] Jiyang Gao, Jared Marx-Kuo, Vaughan McDonald, and Chi Ho Yuen. Sandpile groups of +Cayley graphs of Fr +2, 2022. https://arxiv.org/abs/1912.06919. +[15] Kenneth Hung and Chi Ho Yuen. Critical groups of strongly regular graphs and their +generalizations. Innov. Incidence Geom., 19(3):95–109, 2022. +[16] Kevin Iga. Adinkras: Graphs of Clifford Algebra Representations, Supersymmetry, and +Codes. Adv. Appl. Clifford Algebr., 31(5):Paper No. 76, 2021. +[17] Kevin Iga, Caroline Klivans, Jordan Kostiuk, and Chi Ho Yuen. Eigenvalues and critical +groups of Adinkras. Adv. in Appl. Math., 143:Paper No. 102450, 2023. +[18] Caroline J. Klivans. The mathematics of chip-firing. Discrete Mathematics and its Appli- +cations (Boca Raton). CRC Press, Boca Raton, FL, 2019. +[19] Farbod Shokrieh. The monodromy pairing and discrete logarithm on the Jacobian of finite +graphs. J. Math. Cryptol., 4(1):43–56, 2010. +[20] Richard P. Stanley. Smith normal form in combinatorics. Journal of Combinatorial Theory, +Series A, 144:476–495, 2016. Fifty Years of the Journal of Combinatorial Theory. +[21] Yan X Zhang. Adinkras for mathematicians. Transactions of the American Mathematical +Society, 366(6):3325–3355, 2014. +Chi Ho Yuen: Department of Mathematics, University of Oslo, Oslo, Norway +Email address: chihy@math.uio.no + diff --git a/cNE0T4oBgHgl3EQfnwGk/content/tmp_files/load_file.txt b/cNE0T4oBgHgl3EQfnwGk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2a56b84bef1abcec50f9217e852bb9ac6c5df3b6 --- /dev/null +++ b/cNE0T4oBgHgl3EQfnwGk/content/tmp_files/load_file.txt @@ -0,0 +1,331 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf,len=330 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='02517v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='CO] 6 Jan 2023 THE CRITICAL GROUPS OF ADINKRAS UP TO 2-RANK OF CAYLEY GRAPHS CHI HO YUEN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Adinkras are graphical gadgets introduced by physicists to study supersymmetry, which can be thought of as the Cayley graphs for supersym- metry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Improving the result of Iga et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=', we determine the critical group of an Adinkra given the 2-rank of the Laplacian of the underlying Cay- ley graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' As a corollary, we show that the critical group is independent of the signature of the Adinkra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The proof uses the monodromy pairing on these critical groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Introduction Computing the cokernel, or equivalently, the Smith Normal Form (SNF), of an integer matrix is a recurrent topic in algebraic combinatorics [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' While the definition is elementary, elaborated machinery has been used to approach the task, and the answer to many seemingly simple matrices remain unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' A prominent notion within the topic is the critical group K(G) of a graph G, which is (the torsion part of) the cokernel of the graph Laplacian L(G) [7, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' A more specific direction is to compute the critical groups of graphs coming from algebra, where ideas from representation theory, Gr¨obner theory, and p- adic number theory have been applied [4, 6, 12, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' In this paper, we study the critical groups of Adinkras, which are decorated (colored, dashed) graphs introduced by physicists to encode special supersymme- try algebras [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Roughly speaking, the vertices are particles, and the edges of different colors correspond to the actions of different basis elements of the alge- bra acting on these particles, in which an edge is dashed if the action flips the sign of the particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Therefore, Adinkras can be thought of as a signed analogue of Cayley graphs for supersymmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' In [17], the authors initiated the problem, and proved the following using a “deform to Z[x]-modules” argument: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [17] Let A be an N-colored Adinkra on v vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then the odd component of K(A) is isomorphic to that of (Z/(N2 − N)Z)v/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' However, the 2-Sylow subgroup of K(A) remains open, which seems tricky for its connection with another problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The underlying graph of an Adinkra is a Cayley graph G of Fn 2, and the number of even invariant factors of K(A) equals the corank of L(G) over F2, but this quantity is poorly-understood itself: even the simplest case of hypercube was a conjecture of Reiner before being solved by 1 2 CHI HO YUEN Bai [1], and no good general description is known despite some recent progress [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Nevertheless, we are able to bypass the apparent first hurdle and prove: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Let A be an N-colored Adinkra on v vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Let the 2-rank of the Laplacian of the underlying Cayley graph be v/2 − m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' we have m ≥ 0 by [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then K(A) ∼= (Z/2Z)m ⊕ (Z/ N2−N 2 Z)m ⊕ (Z/(N2 − N)Z)v/2−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Our proof differs from the main strategy in [17], and uses monodromy pairing [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' While the pairing is defined for any critical group, the regularity of Adinkras yields a simple description of it [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' We find a large “orthonormal” subset of K(A), which implies that K(A) contains a large subgroup whose structure is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then we argue that the remaining part only depends on the 2-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' To the best of our knowledge, this is the first time the monodromy pairing is being used to study the structure of specific critical groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' As a corollary, we answer [17, Conjecture 1] in the affirmative: Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' K(A) is independent of the signature of the Adinkra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' It is worth noting that while only the signed (dashed) graph structure of A was used to define K(A), the existence of a compatible edge coloring is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' It is another curious instance in mathematics that admitting extra structure imposes constraints on seemingly irrelevant invariants of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Preliminaries Unless otherwise specified, all (signed) graphs are finite, simple, and connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Adinkras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' A signed graph Gσ is a graph G together with an assignment σ : E(G) → {±} (a signature of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Switching a vertex flips the signs of the edges incident to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' For N ≥ 2, an N-colored Adinkra A is a signed graph with each edge colored by one of N colors1, satisfying the following conditions: (1) the graph is bipartite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' (2) every vertex is incident to exactly one edge of each color;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' (3) for every pair of distinct colors, the graph restricted to edges of these colors is a disjoint union of 4-cycles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' (4) each bi-colored 4-cycle contains an odd number of negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' In Figure 1, on the left is a 3-colored Adinkra supported on Q3, and on the right is a 4-colored Adinkra supported on K4,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Negative edges are represented by dashed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' 1Technically, the data here only defines a Cliffordinkra, as we need to further choose an acyclic orientation of the underlying graph to define an Adinkra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' However, there are no compatibility conditions between the choice and the rest of the data, so we omit the orientation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' CRITICAL GROUPS OF ADINKRAS 3 B F C G A E D H E F G H A B C D Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Two examples of Adinkras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' We have the following theorem concerning the underlying graph of an Adinkra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [8] A graph admits an Adinkra structure if and only if it is the quotient of a hypercube Qt by a doubly even code (a subspace C of Ft 2 where the size of the support of each element is divisible by 4), which is necessarily a Cayley graph of Fn 2, where n = t − dim C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Critical Groups and Their Monodromy Pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The Laplacian of Gσ is L(Gσ) = D−Aσ, where D is the diagonal matrix whose entries are the vertex degrees, and Aσ is the signed adjacency matrix in which a positive (respectively, negative) edge xy is represented by Axy = Ayx = 1 (respectively, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The Laplacian of an ordinary graph G can be viewed/defined as L(Gσ) with σ ≡ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The critical group K(Gσ) is the torsion part of the cokernel of L(Gσ) over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' In the case of Adinkras (indeed, any unbalanced signed graphs), the cokernel itself is finite, whereas it is always of rank 1 for ordinary graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' As two simple observations: (1) switching vertices perserves K(Gσ), and whether the signed graph is an Adinkra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' and (2) the rank of L(Gσ) over F2 (2-rank) is in- dependent of σ, in particular, it is equal to that of L(G), and the number of even invariant factors of K(Gσ) equals v minus (2-rank of L + rank of coker L(Gσ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The critical group of an Adinkra (or more generally, the torsion of the cokernel of any symmetric integer matrix) is equipped with a canonical pairing ⟨·, ·⟩ taking values in Q/Z, known as the monodromy pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' It is related to several other pairings in arithmetic geometry and discrete potential theory [3, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Let x, y ∈ Zv be two vectors representing two elements of K(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Choose a positive integer m such that L(A)f = mx for some f ∈ Zv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then the monodromy pairing between [x], [y] is ⟨[x], [y]⟩ := f T y m ∈ Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [5, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1] The pairing is well-defined, bilinear, and sym- metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' 4 CHI HO YUEN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Proof of the Main Theorem Index the rows and columns of L(A) by V (A) ∼= Fn 2, and denote by {eu : u ∈ V (A)} the standard basis of ZV (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' We first collect some results on the Laplacians and critical groups of Adinkras from [17] that can be obtained in a more elementary manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Let A be an N-colored Adinkra on v vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then L(A) has exactly two distinct eigenvalues N ± √ N of equal multiplicities v/2, and |K(A)| = det(L(A)) = (N2 − N)v/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The invariant factors f1 | f2 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' | fv of L(A) satisfy the relation fifv−i+1 = N2 − N, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Moreover, for i > v/2, (N − 1) | fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The key (and neat) observation is that the signed boundaries of a family of monochromatic edges are “orthonormal” with respect to ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Let A be an Adinkra of N ≥ 3 colors and let uv be an edge of A of sign ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then ⟨eu − ǫev, eu − ǫev⟩ = 2 N ̸= 0 ∈ Q/Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Let xy be another edge of the same color and of sign ǫ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then ⟨eu − ǫev, ex − ǫ′ey⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Without loss of generality, we may assume ǫ = ǫ′ = + by switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' By the first half of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1, L(A) has two distinct eigenvalues, so it satisfies the condition in [15, Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='3)], and we can solve m(eu − ev) = LAf by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1) (N2−N)(eu−ev) = LA[(N−1)eu−(N−1)ev+ � w∈N(u)\\v σ(uw)ew− � w∈N(v)\\u σ(vw)ew], here N(x) is the neighborhood of the vertex x and σ(e) is the sign of the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' By Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='6 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1), ⟨eu − ev, eu − ev⟩ = 2(N−1) N2−N = 2 N , which is non-zero in Q/Z as N ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' For the second statement, {x, y} and {u, v} are necessarily disjoint by (2) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' If x is adjacent to u along a positive edge of color c, (1) of Defini- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2 guarantees that x is not adjacent to v, and (3) and (4) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2 ensure y is adjacent to v along a negative edge of the same color but not to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Now (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1) implies ⟨eu − ev, ex − ey⟩ = 1−1 N2−N = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' the cases when x is adjacent to v and/or the edge is negative are essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The case when x is not adjacent to u nor v is easier as y is not adjacent to u, v either, and the pairing is simply 0−0 N2−N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' □ Next, we state the result from [15] that explains how an orthonormal subset implies a “rectangular” subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='3 ([15, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Let G be a finite abelian group equipped with a monodromy pairing ⟨·, ·⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Suppose there exist g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' , gl ∈ G whose pairwise CRITICAL GROUPS OF ADINKRAS 5 pairings are zero, and for every i, ⟨gi, gi⟩ = µ η for relatively prime µ, η ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then G contains a subgroup isomorphic to (Z/ηZ)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The only Adinkra with parameter N = 2 is the 4-cycle with 1 (or 3) negative edges, in which the theorem can be easily verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' For N ≥ 3, fix a color of the Adinkra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' By switching if necessary, we may assume the v/2 edges of that color are all positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Applying the calculation in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2 to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='3 with η = N/ gcd(2, N), we know that K(A) contains a subgroup isomorphic to (Z/ N gcd(2,N)Z)v/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Hence, by an elementary fact on invariant factors and subgroups (for reference, see [15, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='4]), for every i > v/2, N gcd(2,N) | fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Combining that (N − 1) | fi, ∀i > v/2, we have N2−N gcd(2,N) | fi, which in turn forces each fi with i ≤ v/2 to be either 1 or 2 by the second half of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The number of even invariant factors (necessarily 2) in the first half is m, so the number of invariant factors in the second half that are equal to N2−N 2 is also m, and the remaining non-trivial invariant factors must be N2 − N, the claimed structure of K(A) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Non-generic Adinkras Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='3 is straightforward from the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Proof of Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' From the aforementioned observation, the signature does not affect the 2-rank of the Laplacian, which determines the critical group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' □ We recall some background of the corollary: while Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='4 classifies which graphs admit an Adinkra structure, for a given such graph, there can be multiple (even up to natural notions of isomorphism) Adinkra structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' In particular, while K(A) is invariant under vertex switchings and color-preserving graph au- tomorphisms, there can exist different Adinkra signatures on a graph G = Qt/C that are not equivalent by these two operations, hence the original conjecture is not a vacuous question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Indeed, G admits inequivalent signatures if and only if C contains the all one codeword 1 ∈ Ft 2 [11], and in the language of Cayley graphs, if and only if the sum of generators is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' These Cayley graphs are non-generic in the sense of [14], and they are precisely the Cayley graphs on Fn 2 whose 2-rank drops below 2n−1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=', m > 0 in the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Therefore, the classes of Adinkras (or the underlying graphs thereof) that behave non-trivially in terms of signatures and critical groups turn out to be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' We use this opportunity to mention one more possible instance that the very class of Adinkras is special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' As referred to in the introduction, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1 was proven by deforming the critical group into a Z[x]-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' This could be done by considering the following matrix ˆL(A) over Z[x]: fix an arbitrary color c, replace the diagonal entries of L(A) by x + (N − 1), and replace the off-diagonal entries 6 CHI HO YUEN ±1 corresponding to edges of color c by ±x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Since Z[x] is not a PID, it is not obvious that the SNF of ˆL(A) exists, and it was conjectured in [17] that the SNF exists if and only if K(A) ∼= (Z/(N2 − N)Z)v/2, which we now know the latter is true if and only if A is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' It was only stated in [17] as a fact without proof that the forward direction is true, so we fill in the argument below: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' When A is non-generic, the SNF of ˆL(A) does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' By [17, Corollary 29], the determinant of ˆL(A) is equal to (2(N − 1)x + (N − 1)(N − 2))v/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Since A is non-generic, N must be an even number: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=', if gcd(2, N) = 1, then fi = (N − 1)N for all i > v/2 from the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Suppose the SNF of ˆL(A) exists, and the invariant factors are ˆf1 | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' | ˆfv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Then whenever 2 or x + N−2 2 divides ˆfi for some i ≤ v/2, the same can be said for ˆfj with j ≥ i, a contradiction to the fact that �v i=1 ˆfi = ±2v/2(N −1)v/2(x+ N−2 2 )v/2, and that Z[x] is a UFD with 2, x + N−2 2 not dividing N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Hence, ˆfi | (N − 1) for i ≤ v/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The SNF of L(A) can be obtained from the SNF of ˆL(A) by setting x = 1 (see, for example, [17, Lemma 27]), so the first half of the invariant factors of L(A) must be all odd, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Concluding Remarks In some sense, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2 is the best possible result concerning K(A) unless one is able to make progress on the 2-rank of Cayley graphs, which is a non- trivial problem arguably orthogonal to the combinatorics of Adinkras2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' However, as Adinkras are related to multiple mathematical topics [16, 21], putting our result in the context of those topics would be fruitful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' When studying the critical groups of Cayley graphs or many other graphs from algebra, the results and/or their proofs are often directly related to the algebraic origin of those graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' On the contrary, the works on critical groups of Adinkras so far mostly use the combinatorial axioms to develop alternative algebraic setups for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' So it is interesting to interpret the result here directly using supersymmetry algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' For example, do supersymmetry algebras corresponding to non-generic Adinkras also special in some way?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' On the geometric side, every Adinkra can be canonically embedded to a Rie- mann surface in the sense of Grothendieck’s dessins d’enfants, and some proper- ties of those Riemann surfaces are related to the properties of Adinkras in a deep manner [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Meanwhile, the theory of critical groups is a discrete/tropical 2On the optimistic side, the author does not rule out the possibility of using Adinkras to approach problems in Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' CRITICAL GROUPS OF ADINKRAS 7 analogue of the theory of divisors and Jacobians of algebraic curves [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Compar- ing the two worlds via the embedding is another direction worth looking into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' For example, elements eu − ev’s considered in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='2 are now divisors on the Riemann surface, does the monodromy pairing on K(A) relate to any notion there?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Finally, one can also ask if there are other families of graphs or signed graphs whose critical groups can be approached in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' More generally, can the structure of every critical group be certified by demonstrating an “orthogonal basis” with respect to ⟨·, ·⟩?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' If not, how much information can the method provide?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Acknowledgements The author was supported by the Trond Mohn Foundation project “Algebraic and Topological Cycles in Complex and Tropical Geometries” at the University of Oslo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' He also thanks Kevin Iga for reading an early draft.' metadata={'source': 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+page_content=' Landweber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' An application of cubical cohomology to Adinkras and supersymmetry representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Henri Poincar´e D, 4(3):387–415, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [12] Joshua E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Ducey and Deelan M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Jalil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Integer invariants of abelian Cayley graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=', 445:316–325, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' 8 CHI HO YUEN [13] Michael Faux and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' James Gates, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Adinkras: A graphical technology for supersymmet- ric representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' D (3), 71:065002, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [14] Jiyang Gao, Jared Marx-Kuo, Vaughan McDonald, and Chi Ho Yuen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Sandpile groups of Cayley graphs of Fr 2, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='org/abs/1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='06919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [15] Kenneth Hung and Chi Ho Yuen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Critical groups of strongly regular graphs and their generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Innov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Incidence Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=', 19(3):95–109, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [16] Kevin Iga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Adinkras: Graphs of Clifford Algebra Representations, Supersymmetry, and Codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Clifford Algebr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=', 31(5):Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' 76, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [17] Kevin Iga, Caroline Klivans, Jordan Kostiuk, and Chi Ho Yuen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Eigenvalues and critical groups of Adinkras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Adv.' metadata={'source': 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mathematics of chip-firing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Discrete Mathematics and its Appli- cations (Boca Raton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' CRC Press, Boca Raton, FL, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [19] Farbod Shokrieh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' The monodromy pairing and discrete logarithm on the Jacobian of finite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Cryptol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=', 4(1):43–56, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [20] Richard P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Smith normal form in combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Journal of Combinatorial Theory, Series A, 144:476–495, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Fifty Years of the Journal of Combinatorial Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' [21] Yan X Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Adinkras for mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Transactions of the American Mathematical Society, 366(6):3325–3355, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content=' Chi Ho Yuen: Department of Mathematics, University of Oslo, Oslo, Norway Email address: chihy@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='uio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} +page_content='no' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cNE0T4oBgHgl3EQfnwGk/content/2301.02517v1.pdf'} diff --git a/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf b/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7522d19afa9a5d9edf474ee0ad31adf86eb5c2d7 --- /dev/null +++ b/cdAzT4oBgHgl3EQfLfvk/content/2301.01117v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b627ada203100b1d2e3ccacb7ac6ad026aa81a4c9c8531c211c14adec0166180 +size 275583 diff --git a/cdAzT4oBgHgl3EQfLfvk/vector_store/index.faiss b/cdAzT4oBgHgl3EQfLfvk/vector_store/index.faiss new file mode 100644 index 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a/edFPT4oBgHgl3EQfzjXc/content/tmp_files/2301.13176v1.pdf.txt b/edFPT4oBgHgl3EQfzjXc/content/tmp_files/2301.13176v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..71849cddafc77d2074180f4d9e85eebc5e7f135e --- /dev/null +++ b/edFPT4oBgHgl3EQfzjXc/content/tmp_files/2301.13176v1.pdf.txt @@ -0,0 +1,1537 @@ +Graphene Oxide Photoreduction Recovers Graphene Hot Electron Cooling Dynamics +Alden N. Bradley1∗, Spencer G. Thorp1∗, Gina Mayonado1, Edward Elliott2, Matt W. Graham1 +1. Department of Physics, Oregon State University, Corvallis, OR, 97331, USA and +2.Voxtel Nano, Corvallis, OR, 97330, USA +Reduced graphene oxide (rGO) is a bulk-processable quasi-amorphous 2D material with broad +spectral coverage and fast electronic response. rGO sheets are suspended in a polymer matrix and +sequentially photoreduced while measuring the evolving optical spectra and ultrafast electron relax- +ation dynamics. Photoreduced rGO yields optical absorption spectra that fit with the same Fano +lineshape parameters as monolayer graphene. With increasing photoreduction time, rGO transient +absorption kinetics accelerate monotonically, reaching an optimal point that matches the hot elec- +tron cooling in graphene. All stages of rGO ultrafast kinetics are simulated with a hot-electron +cooling model mediated by disorder-assisted supercollisions. While the rGO room temperature 0.31 +ps−1 electronic cooling rate matches monolayer graphene, subsequent photoreduction can rapidly +increase the rate by 10-12×. Such accelerated supercollision rates imply a reduced mean-free scat- +tering length caused by photoionized point-defects on the rGO sp2 sub-lattice. For visible range +excitations of rGO, photoreduction shows three increasing spectral peaks that match graphene quan- +tum dot (GQD) transitions, while a broad peak from oxygenated defect edge states shrinks. These +three confined GQD states donate their hot carriers to the graphene sub-lattice with a 0.17 ps rise- +time that accelerates with photoreduction. Collectively, many desirable photophysical properties of +2D graphene are replicated through selectively reducing rGO scaffolded within a 3D bulk polymeric +network. +∗ co-authors contributed equally. +I. +INTRODUCTION +Graphene oxides (GO) are a widely-used substitute +for graphene’s remarkable mechanical properties, but its +highly amorphous lattice lacks desirable electronic prop- +erties such as high conductivity, fast photoresponse and +broad spectral coverage. +When GO is incorporated in +certain polymeric networks, we show systematic photore- +duction makes it more graphene-like while maintaining +pristine optical-quality films. GO has oxygenated func- +tional groups attached to the 2D carbon lattice via out- +of-plane bonds that prevent GO sheets from aggregating +in solution phase.1,2 GO can be made more graphene-like +by chemical or photothermal reduction to make reduced +graphene oxide (rGO). Conventionally, these graphene- +like rGO layers aggregate and scatter light strongly, +making their optical properties hard to compare against +monolayer(ml) graphene. Using systematic reduction of +isolated GO-in polymer composites, we show the emer- +gence of spectral lineshapes and extract ultrafast hot- +electron cooling dynamics that are closely analogous to +that of ml-graphene. +GO is often used as a bulk-processable substitute for +graphene for wide-ranging applications, including elec- +tronic sensing, plasmonics, and desalination.3–9 The large +presence of oxygen in GO introduces an effective band +gap (Fig. 1a inset), with a tunable energy determined by +the carbon-to-oxygen ratio. Previous theoretical and ex- +perimental studies suggest bandgaps ranging from ∼0.6- +3.1 eV for GO that can vanish nearly completely as +GO is reduced.10 GO samples reduced via pulsed Xe +arc lamps effectively remove hydroxyl, epoxy, and car- +boxyl groups to increase the size of graphene-like sp2 +regions. The amount of photoreduction changes the ra- +tio of the oxygenated-sp3 to conjugated-sp2 sub-lattice +regions.11–13 Very selective growths and controlled re- +duction are required to realize desired optoelectronic ap- +plications for GO that have included broadband optical +nonlinearity14,15, tunable photoluminescence,16 and res- +onant energy transfer.17 +With widely-varying ratios of oxygen and carbon, the +highly inhomogeneous and amorphous nature of GO and +rGO lattice make a direct comparison to ml-graphene +difficult. +In rGO, individual sp2 graphene-like sub- +lattice regions often become surrounded by sp3 oxidized +domains, forming molecular-like confined regions often +called graphene quantum dots (GQDs) or graphene nan- +oclusters. While the composition of rGO varies greatly, +it can roughly be decomposed into three types of sub- +lattice illustrated in Fig. 1b: (1) extended sp2 hybridized +regions, (2) confined sp2 lattice nanoclusters or GQDs, +and (3) oxidized or sp3 regions. Zhang et. al performed +transient absorption on rGO in solution and found that +the carbon (sp2) and oxidized domains (sp3) could be +treated independently.18,19 Photoexcited carriers in the +spatially-confined sp2 GQDs produce Frenkel excitons +with energies tunable with the size of the GQD con- +jugation network.20,21 The local oxygenated functional +groups at domain edges also create many optically active +defect states within the lattice that are seen in photolu- +minescence studies.22–24 +While some of the mechanical and chemical properties +of GO-based materials are analogous to graphene, the +conditions necessary to replicate graphene-like electronic +behavior in rGO are less clear. Past studies have com- +pared the transient absorption (TA) response of GO and +rGO prepared by chemical reduction in solution25 and +thin films.24,26 This study concerns the optical properties +arXiv:2301.13176v1 [cond-mat.mtrl-sci] 30 Jan 2023 + +2 +of GO and rGO embedded in a transparent polymer film +over six controlled degrees of photoreduction. The TA +relaxation resolves how the ultrafast hot electron cooling +rate is modified at each stage of photoreduction using +tunable probe energies ranging from 1.2 to 2.3 eV. While +the hot electron cooling in graphene is typically modeled +with two rates associated with optical phonon scattering +and disorder-assisted relaxation processes,27–29 In addi- +tion to graphene-like relaxation, prior rGO studies are +dominated by a long, 10-200 ps relaxation component +previously ascribed to electron trapping at defect sites.30 +The results obtained from the succession photoreduc- +tion of GO are modeled with first-principle models of ab- +sorption lineshapes and hot-electron cooling applied pre- +viously to graphene. In Section IV.A, the evolution of the +absorption lineshape with photoreduction is modeled by +competing contributions from graphene-like Fano line- +shape and GO-oxide-related absorption. +Then Section +IV.B applies a hot electron supercollision model to deter- +mine at what stage of photoreduction rGO most closely +matches the dynamics of ml-graphene. Over most visi- +ble and UV excitation energies, Section IV.C shows the +GO-sub-lattice and graphene quantum-dot states domi- +nate both the photoluminescence and ultrafast response. +Lastly, we resolve how photoreduction of GO impacts the +ultrafast rate of acceptor-donor electron transfer from the +photoexcited GQDs to graphene acceptor states. +II. +EXPERIMENTAL METHODS +The GO and rGO polymer samples were fabri- +cated using commercially available chemically exfoliated +graphene oxide sheets (Graphenea) containing ∼53% car- +bon and ∼44% oxygen. The sheets are dispersed in a +N, N-dimethylacrylamide (DMAA) polymer with added +PMMA sites to scaffold the GO and minimize aggrega- +tion. The mixture is cured between two 1 mm thick glass +slides, resulting in a sample thickness of 220 microns. +The sample is then photo-reduced via a pulsed Xenon +arc lamp at a 1 Hz repetition rate. This low frequency +was chosen to prevent gas bubbles from forming during +the reduction process. Absorbance is measured via Cary +IR-UV-Vis spectrometer. Both excitation and emission +photoluminescence are detected with a commercial fluo- +rimeter (Horiba NanoLog). +Both degenerate and non-degenerate pump-probe ex- +periments are conducted with 140 fs pulses from a +Ti:sapphire lasers (Coherent Chameleon) and Optical +Parametric Oscillators (APE OPO Compact). An opti- +cal parametric amplifier is used to tune the output wave- +length. The beam is split into two parts, a strong pump +and a weaker probe power beam with a ratio of ∼10:1. +The intensity of the pump beam is modulated using an +acousto-optic modulator (AOM, Crystal Tech) at 500 +kHz. The polarization of the pump and probe beam is lin- +ear and set parallel to each other. For the non-degenerate +experiment, the pump beam is frequency doubled by a +0 +1 +2 +3 +4 +0.01 +0.1 +1 +DT/T, normalized +Time Delay (ps) + + + + + + + +250 +500 +750 +1000 +1250 +1500 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Transmittance, T +Wavelength (nm) +d. +photo +reduction +GOsolution +rGO +rGO5 +GOo +rGO4 +rGO1 +rGO2 +rGO3 +Epr +rGO5 +GOo +rGO4 +rGO3/2/1 +sp3 +sp3 +sp3 +sp3 +sp3 +sp3 +sp3 +sp2-GQDs +sp2 +graphene +sublattice +sp3 +sp3 +sp3 +c. +b. +Graphene oxide (GO) +Reduced GO (rGO) +most +reduction +a. +M-pt +GOsolution +GOsolution ++polymer +Fano-resonance +0 +1 +2 +3 +4 +5 +0.9 +1.0 +1.1 +1.2 +GO photoreduction time (a.u.) +Relaxation Time t2 (ps) +most +oxidized +sp2 +FIG. 1. (a) +Comparison of GO vs. rGO band with chem- +ical structures. (b) Illustration of the three prominent sub- +lattices types within the rGO structure (sp2, sp2 graphene +quantum dot (GQD) and oxygenated sp3-lattice). (c) Lin- +ear and transient absorption spectra are measured at five +stages of the photoreduction. +With increasing photoreduc- +tion, NIR transmittance decreases to more closely approx- +imate the (renormalized) CVD ml-graphene transmittance +curve. Conversely, as grown GO in solution (gray line) has a +prominent π − π∗ bandgap. (inset) Graphene band structure +highlighting the M-saddle point transition. (d) Correspond- +ing transient transmittance kinetics at Eprobe=1.8 eV show +carrier relaxation accelerates with reduction. (inset) The τ2 +lifetime increases linearly with photoreduction. + +E +kE +元-band +T-band3 +second harmonic generation unit (OPE SHG) prior to +modulation. Alternatively, a white-light supercontinuum +is generated to provide a broadly tunable probe. Both +beams are focused onto the sample by a single lens. The +probe beam waist at the sample is approximately 80 mi- +crons. The transmitted probe-beam is detected by pho- +todiode lock-in amplification (Zurich Instruments, HFLI +and MFLI) at 500 kHz modulation. +To compare the rGO polymer physics to ml-graphene, +similar measurements to the above were carried out using +an ultrafast transient absorption (TA) microscopy setup +with a 1 µm spot size. The ml-graphene was prepared by +chemical vapor deposition (CVD) and wet-transferred to +a thin silicon nitride grid. +The above non-degenerate +pump-probe scheme was used in a collinear geometry +coupled to a 4f -confocal scanning microscope (Olym- +pus BX51W). The absorption spectra of ml-graphene are +taken on the same microscope by coupling in a tunable +Xe-arc illumination source and detecting the full plane +images on a camera (EMCCD, PI-ProEM) camera after +background renormalization. +III. +RESULTS +Spanning the UV to near-IR regions, Fig. 1c plots the +absolute linear transmission of six graphene oxide (GO) +samples in a polymer composite with increasing pho- +tothermal reduction times labeled from rGO1 to rGO5. +Additionally plotted on a renormalized scale, we overlay +the linear absorption spectra of both pristine monolayer +(ml) graphene (black line), and the starting as-grown +commercial GO solution (gray line, GOsolution). +The +GO solution has a clear bandgap, peaking at the molec- +ular π − π∗ transition. Conversely, ml-graphene gives an +expected Fano resonance lineshape peaked at 265 nm, +red-shifted from the M-saddle-point transition labeled in +Fig. 1c (inset).31 The rGOo curve in Fig. 1c is the ‘as- +grown’ GO after incorporation into a hybrid polyacrylic +and PMMA polymer matrix described in the methods. +The absolute absorbance increases monotonically with +GO photothermal reduction time over the NIR and IR +regions plotted (from 0.35 eV to 1.5 eV). Photoreduction +of GO leads to a spectral lineshape that absorbs light +more analogously to CVD monolayer graphene plotted +in Fig. 1c. +In the solution phase and most polymers, GO aggre- +gates as it is reduced, resulting in colloidal mixtures that +strongly scatter light. GO is incorporated in a polymer- +sphere matrix scaffold that makes systematic photore- +duction possible while maintaining pristine optical qual- +ity films. +Thus, we are able to compare the absorp- +tion lineshapes, photoluminescence, and ultrafast hot +electron cooling rates over a wide range of photoreduc- +tion. Interestingly, the more heavily reduced graphene +oxide samples in Fig. 1c have a transmittance lineshape +and slope similar to ml-graphene throughout the near- +infrared (NIR) regions. In the supplementary Fig. S2, +this absorption spectrum is extended out past 3 µm to +the IR-region where the strong similarity to graphene ab- +sorption is maintained. +Figure 1d plots the normalized transient transmission +(∆T/T, semi-log scale) kinetics of sequentially photore- +duced GO/rGO samples acquired with a 1.8 eV degener- +ate pump and probe configuration. As the degree of re- +duction increases, the kinetic relaxation rate accelerates. +The data shown in both Figs. 1 and 2 fits (solid lines) to +a least-squares algorithm requiring three-exponents (τ1, +τ2, and τ3) with pulse deconvolution for the 155 fs laser +autocorrelation response. After GO is incorporated and +stabilized in the polymer matrix, the relaxation dynamics +accelerate monotonically with photoreduction time. In +stark contrast, the as-grown solution of GO (gray line in +Fig. 1d) has much longer TA relaxation dynamics at all +timescales, bearing little resemblance to faster graphene. +At a 1.8 eV visible probe energy, the GO polymer +composite that received no reduction (highest oxygen +content) has the longest TA relaxation kinetics with its +τ3 component comprising 21% of total decay amplitude. +The inset of Fig. 1d shows the τ2 lifetimes all decrease +linearly from ∼1.2 to 0.9 ps with increasing lamp pho- +toreduction time. All samples have a characteristic τ2 +time similar to graphene’s characteristic ∼1 ps decay ex- +pected for 1.8 eV probe, suggesting all five samples ex- +hibit graphene-like hot-electron cooling dynamics. +By +analogy with monolayer graphene, the τ1 would be asso- +ciated with relaxation by optical phonons, and τ2 with +disorder-assisted hot electron cooling.29 The fitting pa- +rameter for the fast and long decays are constant at +τ1 = 0.15 ps and τ3 = 66 ps, and all parameters are +shown in Fig. 2c-d. +Figure 2 plots how the kinetic relaxation rates depend +on the selected probe energy (Epr). Comparing Fig. 2a +at Epr=1.3 eV to Fig. 1d at 1.8 eV, a similar pattern +with photoreduction emerges. However, the longest com- +ponent, τ3 is negligible for all five cases of photothermal +reduction rGO1−5. +In Fig. +2d the slower τ2 lifetime +decreases linearly from 2.5 ps to 1 ps with increasing +photoreduction time. τ1 varies the least with photore- +duction. Interestingly, the most reduced samples relax +even faster compared to monolayer CVD-grown graphene +(black dashed line). Figures 2a show fits to a triexponen- +tial decay curve showing lifetimes of ∼0.4 ps, 1-2.5 ps, +and >30 ps for τ1, τ2 and τ3 respectively. +Regardless of the incident TA probe energy (1.2 to 1.8 +eV), rGO samples relaxed progressively faster as the pho- +toreduction time increased. +Figure 2b shows that TA +dynamics of GO, rGO3, and rGO5 are slower at Epr= +1.3 eV (closed circles, 2.6 eV pump) than the Epr= 1.2 +eV (open circles, 2.2 eV pump) probe energy window. +Interestingly, the most reduced sample, rGO5, always +decays more quickly than ml-graphene. This faster de- +cay relative to graphene suggests that the photothermal +reduction is ultimately damaging the sp2 graphene-sub- +lattice by causing increased disorder and defect sites. +This symmetry-breaking results in low energy disorder + +4 +0 +1 +2 +3 +4 +5 +6 +40 80120 +0.01 +0.1 +1 + + +DT/T (log norm.) +Time Delay (ps) +1.3 eV +1.2 eV + + ml graphene +t2 = 1.9 ps +Eprobe: + 1.2 eV + 1.3 eV + 1.8 eV +0.6 +0.8 +1.0 +Fractional amplitude [A1] +0 +1 +2 +3 +4 +5 +0.0 +0.1 +0.2 +GO photoreduction time (a.u.) +Fractional amplitudes [A2, A3] +0 +1 +2 +3 +4 +5 +1.0 +1.5 +2.0 +2.5 +Relaxtion time, t2 (ps) +1.2 eV +1.3 eV +1.8 eV +GO photoreduction time (a.u.) +0 +2 +4 +100 +200 +0.0 +0.5 +1.0 +DT/T (norm.) + GO + rGO1 + rGO2 + rGO3 + rGO4 + rGO5 + ml graphene +Time Delay (ps) +a. +b. +c. +GO-like +sp2 graphene-like +disordered sp2 +d. +0.0 +0.2 +0.4 +Fast relaxatime time, t1 (ps) +Eprobe: +1.2 eV +1.3 eV +1.8 eV +Eprobe=1.3 eV +Eprobe: +FIG. 2. +(a) ∆T/T relaxation kinetics at Eprobe = 1.3 eV accelerate with sequential GO photoreduction. +Fits show two +exponential lifetimes, with only the most oxidized samples requiring a third lifetime of τ3 = 61±2 ps.(b) The ∆T/T kinetics +for Eprobe = 1.2 eV (open circles) relax faster than at 1.3 eV (closed circles). The rGO3 photoreduction stage most closely +approximates the ml-graphene interband relaxation kinetics shown (dashed line). (c) For each probe energy, the τ1 lifetimes +(top) are roughly constant, whereas the τ2 lifetime (bottom) decrease linearly ∼ 2.5× during with photoreduction to become +even faster than ml-graphene. (d) Amplitudes (A1/2/3) of each lifetime component suggest a composition change with increasing +amplitude from sp2 sub-lattice dynamics. The smallest A3 (blue) amplitude quickly decreases to zero as GO is reduced. +states that have been previously observed in conjugated +carbon systems .32,33 This is further supported by the +qualitative increase in lattice defect states that is evident +by increased emission in IR region of the PL spectra (see +supplemental Fig. S2). +Figures 2c-d contain the results of our exponential fit- +ting lines shown in Fig. 1d and 2a-b (solid lines). The +top panel shows the amplitude of the fast time compo- +nent (∼0.4 ps) at 1.2 eV, 1.3 eV, and 1.8 eV, which +accelerates only moderately as the GO samples are re- +duced. +The middle panel shows the amplitude of the +second (τ2 ≈ 1−2.5 ps, pink) and third (τ3 >30 ps, blue) +time components, which both decrease with reduction. +Importantly, the slow time τ3 component goes to zero in +the limit of heavy reduction and closely resembles the ml- +graphene relaxation. The bottom panel of Fig. 2c shows +the τ2 relaxation time of GO decreases roughly linearly +with photoreduction time. At all probe energies, the τ2 +relaxation time decreases with reduction, with rGO3,4,5 +having lifetimes shorter than that of CVD graphene un- +der the same optical conditions. The CVD ml-graphene +(dashed line in Fig. 1-2) was fit to a τ2 =1.9 ps at 1.2 eV +and 1.1 ps at 1.8 probe energies respectively. +In most heavily oxygenated rGO samples, the longest +τ3 ∼ 61 ps component comprises up to 16% of the total +decay amplitude. Such samples contain many functional +groups, however, the large band gap of the fully oxided +regions is well outside the spectral range of both pump +and probe laser energies. +Instead, graphene quantum +dots (GQD) create gapped sp2 molecule-like regions with +size-tunable bandgaps that are resonant with our probe +beam.22 For rGO3,4,5 samples, Fig. 2d shows that the +τ3 time-component is zero for Eprobe < 1.3 eV, suggest- +ing only graphene-like sp2 sublattice regions are relevant +to the electronic dynamics throughout this near-infrared +probe region. +IV. +ANALYSIS AND DISCUSSION +A. +rGO Fano Lineshape Absorption Analysis +The transmission spectra in Fig. 1a, Fig. S2 and fitted +absorption spectra in Fig. 3 all show lineshapes similar +to ml-graphene throughout the NIR and IR spectral re- +gions from ∼0.4 to 3.5 eV. The absorption maxima of +both ml-graphene and rGO in Fig. 3 (black line) deviate +from the tight-binding model prediction of the graphene +van Hove singularity M-point resonance at ∼5.1 eV.34 +Instead, the graphene absorption is best fit by a Fano + +5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +5.0 +0.0 +0.5 +1.0 +1.5 +Absorbance +Energy (eV) +Fano-lineshape fits: + rGO1, least reduction + rGO5, highest reduction + CVD ml-graphene (norm) +p-p* +n-s* +GQDs: p-p* +E +Fano +n-s* +edge defects +FIG. 3. +Under each linear absorption spectra (solid lines) +the deconvolved Fano resonance lineshape fit is plotted in +dashed lines. +Unlike pristine ml-graphene (black), the two +rGO samples plotted also require two convolved Gaussians +(dash-dot) suggesting molecular-like transition labeled π to π∗ +and edge defect transitions, n to σ∗ (see inset). The resulting +Fano-Gaussian convolved fits (dotted lines) show the graphene +sub-lattice Fano-parameter, q increases with photoreduction +consistent with more lattice disorder. +lineshape with a renormalized peak resonance energy, +Er that is red-shifted from the M-point by energy by +∼= 0.3 − 0.4 eeV.35,36 The asymmetric Fano lineshape +accounts for the ratio of interference between the dis- +crete (M-point) and continuum transition probabilities +through the dimensionless Fano parameter, q.35 Thus, +the tight binding model of the graphene absorption spec- +trum in Fig. 3 is renormalized for effective electron-hole +interaction effects by fitting to the below asymmetric +Fano lineshape, +AF ano(E) = A +� +�� +� +2 +γ (E − Er) + q +�2 +1 + +� +2 +γ (E − Er +�2 +� +�� +(1) +where γ is the Lorentizian homogeneous linewidth and +A is the amplitude scaling constant. +Fig. +3 plots a +hyperspectral measurement of CVD ml-graphene (black +line) with its corresponding Fano lineshape fit (dashed +line), given by equation 1 above. Table I gives the re- +sulting Fano parameters and show excellent agreement +of this work graphene values with the established liter- +ature values.35,37 This provides an essential calibration +base to quantitatively compare against the lineshape fit +of rGO absorption spectra. +Figure 3 shows good agreement between the absorp- +tion spectra of rGO1 and rGO5, and the asymmetric +Fano resonance after it is convolved with two Gaussians +peaks at energies corresponding to the absorption of the +n − σ∗ and π − π∗ transitions. This fitting analysis sug- +gests that the absorption spectrum in rGO can be under- +stood to contain a Fano resonance similar to that of CVD +Sample +Er (eV) +γ (eV) +q +ml-graphene [CVD] +4.80 +1.69 +-3.2 +ml-graphene [exfoliated]35 +4.73 +1.30 +-3.3 +rGO5 [highly reduced] +4.69 +1.68 +-3.2 +rGO1 [barely reduced] +4.62 +2.16 +-50 +TABLE I. Fano fitting parameters for data in Fig. 3 (dashed +lines) show good agreement with our monolayer graphene +data established literature values.35,36 The Fano parameter +q of rGO5 best matches ml-graphene. Two convolved Gaus- +sian for GQD π − π∗ and edge state defects are also required. +ml-graphene. The molecular-like π − π∗ transitions are +illustrated in Fig. 3 (inset), and show graphene quan- +tum dot (GQD) states also contribute to the spectral +weight and are centered near 4.6 eV.38 At 4.3 eV, rGO +also contains sub-gap defect states between the π and +π∗ states, which results from previously reported local +oxygen-based disorder that creates edge defect state (n) +to σ∗ transitions.2,39–42 Due to the heterogeneous oxygen +coverage, these local disorder edge states have a much +broader absorption FWHM. As rGO1 is further reduced, +we observe in Fig. 3 that the peak area of the n−σ∗ Gaus- +sian decreases as oxygen is removed, resulting in fewer +edge states. Both our most oxidized samples (GOo and +GOsolution) did not fit well to a Fano lineshape, suggest- +ing only rGO samples have a graphene-like absorption +lineshape in the IR and NIR regions. +Table I contains a summary of the Fano fitting parame- +ters, showing good agreement between the literature35,37 +and our results for monolayer graphene and rGO5. rGO5 +contains a large absorption from the linear dispersion +near the K and K’ points, where excited carriers cou- +ple strongly to the continuum, similar to monolayer +graphene. For rGO1, the Fano parameter q decreases sig- +nificantly from monolayer graphene, suggesting electron- +hole interaction effects are increasingly screened for tran- +sitions near the van Hove singularity. For GO and lightly +reduced rGO, Table I shows the Fano parameter is many +times larger than highly reduced samples and mono- +layer graphene. This suggests the many edges states in +more oxidized graphene couple strongly to continuum- +like states. +The inset of Fig. 3 shows a qualitative depiction of how +the density of states changes from GO to rGO. As the +samples are reduced, they contain larger area regions of +non-interrupted sp2 carbon, leading to a more graphene- +like distribution of continuum states, resulting in a better +Fano lineshape fit. The two convolved Gaussians show +the effect of reduction on the absorption spectra, with the +amplitude of the n−σ∗ transition decreasing significantly, +suggesting the removal of oxygen functional groups. We +also see that the absorption peak in rGO1 shifts slightly +to lower energy compared to rGO5. This shift has been +theoretically predicted by Roy et al.22, who used DFT +to calculate the band structure of GO at varying oxygen +content, finding that the addition of oxygen decreases +the band gap at the M-point. However, the underlying + +DOSn +元 +@6 +Fano resonance energy (ER in Table I) does not change +with photoreduction. The very large q Fano parameter +required to fit the most oxidized rGO1 samples suggests +the sp2 hybridized regions are not extensively delocalized +and retain a molecular-like character. +B. +Hot-electron cooling rates in reduced graphene +oxide +Figure 4 fits the hot electron cooling TA kinetics in +progressively reduced GO as the TA probe energy is in- +creased from 1.2 (top) to 1.8 eV (bottom). Specifically, +the hot-electron cooling rate (τSC) is extracted. Unlike +the exponential rate τ2 from Fig. 2, τSC is analogous to +the recombination rate as the electron cool near the Fermi +energy, and is independent of probe energy (Eprobe). To +connect the above phenomenological exponential relax- +ation models of GO to this first-principle hot-electron +cooling model, the fits in Figure 4 models our TA re- +laxation kinetics using a hot electron heat dissipation +rate H = Ce(dTe/dt), where Ce and Te are the elec- +tronic heat capacity and temperature respectively. The +top-panel of Fig. 4a contains first-principle hot electron +cooling model fits (solid lines) to the normalized TA ki- +netics of the rGO samples. +Hot electron cooling rates +in rGO can be qualitatively understood by comparing to +CVD ml-graphene kinetics (black dotted line). The low- +est energy probe (Epr=1.2 eV) in the top panel of Fig. +4a shows the hot electron cooling rate response of ml- +graphene (dashed line) is identical to rGO1, rGO2 and +rGO3. Interestingly, rGO4,5 dissipates heat even faster +than CVD ml-graphene. +The mechanism for fast energy dissipation or hot- +electron cooling in graphene has been widely debated in +the past. The optical phonon dissipation model28,43,44 +evolves on the sub-ps relaxation timescale of the τ1 +component. +At longer relaxation times, the disorder- +mediated acoustic phonon decay pathway or supercol- +lision (SC) hot electron cooling model are the primary +factor limiting cooling of the photoexcited hot elec- +tron temperature, Te(t).45 Experimental studies demon- +strate the SC-model45 successfully predicts graphene’s +photocurrent29, optical46 and electrical47 heating re- +sponse. However, the applicability of the SC-model to +more disordered lattice of GO and rGO has not been +considered. +To understand hot electron cooling in rGO, we apply +the acoustic phonon SC-model illustrated in Fig. 4b (in- +set). +In the SC model, hot electron cooling near the +Fermi level occurs without crystal momentum conserva- +tion. Instead, higher-energy (∼ kBTe) acoustic phonons +are emitted with the momentum imbalance, qrecoil ac- +counted for by disorder-induced intrinsic lattice recoil.45 +This SC- hot electron is illustrated in Fig. 4b (inset), +and gives in a faster hot electron cooling rate than a hot- +phonon model that is given by,29,45 +dTe +dt = − H +αTe += −A +α +T 3 +e − T 3 +l +Te +. +(2) +where A/α is the SC rate coefficient, Tl and Te are the +lattice and electron temperatures, respectively. Solving +Eq. 2, Te(t) ∼= +To +1+ATot/α when Te(t) ≫ Tl, where To is +the initial electron temperature. +Since all data shown +is at Tl =292 K, the transient change in Te(t) is small +compared to Tl, or Te(t) − Tl ≪ Tl such that we can +approximate Eq. 2 by expanding the leading terms to +arrive at the room-temperature hot electron tempera- +ture, Te(t) ∼= Tl + (To − Tl)e−t/τSC, to get the expression +τ −1 +SC = 3ATl/α.29 +The TA response is obtained using the hot electron +(or hole) temperature (Te) through analytically fitting to +the transient interband optical conductivity, ∆σ(Eo, t) = +−e2/4ℏ +� +fe/h(Te(t), Epr) − fe/h(Tl, Epr) +� +.34 The Fermi- +Dirac hot-electron occupancy function, fe/h(Te(t), Epr) +at the probe energy (Epr) equations are given in the Sup- +plementary Materials as a change in interband optical +conductivity ∆σ(t, Epr).34,48 In Fig. 4b the hot electron +cooling rates (τ −1 +SC) for rGO are extracted by fitting the +data in Fig. 4a to the analytical SC-model solution (Eq. +2), allowing for two additional exponential components +(τ1 and τ3). This fast component, τ1 ∼= 0.34 ps, averages +over the initial electron thermalization and optic phonon +emission timescale and is discussed elsewhere.49,50 Any +molecular-like π − π∗ transitions present are captured by +τ3 ∼ 61 ps. +The accelerating TA relaxation kinetics in Fig. +4a +are consistent with the idea that photoreduction of GO +creates more disorder and defects on the graphene sub- +lattice. Figure 4b shows an increase in the rate of hot +electron cooling, τ −1 +SC. Unlike the earlier exponential fits, +the rate τ −1 +SC is independent of the probe energy and is +the rate at which the hot-electron Fermi-Dirac distribu- +tion cools. The hot electron cooling time for the com- +parison monolayer CVD -grown graphene (dashed line in +Fig. 4b) at 292 K is 3.1 ps. τ −1 +SC increases by a factor of +∼6 as the samples are reduced. This suggests the xenon +arc lamp used to reduce GO is a largely destructive pro- +cess to underlying sp2 sub-lattice. At the highest level +of photoreduction, Fig. +4b suggests the increased lat- +tice disorder destroys the desired graphene-like extended +lattice by creating to many point-defects. +The τ −1 +SC = 3ATl/α expression is a direct measure of +lattice disorder by the expression +A +α ∼= +2 +3 +λ +kF l +kB +ℏ , where +the mean free scattering path is kF l.45 The electron- +phonon coupling strength can be approximated as λ = +D2 +ρs2 +2EF +π(ℏvF )2 , where both the deformation potential, D and +Fermi energy EF are the experimental variable that in- +crease the hot electron cooling rate. +Figure 4b shows +that A +α ∼= 0.3 ns−1K−1 for rGO1−3, which matches the +monolayer CVD graphene values in literature.46 How- +ever, further photoreduction increases A +α upto 6×, sug- +gesting the graphene sub-lattice is being damaged. If the + +7 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +Hot electon cooling rate (1/tsc) ps-1 + GO photoreduction time (a.u.) + Eprobe= 1.2 eV + Eprobe= 1.3 eV +0 +1 +2 +3 +Disorder parameter (A/a) ns-1 K-1 +a. +0.01 +0.1 +1 +0.01 +0.1 +1 +0 +1 +2 +3 +4 +0.01 +0.1 +1 + + +-DT/T, normalized +Time Delay (ps) +b. +Eprobe =1.2 eV +1.30 eV +ml graphene +1.8 eV +rGO5 +GOo +rGO4 +rGO1 +rGO2 +rGO3 +E +k +qSC +GOo +rGO1-3 +rGO5 +rGO4-5 +graphene +FIG. 4. +(a) TA relaxation kinetics of the six progessively +reduced GO GO samples compared to ml-graphene (black +dashed) at 1.2, 1.3, and 1.8 eV probe energies (top to bot- +tom). Fitted lines now incorporate the SC-hot electron cool- +ing model of Eq. 2. (b) SC-model hot electron cooling rates +(τ −1 +SC) extracted increase sharply for longest photoreduction +times. +For rGO1−3, the disorder parameter A/α is similar +to ml-graphene rate (dashed), and expectantly is invariant to +the laser probe energy of 1.2 eV (black) and 1.3 eV (pink).29 +deformation potential is approximately constant, then +that A/α ∝ EF /kF l, suggesting that the damage of +photoreduction decreases the mean free scattering path +by photoionization, which increase sp2 sub-lattice defect +sites. +Our fitted data in Fig. +4 confirms that acous- +tic phonons supercollisions (SCs) best describe the rate- +limiting heat dissipation kinetics in reduced graphene ox- +ide. Furthermore, Fig. 4b shows how disorder from pho- +todamage to the rGO lattice systematically increases the +hot-electron cooling rate. This controlled change in lat- +tice disorder provides new evidence of the predominant +role of disorder-assisted SC in describing the hot-election +in graphene. +C. +Oxygenated sub-lattice contributions from +graphene quantum dots +Sections IV.A and B above both show the rGO sample +and ml-graphene have remarkably similar lineshape and +hot-electron cooling rates over optical energies that rang- +ing from 0.4 to 1.8 eV. This section focuses one the dif- +ferences that arise in visible and UV range where GQDs +and defect-edge states are also also optically be excited. +Figure 5a plots the PL emission spectra of the least re- +duced, GOo and most reduced, rGO5 samples after a 4.6 +eV excitation. +The main asymmetric peak appears to +shift from ∼2.4 to 2.7 eV with photoreduction. The ex- +perimental emission spectra (dots) are fit (solid lines) us- +ing 4 convolved Gaussian peaks (dotted lines). All peak +energies and FWHM spectral width (except at 2.34 eV) +are found to be approximately invariant to photoreduc- +tion. The peak at 2.7 eV in Fig. 5a corresponds with +emission from the smallest graphene quantum dot states +(labeled GQD1) π∗ − π orbital relaxation. At the lower +energies, both peaks centered near 1.55 eV and 1.80 eV +grow with photoreduction, consistent with emission from +larger graphene quantum dot states labeled GQD2 and +GQD3, respectively. We observe an increase in the emis- +sion intensity from these three sp2 peaks with reduction, +confirming they do not result from oxygen groups. Con- +versely, the emission at 2.3 eV represents the carrier re- +combination in sp3 oxygen (σ∗ − n). The magnitude and +width of this emission decrease with reduction as oxygen +functional groups are removed. +PL from GO and rGO in solution has been widely doc- +umented in the literature, showing that reduction of GO +increases PL intensity at near IR wavelengths while also +blue-shifting the main peak.38,51,52 In accordance with +literature, Fig. 5a shows an increase in PL intensity with +reduction, at peaks centered at 1.80 eV and 1.55 eV. +The PL of the oxygenated GO lattice is known to emit +broadly near 2.4 eV with locally varying oxygen content +responsible for the broader FWHM.26,41 In rGO, PL is +dominated by π∗ − π carrier recombination in regions of +confined graphene quantum dots. As the reduction pro- +cess removes oxygen, formerly isolated sp2 carbon atoms +join together to form conjugated carbon rings, and re- +gions that already contained large area conjugated sp2 +carbon structures increase in size. The observed decreas- +ing area of the peak at 2.3 eV with photoreduction sug- +gests this peak emission is likely due to egde states or +oxygen-defects the boundaries of the sp3 region. +The +newly formed GQD in rGO are ascribed to the increas- + +disorder-assisted +scattering8 +0 +2 +4 +6 +8 +0.0 +0.5 +1.0 +1.5 +2.0 +DT/T (10-6) +Pump Fluence (x1012 photons/cm2) + GO0 + rGO1 + rGO2 + rGO3 + rGO4 + rGO5 +b. +1.6 +1.8 +2.0 +2.2 +2.4 +2.6 +2.8 +3.0 +0 +2 +4 +6 +8 +10 +12 +PL Counts (x104) s-1 +Emisssion Energy (eV) + GO0 + rGO5 +0 +1 +2 +3 +4 +-0.08 +-0.04 +0.00 +DT/T (10-3) +Time Delay (ps) +0.04 +0.08 +0.12 +DT/T (10-3) +GQD1 +Eprobe = 2.3 eV +Eprobe = 1.8 eV +graphene + GQD2 +a. +b. +c. +GQD2 +GQD3 +0 +2 +4 +6 +8 +0.00 +0.05 +0.10 +DT/T (10-3) +Pump Fluence (x1012 photons/cm2) +Eprobe = 1.8 eV (QQD2) +Eprobe= 1.2 eV (graphene) +0 +1 +2 +3 +4 +5 +0.0 +0.1 + 1.8 eV (gr.+ GQD2) + 2.3 eV (gr.+ n-s) +DT/T (10-3) +GO photoreduction time (a.u.) +𝜋∗ +𝜋 +n-s +GQD1 +s* +s +~2.3 eV +ex: 4.6 eV +n: edge +defects +GQD2 +GQD3 +4.6 eV +graphene +FIG. 5. (a) The photoluminescence emission spectra of GOo (green line fit) and rGO5 (red line fit) with 4 convolved Gaussian fit +(dashed lines).Photoreduction increases the PL peaks from graphene quantum dots resonances, labeled GQD1−3. Conversely, +emission from the oxygenated sub-lattice n − σ defect edge state decreases as GO is reduced (see inset for corresponding +transitions). (b)The degenerate TA response near the 1.8 eV GQD2 resonance (top) vs. near the excited state absorption at +2.3 eV (bottom). (inset) TA response increasing with photoreduction. (c) ∆T/T pump power photon fluence dependence of +reduced GO samples using a 1.2 eV probe. Fit are to the graphene SC-hot electron cooling model in Eq. 2. Over a wide range +of incident photon flux, the saturable absorption susceptibility, ∆T/T is invariant to photoreduction suggesting only graphene +hot electrons are probed below ∼1.2 eV. (inset) Conversely at 1.8 eV the ∆T/T changes strongly, suggesting increasing GQD2 +states. +ing PL at 2.7 eV, 1.80 eV and 1.55 eV peaks. +DFT studies by Sk et al.21 show how the bandgap +energy of a GQDs changes with respect to its size and +found that GQDs about 1.3 nm in mean diameter create +Frenkel exciton states near 2.7 eV, while slightly larger 2 +nm GQDs emit around 1.8 eV. rGO contains an ensem- +ble of GQDs of various sizes separated by oxygenated +regions. Reduction removes oxygen, gradually increasing +the GQD size, evidenced by the increased PL in rGO at +1.55 and 1.80 eV. +Figure 5a inset contains a qualitative depiction of the +bands and energy levels in GO. The optical response of +graphene is determined by the π and π∗ states, which lie +between the σ − σ∗ gap in GO.41,53 Oxygen functional +groups break the symmetry of the pristine graphene lat- +tice, resulting in localized defect states that exist in the +π−π∗ gap. Since the gap between σ states is much larger +than 2.4 eV, this emission is suggested as n−σ transition +(dashed purple arrow). In both GO and rGO, emission +at 2.7 eV dominates the PL spectra, which was shown to +result from π states in isolated sp2 domains (gray dashed +arrow).38 Emission at lower energies comes from a broad +range of GQD states and the local disorder states. +Figure 5b shows the degenerate transient absorption +response of the samples at 1.8 eV and 2.5 eV, respec- +tively. At 1.8 eV, we observe a saturable absorption sig- +nal containing a long component that slowly goes away +with reduction. +At 2.5 eV, we see a reverse saturable +absorption response, which decays extremely quickly in +all samples. A similar transition has been previously ob- +served by Bhattacharya et. al, who saw that a sign flip in +the pump-probe response occurred near 2.3 eV.54 Since +the most reduced samples have the largest reverse sat- +urable absorption response, we can rule out excited state +absorption from oxygen groups as the cause of the sign +flip. We attribute this sign-change to absorption from the +interband transition in graphene, which has been previ- +ously documented to exhibit a sign flip for high pump +fluences at this energy.48,55 We do not see a change in +sign when probing the oxygen states at 1.8 eV, further +confirming the sp2 nature of the peak labeled GQD2. +Figure 5c shows the 1.2 eV probe energy pump flu- +ence dependence. +At low pump fluences, the TA re- +sponse of all samples exhibits a linear dependence on the +pump fluence. Above incident photon flux of ∼ 4 × 1012 +photon/cm2, a sublinear trend is observed that is fit to +the Eq. 2 hot electron cooling model TA response. The +nonlinear saturation effect fits to the expected nonlinear +Fermi-Dirac filling factor. +Notably, the more oxidized +GO1 and rGO2 samples have the most nearly linear be- +haviors, consistent with the expected smaller confined +sp2 sub-lattice regions. Conversely, Fig. 5c(inset) shows +the pump power dependence for the differential trans- +mission at 1.8 eV pump and probe. GO displaying the +smallest response, which increases with reduction until +rGO3. +The response saturates for the three most re- +duced samples as shown in the inset of Fig. 5b. This +trend matches the absorption spectra at 1.8 eV, where +the absorption increases monotonically with reduction, +with the exception of the ∆T/T response saturating for +the most reduced samples. +The pump dependence gives us insight into how the +probe response changes with lattice temperature. At low +pump powers, the 1.2 eV probe has the same magni- +tude for all samples, suggesting that even oxidized sam- +ples have large regions of graphene-like sp2 hybridization. + +9 +The 1.8 eV data remains linear overall pump fluences +but has a large dependence on the amount of reduction. +While the 1.2 eV data probes graphene-like states, the +1.8 eV data primarily probes the confined GQD2 states +that lead to longer lifetimes and a strong dependence on +photoreduction. The size and population of these GQD +states depend heavily on the oxygen content. As shown +in Figure 5c(inset), reduction increases the transient re- +sponse, which suggests that reduction increases the pop- +ulation of sp2 GQD2 states that absorbs at 1.8 eV. This +trend matches the increase in PL seen at 1.8 eV after +photoreduction. +a. +b. +c. + GQD +donor +graphene +acceptor +0.0 +0.5 +1.0 +3.0 6.0 +0.0 +0.5 +1.0 +DT/T, Normalized +Time Delay (ps) +2.5 eV +pump +1.2 eV +probe +rGO5 +rGO1 +rGO3 +GOo +E +k +photoreduction +FIG. 6. (a) Normalized transient absorption kinetics shows a +170 delayed rise for GO that systematically accelerates with +successive photoreduction. +(b) This rise is assigned to an +acceptor-donor relationship between the 2.5 eV pump of GQD +states and the 1.2 eV probe of the accepting graphene states. +(c) Band illustration of rGO depicts charge transfer described +from confined GQDs to larger sp2 graphene-like regions. +D. +Donor-acceptor electronic transfer in rGO +Using non-degenerate TA spectroscopy, we can excite +molecular-like GQDs at high energies and probe the elec- +tron transfer rate to graphene at lower energy states. +Figure 6a shows the normalized TA relaxation at 2.5 +eV pump 1.2 eV probe near time zero, which shows a +clear delayed rise in the most oxidized samples. +Con- +versely, the most reduced samples show a rise limited by +the laser cross-correlation. +This delayed rising kinetic +edge is indicative of an acceptor-donor electron relation- +ship. Charge transfer has been documented in GO, where +photoexcited charges on a different molecular species +are transferred to GO.56–58 Figure 6b illustrates charge +transfer between molecular GQDs and larger graphene- +like regions. When the pump moved to longer energies +(e.g. 1.8 eV in Fig. 1c), the delayed rise is no longer +seen because the population of GQD donors is too small +relative to graphene. +Figure 6b depicts the charge transfer process that is +responsible for the observed delayed rise. Carriers pho- +toexcited in the confined GQD states are localized by the +surrounding oxygen functional groups. In GO, the large +density of oxygenated regions results in a weaker coupling +between confined GQDs and graphene submetallic sub- +lattice regions, leading to the observed delayed rise. In +the photoreduced samples, carriers excited into sp2 GQD +states are now closer to extendend graphene regions, and +so the delayed acceptor-donor electron transfer is not ob- +served to lower energy states. +Figure 6c gives a qualitative description of the struc- +ture and acceptor-donor electron transfer process in rGO. +Our graphene oxide begins with ∼44% oxygen content, +these oxygen functional groups interrupt the delocalized +π-orbitals and prohibit hopping between carbon sites. +Reduction removes oxygen, which decreases the mean +distance from a confined GQD donor and graphene-like +sp2 sublattice region. Such changes to the effective per- +colation network of the sp2 sublattice have previously +been shown to also increase GO carrier mobility and +conductivity59,60. The longer dynamics in GO are caused +by excited carriers being more isolated by larger oxy- +genated regions as shown in Fig. 6c, which limit possible +relaxation pathways. In rGO, some of the oxygen has +been removed, recovering large-area graphene-like do- +mains which decay more quickly than pristine graphene. +V. +CONCLUSIONS +The +highly +variable +composition +of +the +quasi- +amorphous GO 2D lattice makes a systematic compar- +ison against monolayer graphene a challenge. +To help +overcome this challenge, GO is suspended in a poly- +meric network scaffold where five successive photoreduc- +tions (rGO1−5) were possible without any evidence of +inter-layer aggregation. +Ultimately, this yielded opti- +cal quality rGO films with an absorption lineshape that +fits to ml-graphene Fano resonance lineshape parame- +ters. Likewise this step-wise photoreduction accelerates +the hot electron relaxation kinetics monotonically over +each of the variable probe energy windows studied from +1.2 to 2.5 eV. At intermediate photoreduction times or +rGO2−3, Fig. 4 shows that a hot electron cooling model +of disorder-assisted supercollision matches the τSC =3.1 +ps hot electron cooling of monolayer graphene. Figure +4b shows the recovery of ultrafast hot electron relaxation +rates similar to monolayer-graphene in moderately re- +duced samples(rGO1−3 ), suggesting a largely uninter- +rupted sp2 bonded network analogous to graphene. +Under extreme photoreduction or using UV-Vis optical + +Confined +Metallic +region +region0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +0 +010 +excitation, the optical properties of rGO begin to deviate +strongly from graphene. Owing to increasing local dis- +order and broken lattice symmetry, extreme photother- +mal reduction yields hot electron cooling rates that are +faster than pristine graphene. Subsequent photoreduc- +tion accelerates the extracted hot electron cooling rate +10-12x, revealing how photodamage induces local disor- +der to mediate faster hot electron cooling. On longer, +>50 ps timescales, rGO also exhibits a slower decay re- +sponse than graphene owing to many isolated graphene +quantum dot (GQD) regions and oxygenated edge trap +states which serve to delay the ground state recovery. +Using probe energies in the visible wavelength range at +1.8 eV, Figs. 1c and 4 shows that photothermal reduc- +tion does not recover pristine graphene properties, as ev- +idenced by the slower decay kinetics of all rGO samples +relative to graphene. The prevalence of isolated GQDs +regions and oxygenated-edge trap states each create fur- +ther bottlenecks of electronic relaxation that slow the +effective relaxation. Fortunately, we find these long life- +times of rGO are no longer oberved below 1.3 eV optical +excitations, as there are no discernible GQD sub-lattice +states large enough to creae a resonance at these energies. +Collectively, these results show many of the desirable op- +toelectronics properties of 2D graphene can be replicated +using selectively reduced graphene oxide suspended in a +3D bulk polymeric network. This study lends itself to +large-scale processing of rGO thin films and applications +in high-speed optoelectronics and photonic switching ap- +plications. +ACKNOWLEDGMENTS +This material is based upon work supported by the +Office of the Under Secretary of Defense for Research and +Engineering under award number FA9550-22-1-0276, and +the DEVCOM Army Research Laboratory award number +W56HZV-16-C-0147. +Supplementary Materials: Details on sample char- +acteristics, data modeling methods, and further absorp- +tion and PL spectral data show similar graphene-like +propertis out to the mid-IR regions as far as 0.5 eV. +Data Availability Statement:The data that sup- +port the findings of this study are available from the cor- +responding author upon reasonable request. +1 G. Yang, L. Li, W. B. Lee, +and M. C. Ng, Science and +Technology of Advanced Materials 19, 613 (2018). +2 K. A. Mkhoyan, A. W. Contryman, J. Silcox, D. A. Stew- +art, G. Eda, C. Mattevi, S. Miller, +and M. 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Hu, Materials Today 21, 186 (2018). + diff --git a/edFPT4oBgHgl3EQfzjXc/content/tmp_files/load_file.txt b/edFPT4oBgHgl3EQfzjXc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..efa5cb44c2cd5e8bab57522a70e69c79d13aaf7e --- /dev/null +++ b/edFPT4oBgHgl3EQfzjXc/content/tmp_files/load_file.txt @@ -0,0 +1,1259 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf,len=1258 +page_content='Graphene Oxide Photoreduction Recovers Graphene Hot Electron Cooling Dynamics Alden N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Bradley1∗, Spencer G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Thorp1∗, Gina Mayonado1, Edward Elliott2, Matt W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Graham1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Department of Physics, Oregon State University, Corvallis, OR, 97331, USA and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='Voxtel Nano, Corvallis, OR, 97330, USA Reduced graphene oxide (rGO) is a bulk-processable quasi-amorphous 2D material with broad spectral coverage and fast electronic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' rGO sheets are suspended in a polymer matrix and sequentially photoreduced while measuring the evolving optical spectra and ultrafast electron relax- ation dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Photoreduced rGO yields optical absorption spectra that fit with the same Fano lineshape parameters as monolayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' With increasing photoreduction time, rGO transient absorption kinetics accelerate monotonically, reaching an optimal point that matches the hot elec- tron cooling in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' All stages of rGO ultrafast kinetics are simulated with a hot-electron cooling model mediated by disorder-assisted supercollisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' While the rGO room temperature 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='31 ps−1 electronic cooling rate matches monolayer graphene, subsequent photoreduction can rapidly increase the rate by 10-12×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Such accelerated supercollision rates imply a reduced mean-free scat- tering length caused by photoionized point-defects on the rGO sp2 sub-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' For visible range excitations of rGO, photoreduction shows three increasing spectral peaks that match graphene quan- tum dot (GQD) transitions, while a broad peak from oxygenated defect edge states shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' These three confined GQD states donate their hot carriers to the graphene sub-lattice with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='17 ps rise- time that accelerates with photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Collectively, many desirable photophysical properties of 2D graphene are replicated through selectively reducing rGO scaffolded within a 3D bulk polymeric network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' ∗ co-authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' INTRODUCTION Graphene oxides (GO) are a widely-used substitute for graphene’s remarkable mechanical properties, but its highly amorphous lattice lacks desirable electronic prop- erties such as high conductivity, fast photoresponse and broad spectral coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' When GO is incorporated in certain polymeric networks, we show systematic photore- duction makes it more graphene-like while maintaining pristine optical-quality films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' GO has oxygenated func- tional groups attached to the 2D carbon lattice via out- of-plane bonds that prevent GO sheets from aggregating in solution phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1,2 GO can be made more graphene-like by chemical or photothermal reduction to make reduced graphene oxide (rGO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Conventionally, these graphene- like rGO layers aggregate and scatter light strongly, making their optical properties hard to compare against monolayer(ml) graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Using systematic reduction of isolated GO-in polymer composites, we show the emer- gence of spectral lineshapes and extract ultrafast hot- electron cooling dynamics that are closely analogous to that of ml-graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' GO is often used as a bulk-processable substitute for graphene for wide-ranging applications, including elec- tronic sensing, plasmonics, and desalination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3–9 The large presence of oxygen in GO introduces an effective band gap (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1a inset), with a tunable energy determined by the carbon-to-oxygen ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Previous theoretical and ex- perimental studies suggest bandgaps ranging from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 eV for GO that can vanish nearly completely as GO is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='10 GO samples reduced via pulsed Xe arc lamps effectively remove hydroxyl, epoxy, and car- boxyl groups to increase the size of graphene-like sp2 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The amount of photoreduction changes the ra- tio of the oxygenated-sp3 to conjugated-sp2 sub-lattice regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='11–13 Very selective growths and controlled re- duction are required to realize desired optoelectronic ap- plications for GO that have included broadband optical nonlinearity14,15, tunable photoluminescence,16 and res- onant energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='17 With widely-varying ratios of oxygen and carbon, the highly inhomogeneous and amorphous nature of GO and rGO lattice make a direct comparison to ml-graphene difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In rGO, individual sp2 graphene-like sub- lattice regions often become surrounded by sp3 oxidized domains, forming molecular-like confined regions often called graphene quantum dots (GQDs) or graphene nan- oclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' While the composition of rGO varies greatly, it can roughly be decomposed into three types of sub- lattice illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1b: (1) extended sp2 hybridized regions, (2) confined sp2 lattice nanoclusters or GQDs, and (3) oxidized or sp3 regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Zhang et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' al performed transient absorption on rGO in solution and found that the carbon (sp2) and oxidized domains (sp3) could be treated independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='18,19 Photoexcited carriers in the spatially-confined sp2 GQDs produce Frenkel excitons with energies tunable with the size of the GQD con- jugation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='20,21 The local oxygenated functional groups at domain edges also create many optically active defect states within the lattice that are seen in photolu- minescence studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='22–24 While some of the mechanical and chemical properties of GO-based materials are analogous to graphene, the conditions necessary to replicate graphene-like electronic behavior in rGO are less clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Past studies have com- pared the transient absorption (TA) response of GO and rGO prepared by chemical reduction in solution25 and thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='24,26 This study concerns the optical properties arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='13176v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='mtrl-sci] 30 Jan 2023 2 of GO and rGO embedded in a transparent polymer film over six controlled degrees of photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The TA relaxation resolves how the ultrafast hot electron cooling rate is modified at each stage of photoreduction using tunable probe energies ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' While the hot electron cooling in graphene is typically modeled with two rates associated with optical phonon scattering and disorder-assisted relaxation processes,27–29 In addi- tion to graphene-like relaxation, prior rGO studies are dominated by a long, 10-200 ps relaxation component previously ascribed to electron trapping at defect sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='30 The results obtained from the succession photoreduc- tion of GO are modeled with first-principle models of ab- sorption lineshapes and hot-electron cooling applied pre- viously to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='A, the evolution of the absorption lineshape with photoreduction is modeled by competing contributions from graphene-like Fano line- shape and GO-oxide-related absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Then Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='B applies a hot electron supercollision model to deter- mine at what stage of photoreduction rGO most closely matches the dynamics of ml-graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Over most visi- ble and UV excitation energies, Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='C shows the GO-sub-lattice and graphene quantum-dot states domi- nate both the photoluminescence and ultrafast response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Lastly, we resolve how photoreduction of GO impacts the ultrafast rate of acceptor-donor electron transfer from the photoexcited GQDs to graphene acceptor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' EXPERIMENTAL METHODS The GO and rGO polymer samples were fabri- cated using commercially available chemically exfoliated graphene oxide sheets (Graphenea) containing ∼53% car- bon and ∼44% oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The sheets are dispersed in a N, N-dimethylacrylamide (DMAA) polymer with added PMMA sites to scaffold the GO and minimize aggrega- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The mixture is cured between two 1 mm thick glass slides, resulting in a sample thickness of 220 microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The sample is then photo-reduced via a pulsed Xenon arc lamp at a 1 Hz repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This low frequency was chosen to prevent gas bubbles from forming during the reduction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Absorbance is measured via Cary IR-UV-Vis spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Both excitation and emission photoluminescence are detected with a commercial fluo- rimeter (Horiba NanoLog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Both degenerate and non-degenerate pump-probe ex- periments are conducted with 140 fs pulses from a Ti:sapphire lasers (Coherent Chameleon) and Optical Parametric Oscillators (APE OPO Compact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' An opti- cal parametric amplifier is used to tune the output wave- length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The beam is split into two parts, a strong pump and a weaker probe power beam with a ratio of ∼10:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The intensity of the pump beam is modulated using an acousto-optic modulator (AOM, Crystal Tech) at 500 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The polarization of the pump and probe beam is lin- ear and set parallel to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' For the non-degenerate experiment, the pump beam is frequency doubled by a 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 1 DT/T, normalized Time Delay (ps) 250 500 750 1000 1250 1500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 Transmittance, T Wavelength (nm) d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' photo reduction GOsolution rGO rGO5 GOo rGO4 rGO1 rGO2 rGO3 Epr rGO5 GOo rGO4 rGO3/2/1 sp3 sp3 sp3 sp3 sp3 sp3 sp3 sp2-GQDs sp2 graphene sublattice sp3 sp3 sp3 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Graphene oxide (GO) Reduced GO (rGO) most reduction a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' M-pt GOsolution GOsolution +polymer Fano-resonance 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 GO photoreduction time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=') Relaxation Time t2 (ps) most oxidized sp2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (a) Comparison of GO vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' rGO band with chem- ical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (b) Illustration of the three prominent sub- lattices types within the rGO structure (sp2, sp2 graphene quantum dot (GQD) and oxygenated sp3-lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (c) Lin- ear and transient absorption spectra are measured at five stages of the photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' With increasing photoreduc- tion, NIR transmittance decreases to more closely approx- imate the (renormalized) CVD ml-graphene transmittance curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Conversely, as grown GO in solution (gray line) has a prominent π − π∗ bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (inset) Graphene band structure highlighting the M-saddle point transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (d) Correspond- ing transient transmittance kinetics at Eprobe=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV show carrier relaxation accelerates with reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (inset) The τ2 lifetime increases linearly with photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' E kE 元-band T-band3 second harmonic generation unit (OPE SHG) prior to modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Alternatively, a white-light supercontinuum is generated to provide a broadly tunable probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Both beams are focused onto the sample by a single lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The probe beam waist at the sample is approximately 80 mi- crons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The transmitted probe-beam is detected by pho- todiode lock-in amplification (Zurich Instruments, HFLI and MFLI) at 500 kHz modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' To compare the rGO polymer physics to ml-graphene, similar measurements to the above were carried out using an ultrafast transient absorption (TA) microscopy setup with a 1 µm spot size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The ml-graphene was prepared by chemical vapor deposition (CVD) and wet-transferred to a thin silicon nitride grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The above non-degenerate pump-probe scheme was used in a collinear geometry coupled to a 4f -confocal scanning microscope (Olym- pus BX51W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The absorption spectra of ml-graphene are taken on the same microscope by coupling in a tunable Xe-arc illumination source and detecting the full plane images on a camera (EMCCD, PI-ProEM) camera after background renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' RESULTS Spanning the UV to near-IR regions, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1c plots the absolute linear transmission of six graphene oxide (GO) samples in a polymer composite with increasing pho- tothermal reduction times labeled from rGO1 to rGO5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Additionally plotted on a renormalized scale, we overlay the linear absorption spectra of both pristine monolayer (ml) graphene (black line), and the starting as-grown commercial GO solution (gray line, GOsolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The GO solution has a clear bandgap, peaking at the molec- ular π − π∗ transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Conversely, ml-graphene gives an expected Fano resonance lineshape peaked at 265 nm, red-shifted from the M-saddle-point transition labeled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1c (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='31 The rGOo curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1c is the ‘as- grown’ GO after incorporation into a hybrid polyacrylic and PMMA polymer matrix described in the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The absolute absorbance increases monotonically with GO photothermal reduction time over the NIR and IR regions plotted (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='35 eV to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Photoreduction of GO leads to a spectral lineshape that absorbs light more analogously to CVD monolayer graphene plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In the solution phase and most polymers, GO aggre- gates as it is reduced, resulting in colloidal mixtures that strongly scatter light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' GO is incorporated in a polymer- sphere matrix scaffold that makes systematic photore- duction possible while maintaining pristine optical qual- ity films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Thus, we are able to compare the absorp- tion lineshapes, photoluminescence, and ultrafast hot electron cooling rates over a wide range of photoreduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Interestingly, the more heavily reduced graphene oxide samples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1c have a transmittance lineshape and slope similar to ml-graphene throughout the near- infrared (NIR) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In the supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' S2, this absorption spectrum is extended out past 3 µm to the IR-region where the strong similarity to graphene ab- sorption is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 1d plots the normalized transient transmission (∆T/T, semi-log scale) kinetics of sequentially photore- duced GO/rGO samples acquired with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV degener- ate pump and probe configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' As the degree of re- duction increases, the kinetic relaxation rate accelerates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The data shown in both Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1 and 2 fits (solid lines) to a least-squares algorithm requiring three-exponents (τ1, τ2, and τ3) with pulse deconvolution for the 155 fs laser autocorrelation response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' After GO is incorporated and stabilized in the polymer matrix, the relaxation dynamics accelerate monotonically with photoreduction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In stark contrast, the as-grown solution of GO (gray line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1d) has much longer TA relaxation dynamics at all timescales, bearing little resemblance to faster graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV visible probe energy, the GO polymer composite that received no reduction (highest oxygen content) has the longest TA relaxation kinetics with its τ3 component comprising 21% of total decay amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1d shows the τ2 lifetimes all decrease linearly from ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='9 ps with increasing lamp pho- toreduction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' All samples have a characteristic τ2 time similar to graphene’s characteristic ∼1 ps decay ex- pected for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV probe, suggesting all five samples ex- hibit graphene-like hot-electron cooling dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' By analogy with monolayer graphene, the τ1 would be asso- ciated with relaxation by optical phonons, and τ2 with disorder-assisted hot electron cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='29 The fitting pa- rameter for the fast and long decays are constant at τ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='15 ps and τ3 = 66 ps, and all parameters are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2c-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 2 plots how the kinetic relaxation rates depend on the selected probe energy (Epr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2a at Epr=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1d at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV, a similar pattern with photoreduction emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' However, the longest com- ponent, τ3 is negligible for all five cases of photothermal reduction rGO1−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2d the slower τ2 lifetime decreases linearly from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 ps to 1 ps with increasing photoreduction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' τ1 varies the least with photore- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Interestingly, the most reduced samples relax even faster compared to monolayer CVD-grown graphene (black dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figures 2a show fits to a triexponen- tial decay curve showing lifetimes of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 ps, 1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 ps, and >30 ps for τ1, τ2 and τ3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Regardless of the incident TA probe energy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV), rGO samples relaxed progressively faster as the pho- toreduction time increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 2b shows that TA dynamics of GO, rGO3, and rGO5 are slower at Epr= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV (closed circles, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 eV pump) than the Epr= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV (open circles, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV pump) probe energy window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Interestingly, the most reduced sample, rGO5, always decays more quickly than ml-graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This faster de- cay relative to graphene suggests that the photothermal reduction is ultimately damaging the sp2 graphene-sub- lattice by causing increased disorder and defect sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This symmetry-breaking results in low energy disorder 4 0 1 2 3 4 5 6 40 80120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 1 DT/T (log norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=') Time Delay (ps) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV ml graphene t2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='9 ps Eprobe: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 Fractional amplitude [A1] 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 GO photoreduction time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=') Fractional amplitudes [A2, A3] 0 1 2 3 4 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 Relaxtion time, t2 (ps) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV GO photoreduction time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=') 0 2 4 100 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 DT/T (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=') GO rGO1 rGO2 rGO3 rGO4 rGO5 ml graphene Time Delay (ps) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' GO-like sp2 graphene-like disordered sp2 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 Fast relaxatime time, t1 (ps) Eprobe: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV Eprobe=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV Eprobe: FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (a) ∆T/T relaxation kinetics at Eprobe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV accelerate with sequential GO photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Fits show two exponential lifetimes, with only the most oxidized samples requiring a third lifetime of τ3 = 61±2 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (b) The ∆T/T kinetics for Eprobe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV (open circles) relax faster than at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV (closed circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The rGO3 photoreduction stage most closely approximates the ml-graphene interband relaxation kinetics shown (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (c) For each probe energy, the τ1 lifetimes (top) are roughly constant, whereas the τ2 lifetime (bottom) decrease linearly ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5× during with photoreduction to become even faster than ml-graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (d) Amplitudes (A1/2/3) of each lifetime component suggest a composition change with increasing amplitude from sp2 sub-lattice dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The smallest A3 (blue) amplitude quickly decreases to zero as GO is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' states that have been previously observed in conjugated carbon systems .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='32,33 This is further supported by the qualitative increase in lattice defect states that is evident by increased emission in IR region of the PL spectra (see supplemental Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figures 2c-d contain the results of our exponential fit- ting lines shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1d and 2a-b (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The top panel shows the amplitude of the fast time compo- nent (∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 ps) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV, which accelerates only moderately as the GO samples are re- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The middle panel shows the amplitude of the second (τ2 ≈ 1−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 ps, pink) and third (τ3 >30 ps, blue) time components, which both decrease with reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Importantly, the slow time τ3 component goes to zero in the limit of heavy reduction and closely resembles the ml- graphene relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2c shows the τ2 relaxation time of GO decreases roughly linearly with photoreduction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At all probe energies, the τ2 relaxation time decreases with reduction, with rGO3,4,5 having lifetimes shorter than that of CVD graphene un- der the same optical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The CVD ml-graphene (dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1-2) was fit to a τ2 =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='9 ps at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 ps at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 probe energies respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In most heavily oxygenated rGO samples, the longest τ3 ∼ 61 ps component comprises up to 16% of the total decay amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Such samples contain many functional groups, however, the large band gap of the fully oxided regions is well outside the spectral range of both pump and probe laser energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Instead, graphene quantum dots (GQD) create gapped sp2 molecule-like regions with size-tunable bandgaps that are resonant with our probe beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='22 For rGO3,4,5 samples, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2d shows that the τ3 time-component is zero for Eprobe < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV, suggest- ing only graphene-like sp2 sublattice regions are relevant to the electronic dynamics throughout this near-infrared probe region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' ANALYSIS AND DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' rGO Fano Lineshape Absorption Analysis The transmission spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1a, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' S2 and fitted absorption spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3 all show lineshapes similar to ml-graphene throughout the NIR and IR spectral re- gions from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The absorption maxima of both ml-graphene and rGO in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3 (black line) deviate from the tight-binding model prediction of the graphene van Hove singularity M-point resonance at ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='34 Instead, the graphene absorption is best fit by a Fano 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 Absorbance Energy (eV) Fano-lineshape fits: rGO1, least reduction rGO5, highest reduction CVD ml-graphene (norm) p-p* n-s* GQDs: p-p* E Fano n-s* edge defects FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Under each linear absorption spectra (solid lines) the deconvolved Fano resonance lineshape fit is plotted in dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Unlike pristine ml-graphene (black), the two rGO samples plotted also require two convolved Gaussians (dash-dot) suggesting molecular-like transition labeled π to π∗ and edge defect transitions, n to σ∗ (see inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The resulting Fano-Gaussian convolved fits (dotted lines) show the graphene sub-lattice Fano-parameter, q increases with photoreduction consistent with more lattice disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' lineshape with a renormalized peak resonance energy, Er that is red-shifted from the M-point by energy by ∼= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 eeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='35,36 The asymmetric Fano lineshape accounts for the ratio of interference between the dis- crete (M-point) and continuum transition probabilities through the dimensionless Fano parameter, q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='35 Thus, the tight binding model of the graphene absorption spec- trum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3 is renormalized for effective electron-hole interaction effects by fitting to the below asymmetric Fano lineshape, AF ano(E) = A � �� � 2 γ (E − Er) + q �2 1 + � 2 γ (E − Er �2 � �� (1) where γ is the Lorentizian homogeneous linewidth and A is the amplitude scaling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3 plots a hyperspectral measurement of CVD ml-graphene (black line) with its corresponding Fano lineshape fit (dashed line), given by equation 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Table I gives the re- sulting Fano parameters and show excellent agreement of this work graphene values with the established liter- ature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='35,37 This provides an essential calibration base to quantitatively compare against the lineshape fit of rGO absorption spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 3 shows good agreement between the absorp- tion spectra of rGO1 and rGO5, and the asymmetric Fano resonance after it is convolved with two Gaussians peaks at energies corresponding to the absorption of the n − σ∗ and π − π∗ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This fitting analysis sug- gests that the absorption spectrum in rGO can be under- stood to contain a Fano resonance similar to that of CVD Sample Er (eV) γ (eV) q ml-graphene [CVD] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='69 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 ml-graphene [exfoliated]35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 rGO5 [highly reduced] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 rGO1 [barely reduced] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='16 50 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Fano fitting parameters for data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3 (dashed lines) show good agreement with our monolayer graphene data established literature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='35,36 The Fano parameter q of rGO5 best matches ml-graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Two convolved Gaus- sian for GQD π − π∗ and edge state defects are also required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' ml-graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The molecular-like π − π∗ transitions are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3 (inset), and show graphene quan- tum dot (GQD) states also contribute to the spectral weight and are centered near 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='38 At 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV, rGO also contains sub-gap defect states between the π and π∗ states, which results from previously reported local oxygen-based disorder that creates edge defect state (n) to σ∗ transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2,39–42 Due to the heterogeneous oxygen coverage, these local disorder edge states have a much broader absorption FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' As rGO1 is further reduced, we observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3 that the peak area of the n−σ∗ Gaus- sian decreases as oxygen is removed, resulting in fewer edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Both our most oxidized samples (GOo and GOsolution) did not fit well to a Fano lineshape, suggest- ing only rGO samples have a graphene-like absorption lineshape in the IR and NIR regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Table I contains a summary of the Fano fitting parame- ters, showing good agreement between the literature35,37 and our results for monolayer graphene and rGO5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' rGO5 contains a large absorption from the linear dispersion near the K and K’ points, where excited carriers cou- ple strongly to the continuum, similar to monolayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' For rGO1, the Fano parameter q decreases sig- nificantly from monolayer graphene, suggesting electron- hole interaction effects are increasingly screened for tran- sitions near the van Hove singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' For GO and lightly reduced rGO, Table I shows the Fano parameter is many times larger than highly reduced samples and mono- layer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This suggests the many edges states in more oxidized graphene couple strongly to continuum- like states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 3 shows a qualitative depiction of how the density of states changes from GO to rGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' As the samples are reduced, they contain larger area regions of non-interrupted sp2 carbon, leading to a more graphene- like distribution of continuum states, resulting in a better Fano lineshape fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The two convolved Gaussians show the effect of reduction on the absorption spectra, with the amplitude of the n−σ∗ transition decreasing significantly, suggesting the removal of oxygen functional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' We also see that the absorption peak in rGO1 shifts slightly to lower energy compared to rGO5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This shift has been theoretically predicted by Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='22, who used DFT to calculate the band structure of GO at varying oxygen content, finding that the addition of oxygen decreases the band gap at the M-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' However, the underlying DOSn 元 @6 Fano resonance energy (ER in Table I) does not change with photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The very large q Fano parameter required to fit the most oxidized rGO1 samples suggests the sp2 hybridized regions are not extensively delocalized and retain a molecular-like character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Hot-electron cooling rates in reduced graphene oxide Figure 4 fits the hot electron cooling TA kinetics in progressively reduced GO as the TA probe energy is in- creased from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 (top) to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Specifically, the hot-electron cooling rate (τSC) is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Unlike the exponential rate τ2 from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2, τSC is analogous to the recombination rate as the electron cool near the Fermi energy, and is independent of probe energy (Eprobe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' To connect the above phenomenological exponential relax- ation models of GO to this first-principle hot-electron cooling model, the fits in Figure 4 models our TA re- laxation kinetics using a hot electron heat dissipation rate H = Ce(dTe/dt), where Ce and Te are the elec- tronic heat capacity and temperature respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The top-panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4a contains first-principle hot electron cooling model fits (solid lines) to the normalized TA ki- netics of the rGO samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Hot electron cooling rates in rGO can be qualitatively understood by comparing to CVD ml-graphene kinetics (black dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The low- est energy probe (Epr=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV) in the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4a shows the hot electron cooling rate response of ml- graphene (dashed line) is identical to rGO1, rGO2 and rGO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Interestingly, rGO4,5 dissipates heat even faster than CVD ml-graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The mechanism for fast energy dissipation or hot- electron cooling in graphene has been widely debated in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The optical phonon dissipation model28,43,44 evolves on the sub-ps relaxation timescale of the τ1 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At longer relaxation times, the disorder- mediated acoustic phonon decay pathway or supercol- lision (SC) hot electron cooling model are the primary factor limiting cooling of the photoexcited hot elec- tron temperature, Te(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='45 Experimental studies demon- strate the SC-model45 successfully predicts graphene’s photocurrent29, optical46 and electrical47 heating re- sponse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' However, the applicability of the SC-model to more disordered lattice of GO and rGO has not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' To understand hot electron cooling in rGO, we apply the acoustic phonon SC-model illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4b (in- set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In the SC model, hot electron cooling near the Fermi level occurs without crystal momentum conserva- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Instead, higher-energy (∼ kBTe) acoustic phonons are emitted with the momentum imbalance, qrecoil ac- counted for by disorder-induced intrinsic lattice recoil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='45 This SC- hot electron is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4b (inset), and gives in a faster hot electron cooling rate than a hot- phonon model that is given by,29,45 dTe dt = − H αTe = −A α T 3 e − T 3 l Te .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (2) where A/α is the SC rate coefficient, Tl and Te are the lattice and electron temperatures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2, Te(t) ∼= To 1+ATot/α when Te(t) ≫ Tl, where To is the initial electron temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Since all data shown is at Tl =292 K, the transient change in Te(t) is small compared to Tl, or Te(t) − Tl ≪ Tl such that we can approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2 by expanding the leading terms to arrive at the room-temperature hot electron tempera- ture, Te(t) ∼= Tl + (To − Tl)e−t/τSC, to get the expression τ −1 SC = 3ATl/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='29 The TA response is obtained using the hot electron (or hole) temperature (Te) through analytically fitting to the transient interband optical conductivity, ∆σ(Eo, t) = −e2/4ℏ � fe/h(Te(t), Epr) − fe/h(Tl, Epr) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='34 The Fermi- Dirac hot-electron occupancy function, fe/h(Te(t), Epr) at the probe energy (Epr) equations are given in the Sup- plementary Materials as a change in interband optical conductivity ∆σ(t, Epr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='34,48 In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4b the hot electron cooling rates (τ −1 SC) for rGO are extracted by fitting the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4a to the analytical SC-model solution (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2), allowing for two additional exponential components (τ1 and τ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This fast component, τ1 ∼= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='34 ps, averages over the initial electron thermalization and optic phonon emission timescale and is discussed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='49,50 Any molecular-like π − π∗ transitions present are captured by τ3 ∼ 61 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The accelerating TA relaxation kinetics in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4a are consistent with the idea that photoreduction of GO creates more disorder and defects on the graphene sub- lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 4b shows an increase in the rate of hot electron cooling, τ −1 SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Unlike the earlier exponential fits, the rate τ −1 SC is independent of the probe energy and is the rate at which the hot-electron Fermi-Dirac distribu- tion cools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The hot electron cooling time for the com- parison monolayer CVD -grown graphene (dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4b) at 292 K is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' τ −1 SC increases by a factor of ∼6 as the samples are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This suggests the xenon arc lamp used to reduce GO is a largely destructive pro- cess to underlying sp2 sub-lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At the highest level of photoreduction, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4b suggests the increased lat- tice disorder destroys the desired graphene-like extended lattice by creating to many point-defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The τ −1 SC = 3ATl/α expression is a direct measure of lattice disorder by the expression A α ∼= 2 3 λ kF l kB ℏ , where the mean free scattering path is kF l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='45 The electron- phonon coupling strength can be approximated as λ = D2 ρs2 2EF π(ℏvF )2 , where both the deformation potential, D and Fermi energy EF are the experimental variable that in- crease the hot electron cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 4b shows that A α ∼= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 ns−1K−1 for rGO1−3, which matches the monolayer CVD graphene values in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='46 How- ever, further photoreduction increases A α upto 6×, sug- gesting the graphene sub-lattice is being damaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' If the 7 0 1 2 3 4 5 0 1 2 3 Hot electon cooling rate (1/tsc) ps-1 GO photoreduction time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=') Eprobe= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV Eprobe= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV 0 1 2 3 Disorder parameter (A/a) ns-1 K-1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 1 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 1 DT/T, normalized Time Delay (ps) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Eprobe =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='30 eV ml graphene 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV rGO5 GOo rGO4 rGO1 rGO2 rGO3 E k qSC GOo rGO1-3 rGO5 rGO4-5 graphene FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (a) TA relaxation kinetics of the six progessively reduced GO GO samples compared to ml-graphene (black dashed) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV probe energies (top to bot- tom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Fitted lines now incorporate the SC-hot electron cool- ing model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (b) SC-model hot electron cooling rates (τ −1 SC) extracted increase sharply for longest photoreduction times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' For rGO1−3, the disorder parameter A/α is similar to ml-graphene rate (dashed), and expectantly is invariant to the laser probe energy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV (black) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='29 deformation potential is approximately constant, then that A/α ∝ EF /kF l, suggesting that the damage of photoreduction decreases the mean free scattering path by photoionization, which increase sp2 sub-lattice defect sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Our fitted data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4 confirms that acous- tic phonons supercollisions (SCs) best describe the rate- limiting heat dissipation kinetics in reduced graphene ox- ide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4b shows how disorder from pho- todamage to the rGO lattice systematically increases the hot-electron cooling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This controlled change in lat- tice disorder provides new evidence of the predominant role of disorder-assisted SC in describing the hot-election in graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Oxygenated sub-lattice contributions from graphene quantum dots Sections IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='A and B above both show the rGO sample and ml-graphene have remarkably similar lineshape and hot-electron cooling rates over optical energies that rang- ing from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This section focuses one the dif- ferences that arise in visible and UV range where GQDs and defect-edge states are also also optically be excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 5a plots the PL emission spectra of the least re- duced, GOo and most reduced, rGO5 samples after a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 eV excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The main asymmetric peak appears to shift from ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='7 eV with photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The ex- perimental emission spectra (dots) are fit (solid lines) us- ing 4 convolved Gaussian peaks (dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' All peak energies and FWHM spectral width (except at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='34 eV) are found to be approximately invariant to photoreduc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The peak at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='7 eV in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 5a corresponds with emission from the smallest graphene quantum dot states (labeled GQD1) π∗ − π orbital relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At the lower energies, both peaks centered near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='55 eV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='80 eV grow with photoreduction, consistent with emission from larger graphene quantum dot states labeled GQD2 and GQD3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' We observe an increase in the emis- sion intensity from these three sp2 peaks with reduction, confirming they do not result from oxygen groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Con- versely, the emission at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV represents the carrier re- combination in sp3 oxygen (σ∗ − n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The magnitude and width of this emission decrease with reduction as oxygen functional groups are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' PL from GO and rGO in solution has been widely doc- umented in the literature, showing that reduction of GO increases PL intensity at near IR wavelengths while also blue-shifting the main peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='38,51,52 In accordance with literature, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 5a shows an increase in PL intensity with reduction, at peaks centered at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='80 eV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='55 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The PL of the oxygenated GO lattice is known to emit broadly near 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 eV with locally varying oxygen content responsible for the broader FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='26,41 In rGO, PL is dominated by π∗ − π carrier recombination in regions of confined graphene quantum dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' As the reduction pro- cess removes oxygen, formerly isolated sp2 carbon atoms join together to form conjugated carbon rings, and re- gions that already contained large area conjugated sp2 carbon structures increase in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The observed decreas- ing area of the peak at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV with photoreduction sug- gests this peak emission is likely due to egde states or oxygen-defects the boundaries of the sp3 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The newly formed GQD in rGO are ascribed to the increas- disorder-assisted scattering8 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 DT/T (10-6) Pump Fluence (x1012 photons/cm2) GO0 rGO1 rGO2 rGO3 rGO4 rGO5 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0 2 4 6 8 10 12 PL Counts (x104) s-1 Emisssion Energy (eV) GO0 rGO5 0 1 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='00 DT/T (10-3) Time Delay (ps) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='12 DT/T (10-3) GQD1 Eprobe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV Eprobe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV graphene + GQD2 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' GQD2 GQD3 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='10 DT/T (10-3) Pump Fluence (x1012 photons/cm2) Eprobe = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV (QQD2) Eprobe= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV (graphene) 0 1 2 3 4 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV (gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='+ GQD2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV (gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='+ n-s) DT/T (10-3) GO photoreduction time (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=') 𝜋∗ 𝜋 n-s GQD1 s* s ~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV ex: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 eV n: edge defects GQD2 GQD3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='6 eV graphene FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (a) The photoluminescence emission spectra of GOo (green line fit) and rGO5 (red line fit) with 4 convolved Gaussian fit (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='Photoreduction increases the PL peaks from graphene quantum dots resonances, labeled GQD1−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Conversely, emission from the oxygenated sub-lattice n − σ defect edge state decreases as GO is reduced (see inset for corresponding transitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (b)The degenerate TA response near the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV GQD2 resonance (top) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' near the excited state absorption at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (inset) TA response increasing with photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (c) ∆T/T pump power photon fluence dependence of reduced GO samples using a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Fit are to the graphene SC-hot electron cooling model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Over a wide range of incident photon flux, the saturable absorption susceptibility, ∆T/T is invariant to photoreduction suggesting only graphene hot electrons are probed below ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (inset) Conversely at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV the ∆T/T changes strongly, suggesting increasing GQD2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' ing PL at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='7 eV, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='80 eV and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='55 eV peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' DFT studies by Sk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='21 show how the bandgap energy of a GQDs changes with respect to its size and found that GQDs about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 nm in mean diameter create Frenkel exciton states near 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='7 eV, while slightly larger 2 nm GQDs emit around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' rGO contains an ensem- ble of GQDs of various sizes separated by oxygenated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Reduction removes oxygen, gradually increasing the GQD size, evidenced by the increased PL in rGO at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='55 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='80 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 5a inset contains a qualitative depiction of the bands and energy levels in GO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The optical response of graphene is determined by the π and π∗ states, which lie between the σ − σ∗ gap in GO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='41,53 Oxygen functional groups break the symmetry of the pristine graphene lat- tice, resulting in localized defect states that exist in the π−π∗ gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Since the gap between σ states is much larger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='4 eV, this emission is suggested as n−σ transition (dashed purple arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In both GO and rGO, emission at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='7 eV dominates the PL spectra, which was shown to result from π states in isolated sp2 domains (gray dashed arrow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='38 Emission at lower energies comes from a broad range of GQD states and the local disorder states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 5b shows the degenerate transient absorption response of the samples at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV, we observe a saturable absorption sig- nal containing a long component that slowly goes away with reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV, we see a reverse saturable absorption response, which decays extremely quickly in all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' A similar transition has been previously ob- served by Bhattacharya et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' al, who saw that a sign flip in the pump-probe response occurred near 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='54 Since the most reduced samples have the largest reverse sat- urable absorption response, we can rule out excited state absorption from oxygen groups as the cause of the sign flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' We attribute this sign-change to absorption from the interband transition in graphene, which has been previ- ously documented to exhibit a sign flip for high pump fluences at this energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='48,55 We do not see a change in sign when probing the oxygen states at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV, further confirming the sp2 nature of the peak labeled GQD2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 5c shows the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV probe energy pump flu- ence dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At low pump fluences, the TA re- sponse of all samples exhibits a linear dependence on the pump fluence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Above incident photon flux of ∼ 4 × 1012 photon/cm2, a sublinear trend is observed that is fit to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2 hot electron cooling model TA response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The nonlinear saturation effect fits to the expected nonlinear Fermi-Dirac filling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Notably, the more oxidized GO1 and rGO2 samples have the most nearly linear be- haviors, consistent with the expected smaller confined sp2 sub-lattice regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Conversely, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 5c(inset) shows the pump power dependence for the differential trans- mission at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV pump and probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' GO displaying the smallest response, which increases with reduction until rGO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The response saturates for the three most re- duced samples as shown in the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This trend matches the absorption spectra at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV, where the absorption increases monotonically with reduction, with the exception of the ∆T/T response saturating for the most reduced samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The pump dependence gives us insight into how the probe response changes with lattice temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At low pump powers, the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV probe has the same magni- tude for all samples, suggesting that even oxidized sam- ples have large regions of graphene-like sp2 hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 9 The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV data remains linear overall pump fluences but has a large dependence on the amount of reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' While the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV data probes graphene-like states, the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV data primarily probes the confined GQD2 states that lead to longer lifetimes and a strong dependence on photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The size and population of these GQD states depend heavily on the oxygen content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' As shown in Figure 5c(inset), reduction increases the transient re- sponse, which suggests that reduction increases the pop- ulation of sp2 GQD2 states that absorbs at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This trend matches the increase in PL seen at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV after photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' GQD donor graphene acceptor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='0 DT/T, Normalized Time Delay (ps) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV pump 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV probe rGO5 rGO1 rGO3 GOo E k photoreduction FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (a) Normalized transient absorption kinetics shows a 170 delayed rise for GO that systematically accelerates with successive photoreduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (b) This rise is assigned to an acceptor-donor relationship between the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV pump of GQD states and the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV probe of the accepting graphene states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' (c) Band illustration of rGO depicts charge transfer described from confined GQDs to larger sp2 graphene-like regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Donor-acceptor electronic transfer in rGO Using non-degenerate TA spectroscopy, we can excite molecular-like GQDs at high energies and probe the elec- tron transfer rate to graphene at lower energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 6a shows the normalized TA relaxation at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV pump 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 eV probe near time zero, which shows a clear delayed rise in the most oxidized samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Con- versely, the most reduced samples show a rise limited by the laser cross-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This delayed rising kinetic edge is indicative of an acceptor-donor electron relation- ship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Charge transfer has been documented in GO, where photoexcited charges on a different molecular species are transferred to GO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='56–58 Figure 6b illustrates charge transfer between molecular GQDs and larger graphene- like regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' When the pump moved to longer energies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1c), the delayed rise is no longer seen because the population of GQD donors is too small relative to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 6b depicts the charge transfer process that is responsible for the observed delayed rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Carriers pho- toexcited in the confined GQD states are localized by the surrounding oxygen functional groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In GO, the large density of oxygenated regions results in a weaker coupling between confined GQDs and graphene submetallic sub- lattice regions, leading to the observed delayed rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In the photoreduced samples, carriers excited into sp2 GQD states are now closer to extendend graphene regions, and so the delayed acceptor-donor electron transfer is not ob- served to lower energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 6c gives a qualitative description of the struc- ture and acceptor-donor electron transfer process in rGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Our graphene oxide begins with ∼44% oxygen content, these oxygen functional groups interrupt the delocalized π-orbitals and prohibit hopping between carbon sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Reduction removes oxygen, which decreases the mean distance from a confined GQD donor and graphene-like sp2 sublattice region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Such changes to the effective per- colation network of the sp2 sublattice have previously been shown to also increase GO carrier mobility and conductivity59,60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The longer dynamics in GO are caused by excited carriers being more isolated by larger oxy- genated regions as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 6c, which limit possible relaxation pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' In rGO, some of the oxygen has been removed, recovering large-area graphene-like do- mains which decay more quickly than pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' CONCLUSIONS The highly variable composition of the quasi- amorphous GO 2D lattice makes a systematic compar- ison against monolayer graphene a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' To help overcome this challenge, GO is suspended in a poly- meric network scaffold where five successive photoreduc- tions (rGO1−5) were possible without any evidence of inter-layer aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Ultimately, this yielded opti- cal quality rGO films with an absorption lineshape that fits to ml-graphene Fano resonance lineshape parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Likewise this step-wise photoreduction accelerates the hot electron relaxation kinetics monotonically over each of the variable probe energy windows studied from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='2 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' At intermediate photoreduction times or rGO2−3, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 4 shows that a hot electron cooling model of disorder-assisted supercollision matches the τSC =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='1 ps hot electron cooling of monolayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Figure 4b shows the recovery of ultrafast hot electron relaxation rates similar to monolayer-graphene in moderately re- duced samples(rGO1−3 ), suggesting a largely uninter- rupted sp2 bonded network analogous to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Under extreme photoreduction or using UV-Vis optical Confined Metallic region region0 0 0 0 0 0 0 0 0 0 0 0 010 excitation, the optical properties of rGO begin to deviate strongly from graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Owing to increasing local dis- order and broken lattice symmetry, extreme photother- mal reduction yields hot electron cooling rates that are faster than pristine graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Subsequent photoreduc- tion accelerates the extracted hot electron cooling rate 10-12x, revealing how photodamage induces local disor- der to mediate faster hot electron cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' On longer, >50 ps timescales, rGO also exhibits a slower decay re- sponse than graphene owing to many isolated graphene quantum dot (GQD) regions and oxygenated edge trap states which serve to delay the ground state recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Using probe energies in the visible wavelength range at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='8 eV, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1c and 4 shows that photothermal reduc- tion does not recover pristine graphene properties, as ev- idenced by the slower decay kinetics of all rGO samples relative to graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' The prevalence of isolated GQDs regions and oxygenated-edge trap states each create fur- ther bottlenecks of electronic relaxation that slow the effective relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Fortunately, we find these long life- times of rGO are no longer oberved below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='3 eV optical excitations, as there are no discernible GQD sub-lattice states large enough to creae a resonance at these energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Collectively, these results show many of the desirable op- toelectronics properties of 2D graphene can be replicated using selectively reduced graphene oxide suspended in a 3D bulk polymeric network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' This study lends itself to large-scale processing of rGO thin films and applications in high-speed optoelectronics and photonic switching ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' ACKNOWLEDGMENTS This material is based upon work supported by the Office of the Under Secretary of Defense for Research and Engineering under award number FA9550-22-1-0276, and the DEVCOM Army Research Laboratory award number W56HZV-16-C-0147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Supplementary Materials: Details on sample char- acteristics, data modeling methods, and further absorp- tion and PL spectral data show similar graphene-like propertis out to the mid-IR regions as far as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content='5 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Data Availability Statement:The data that sup- port the findings of this study are available from the cor- responding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Yang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Lee, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Ng, Science and Technology of Advanced Materials 19, 613 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' 2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Mkhoyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Lacey, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Xie, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Li, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Danner, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} +page_content=' Hu, Materials Today 21, 186 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edFPT4oBgHgl3EQfzjXc/content/2301.13176v1.pdf'} diff --git a/fNAzT4oBgHgl3EQfMPtb/content/tmp_files/2301.01128v1.pdf.txt b/fNAzT4oBgHgl3EQfMPtb/content/tmp_files/2301.01128v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9e3e290a6b9065aa5b605a742d10d0961cdd47b --- /dev/null +++ b/fNAzT4oBgHgl3EQfMPtb/content/tmp_files/2301.01128v1.pdf.txt @@ -0,0 +1,624 @@ +arXiv:2301.01128v1 [math.AP] 3 Jan 2023 +NON-NORMALIZED SOLUTIONS TO THE HOROSPHERICAL +MINKOWSKI PROBLEM +LI CHEN +Abstract. Recently, the horospherical p-Minkowski problem in hyperbolic space was +proposed as a counterpart of Lp Minkowski problem in Euclidean space. +Through +designing a new volume preserving curvature flow, the existence of normalized even +solution to the horospherical p-Minkowski problem was solved for all p ∈ R. However, +due to the lack of homogeneity of the horospherical p-surface area measure, it is difficult +to remove the normalizing factor. In this paper, we overcome this difficulty for −n ≤ +p < n by the degree-theoretic approach. +1. Introduction +The classical Minkowski problem and its extension Lp Minkowski problem [20, 3] play +very important roles in the study of convex bodies in Euclidean space (see also [21]). It +is a dream to develop similar problems in hyperbolic space. Recently, Andrews-Chen- +Wei declared interesting formal similarities between the geometry of h-convex domains +in hyperbolic space and that of convex Euclidean bodies (see section 5 in [2]). Along +the lines of Andrews-Chen-Wei, Li-Xu [16] introduced the horospherical p-surface area +measure by use of the hyperbolic p-sum which they defined and proposed the associated +horospherical p-Minkowski problem. We will briefly describe them below followed by +Section 5 in [2] and Section 2 [16]. +We shall work in the Hyperboloid model of Hn+1. For that, consider the Minkowski +space Rn+1,1 with canonical coordinates X = (X1, ..., Xn+1, X0) and the Lorentzian +metric +⟨X, X⟩ = +n+1 +� +i=1 +(Xi)2 − (X0)2. +Hn+1 is the future time-like hyperboloid in Minkowski space Rn+1,1, i.e. +Hn+1 = +� +X = (X1, · · ·, Xn+1, X0) ∈ Rn+1,1 : ⟨X, X⟩ = −1, X0 > 0 +� +. +2010 Mathematics Subject Classification. Primary 35J96, 52A39; Secondary 53A05. +Key words and phrases. Minkowski type problem; h-convex; Mong-Amp´ere type equation. +1 + +2 +LI CHEN +In the Hyperboloid model of Hn+1, the horospheres are hypersurfaces in Hn+1 whose +constant principal curvatures equal to 1 everywhere. +They can be parameterized by +Sn × R +Hx(r) = {X ∈ Hn+1 : X · (x, 1) = −er}, +where r represents the signed geodesic distance s from the “north pole” N = (0, 1) ∈ +Hn+1. The horo-ball which is the interior of the horosphere is denoted by +Bx(r) = {X ∈ Hn+1 : 0 > X · (x, 1) > −er}. +If we use the Poincar´e ball model Bn+1 of Hn+1, then Bx(r) corresponds to an (n + 1)- +dimensional ball which tangents to ∂Bn+1 at x. Furthermore, Bx(r) contains the origin +for r > 0. +Definition 1.1. A compact domain Ω ⊂ Hn+1 (or its boundary ∂Ω) is horospherically +convex (or h-convex for short) if every boundary point p of ∂Ω has a supporting horo-ball, +i.e. a horo-ball B such that Ω ⊂ B and p ∈ ∂B. +For a smooth compact domain Ω, we say Ω (or ∂Ω) is uniformly h-convex if its +principal curvatures are greater than 1. +Definition 1.2. Let Ω ⊂ Hn+1 be a h-convex compact domain. For each X ∈ ∂Ω, ∂Ω +has a supporting horo-ball Bx(r) for some r ∈ R and x ∈ Sn. Then the horospherical +Gauss map +G : ∂Ω → Sn +is defined by +G(X) = x. +Let Ω be a h-convex compact domain in Hn+1. Then for each x ∈ Sn we define the +horospherical support function of Ω (or ∂Ω) in direction x by +u(x) := inf{s ∈ R : Ω ⊂ Bx(s)}. +We also have the alternative characterisation +u(x) = sup{log(−⟨X, (x, 1)⟩) : X ∈ Ω}. +(1.1) + +MINKOWSKI PROBLEM +3 +The support function completely determines a h-convex domain Ω, as an intersection of +horo-balls: +Ω = +� +x∈Sn +Bx(u(x)). +(1.2) +If the compact domain Ω is uniformly h-convex, then G is a diffeomorphism from ∂Ω +to Sn. Then, X = G−1 is a smooth embedding and X can be written in terms of the +support function u, as follows: +X(x) = 1 +2ϕ(−x, 1) + 1 +2 +�|Dϕ|2 +ϕ ++ 1 +ϕ +� +(x, 1) − (Dϕ, 0), +(1.3) +where u = log ϕ is the horospherical support function of the domain Ω, D is the Levi- +Civita connection of the standard metric σ of Sn. Then, after choosing normal coor- +dinates around x on Sn+1, the area element dµ of Ω at X(x) = G−1(x) can be given +by +dµ = +� +det⟨∂iX, ∂jX⟩dσ = det A[ϕ]dσ, +(1.4) +where +A[ϕ] = D2ϕ − 1 +2 +|Dϕ|2 +ϕ +I + 1 +2 +� +ϕ − 1 +ϕ +� +I +and I is the identity matrix. Moreover, the compact domain Ω is uniformly h-convex if +and only if the matrix A[ϕ] is positive definite. +Until now, we have seen many interesting similarities between the geometry of h- +convex domains in hyperbolic space and that of convex Euclidean bodies. Recently, +Li-Xu [16] developed deeply this similarities. In particular, they introduced a sum of +two sets in hyperbolic space which they called the hyperbolic p-sum (see Definition 1.1 +[16]). +Definition 1.3. Let 1 +2 ≤ p ≤ 2, a ≥ 0, b ≥ 0 and a + b ≥ 1, and let K and L be two +smooth uniformly h-convex compact domains with the horospherical support functions +uK(x) and uL(x) respectively. The hyperbolic p-sum Ω := a · K +p b · L of K and L is +defined by the h-convex compact domain with the horospherical support function +uΩ(x) := 1 +p log +� +aepuK(x) + bepuL(x)� +. +They also gave a pointwise definition of the hyperbolic p-sum (see Definition 1.3 in +[16]). Then, they calculated the variation of the volume along the hyperbolic p-sum (see + +4 +LI CHEN +Lemma 5.2 in [16]) +lim +t→0+ +Vol(K +p t · L) − Vol(K) +t += 1 +p +� +Sn ϕp +LdSp(K, x), +where the horospherical p-surface area measure of a smooth uniformly h-convex compact +domain K ⊂ Hn+1 is defined by (see Definition 5.2 [16]) +dSp(K, ·) = ϕ−p +K det(A[ϕK])dσ, +uK = log ϕK and uL = log ϕL are the horospherical support functions of smooth uni- +formly h-convex compact domains K and L respectively. In particular, p = 0, dSp(K, ·) +is just the surface area measure (1.4) of K. +Then, Li-Xu proposed the associated horospherical p-Minkowski problem (see Problem +5.1 [16]): +Problem 1.1. For a given smooth positive function f(x) defined on Sn, we ask the +sufficient and necessary conditions for f(x), such that there exists a smooth uniformly +h-convex compact domain K ⊂ Hn+1 satisfying +dSp(K, x) = f(x)dσ, +which is equivalent to find a smooth positive solution ϕ(x) with A[ϕ(x)] > 0 for all +x ∈ Sn (or the uniformly h-convex solution for short) to the equation +ϕ−p(x)det(A[ϕ(x)]) = f(x), +(1.5) +where u = logϕ is the support function of some horo-convex domain in Hn+1. +In particular, for p = 0, Problem 1.1 is just the prescribed surface area measure +problem for horo-convex domains in Hn+1. For p = −n, Problem 1.1 is just the prescribed +shifted Gauss curvature problem (see Section 7 in [16]) or the n-th symmetric Christoffel +problem in Hn+1 (see Page 26 in [5]). +Through designing a new volume preserving curvature flow, Li-Xu solved the existence +of normalized even solution to the horospherical p-Minkowski problem 1.1 for all p ∈ R. +A function g : Sn → R is called even if g(x) = g(−x) for all x ∈ Sn. +Theorem 1.2 (Li-Xu [16]). Given a smooth positive even function f defined on Sn, for +any p ∈ R, there exists a smooth even and uniformly h-convex solution to the equation +ϕ−pdet(A[ϕ]) = γf +(1.6) +for some γ > 0. + +MINKOWSKI PROBLEM +5 +Due to the lack of homogeneity of the horospherical p-surface area measure, it is very +difficult to remove the normalizing factor γ in Theorem 1.2. This difficulty also appears +in Orlicz-Minkowski-type problem [19] and the Gaussian Minkowski type problem [11, +18, 6, 7]. In the latter problem, a degree-theoretic approach was used to overcome this +difficulty in the even case. +In this paper, we can obtain a non-normalized solution (without the normalizing factor +γ) to Problem 1.1 for −n ≤ p < n by the degree-theoretic approach. +Theorem 1.3. Assume −n ≤ p < n, there exists a smooth even and uniformly h-convex +solution to the equation (1.5) for any smooth positive even function f defined on Sn. +It should be noted that the continuity method was not applied to prove existence. +The reason is that the continuity method requires that the corresponding linearized +equation is invertible on any admissible solution and this result is not available for +the equation (1.5) studied in this paper. However, within the framework of the degree +theory developed in [17] it suffices to know invertibility on constant solutions which is +guaranteed by the uniqueness of even solutions to the equation (1.5) for −n ≤ p < n +(see Theorem 8.1 (6)(7) in [16]). For other ranges p ≥ n or p < −n, either the constant +solution may not be unique or its uniqueness is unknown. (see Theorem 8.1 in [16]). So, +it is difficult to prove existence by the degree-theoretic approach, although the a prior +estimates for solutions to the equation (1.5) can be established for all p < n. +Remark 1.4. It is interesting to consider the uniqueness of solutions to the equation +(1.5) when f is not constant. In contrast with Lp Minkowski problem in Euclidean space +for which the uniqueness of solutions is guaranteed by Lp Brunn-Minkowski inequality +for p ≥ 1 [20], the uniqueness of solutions to the horospherical p-Minkowski problem +1.1 is unknown due to the lack of analogous inequality in hyperbolic space (see related +discussions in Section 9 [16]). +The present paper is built up as follows. In Sect. 2, we obtain the a priori estimates. +We will prove Theorem 1.3 in Sect. 3. +2. The a priori estimates +For convenience, in the following of this paper, we always assume that f is a smooth +positive, even function on Sn and ϕ is a smooth even, uniformly h-convex solution to + +6 +LI CHEN +the equation (1.5). Moreover, let Ω be the uniformly h-convex compact domain in Hn+1 +with the horospherical support function u = log ϕ. Clearly, Ω is symmetry with the +origin and ϕ(x) > 1 for x ∈ Sn. +The following easy and important equality is key for the C0 estimate. +Lemma 2.1. We have +1 +2 +� +max +Sn ϕ + +1 +maxSn ϕ +� +≤ min +Sn ϕ. +(2.1) +Proof. The inequality can be found in the proof in Lemma 7.2 in [16]. For completeness, +we give a proof here. Assume that u(x1) = maxSn u and denote X(x1) = G−1(x1) as +before. Then, we have for any x ∈ Sn by the definition of the horospherical support +function (1.1) +−⟨X(x1), (x, 1)⟩ ≤ ϕ(x), +∀x ∈ Sn. +Substituting the expression (1.3) for X into the above equality yields +1 +2ϕ(x1)(1 + ⟨x1, x⟩) + 1 +2 +1 +ϕ(x1)(1 − ⟨x1, x⟩) ≤ ϕ(x), +(2.2) +where we used the fact Dϕ(x1) = 0. Note that ϕ(x1) ≥ 1, we find from (2.2) +1 +2 +� +ϕ(x1) + +1 +ϕ(x1) +� +≤ ϕ(x) +for +⟨x, x1⟩ ≥ 0. +(2.3) +Since Ω is origin symmetric, we can assume that the minimum point x0 of u(x) satisfies +⟨x0, x1⟩ ≥ 0. Thus, the equality (2.1) follows that from (2.3). +□ +Remark 2.2. It is very interesting to compare the inequality (2.1) with the the analogous +inequality for convex bodies in Euclidean space. In fact, for an origin-symmetric convex +body in Euclidean space, the definition of its support function gives +|⟨x1, x0⟩| max +Sn h ≤ min +Sn h, +(2.4) +where h is the support function of the convex body, x1 and x0 are the maximum and +minimum points of h respectively. Clearly, the inequality (2.1) in hyperbolic space is +better than (2.4), since the term |⟨x1, x0⟩| in (2.4) may not be bounded from below. +Now we begin to consider the C0-estimate. Similar estimate is obtained in Lemma +7.2 [16] when the volume of h-convex domain is fixed. But for our case, the volume is +not fixed, we use the maximum principle to get the C0-estimate. + +MINKOWSKI PROBLEM +7 +Lemma 2.3. If p < n, we have +0 < 1 +C ≤ u(x) ≤ C, +∀ x ∈ Sn, +(2.5) +where C is a positive constant depending on p, n and f. +Proof. Using the maximum principle, we have from the equation (1.5) +(max +Sn ϕ)n−p 1 +2n +� +1 − +1 +(maxSn ϕ)2 +�n +≥ C > 0 +and +(min +Sn ϕ)n−p 1 +2n +� +1 − +1 +(minSn ϕ)2 +�n +≤ C. +Since the function +g(x) = xn−p 1 +2n +� +1 − 1 +x2 +�n +is increasing in [1, +∞) for p < n, g(1) = 0 and g(+∞) = +∞, we obtain +min +Sn ϕ ≤ C, +and +max +Sn ϕ ≥ C > 1. +(2.6) +Combining (2.6) and (2.1), we find +1 < C ≤ min +Sn ϕ ≤ max +Sn ϕ ≤ C′, +which implies that +0 < 1 +C ≤ min +Sn u ≤ max +Sn u ≤ C. +So, we complete the proof. +□ +As a corollary, we have the gradient estimate from Lemma 7.3 in [16]. +Corollary 2.1. We have +|Dϕ(x)| ≤ C, +∀ x ∈ Sn, +(2.7) +where C is a positive constant depending only on the constant in Lemma 2.3. +We give some notations before considering the C2 estimate. Denote by +Uij = ϕij − 1 +2 +|Dϕ|2 +ϕ +δij + 1 +2(ϕ − 1 +ϕ)δij +and +F(U) = det U, +F ij = ∂F +∂Uij +, +F ij,kl = +∂2F +∂Uij∂Ukl +. + +8 +LI CHEN +Lemma 2.4. We have for 1 ≤ i ≤ n +λi(U(x)) ≤ C, +∀ x ∈ Sn, +where λ1(U), ..., λn(U) are eigenvalues of the matrix U and C is a positive constant +depending only on the constant in Lemma 2.3 and Corollary 2.1. +Proof. Since +λi(U) ≤ trU = ∆ϕ − n +ϕ|Dϕ|2 + n +2(ϕ − 1 +ϕ), +∀1 ≤ i ≤ n, +it is sufficient to prove ∆ϕ ≤ C in view of C0 estimate (2.5) and C1 estimate (2.7). +Moreover, these two estimates together with the positivity of the matrix U imply +λi(D2ϕ) ≥ −C for 1 ≤ i ≤ n. Thus, +|λi(D2ϕ)| ≤ C∆ϕ, +∀1 ≤ i ≤ n. +(2.8) +We take the auxiliary function +W(x) = ∆ϕ. +Assume x0 is the maximum point of W. After an appropriate choice of the normal frame +at x0, we further assume Uij, hence ϕij and F ij is diagonal at the point x0. Then, +(2.9) +Wi(x0) = +� +k +ϕkki = 0, +and +(2.10) +Wii(x0) = +� +k +ϕkkii ≤ 0. +From the positivity of F ij and (2.10), we arrive at x0 if ∆ϕ is large enough +0 +≥ +� +i +F iiWii = +� +i +F ii � +k +ϕkkii ≥ +� +i +F ii � +k +� +ϕiikk − C∆ϕ +� +, +where we use Ricci identity and the equality (2.8) to get the last inequality. Thus it +follows from the definition of U, (2.5), (2.7) and (2.9), +0 +≥ +� +i +F ii � +q +� +Uiiqq + +� 1 +2ϕ|Dϕ|2� +qq − 1 +2 +� +ϕ − 1 +ϕ +� +qq +� +− C +� +i +F ii∆ϕ +≥ +� +i +F ii � +q +� +Uiiqq + 1 +ϕ(ϕqq)2 − C∆ϕ − C +� +− C +� +i +F ii∆ϕ. +(2.11) +Differentiating the equation (1.5) twice gives +F iiUiiqq + F ij,klUijqUklq = (ϕpf)qq, + +MINKOWSKI PROBLEM +9 +which yields +F iiUiiqq ≥ −C∆ϕ − C +(2.12) +in view of C0 estimate (2.5) and C1 estimate (2.7). Substituting (2.12) into (2.11) and +using (∆ϕ)2 ≤ n � +q(ϕqq)2, we have +0 ≥ +� +C(∆ϕ)2 − C∆ϕ − C +� � +i +F ii − C∆ϕ − C. +(2.13) +Note that +� +i +F ii ≥ n(det U)1− 1 +n = n(ϕpf)1− 1 +n ≥ C. +(2.14) +Then we conclude at x0 from the inequality (2.14) +C ≥ |∆ϕ|2 +if ∆ϕ is chosen large enough. So, we complete the proof. +□ +3. The proof of the main theorem +In this section, we use the degree theory for nonlinear elliptic equation developed in +[17] to prove Theorem 1.3. Such approach was also used in prescribed curvature problem +[1, 14, 12, 13], prescribed curvature problem for for convex hypersurfaces with group +invariance assumptions [8, 9] and the Gaussian Minkowski type problem [11, 18, 6, 7]. +For the use of the degree theory, the uniqueness of constant solutions to the equation +(1.5) is important for us. Fortunately, this fact can be guaranteed by Theorem 8.1 (6)(7) +in [16] for −n ≤ p < n. We summarize it as follows. +Lemma 3.1. For −n ≤ p < n, there exist a unique even solution ϕ = c to the equation +ϕ−p(x)det(A[ϕ(x)]) = γ +(3.1) +for any γ > 0, where c is the unique positive solution to the equation +c−p�1 +2(c − c−1) +�n += γ. +(3.2) +Proof. For −n < p < n, Theorem 8.1 (6) in [16] tells us that there exist a unique solution +ϕ = c to the equation (3.1) for any γ > 0, where c is the unique positive solution to the +equation (3.2). + +10 +LI CHEN +For p = −n, Theorem 8.1 (7) in [16] tells us that the solutions to the equation (3.1) +for any γ > 0 are given by +ϕ(x) = +� +1 + 2γ +1 +n +0 +� 1 +2�� +|x0|2 + 1 − ⟨x0, x⟩ +� +, +where x0 ∈ Rn+1. Clearly, ϕ(x) = +� +1 + 2γ +1 +n +0 +� 1 +2 is the unique even solution. Thus, the +conclusion holds true. +□ +Now, we begin to use the degree theory to prove Theorem 1.3. After establishing the +a priori estimates (2.5), (2.7) and (2.4) for p < n, we know that the equation (1.5) is +uniformly elliptic, i.e. +λi(U(x)) ≥ C > 0, +∀ x ∈ Sn, +(3.3) +where λ1(U), ..., λn(U) are eigenvalues of the matrix U . From Evans-Krylov estimates +[4, 15], and Schauder estimates [10], we have +|ϕ|C4,α(Sn) ≤ C +(3.4) +for any smooth, even and uniformly h-convex solution u to the equation (1.5). We define +B2,α(Sn) = {ϕ ∈ C2,α(Sn) : u = log ϕ is even} +and +B4,α +0 (Sn) = {ϕ ∈ C4,α(Sn) : u = log ϕ is h-convex and even}. +Let us consider +L(·, t) : B4,α +0 (Sn) → B2,α(Sn), +which is defined by +L(ϕ, t) = det U − ϕp[(1 − t)γ + tf], +where the constant γ will be chosen later and U is denoted as before +U = D2ϕ − 1 +2 +|Dϕ|2 +ϕ +I + 1 +2 +� +ϕ − 1 +ϕ +� +I. +Let +OR = {ϕ ∈ B4,α +0 (Sn) : 1 + 1 +R < ϕ, 1 +RI < U, |ϕ|C4,α(Sn) < R}, +which clearly is an open set of B4,α +0 (Sn). Moreover, if R is sufficiently large, L(ϕ, t) = 0 +has no solution on ∂OR by the a priori estimates established in (2.5), (3.3) and (3.4). + +MINKOWSKI PROBLEM +11 +Therefore the degree deg(L(·, t), OR, 0) is well-defined for 0 ≤ t ≤ 1. Using the homo- +topic invariance of the degree (Proposition 2.2 in [17]), we have +deg(L(·, 1), OR, 0) = deg(L(·, 0), OR, 0). +(3.5) +For −n ≤ p < n, Lemma 3.1 tells us that ϕ = c is the unique even solution for +L(ϕ, 0) = 0 in OR. Direct calculation show that the linearized operator of L at ϕ = c is +Lc(ψ) = ∆Snψ + 1 +2 +� +1 + 1 +c2 − pcp−1γ +� +ψ, +where γ is given by the equation (3.2). Since spherical Laplacian has a discrete spectrum, +we choose c = c0 such that Lc0 is an invertible operator. Then we have by Proposition +2.3 in [17] +deg(L(·, 0), OR, 0) = deg(Lc0, OR, 0) = ±1, +where the last inequality follows from Proposition 2.4 in [17] . Therefore, it follows from +(3.5) +deg(L(·, 1), OR; 0) = deg(L(·, 0), OR, 0) = ±1. +So, we obtain a solution at t = 1. This completes the proof of Theorem 1.3. +References +[1] F. Andrade, J. Barbosa and J. de Lira, Closed Weingarten hypersurfaces in warped product man- +ifolds, Indiana Univ. Math. J., 58 (2009), 1691-1718. +[2] B. Andrews, X. Chen and Y. Wei, Volume preserving flow and Alexandrov-Fenchel type inequalities +in hyperbolic space, to appear in J. Eur. Math. Soc.(JEMS), arXiv:1805.11776v1. +[3] K.-S. Chou and X.-J. Wang. The Lp-Minkowski problem and the Minkowski problem in centroaffine +geometry. Adv. Math., 205:33-83, 2006. +[4] L. Evans, Classical solutions of fully nonlinear, convex, second-order elliptic equations, Comm. +Pure Appl. Math., 35 (1982), 333-363. +[5] J. Espinar, J. G´alvez, and P. Mira, Hypersurfaces in Hn+1 and conformally invariant equations: +the generalized Christoffel and Nirenberg problems, J. Eur. Math. Soc., 11 (2009), 903-939. +[6] Y. Feng, W. Liu and L. Xu, Existence of non-symmetric solutions to the Gaussian Minkowski +problem, J. Geom. Anal. (2022), accepted. +[7] Y. Feng, S. Hu and L. Xu, On the Lp Gaussian Minkowski problem, arXiv:2211.10956. +[8] B. Guan and P. Guan, Convex Hypersurfaces of Prescribed Curvatures. Annual of Mathematics, +Vol 256, No. 2 (2002), 655-673. +[9] P. Guan and X. Zhang, A class of curvature type equations Pure and Applied Math Quarterly, +Vol. 17, No. 3 (2021), 865-907. +[10] D. Gilbarg and N. Trudinger, Elliptic partial differential equations of second order, Springer, 2015. +[11] Y. Huang, D. Xi and Y. Zhao, The Minkowski problem in Gaussian probability space, Adv. Math. +385 2021, 107769. + +12 +LI CHEN +[12] Q. Jin and Y. Li, Starshaped compact hypersurfaces with prescribed k-th mean curvature in +hyperbolic space, Discrete Contin. Dyn. Syst., 15 (2006), 367-377. +[13] Q. Li and W. Sheng, Closed hypersurfaces with prescribed Weingarten curvature in Riemannian +manifolds, Calc. Var. Partial Differential Equations, 48 (2013), 41-66. +[14] Y. Y. Li and V. I. Oliker, Starshaped compact hypersurfaces with prescribed m-th mean curvature +in elliptic space. J. Partial Differential Equations, 15(2002), no. 3, 68-80. +[15] N. Krylov, Boundedly inhomogeneous elliptic and parabolic equations in a domain, Izv. Akad. +Nauk SSSR Ser. Mat., 47 (1983), 75-108. +[16] H. Li, and B. Xu, Hyperbolic p-sum and horospherical p-Brunn-Minkowski theory in Hyperbolic +Space, arXiv:2211.06875v1. +[17] Y. Li, Degree theory for second order nonlinear elliptic operators and its applications, Comm. +Partial Differential Equations, 14 (1989), 1541-1578. +[18] J. Liu, The Lp-Gaussian Minkowski problem, Calc. Var. Partial Differential Equations, 61 (2022), +1-23. +[19] Y. Liu and J. Lu, A flow method for the dual Orlicz-Minkowski problem, Trans. Amer. Math. Soc. +373(2020), 5833-5853. +[20] E. Lutwak, The Brunn-Minkowski-Firey theory.I. Mixed volumes and the Minkowski problem, J. +Differential Geom. 38 (1993), no. 1, 131-150. +[21] R. Schneider, Convex bodies: the Brunn-Minkowski theory, Second edition, 151 Cambridge Uni- +versity Press, 2013. +Faculty of Mathematics and Statistics, Hubei Key Laboratory of Applied Mathemat- +ics, Hubei University, Wuhan 430062, P.R. China +Email address: chenli@hubu.edu.cn + diff --git a/fNAzT4oBgHgl3EQfMPtb/content/tmp_files/load_file.txt b/fNAzT4oBgHgl3EQfMPtb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d9044e010ad8ecd766b54775866a89899b847f4 --- /dev/null +++ b/fNAzT4oBgHgl3EQfMPtb/content/tmp_files/load_file.txt @@ -0,0 +1,415 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf,len=414 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='01128v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='AP] 3 Jan 2023 NON-NORMALIZED SOLUTIONS TO THE HOROSPHERICAL MINKOWSKI PROBLEM LI CHEN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Recently, the horospherical p-Minkowski problem in hyperbolic space was proposed as a counterpart of Lp Minkowski problem in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Through designing a new volume preserving curvature flow, the existence of normalized even solution to the horospherical p-Minkowski problem was solved for all p ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' However, due to the lack of homogeneity of the horospherical p-surface area measure, it is difficult to remove the normalizing factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In this paper, we overcome this difficulty for −n ≤ p < n by the degree-theoretic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Introduction The classical Minkowski problem and its extension Lp Minkowski problem [20, 3] play very important roles in the study of convex bodies in Euclidean space (see also [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' It is a dream to develop similar problems in hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Recently, Andrews-Chen- Wei declared interesting formal similarities between the geometry of h-convex domains in hyperbolic space and that of convex Euclidean bodies (see section 5 in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Along the lines of Andrews-Chen-Wei, Li-Xu [16] introduced the horospherical p-surface area measure by use of the hyperbolic p-sum which they defined and proposed the associated horospherical p-Minkowski problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We will briefly describe them below followed by Section 5 in [2] and Section 2 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We shall work in the Hyperboloid model of Hn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For that, consider the Minkowski space Rn+1,1 with canonical coordinates X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=', Xn+1, X0) and the Lorentzian metric ⟨X, X⟩ = n+1 � i=1 (Xi)2 − (X0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Hn+1 is the future time-like hyperboloid in Minkowski space Rn+1,1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Hn+1 = � X = (X1, · · ·, Xn+1, X0) ∈ Rn+1,1 : ⟨X, X⟩ = −1, X0 > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Primary 35J96, 52A39;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Secondary 53A05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Minkowski type problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' h-convex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Mong-Amp´ere type equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 1 2 LI CHEN In the Hyperboloid model of Hn+1, the horospheres are hypersurfaces in Hn+1 whose constant principal curvatures equal to 1 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' They can be parameterized by Sn × R Hx(r) = {X ∈ Hn+1 : X · (x, 1) = −er}, where r represents the signed geodesic distance s from the “north pole” N = (0, 1) ∈ Hn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' The horo-ball which is the interior of the horosphere is denoted by Bx(r) = {X ∈ Hn+1 : 0 > X · (x, 1) > −er}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' If we use the Poincar´e ball model Bn+1 of Hn+1, then Bx(r) corresponds to an (n + 1)- dimensional ball which tangents to ∂Bn+1 at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Furthermore, Bx(r) contains the origin for r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' A compact domain Ω ⊂ Hn+1 (or its boundary ∂Ω) is horospherically convex (or h-convex for short) if every boundary point p of ∂Ω has a supporting horo-ball, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' a horo-ball B such that Ω ⊂ B and p ∈ ∂B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For a smooth compact domain Ω, we say Ω (or ∂Ω) is uniformly h-convex if its principal curvatures are greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Let Ω ⊂ Hn+1 be a h-convex compact domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For each X ∈ ∂Ω, ∂Ω has a supporting horo-ball Bx(r) for some r ∈ R and x ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then the horospherical Gauss map G : ∂Ω → Sn is defined by G(X) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Let Ω be a h-convex compact domain in Hn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then for each x ∈ Sn we define the horospherical support function of Ω (or ∂Ω) in direction x by u(x) := inf{s ∈ R : Ω ⊂ Bx(s)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We also have the alternative characterisation u(x) = sup{log(−⟨X, (x, 1)⟩) : X ∈ Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) MINKOWSKI PROBLEM 3 The support function completely determines a h-convex domain Ω, as an intersection of horo-balls: Ω = � x∈Sn Bx(u(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2) If the compact domain Ω is uniformly h-convex, then G is a diffeomorphism from ∂Ω to Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then, X = G−1 is a smooth embedding and X can be written in terms of the support function u, as follows: X(x) = 1 2ϕ(−x, 1) + 1 2 �|Dϕ|2 ϕ + 1 ϕ � (x, 1) − (Dϕ, 0), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3) where u = log ϕ is the horospherical support function of the domain Ω, D is the Levi- Civita connection of the standard metric σ of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then, after choosing normal coor- dinates around x on Sn+1, the area element dµ of Ω at X(x) = G−1(x) can be given by dµ = � det⟨∂iX, ∂jX⟩dσ = det A[ϕ]dσ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4) where A[ϕ] = D2ϕ − 1 2 |Dϕ|2 ϕ I + 1 2 � ϕ − 1 ϕ � I and I is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Moreover, the compact domain Ω is uniformly h-convex if and only if the matrix A[ϕ] is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Until now, we have seen many interesting similarities between the geometry of h- convex domains in hyperbolic space and that of convex Euclidean bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Recently, Li-Xu [16] developed deeply this similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In particular, they introduced a sum of two sets in hyperbolic space which they called the hyperbolic p-sum (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Let 1 2 ≤ p ≤ 2, a ≥ 0, b ≥ 0 and a + b ≥ 1, and let K and L be two smooth uniformly h-convex compact domains with the horospherical support functions uK(x) and uL(x) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' The hyperbolic p-sum Ω := a · K +p b · L of K and L is defined by the h-convex compact domain with the horospherical support function uΩ(x) := 1 p log � aepuK(x) + bepuL(x)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' They also gave a pointwise definition of the hyperbolic p-sum (see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then, they calculated the variation of the volume along the hyperbolic p-sum (see 4 LI CHEN Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2 in [16]) lim t→0+ Vol(K +p t · L) − Vol(K) t = 1 p � Sn ϕp LdSp(K, x), where the horospherical p-surface area measure of a smooth uniformly h-convex compact domain K ⊂ Hn+1 is defined by (see Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2 [16]) dSp(K, ·) = ϕ−p K det(A[ϕK])dσ, uK = log ϕK and uL = log ϕL are the horospherical support functions of smooth uni- formly h-convex compact domains K and L respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In particular, p = 0, dSp(K, ·) is just the surface area measure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4) of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then, Li-Xu proposed the associated horospherical p-Minkowski problem (see Problem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 [16]): Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For a given smooth positive function f(x) defined on Sn, we ask the sufficient and necessary conditions for f(x), such that there exists a smooth uniformly h-convex compact domain K ⊂ Hn+1 satisfying dSp(K, x) = f(x)dσ, which is equivalent to find a smooth positive solution ϕ(x) with A[ϕ(x)] > 0 for all x ∈ Sn (or the uniformly h-convex solution for short) to the equation ϕ−p(x)det(A[ϕ(x)]) = f(x), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) where u = logϕ is the support function of some horo-convex domain in Hn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In particular, for p = 0, Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 is just the prescribed surface area measure problem for horo-convex domains in Hn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For p = −n, Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 is just the prescribed shifted Gauss curvature problem (see Section 7 in [16]) or the n-th symmetric Christoffel problem in Hn+1 (see Page 26 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Through designing a new volume preserving curvature flow, Li-Xu solved the existence of normalized even solution to the horospherical p-Minkowski problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 for all p ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' A function g : Sn → R is called even if g(x) = g(−x) for all x ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2 (Li-Xu [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Given a smooth positive even function f defined on Sn, for any p ∈ R, there exists a smooth even and uniformly h-convex solution to the equation ϕ−pdet(A[ϕ]) = γf (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='6) for some γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' MINKOWSKI PROBLEM 5 Due to the lack of homogeneity of the horospherical p-surface area measure, it is very difficult to remove the normalizing factor γ in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' This difficulty also appears in Orlicz-Minkowski-type problem [19] and the Gaussian Minkowski type problem [11, 18, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In the latter problem, a degree-theoretic approach was used to overcome this difficulty in the even case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In this paper, we can obtain a non-normalized solution (without the normalizing factor γ) to Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 for −n ≤ p < n by the degree-theoretic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Assume −n ≤ p < n, there exists a smooth even and uniformly h-convex solution to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) for any smooth positive even function f defined on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' It should be noted that the continuity method was not applied to prove existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' The reason is that the continuity method requires that the corresponding linearized equation is invertible on any admissible solution and this result is not available for the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' However, within the framework of the degree theory developed in [17] it suffices to know invertibility on constant solutions which is guaranteed by the uniqueness of even solutions to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) for −n ≤ p < n (see Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 (6)(7) in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For other ranges p ≥ n or p < −n, either the constant solution may not be unique or its uniqueness is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (see Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' So, it is difficult to prove existence by the degree-theoretic approach, although the a prior estimates for solutions to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) can be established for all p < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' It is interesting to consider the uniqueness of solutions to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) when f is not constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In contrast with Lp Minkowski problem in Euclidean space for which the uniqueness of solutions is guaranteed by Lp Brunn-Minkowski inequality for p ≥ 1 [20], the uniqueness of solutions to the horospherical p-Minkowski problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 is unknown due to the lack of analogous inequality in hyperbolic space (see related discussions in Section 9 [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' The present paper is built up as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 2, we obtain the a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3 in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' The a priori estimates For convenience, in the following of this paper, we always assume that f is a smooth positive, even function on Sn and ϕ is a smooth even, uniformly h-convex solution to 6 LI CHEN the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Moreover, let Ω be the uniformly h-convex compact domain in Hn+1 with the horospherical support function u = log ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Clearly, Ω is symmetry with the origin and ϕ(x) > 1 for x ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' The following easy and important equality is key for the C0 estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We have 1 2 � max Sn ϕ + 1 maxSn ϕ � ≤ min Sn ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' The inequality can be found in the proof in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2 in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For completeness, we give a proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Assume that u(x1) = maxSn u and denote X(x1) = G−1(x1) as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then, we have for any x ∈ Sn by the definition of the horospherical support function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) −⟨X(x1), (x, 1)⟩ ≤ ϕ(x), ∀x ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Substituting the expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3) for X into the above equality yields 1 2ϕ(x1)(1 + ⟨x1, x⟩) + 1 2 1 ϕ(x1)(1 − ⟨x1, x⟩) ≤ ϕ(x), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2) where we used the fact Dϕ(x1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Note that ϕ(x1) ≥ 1, we find from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2) 1 2 � ϕ(x1) + 1 ϕ(x1) � ≤ ϕ(x) for ⟨x, x1⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3) Since Ω is origin symmetric, we can assume that the minimum point x0 of u(x) satisfies ⟨x0, x1⟩ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Thus, the equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) follows that from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' It is very interesting to compare the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) with the the analogous inequality for convex bodies in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' In fact, for an origin-symmetric convex body in Euclidean space, the definition of its support function gives |⟨x1, x0⟩| max Sn h ≤ min Sn h, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4) where h is the support function of the convex body, x1 and x0 are the maximum and minimum points of h respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Clearly, the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) in hyperbolic space is better than (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4), since the term |⟨x1, x0⟩| in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4) may not be bounded from below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Now we begin to consider the C0-estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Similar estimate is obtained in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2 [16] when the volume of h-convex domain is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' But for our case, the volume is not fixed, we use the maximum principle to get the C0-estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' MINKOWSKI PROBLEM 7 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' If p < n, we have 0 < 1 C ≤ u(x) ≤ C, ∀ x ∈ Sn, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) where C is a positive constant depending on p, n and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Using the maximum principle, we have from the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) (max Sn ϕ)n−p 1 2n � 1 − 1 (maxSn ϕ)2 �n ≥ C > 0 and (min Sn ϕ)n−p 1 2n � 1 − 1 (minSn ϕ)2 �n ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Since the function g(x) = xn−p 1 2n � 1 − 1 x2 �n is increasing in [1, +∞) for p < n, g(1) = 0 and g(+∞) = +∞, we obtain min Sn ϕ ≤ C, and max Sn ϕ ≥ C > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='6) Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1), we find 1 < C ≤ min Sn ϕ ≤ max Sn ϕ ≤ C′, which implies that 0 < 1 C ≤ min Sn u ≤ max Sn u ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' So, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' □ As a corollary, we have the gradient estimate from Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3 in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We have |Dϕ(x)| ≤ C, ∀ x ∈ Sn, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='7) where C is a positive constant depending only on the constant in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We give some notations before considering the C2 estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Denote by Uij = ϕij − 1 2 |Dϕ|2 ϕ δij + 1 2(ϕ − 1 ϕ)δij and F(U) = det U, F ij = ∂F ∂Uij , F ij,kl = ∂2F ∂Uij∂Ukl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 8 LI CHEN Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We have for 1 ≤ i ≤ n λi(U(x)) ≤ C, ∀ x ∈ Sn, where λ1(U), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=', λn(U) are eigenvalues of the matrix U and C is a positive constant depending only on the constant in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Since λi(U) ≤ trU = ∆ϕ − n ϕ|Dϕ|2 + n 2(ϕ − 1 ϕ), ∀1 ≤ i ≤ n, it is sufficient to prove ∆ϕ ≤ C in view of C0 estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) and C1 estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Moreover, these two estimates together with the positivity of the matrix U imply λi(D2ϕ) ≥ −C for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Thus, |λi(D2ϕ)| ≤ C∆ϕ, ∀1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='8) We take the auxiliary function W(x) = ∆ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Assume x0 is the maximum point of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' After an appropriate choice of the normal frame at x0, we further assume Uij, hence ϕij and F ij is diagonal at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='9) Wi(x0) = � k ϕkki = 0, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='10) Wii(x0) = � k ϕkkii ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' From the positivity of F ij and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='10), we arrive at x0 if ∆ϕ is large enough 0 ≥ � i F iiWii = � i F ii � k ϕkkii ≥ � i F ii � k � ϕiikk − C∆ϕ � , where we use Ricci identity and the equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='8) to get the last inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Thus it follows from the definition of U, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='9), 0 ≥ � i F ii � q � Uiiqq + � 1 2ϕ|Dϕ|2� qq − 1 2 � ϕ − 1 ϕ � qq � − C � i F ii∆ϕ ≥ � i F ii � q � Uiiqq + 1 ϕ(ϕqq)2 − C∆ϕ − C � − C � i F ii∆ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='11) Differentiating the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) twice gives F iiUiiqq + F ij,klUijqUklq = (ϕpf)qq, MINKOWSKI PROBLEM 9 which yields F iiUiiqq ≥ −C∆ϕ − C (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='12) in view of C0 estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) and C1 estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='12) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='11) and using (∆ϕ)2 ≤ n � q(ϕqq)2, we have 0 ≥ � C(∆ϕ)2 − C∆ϕ − C � � i F ii − C∆ϕ − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='13) Note that � i F ii ≥ n(det U)1− 1 n = n(ϕpf)1− 1 n ≥ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='14) Then we conclude at x0 from the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='14) C ≥ |∆ϕ|2 if ∆ϕ is chosen large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' So, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' The proof of the main theorem In this section, we use the degree theory for nonlinear elliptic equation developed in [17] to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Such approach was also used in prescribed curvature problem [1, 14, 12, 13], prescribed curvature problem for for convex hypersurfaces with group invariance assumptions [8, 9] and the Gaussian Minkowski type problem [11, 18, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For the use of the degree theory, the uniqueness of constant solutions to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) is important for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Fortunately, this fact can be guaranteed by Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 (6)(7) in [16] for −n ≤ p < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We summarize it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For −n ≤ p < n, there exist a unique even solution ϕ = c to the equation ϕ−p(x)det(A[ϕ(x)]) = γ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) for any γ > 0, where c is the unique positive solution to the equation c−p�1 2(c − c−1) �n = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' For −n < p < n, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 (6) in [16] tells us that there exist a unique solution ϕ = c to the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) for any γ > 0, where c is the unique positive solution to the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 10 LI CHEN For p = −n, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 (7) in [16] tells us that the solutions to the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1) for any γ > 0 are given by ϕ(x) = � 1 + 2γ 1 n 0 � 1 2�� |x0|2 + 1 − ⟨x0, x⟩ � , where x0 ∈ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Clearly, ϕ(x) = � 1 + 2γ 1 n 0 � 1 2 is the unique even solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Thus, the conclusion holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' □ Now, we begin to use the degree theory to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' After establishing the a priori estimates (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4) for p < n, we know that the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) is uniformly elliptic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' λi(U(x)) ≥ C > 0, ∀ x ∈ Sn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3) where λ1(U), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=', λn(U) are eigenvalues of the matrix U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' From Evans-Krylov estimates [4, 15], and Schauder estimates [10], we have |ϕ|C4,α(Sn) ≤ C (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4) for any smooth, even and uniformly h-convex solution u to the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' We define B2,α(Sn) = {ϕ ∈ C2,α(Sn) : u = log ϕ is even} and B4,α 0 (Sn) = {ϕ ∈ C4,α(Sn) : u = log ϕ is h-convex and even}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Let us consider L(·, t) : B4,α 0 (Sn) → B2,α(Sn), which is defined by L(ϕ, t) = det U − ϕp[(1 − t)γ + tf], where the constant γ will be chosen later and U is denoted as before U = D2ϕ − 1 2 |Dϕ|2 ϕ I + 1 2 � ϕ − 1 ϕ � I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Let OR = {ϕ ∈ B4,α 0 (Sn) : 1 + 1 R < ϕ, 1 RI < U, |ϕ|C4,α(Sn) < R}, which clearly is an open set of B4,α 0 (Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Moreover, if R is sufficiently large, L(ϕ, t) = 0 has no solution on ∂OR by the a priori estimates established in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' MINKOWSKI PROBLEM 11 Therefore the degree deg(L(·, t), OR, 0) is well-defined for 0 ≤ t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Using the homo- topic invariance of the degree (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2 in [17]), we have deg(L(·, 1), OR, 0) = deg(L(·, 0), OR, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) For −n ≤ p < n, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='1 tells us that ϕ = c is the unique even solution for L(ϕ, 0) = 0 in OR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Direct calculation show that the linearized operator of L at ϕ = c is Lc(ψ) = ∆Snψ + 1 2 � 1 + 1 c2 − pcp−1γ � ψ, where γ is given by the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Since spherical Laplacian has a discrete spectrum, we choose c = c0 such that Lc0 is an invertible operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Then we have by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3 in [17] deg(L(·, 0), OR, 0) = deg(Lc0, OR, 0) = ±1, where the last inequality follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='4 in [17] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' Therefore, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='5) deg(L(·, 1), OR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' 0) = deg(L(·, 0), OR, 0) = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' So, we obtain a solution at t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fNAzT4oBgHgl3EQfMPtb/content/2301.01128v1.pdf'} +page_content=' References [1] F.' metadata={'source': 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+San Francisco, California, USA +ABSTRACT +Learning feature interactions is crucial to success for large-scale +CTR prediction in recommender systems and Ads ranking. Re- +searchers and practitioners extensively proposed various neural +network architectures for searching and modeling feature interac- +tions. However, we observe that different datasets favor different +neural network architectures and feature interaction types, sug- +gesting that different feature interaction learning methods may +have their own unique advantages. Inspired by this observation, +we propose AdaEnsemble: a Sparsely-Gated Mixture-of-Experts +(SparseMoE) architecture that can leverage the strengths of het- +erogeneous feature interaction experts and adaptively learns the +routing to a sparse combination of experts for each example, al- +lowing us to build a dynamic hierarchy of the feature interactions +of different types and orders. To further improve the prediction +accuracy and inference efficiency, we incorporate the dynamic +early exiting mechanism for feature interaction depth selection. +The AdaEnsemble can adaptively choose the feature interaction +depth and find the corresponding SparseMoE stacking layer to exit +and compute prediction from. Therefore, our proposed architecture +inherits the advantages of the exponential combinations of sparsely +gated experts within SparseMoE layers and further dynamically se- +lects the optimal feature interaction depth without executing deeper +layers. We implement the proposed AdaEnsemble and evaluate its +performance on real-world datasets. Extensive experiment results +demonstrate the efficiency and effectiveness of AdaEnsemble over +state-of-the-art models. +CCS CONCEPTS +• Computing methodologies; • Machine learning; • Machine +learning approaches; • Neural networks; +KEYWORDS +CTR prediction, Recommendation System, Feature Interaction, Mix- +ture of Experts, Dynamic Inference, Early Exiting, AutoML, Deep +Neural Network +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than ACM +must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, +to post on servers or to redistribute to lists, requires prior specific permission and/or a +fee. Request permissions from permissions@acm.org. +Conference’17, July 2017, Washington, DC, USA +© 2022 Association for Computing Machinery. +ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00 +https://doi.org/XXXXXXX.XXXXXXX +ACM Reference Format: +Yachen Yan and Liubo Li. 2022. AdaEnsemble: Learning Adaptively Sparse +Structured Ensemble Network for Click-Through Rate Prediction. In Pro- +ceedings of ACM Conference (Conference’17). ACM, New York, NY, USA, +9 pages. https://doi.org/XXXXXXX.XXXXXXX +1 +INTRODUCTION +Click-through rate (CTR) prediction model [25] is an essential com- +ponent for the large-scale search ranking, online advertising and +recommendation system [4, 9, 20, 37]. +Many deep learning-based models have been proposed for CTR +prediction problems in the industry. They have become dominant +in learning the useful feature interactions of the mixed-type input +in an end-to-end fashion[37]. Although most of the existing meth- +ods can effectively capture higher-order feature interactions, we +observe that their performance varies for different datasets. We +believe this is due to their inductive bias: different methods learn +different types of feature interactions and favor different datasets. +While every existing method focuses on automatically model- +ing different types of feature interactions, there have been few +attempts to model different types of interactions jointly and dy- +namically. We believe that ensembling different interaction mod- +ules to create heterogeneous feature interactions can complement +the non-overlapping knowledge that each interaction learning ap- +proach learned, as opposed to the homogeneous interaction model- +ing method, which restricts the types of feature interactions to be +learned. For utilizing various interaction modules to learn different +types of feature interactions, we use Sparsely-Gated Mixture-of- +Experts (SpasrseMoE) architecture to enrich the model capacity +while achieving computational efficiency through conditional com- +putation. +We propose AdaEnsemble: a Sparsely-Gated Mixture-of-Experts +(SparseMoE) hierarchical architecture to ensemble different inter- +action learning modules and dynamically select optimal feature +interaction depth. Within each SparseMoE layer of AdaEnsemble, +there is a collection of interaction learning experts, and a trainable +gating network determines a sparse combination of these experts +to use for each example. Within the Depth Selecting Controller, a +trainable gating network will choose the feature interaction depth +for each example and recursively propagate feature interaction rep- +resentations through SparseMoE layers to the corresponding depth +for computing the prediction. Through these conditional compu- +tation mechanisms, we enlarged the model capacity exponentially +without increasing inference cost. The main contributions of this +paper can be summarized as follows: +• We designed a novel model architecture called AdaEnsemble to +ensemble various types of feature interaction learning modules +arXiv:2301.08353v1 [cs.IR] 6 Jan 2023 + +Conference’17, July 2017, Washington, DC, USA +Yachen and Liubo +by Sparsely-Gated Mixture-of-Experts (SparsseMoE). Through +utilizing MoE layers recursively with residual connections and +normalization, AdaEnsemble can model different types of inter- +actions jointly and dynamically. +• We designed an efficient and effective Depth Selecting Con- +troller to adaptively choose the optimal feature interaction depth. +Through utilizing this controller, AdaEnsemble can dynamically +determine the layer for early exiting to improve prediction accu- +racy and inference efficiency. +• We designed a bi-level optimization algorithm for iteratively +training the modeling network and gating network. +• We conduct extensive experiments on real-world datasets and +study the learning patterns of AdaEnsemble. +2 +RELATED WORK +2.1 +Feature Interaction Modeling +Learning the feature interactions is the key topic in CTR predic- +tion problems and has been widely discussed in literature. Various +hybrid network architectures [4, 8, 16, 22, 23, 31, 32] utilize the feed- +forward neural network with non-linear activation function as its +core component, to learn implicit interactions. The complement of +the implicit interaction modeling improves the performance of the +network that only models the explicit interactions [3]. +Another group of models focuses on exploring bit-wise/vector- +wise feature interactions. Deep & Cross Network (DCN) [31] and +its improved version DCN V2 [32] explores the feature interactions +at the bit-wise level explicitly in a recursive fashion. Deep Factor- +ization Machine (DeepFM) [8] utilizes factorization machine layer +to model the pairwise vector-wise interactions. Product Neural Net- +work (PNN) [22, 23] introduces the inner product layer and the outer +product layer to learn vector-wise interactions and bit-wise inter- +actions, respectively. xDeepFM [16] learns the explicit vector-wise +interaction using Compressed Interaction Network (CIN), which has +an RNN-like architecture and learns vector-wise interactions using +Hadamard product. FiBiNET [11] utilizes Squeeze-and-Excitation +network to dynamically learn the importance of features and model +the feature interactions via bilinear function. AutoInt [28] leverages +the Transformer [30] architecture to learn different orders of feature +combinations of input features. xDeepInt [35] introduces polyno- +mial interaction layer to recursively learn higher-order vector-wise +and bit-wise interactions jointly with controlled degree, dispensing +with jointly-trained DNN and nonlinear activation functions. +2.2 +Sparse Mixture-of-Experts Network +The Sparsely-Gated MoE model [2, 27] combines multiple experts +and a trainable gating network that selects a subset of experts +for each example. This network architecture can be viewed as a +dynamic sparsity structure that maintains all weights but intro- +duces sparsity into the model through conditional computation. +The SparseMoE is widely used in natural language processing re- +search area. Most of the discussions focus on improving the rout- +ing mechanism for the experts. Switch Transformer [7] simplifies +the top-1 routing algorithm. GShard [14] uses group-level top-2 +routing. Both are trained with load balancing losses and improve +language models with reduced communication and computational +costs. BASE Layers [15] treated routing as a linear assignment +Embedding Layer +Categorical +Feature +Bucktized + Numeric Feature +1st Sparse MoE Layer +2nd Sparse MoE Layer +l-th Sparse MoE Layer +Input Feature Map +Add & Normalize +Add & Normalize +Add & Normalize +--- +Depth Selecting +Network +1st +Estimator +2nd +Estimator +l-th +Estimator +Figure 1: The Architecture of AdaEnsemble +In this example, the depth selecting network selects the 2nd layer to exit +and compute the final prediction, therefore the deeper layers was not +activated and plotted translucent in the figure. +problem and removed the need for load balancing auxiliary losses. +M6-T [36] splits experts into different groups and applies k top- +1 routing procedures. Some literature explore the training of the +Sparsely-Gated MoE. EvoMoE [21] decouples the training of ex- +perts and the sparse gate by training all experts at first and then +gradually and adaptively becomes sparser while routes to fewer +experts for learning the sparse gate. ST-MoE [40] further studies the +training instabilities and uncertain quality issue of the MoE model. +X-MoE [5] proposed a dimension reduction and L2 normalization +to solve the representation collapse in the training of MoE model. +2.3 +Early-Exiting Network +The idea of early-exiting for the neural network was firstly pro- +posed by BranchyNet [29] for computer vision. This technique +is also applied to NLP tasks, DeeBERT [33], FastBERT [18], and +PABEE [39] was later introduced for improving inference efficiency +of Transformer-Based BERT models. +For the early-exiting mechanism, BranchyNet [29], DeeBERT [33], +FastBERT [18] and SDN [12] use the entropy-based or confidence- +based criteria. While using entropy-based or confidence-based cri- +teria is straightforward and effective, it takes advantage of the fact +that the model’s output is a probability distribution in multi-class +classification tasks. This technique generally cannot be applied +to binary classification and regression tasks. On the other hand, +BERxiT [34] and Epnet [6] use learned modules for early-exiting. + +Research Track Paper +Conference’17, July 2017, Washington, DC, USA +3 +PROPOSED MODEL: ADAENSEMBLE +In this section, we give an overview of the architectures of AdaEnsem- +ble. First, we introduce the feature processing and embedding layer, +which maps continuous features and high-dimensional categorical +features onto a dense embedding vector. Second, we introduce the +feature interaction experts we considered for jointly learning the +hierarchy of the deep feature representations. Third, we present the +sparse mixture-of-experts (SparseMoE) layer, which ensemble mul- +tiple interaction experts dynamically, and the estimator associated +with each SparseMoE Layer. Fourth, we discuss how to automat- +ically and dynamically select the feature interaction depth based +on the Depth Selecting Controller. Finally, a bi-level optimization +algorithm will be provided for the training. +3.1 +Embedding Layer +In large-scale CTR prediction tasks, inputs include both continuous +and categorical features. Categorical features are often directly +encoded by one-hot encoding, which results in an excessively high- +dimensional and sparse feature space. +Suppose we have 𝐹 fields. In our feature processing step, we +bucketize all the continuous features to equal frequency bins, then +embed the bucketized continuous features and categorical features +embed each feature onto a dense embedding vector 𝑒𝑖 of the same +dimension 𝐷. +e𝑖 = x𝑖V𝑖, +where 𝑒𝑖 ∈ 𝑅𝐷, V𝑖 is an embedding matrix for the 𝑖-th field, and +x𝑖 is the corresponding one-hot vector. Lastly, we concatenate 𝐹 +embedding vectors and denote the output of embedding layer 𝑋0 ∈ +𝑅𝐹×𝐷 as the input feature map: +𝑋0 = [𝑒1,𝑒2, · · · ,𝑒𝐹 ]⊺. +(1) +3.2 +Feature Interaction Experts +We considered several types of feature interaction experts in our +model: Dense Layer, Convolution Layer, Multi-Head Self-Attention +Layer, Polynomial Interaction Layer, and Cross Layer. Essentially, +any feature interaction learning layer can be included in our frame- +work, and the residual connection and normalization will be applied +to their ensembles. Now we introduce these feature interaction ex- +perts included in our framework. Note that our proposed framework +is general and can use arbitrary feature interaction modules. The +potential feature interaction experts can be used are not limited to +the following. +3.2.1 +Dense Layer. Dense Layer is also known as fully connected +layer and is the most widely used module for modeling implicit +feature interactions. In this paper, we use the dense layer with +non-linear activation function for learning the deep feature rep- +resentations. Given an input of embedding 𝑋𝑙−1, the output of +embedding 𝑋𝑙 is obtained from: +𝑋𝑙 = 𝜎(𝑊𝑙 · 𝑋𝑙−1) +(2) +where 𝜎 denotes activation function and𝑊𝑙 denotes the weights +of the 𝑙-th dense layer. +3.2.2 +Convolution Layer. Convolution layers are widely used for +computer vision problems. In this paper, we applied 1D convolution +as one of the interaction experts. Here we utilize a dense layer ahead +of the convolution layer for fusing the inputs embeddings first, as +the convolution layer is locally connected. Given the embedding +𝑋𝑙−1 as input, the output of embedding 𝑋𝑙 is obtained from: +𝑋𝑙 = Dense(Pooling(Conv1D(Reshape(𝑋𝑙−1)))) +(3) +Here we first reshape the input embedding and then apply 1D +convolution followed by a pooling layer. Finally, we use a dense +layer to project the output to the desired dimension. +3.2.3 +Multi-Head Self-Attention Layer. Multi-Head Self-Attention +Layer [30] is widely used in transformer networks for its superior +performance in natural language processing and has started to be +popular in the computer vision research area. We consider utiliz- +ing Multi-Head Self-Attention Layer for modeling the dependency +between features and forming meaningful higher-order features. +Given an input of embedding 𝑋𝑙−1, the output of embedding 𝑋𝑙 is +obtained from: +𝑋𝑙 = Dense(MultiHeadSelfAttention(Reshape(𝑋𝑙−1)) +(4) +Here we first reshape the input embedding and then apply Multi- +Head Self-Attention Layer followed by a dense layer to project the +output to the desired dimension. +3.2.4 +Polynomial Interaction Layer. Polynomial Interaction Net- +work [35] is designed to capture bounded degree feature interac- +tions explicitly. In this paper, we adopt the PIN layer as one of our +feature interaction learning experts. Given an input of embedding +𝑋𝑙−1, the mathematical representation of the 𝑙-th PIN layer’s output +is given by: +𝑋𝑙 = 𝑋𝑙−1 ◦ (𝑊𝑙 · 𝑋0) +(5) +where ◦ denotes the Hadamard product and𝑊 denotes the kernel +weights of the PIN layer. We omit the residual connection from the +original paper in the above equation as the residual connection will +be used across the MoE layers. +3.2.5 +Cross Layer. Deep Cross Network [32] is later proposed to +explore the feature interactions in a recursive fashion. Given an +input of embedding 𝑋𝑙−1, the output of embedding 𝑋𝑙 is obtained +from: +𝑋𝑙 = 𝑋0 ◦ (𝑊𝑙 · 𝑋𝑙−1) + 𝑏𝑙 +(6) +Where 𝑊 and 𝑏 denote the weight matrix and bias vector in the +𝑙-th DCN layer. We also omit the residual connection of the original +implementation in the above equation, as we will use the residual +connection across the MoE layers. +3.3 +Sparse Mixture-of-Experts Layer +The Sparse Mixture-of-Experts layer ensembles aforementioned +heterogeneous feature interaction experts and consists of several +other essential parts to make the overall model can be stably trained. + +Conference’17, July 2017, Washington, DC, USA +Yachen and Liubo +Sparse MoE Layer +Expert 1 +Expert 2 +Expert 3 +Expert n-1 +Expert n +Input +Embedding +Output +Embedding +Gating +Network +Sparse +Dispatcher +non-zero index +non-zero value +.... +Figure 2: The architecture of Sparse Mixture-of-Experts +Layer +Input +Embedding +Feed-Forward Network +e1 +e2 +e3 +e4 +Expert +Embedding +Cosine +Similarity +Learnable Temperature +Re-Scaling +Top-K +L2 Normalization +L2 Normalization +Routing Score +Softmax +Noise Injection +Figure 3: The Noisy Gating Network within Sparse Mixture- +of-Experts Layer +3.3.1 +Noisy Gating Network. The gating network essentially com- +putes the gating value for selecting and weighting the output em- +bedding of each expert. +For the input embedding of gating network 𝑋0, it firstly pro- +cessed by the gating network: a two-layer feed-forward network, +i.e. a dimension reduction layer with reduction ratio 𝑟 [10], a non- +linear activation function and then a dense layer projecting to +hidden state ℎ ∈ 𝑅𝑑. Additionally, we applied multiplicative jitter +noise for introducing exploration and promoting load balancing +between different experts. +ℎ = FFN(𝑋0 ◦ RandomUniform(1.0 − eps, 1.0 + eps)) +(7) +After projecting the input embedding to hidden state ℎ ∈ 𝑅𝑑, +we apply the 𝐿2 normalization to both hidden state ℎ ∈ 𝑅𝑑 and +learnable expert embeddings 𝑒𝑗 ∈ 𝑅𝑑, where 𝑗 is the index of +expert. Then, we compute the cosine similarity between the hidden +state and expert embedding as the initial routing score. Here we +encourage the uniformity of representations to avoid dominated +experts issue. +𝑠𝑗 = +ℎ · 𝑒𝑗 +∥ℎ∥∥𝑒𝑗 ∥ +(8) +Finally, we use a learnable temperature scalar 𝜏 to re-scale the +routing scores to the range [−1, +1]. +𝑔𝑗 = 𝑠𝑗/𝜏 +(9) +For the computed routing score 𝑔, we only keep the top k values +and set the rest to −∞, resulting in the corresponding softmax +gating values equal 0. The 𝑖-th element of the output of the gating +network is +𝐸𝑥𝑝𝑒𝑟𝑡𝐺(𝑥)𝑖 = +exp +� +TopK(𝑔,𝑘)𝑖 +� +�𝑁 +𝑗=1 exp +� +TopK(𝑔,𝑘)𝑗 +� , +(10) +where +TopK(𝑔,𝑘)𝑗 = +� +𝑔𝑗 +if 𝑔𝑗 is in the top 𝑘 elements of 𝑔 +−∞ +otherwise. +(11) +These gating values will be used by the sparse dispatcher for +routing examples to different experts. This is the essential step for +achieving sparsity of our Sparse Mixture-of-Experts layer. Note +that the 𝐺(𝑥) is differentiable regardless the value of 𝑘[7]. +3.3.2 +Annealing Top-K Gating. We also introduce annealing mech- +anism to the Top-K operation. We starts with 𝑘 value equal to the +number of experts, which means that we starts as a fully dense gate +that routes examples to all experts. Then we gradually decrease +the 𝑘 and route examples to fewer experts, to adaptively make +the structure sparser and continuously improving the computation +efficiency. +By annealing of the 𝑘 value, we start to train our architecture +with a dense structure which allows us to thoroughly learn all ex- +perts and adjust the gating network in the correct direction at the +beginning. Therefore, we can control the sparsity of our architec- +ture while training to not only accelerate the convergence of the +gating network but also benefit the experts’ specialty for learning +particular types of feature interactions. +3.3.3 +Sparse Dispatcher. The sparse dispatcher takes the examples +gating values and experts as input. It firstly dispatches the examples +to the experts corresponding to the non-zero gating values, and lets +experts generate the output embeddings. The output𝑦 of the Sparse +Mixture-of-Experts layer is the linearly weighted combination of +expert output embeddings by the non-zero gating values. + +Research Track Paper +Conference’17, July 2017, Washington, DC, USA +𝑦 = +∑︁ +𝑗 ∈𝜙 +𝐸𝑥𝑝𝑒𝑟𝑡𝐺𝑗 (𝑥)𝐸𝑗 (𝑥) +(12) +Where 𝜙 denotes the selected non-zero indices. We save com- +putation based on the sparsity of 𝐺(𝑥). Wherever 𝐺(𝑥)𝑗 = 0, we +don’t pass the expert to the corresponding expert and do not need +to compute expert embedding 𝐸𝑗 (𝑥). +3.3.4 +Load Distribution Regularization. As stated in the previous +research [5, 7, 27, 40], the gating network tends to select only a +few experts if no regularization is applied, especially when certain +experts are easier to train than other experts. This phenomenon +is self-reinforcing, since the selected experts are trained more and +will be selected more frequently by the gating network. Therefore, +the load balancing loss is applied to enforce the uniform expert +routing. +𝐿balance = 𝜆 · 𝑁 · +𝑁 +∑︁ +𝑗=1 +𝑓𝑗 · 𝑃𝑗 +(13) +where 𝑁 is the number of experts, 𝑓𝑗 is the fraction of examples +dispatched to expert j, 𝑃𝑗 is the average of the router probability +allocated for expert j, and 𝜆 is the coefficient for the regularization +term. +𝑓𝑗 = 1 +𝐵 +∑︁ +𝑥 ∈B +1{argmax 𝑝(𝑥) = 𝑗} +(14) +𝑃𝑗 = 1 +𝐵 +∑︁ +𝑥 ∈B +𝑝𝑗 (𝑥) +(15) +While the default load balancing loss is applicable and effective +when experts are of the same type, AdaEnsemble is using hetero- +geneous feature interaction experts, and the optimal load for each +expert is not uniform. Therefore, we apply the below load distribu- +tion regularization to encourage the expected load distribution of +heterogeneous experts. +𝐿distribution = 𝜆 · +𝑁 +∑︁ +𝑗=1 +𝑓𝑗 · 𝑃𝑗 +𝑤𝑗 +(16) +where 𝑤𝑗 is the expected load fraction of examples dispatched +to expert j, and naturally �𝑁 +𝑗=1 𝑤𝑗 = 1. In practice, the 𝜆 should +be sufficiently large to prevent expert selection self-reinforcing +phenomenon at the initial training stage while not overwhelming +the primary LogLoss objective. +3.4 +Estimator Layer +The output of the Sparse Mixture-of-Experts layer is a feature map +that consists of feature interactions of different degrees and types, +including raw input feature map reserved by residual connections +and higher-order feature interactions jointly learned by experts. +For the final prediction, we merely use the formula as follows: +ˆ𝑦 = 𝜎(𝑊𝑙𝑋𝑙 + 𝑏𝑙) +(17) +where 𝜎 is the sigmoid function, 𝑊𝑙 ∈ 𝑅1×𝐹 is a feature map aggre- +gation vector that linearly combines all the learned feature interac- +tions in the feature map, 𝑏 ∈ 𝑅 is the bias. +3.5 +Depth Selecting Controller +3.5.1 +Depth Selecting Network. The Depth Selecting Network is +essentially the same configuration as the aforementioned Noisy Gat- +ing Network for SparseMoE layer. We denote it by 𝐷𝑒𝑝𝑡ℎ𝐺(𝑥). The +outputs of 𝐷𝑒𝑝𝑡ℎ𝐺(𝑥) are [𝑔𝑑𝑒𝑝𝑡ℎ +1 +,𝑔𝑑𝑒𝑝𝑡ℎ +2 +, · · · ,𝑔𝑑𝑒𝑝𝑡ℎ +𝐿 +], indicating +each example’s optimal forward propagation depth. The 𝑙-th unit +denotes the probability of selecting the 𝑙-th MoE layer to exit. The +optimal depth is automatically selected as the one corresponding +to the largest probability. In contrast to the expert selection, when +choosing the optimal depth of each example for the dynamic infer- +ence, we only keep the top-1 depth index from the output units of +the Depth Selecting Network. Note that we can also apply the load +distribution regularization to encourage the examples’ propagation +depth distribution. +3.5.2 +Dynamic Propagation Mechanism. With the depth gates𝑔𝑑𝑒𝑝𝑡ℎ +𝑙 +∈ +[0, 1] computed by Depth Selecting Network, we obtain the opti- +mal depth for each example. If 𝑔𝑑𝑒𝑝𝑡ℎ +𝑙 += 0, we recursively forward +propagate examples through MoE layers and compute deeper repre- +sentation until 𝑔𝑑𝑒𝑝𝑡ℎ +𝑙 += 1 or reaching the final layer. If 𝑔𝑑𝑒𝑝𝑡ℎ +𝑙 += 1, +the forward propagation will be stopped and the corresponding +𝑙-th estimator will compute the prediction. To efficiently process a +batch of examples with different optimal propagation depths, we +utilize algorithm 1 for dynamic forward propagation. +Algorithm 1 Dynamic Propagation +1: DepthGates ← DepthSelectingNetwork(x) +2: �𝑦 ← DynamicPropagation(x, DepthGates, depth=0) +3: return �𝑦 +4: +5: function DynamicPropagation(Inputs, Gates, Depth) +6: +Outputs = MoE(Inputs) +7: +Depth += 1 +8: +if Depth == Number of Layer then +9: +�𝑦 = Estimator(Outputs) +10: +else +11: +g = Gates[:, Depth] +12: +Outputskeep, Outputsexit = Dispatch(Outputs, g) +13: +Gateskeep, _ = Dispatch(Gates, g) +14: +15: +�𝑦keep = DynamicPropagation(Outputskeep, Gateskeep, Depth) +16: +�𝑦exit = Estimator(Outputsexit) +17: +�𝑦 = Combine(�𝑦keep, �𝑦exit) +18: +end if +19: +return �𝑦 +20: end function +3.6 +Training +3.6.1 +Training Objective. The loss function we use a linearly weighted +combination of the Log Loss and the auxiliary load distribution +regularization, +𝐿𝑜𝑠𝑠 = 𝐿LogLoss + 𝜆1𝐿expert +distribution + 𝜆2𝐿depth +distribution +(18) +where 𝜆1 and 𝜆2 are the coefficients for weighting the load distri- +bution regularization. + +Conference’17, July 2017, Washington, DC, USA +Yachen and Liubo +3.6.2 +Bi-Level Optimization. The optimization task for training the +AdaEnsemble is to jointly optimize the parameters𝑊 , which stands +for the expert layers and estimator layers, and 𝛼, which represents +the expert gating network and depth selecting network. Inspired +by the DARTS [17], we apply bi-level optimization algorithm for +training our model, where 𝛼 is the upper-level parameters and 𝑊 +is the lower-level parameters. We apply algorithm 2 to optimize 𝑊 +and 𝛼 alternatively and iteratively. +Algorithm 2 Bi-Level Optimization for AdaEnsemble +Input: training examples with corresponding labels, step size 𝑡 +Output: well-learned parameters W∗ and 𝛼∗ +1: while not converged do +2: +Sample a mini-batch of validation data +3: +Updating 𝛼 by descending ∇𝛼 L𝑣𝑎𝑙 +�W − 𝜉 ∇WL𝑡𝑟𝑎𝑖𝑛 (W, 𝛼), 𝛼� +4: +(𝜉 = 0 for first-order approximation) +5: +for 𝑖 ← 1,𝑡 do +6: +Sample a mini-batch of training data +7: +Update W by descending ∇WL𝑡𝑟𝑎𝑖𝑛 (W, 𝛼) +8: +end for +9: end while +3.7 +Discussion on AdaEnsemble +The combination of sparse experts routing at each SparseMoE layer +and the depth selecting controller brings two merits to the proposed +model. On one hand, the stacked sparseMoE layers allow the pro- +posed model to leverage the exponential combinations of sparsely +gated experts, which brings in more predicting power. On the other +hand, the depth selecting controller enables the proposed model +to learn the instance-ware model depth. It improves the efficiency +during model serving. In the next section, we will illustrate the +effectiveness of the proposed model through some experimental +studies. +4 +EXPERIMENTS +In this section, we focus on evaluating the effectiveness of our +proposed models and seeking answers to the following research +questions:: +• Q1: How does our proposed AdaEnsemble perform compared to +each baseline in the CTR prediction problem? +• Q2: How does the SparseMoE layer perform compared to Dense- +MoE, which utilizes all feature interaction experts? Does the +cascade of SparseMoE layers effectively capture different types +of feature interactions? +• Q3: How does the depth selecting controller perform compared +to a full-depth network? Does the early exiting mechanism +achieve both effectiveness and efficiency? +• Q4: How do different hyper-parameter settings influence the +performance of AdaEnsemble? +4.1 +Experiment Setup +4.1.1 +Datasets. We evaluate our proposed model on three public +real-world datasets widely used for research. +1. Criteo.1 Criteo dataset is from Kaggle competition in 2014. +Criteo AI Lab officially released this dataset after, for academic +use. This dataset contains 13 numerical features and 26 categor- +ical features. We discretize all the numerical features to integers +by transformation function ⌊𝐿𝑜𝑔 �𝑉 2�⌋ and treat them as categor- +ical features, which is conducted by the winning team of Criteo +competition. +2. Avazu.2 Avazu dataset is from Kaggle competition in 2015. +Avazu provided 10 days of click-through data. We use 21 features +in total for modeling. All the features in this dataset are categorical +features. +3. iPinYou.3 iPinYou dataset is from iPinYou Global RTB(Real- +Time Bidding) Bidding Algorithm Competition in 2013. We follow +the data processing steps of [38] and consider all 16 categorical +features. +For all the datasets, we randomly split the examples into three +parts: 70% is for training, 10% is for validation, and 20% is for test- +ing. We also remove each categorical features’ infrequent levels +appearing less than 20 times to reduce sparsity issue. Note that +we want to compare the effectiveness and efficiency on learning +higher-order feature interactions automatically, so we do not do any +feature engineering but only feature transformation, e.g., numerical +feature bucketing and categorical feature frequency thresholding. +4.1.2 +Evaluation Metrics. We use AUC and LogLoss to evaluate +the performance of the models. +LogLoss LogLoss is both our loss function and evaluation metric. +It measures the average distance between predicted probability and +true label of all the examples. +AUC Area Under the ROC Curve (AUC) measures the probabil- +ity that a randomly chosen positive example ranked higher by the +model than a randomly chosen negative example. AUC only con- +siders the relative order between positive and negative examples. +A higher AUC indicates better ranking performance. +4.1.3 +Competing Models. We compare AdaEnsemble with follow- +ing models: LR (Logistic Regression) [19, 20], FM (Factorization Ma- +chine) [24], DNN (Multilayer Perceptron), Wide & Deep [4], Deep- +Crossing [26], DCN (Deep & Cross Network) [31], PNN (with both +inner product layer and outer product layer) [22, 23], DeepFM [8], +xDeepFM [16], AutoInt [28], FiBiNET [11], xDeepInt[35] and DCN +V2 [32]. Some of the models are state-of-the-art models for CTR +prediction problem and are widely used in the industry. +4.1.4 +Reproducibility. We implement all the models using Tensor- +flow [1]. The mini-batch size is 4096, and the embedding dimension +is 16 for all the features. For optimization, we employ Adam [13] +with learning rate is tuned from 10−4 to 10−3 for all the neural +network models, and we apply FTRL [19, 20] with learning rate +tuned from 10−2 to 10−1 for both LR and FM. For regularization, we +choose L2 regularization with 𝜆 ranging from 10−4 to 10−3 for dense +layer. Grid-search for each competing model’s hyper-parameters +is conducted on the validation dataset. The number of dense or +interaction layers is from 1 to 4. The number of neurons ranges +1https://www.kaggle.com/c/criteo-display-ad-challenge +2https://www.kaggle.com/c/avazu-ctr-prediction +3http://contest.ipinyou.com/ + +Research Track Paper +Conference’17, July 2017, Washington, DC, USA +from 128 to 1024. All the models are trained with early stopping +and are evaluated every 2000 training steps. +The setup is as follows for the hyper-parameters search of AdaEnsem- +ble: The number of recursive feature interaction layers 𝑙 is searched +from 1 to 4. For the number of selected experts 𝑘 per SparseMoE +layer, the searched values are from 1 to 3. For the reduction ratio +for both the expert gating network and depth selecting network, +we search from 4 to 16. We use G-FTRL optimizer for embedding +table and Adam for the model weights. For AdaEnsemble, as the +performance will be generally better when using more experts or +layers, we only report the one with fewer experts or layers used if +its AUC difference is within 0.02% compared to the ones using one +more expert or layer. +4.2 +Model Performance Comparison (Q1) +Table 1: Performance Comparison of Different Algorithms +on Criteo, Avazu and iPinYou Dataset. +Criteo +Avazu +iPinYou +Model +AUC +LogLoss +AUC +LogLoss +AUC +LogLoss +LR +0.7924 +0.4577 +0.7533 +0.3952 +0.7692 +0.005605 +FM +0.8030 +0.4487 +0.7652 +0.3889 +0.7737 +0.005576 +DNN +0.8051 +0.4461 +0.7627 +0.3895 +0.7732 +0.005749 +Wide&Deep +0.8062 +0.4451 +0.7637 +0.3889 +0.7763 +0.005589 +DeepFM +0.8069 +0.4445 +0.7665 +0.3879 +0.7749 +0.005609 +DeepCrossing +0.8068 +0.4456 +0.7628 +0.3891 +0.7706 +0.005657 +DCN +0.8056 +0.4457 +0.7661 +0.3880 +0.7758 +0.005682 +PNN +0.8083 +0.4433 +0.7663 +0.3882 +0.7783 +0.005584 +xDeepFM +0.8077 +0.4439 +0.7668 +0.3878 +0.7772 +0.005664 +AutoInt +0.8053 +0.4462 +0.7650 +0.3883 +0.7732 +0.005758 +FiBiNET +0.8082 +0.4439 +0.7652 +0.3886 +0.7756 +0.005679 +xDeepInt +0.8111 +0.4408 +0.7672 +0.3876 +0.7790 +0.005567 +DCN V2 +0.8086 +0.4433 +0.7662 +0.3882 +0.7765 +0.005593 +AdaEnsemble +0.8132 +0.4394 +0.7687 +0.3865 +0.7807 +0.005550 +The overall performance of different model architectures is listed +in Table 1. We have the following observations in terms of model +effectiveness: +• FM brings the most significant relative boost in performance +while we increase model complexity compared to LR baseline. +This reveals the importance of learning feature interactions. +• Models with more than two feature interaction modules gen- +erally perform better than models with only a single feature +interaction module, indicating the importance of jointly learned +feature interaction representation. +• The optimal feature interaction depth varies by feature inter- +action module type and when combined with different module +types, indicating the necessity for dynamically combining differ- +ent feature interactions on different interaction depths. +• AdaEnsemble achieves the best prediction performance among +all models. Our model’s superior performance could be attrib- +uted to the fact that AdaEnsemble jointly model various types +of feature interactions by adaptively selecting the feature inter- +action experts combination and determining the optimal feature +interaction depth by the controller. +4.3 +Feature Interaction Expert Selection +Analysis (Q2) +We compare the model performance and FLOPs between the Dense- +MoE and SparseMoE layers in AdaEnsemble architecture. We also +include the performance of different multi-layer single expert mod- +els and their ensemble. All the performance of above methods are +listed in Table 2. We also draw the alluvial diagram Figure 4 to illus- +trate the dependency of each SparseMoE layer’s expert selection. +The color of the flow is clustered by the frequency of the expert com- +bination. Based on the above observations, we developed following +understandings: +• Utilizing different feature interaction experts result in better per- +formance than single expert models in general. SparseMoE layer +achieves a better tradeoff between accuracy and computation +efficiency. +• Only utilizing one expert per SparseMoE layer generally hurts +the model performance as the model cannot ensemble different +types of feature interactions. +• When utilizing more than one expert per SparseMoE layer, even +though only a subset of feature interaction experts are selected, +SparseMoE can still effectively capture the most significant fea- +ture interactions of different depths and maintain similar perfor- +mance as the DenseMoE layer, while including more experts can +also result in more computational cost. +• Figure 4 shows that the SparseMoE layers dynamically utilize a +different combination of experts across different layers to capture +the complex feature interactions effectively. That also explains +why fusing different feature interactions is crucial for prediction +accuracy. +Table 2: Performance Comparison of SparseMoE and Dense- +MoE on Criteo Dataset. +AUC +LogLoss +FLOPs +SparseMoE(k=1) +0.8096 +0.4423 +2.26M +SparseMoE(k=2) +0.8121 +0.4400 +4.14M +SparseMoE(k=3) +0.8132 +0.4394 +6.02M +SparseMoE(k=4) +0.8133 +0.4393 +7.09M +DenseMoE +0.8133 +0.4392 +9.78M +Ensemble +0.8120 +0.4401 +12.15M +Dense Expert Only +0.8050 +0.4463 +3.71M +Cross Expert Only +0.8086 +0.4433 +3.36M +Polynomial Expert Only +0.8111 +0.4408 +3.32M +CNN Expert Only +0.8022 +0.4501 +1.11M +MHSA Expert Only +0.8051 +0.4465 +2.17M +4.4 +Depth Selection Analysis (Q3) +We compare the model performance between the AdaEnsemble +with and without depth selecting controller to investigate whether +the model achieves the harmony between prediction accuracy and +inference efficiency with respect to depth selection. The perfor- +mance of the different types of MoE layers and ensemble result is +listed in Table 3. + +Conference’17, July 2017, Washington, DC, USA +Yachen and Liubo +CNN +Cross +Dense +MHSA +Poly +CNN +Cross +Dense +MHSA +Poly +CNN +Cross +Dense +MHSA +Poly +CNN +Cross +Dense +MHSA +Poly +Layer1 +Layer2 +Layer3 +Layer4 +Figure 4: The Alluvial diagram for illustrating the depen- +dency of each SparseMoE layer’s expert selection +Each vertical axis represents a SparseMoE layer and the proportion of an +expert being used. The horizontal flows indicate the dependency and +relation of each SparseMoE layer’s expert selection. The proportion of the +expert combination was represented by the width of the flows and further +clustered to different colors. +With the incorporation of the depth selecting controller, we can +observe that our model can significantly improve training complex- +ity and inference efficiency (measured in FLOPs) while achieving +slightly better performance than the full-depth model. We think +the full-depth model is easier to overfit compared to AdaEnsem- +ble, thus resulting in slightly worse accuracy performance. The +AdaEnsemble with depth selecting controller adaptively selects +feature interaction depth per example basis, thus achieving better +trade-offs between prediction accuracy and inference efficiency. +The distribution of per example forward propagation depth is listed +in Table 4. +Table 3: Performance Comparison of AdaEnsemble with and +without controller on Criteo Dataset. +AUC +LogLoss +FLOPs +w/ controller +0.8132 +0.4394 +6.02M +w/o controller +0.8128 +0.4396 +8.58M +Table 4: AdaEnsemble Propagation Depth on Criteo Dataset. +Layer 1 +Layer 2 +Layer 3 +Layer 4 +Fraction +6.53% +19.36% +66.43% +7.68% +4.5 +Hyper-Parameter Study (4) +In order to have deeper insights into the proposed model, we con- +duct experiments on the Criteo dataset and compare model perfor- +mance on different hyper-parameter settings. This section evaluates +the model performance change with respect to hyper-parameters +that include: 1) depth of SparseMoE layers; 2) number of selected +experts in SparseMoE layers; +0.440 +0.441 +0.442 +1 +2 +3 +4 +5 +LogLoss +0.810 +0.811 +0.812 +0.813 +1 +2 +3 +4 +5 +Depth +AUC +(a) Layer Depth +0.440 +0.441 +0.442 +1 +2 +3 +4 +LogLoss +0.810 +0.811 +0.812 +0.813 +1 +2 +3 +4 +Number of Experts +AUC +(b) Number of Experts +Figure 5: Logloss and AUC v.s. feature interaction depth and +number of experts. +4.5.1 +Depth. The depth of SparsMoE layers 𝑙 determines the max- +imum order of feature interactions learned. In this experiment, we +set the number of selected experts 𝑘 as 3, which is generally a good +choice for the Criteo dataset. +Figure 5a shows the performance v.s. the depth𝑙 of the AdaEnsem- +ble on Criteo dataset. We observe that the performance keeps in- +creasing until we increase the depth up to 4. This aligns with our +understanding of the performance v.s. model complexity. Note that +we still let the controller determine the interaction depth per ex- +ample; the depth here is to control the maximum depth and model +complexity. +4.5.2 +Number of Experts. The number of selected experts of SparsMoE +layers 𝑘 determines the number of selected feature interactions ex- +perts per SparseMoE layer. In this experiment, we set the depth of +AdaEnsemble 𝑙 as 4, which is best for the Criteo dataset. +Figure 5b shows the performance v.s. the number of experts 𝑘 for +AdaEnsemble on Criteo dataset. We observe that the performance +keeps increasing until𝑘 equals 3. This indicates that the incremental +gain diminishes while we increase the number of experts selected +in SparseMoE layers. +5 +CONCLUSION +In this paper, we proposed a new CTR model which ensembles +the different interaction learning experts using the Sparse-Gated +Mixture-of-Experts (SparseMoE) hierarchical architecture. We also +introduce the Depth Selecting Controller for selecting the optimal +depth for each example. Based on these two conditional computa- +tion mechanisms, our model will select a subset of experts and an +optimal depth for each example. It enlarged the model capacity ex- +ponentially without increasing inference cost. Our comprehensive +experiments have demonstrated the effectiveness and efficiency of +our method. +In further work, We would like to study how to effectively ex- +tend our approach to user behavior sequence. While learning the +sparse ensemble of different models, we expect our approach can +dynamically select the optimal expert for different behaviors in the +user behavior sequence data. + +Research Track Paper +Conference’17, July 2017, Washington, DC, USA +REFERENCES +[1] Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey +Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. +2016. 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Designing effective sparse expert models. +arXiv preprint arXiv:2202.08906 (2022). + diff --git a/i9E_T4oBgHgl3EQf4hzI/content/tmp_files/load_file.txt b/i9E_T4oBgHgl3EQf4hzI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1401343855b8f65fda3b802b1fb98fd512ac6287 --- /dev/null +++ b/i9E_T4oBgHgl3EQf4hzI/content/tmp_files/load_file.txt @@ -0,0 +1,721 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf,len=720 +page_content='AdaEnsemble: Learning Adaptively Sparse Structured Ensemble Network for Click-Through Rate Prediction Yachen Yan yachen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='yan@creditkarma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='com Credit Karma San Francisco, California, USA Liubo Li liubo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='li@creditkarma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='com Credit Karma San Francisco, California, USA ABSTRACT Learning feature interactions is crucial to success for large-scale CTR prediction in recommender systems and Ads ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Re- searchers and practitioners extensively proposed various neural network architectures for searching and modeling feature interac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' However, we observe that different datasets favor different neural network architectures and feature interaction types, sug- gesting that different feature interaction learning methods may have their own unique advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Inspired by this observation, we propose AdaEnsemble: a Sparsely-Gated Mixture-of-Experts (SparseMoE) architecture that can leverage the strengths of het- erogeneous feature interaction experts and adaptively learns the routing to a sparse combination of experts for each example, al- lowing us to build a dynamic hierarchy of the feature interactions of different types and orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' To further improve the prediction accuracy and inference efficiency, we incorporate the dynamic early exiting mechanism for feature interaction depth selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The AdaEnsemble can adaptively choose the feature interaction depth and find the corresponding SparseMoE stacking layer to exit and compute prediction from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Therefore, our proposed architecture inherits the advantages of the exponential combinations of sparsely gated experts within SparseMoE layers and further dynamically se- lects the optimal feature interaction depth without executing deeper layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We implement the proposed AdaEnsemble and evaluate its performance on real-world datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Extensive experiment results demonstrate the efficiency and effectiveness of AdaEnsemble over state-of-the-art models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' CCS CONCEPTS Computing methodologies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' • Machine learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' • Machine learning approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' • Neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' KEYWORDS CTR prediction, Recommendation System, Feature Interaction, Mix- ture of 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='XXXXXXX ACM Reference Format: Yachen Yan and Liubo Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' AdaEnsemble: Learning Adaptively Sparse Structured Ensemble Network for Click-Through Rate Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In Pro- ceedings of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' ACM, New York, NY, USA, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='org/XXXXXXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='XXXXXXX 1 INTRODUCTION Click-through rate (CTR) prediction model [25] is an essential com- ponent for the large-scale search ranking, online advertising and recommendation system [4, 9, 20, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Many deep learning-based models have been proposed for CTR prediction problems in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' They have become dominant in learning the useful feature interactions of the mixed-type input in an end-to-end fashion[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Although most of the existing meth- ods can effectively capture higher-order feature interactions, we observe that their performance varies for different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We believe this is due to their inductive bias: different methods learn different types of feature interactions and favor different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' While every existing method focuses on automatically model- ing different types of feature interactions, there have been few attempts to model different types of interactions jointly and dy- namically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We believe that ensembling different interaction mod- ules to create heterogeneous feature interactions can complement the non-overlapping knowledge that each interaction learning ap- proach learned, as opposed to the homogeneous interaction model- ing method, which restricts the types of feature interactions to be learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For utilizing various interaction modules to learn different types of feature interactions, we use Sparsely-Gated Mixture-of- Experts (SpasrseMoE) architecture to enrich the model capacity while achieving computational efficiency through conditional com- putation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We propose AdaEnsemble: a Sparsely-Gated Mixture-of-Experts (SparseMoE) hierarchical architecture to ensemble different inter- action learning modules and dynamically select optimal feature interaction depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Within each SparseMoE layer of AdaEnsemble, there is a collection of interaction learning experts, and a trainable gating network determines a sparse combination of these experts to use for each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Within the Depth Selecting Controller, a trainable gating network will choose the feature interaction depth for each example and recursively propagate feature interaction rep- resentations through SparseMoE layers to the corresponding depth for computing the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Through these conditional compu- tation mechanisms, we enlarged the model capacity exponentially without increasing inference cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The main contributions of this paper can be summarized as follows: We designed a novel model architecture called AdaEnsemble to ensemble various types of feature interaction learning modules arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='08353v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='IR] 6 Jan 2023 Conference’17, July 2017, Washington, DC, USA Yachen and Liubo by Sparsely-Gated Mixture-of-Experts (SparsseMoE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Through utilizing MoE layers recursively with residual connections and normalization, AdaEnsemble can model different types of inter- actions jointly and dynamically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We designed an efficient and effective Depth Selecting Con- troller to adaptively choose the optimal feature interaction depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Through utilizing this controller, AdaEnsemble can dynamically determine the layer for early exiting to improve prediction accu- racy and inference efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We designed a bi-level optimization algorithm for iteratively training the modeling network and gating network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We conduct extensive experiments on real-world datasets and study the learning patterns of AdaEnsemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Feature Interaction Modeling Learning the feature interactions is the key topic in CTR predic- tion problems and has been widely discussed in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Various hybrid network architectures [4, 8, 16, 22, 23, 31, 32] utilize the feed- forward neural network with non-linear activation function as its core component, to learn implicit interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The complement of the implicit interaction modeling improves the performance of the network that only models the explicit interactions [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Another group of models focuses on exploring bit-wise/vector- wise feature interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Deep & Cross Network (DCN) [31] and its improved version DCN V2 [32] explores the feature interactions at the bit-wise level explicitly in a recursive fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Deep Factor- ization Machine (DeepFM) [8] utilizes factorization machine layer to model the pairwise vector-wise interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Product Neural Net- work (PNN) [22, 23] introduces the inner product layer and the outer product layer to learn vector-wise interactions and bit-wise inter- actions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' xDeepFM [16] learns the explicit vector-wise interaction using Compressed Interaction Network (CIN), which has an RNN-like architecture and learns vector-wise interactions using Hadamard product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' FiBiNET [11] utilizes Squeeze-and-Excitation network to dynamically learn the importance of features and model the feature interactions via bilinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' AutoInt [28] leverages the Transformer [30] architecture to learn different orders of feature combinations of input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' xDeepInt [35] introduces polyno- mial interaction layer to recursively learn higher-order vector-wise and bit-wise interactions jointly with controlled degree, dispensing with jointly-trained DNN and nonlinear activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Sparse Mixture-of-Experts Network The Sparsely-Gated MoE model [2, 27] combines multiple experts and a trainable gating network that selects a subset of experts for each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This network architecture can be viewed as a dynamic sparsity structure that maintains all weights but intro- duces sparsity into the model through conditional computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The SparseMoE is widely used in natural language processing re- search area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Most of the discussions focus on improving the rout- ing mechanism for the experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Switch Transformer [7] simplifies the top-1 routing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' GShard [14] uses group-level top-2 routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Both are trained with load balancing losses and improve language models with reduced communication and computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' BASE Layers [15] treated routing as a linear assignment Embedding Layer Categorical Feature Bucktized Numeric Feature 1st Sparse MoE Layer 2nd Sparse MoE Layer l-th Sparse MoE Layer Input Feature Map Add & Normalize Add & Normalize Add & Normalize --- Depth Selecting Network 1st Estimator 2nd Estimator l-th Estimator Figure 1: The Architecture of AdaEnsemble In this example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' the depth selecting network selects the 2nd layer to exit and compute the final prediction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' therefore the deeper layers was not activated and plotted translucent in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' problem and removed the need for load balancing auxiliary losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' M6-T [36] splits experts into different groups and applies k top- 1 routing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Some literature explore the training of the Sparsely-Gated MoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' EvoMoE [21] decouples the training of ex- perts and the sparse gate by training all experts at first and then gradually and adaptively becomes sparser while routes to fewer experts for learning the sparse gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' ST-MoE [40] further studies the training instabilities and uncertain quality issue of the MoE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' X-MoE [5] proposed a dimension reduction and L2 normalization to solve the representation collapse in the training of MoE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3 Early-Exiting Network The idea of early-exiting for the neural network was firstly pro- posed by BranchyNet [29] for computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This technique is also applied to NLP tasks, DeeBERT [33], FastBERT [18], and PABEE [39] was later introduced for improving inference efficiency of Transformer-Based BERT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For the early-exiting mechanism, BranchyNet [29], DeeBERT [33], FastBERT [18] and SDN [12] use the entropy-based or confidence- based criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' While using entropy-based or confidence-based cri- teria is straightforward and effective, it takes advantage of the fact that the model’s output is a probability distribution in multi-class classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This technique generally cannot be applied to binary classification and regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' On the other hand, BERxiT [34] and Epnet [6] use learned modules for early-exiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Research Track Paper Conference’17, July 2017, Washington, DC, USA 3 PROPOSED MODEL: ADAENSEMBLE In this section, we give an overview of the architectures of AdaEnsem- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' First, we introduce the feature processing and embedding layer, which maps continuous features and high-dimensional categorical features onto a dense embedding vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Second, we introduce the feature interaction experts we considered for jointly learning the hierarchy of the deep feature representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Third, we present the sparse mixture-of-experts (SparseMoE) layer, which ensemble mul- tiple interaction experts dynamically, and the estimator associated with each SparseMoE Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Fourth, we discuss how to automat- ically and dynamically select the feature interaction depth based on the Depth Selecting Controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Finally, a bi-level optimization algorithm will be provided for the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Embedding Layer In large-scale CTR prediction tasks, inputs include both continuous and categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Categorical features are often directly encoded by one-hot encoding, which results in an excessively high- dimensional and sparse feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Suppose we have 𝐹 fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In our feature processing step, we bucketize all the continuous features to equal frequency bins, then embed the bucketized continuous features and categorical features embed each feature onto a dense embedding vector 𝑒𝑖 of the same dimension 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' e𝑖 = x𝑖V𝑖, where 𝑒𝑖 ∈ 𝑅𝐷, V𝑖 is an embedding matrix for the 𝑖-th field, and x𝑖 is the corresponding one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Lastly, we concatenate 𝐹 embedding vectors and denote the output of embedding layer 𝑋0 ∈ 𝑅𝐹×𝐷 as the input feature map: 𝑋0 = [𝑒1,𝑒2, · · · ,𝑒𝐹 ]⊺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' (1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Feature Interaction Experts We considered several types of feature interaction experts in our model: Dense Layer, Convolution Layer, Multi-Head Self-Attention Layer, Polynomial Interaction Layer, and Cross Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Essentially, any feature interaction learning layer can be included in our frame- work, and the residual connection and normalization will be applied to their ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Now we introduce these feature interaction ex- perts included in our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Note that our proposed framework is general and can use arbitrary feature interaction modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The potential feature interaction experts can be used are not limited to the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Dense Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Dense Layer is also known as fully connected layer and is the most widely used module for modeling implicit feature interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In this paper, we use the dense layer with non-linear activation function for learning the deep feature rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Given an input of embedding 𝑋𝑙−1, the output of embedding 𝑋𝑙 is obtained from: 𝑋𝑙 = 𝜎(𝑊𝑙 · 𝑋𝑙−1) (2) where 𝜎 denotes activation function and𝑊𝑙 denotes the weights of the 𝑙-th dense layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Convolution Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Convolution layers are widely used for computer vision problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In this paper, we applied 1D convolution as one of the interaction experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Here we utilize a dense layer ahead of the convolution layer for fusing the inputs embeddings first, as the convolution layer is locally connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Given the embedding 𝑋𝑙−1 as input, the output of embedding 𝑋𝑙 is obtained from: 𝑋𝑙 = Dense(Pooling(Conv1D(Reshape(𝑋𝑙−1)))) (3) Here we first reshape the input embedding and then apply 1D convolution followed by a pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Finally, we use a dense layer to project the output to the desired dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3 Multi-Head Self-Attention Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Multi-Head Self-Attention Layer [30] is widely used in transformer networks for its superior performance in natural language processing and has started to be popular in the computer vision research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We consider utiliz- ing Multi-Head Self-Attention Layer for modeling the dependency between features and forming meaningful higher-order features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Given an input of embedding 𝑋𝑙−1, the output of embedding 𝑋𝑙 is obtained from: 𝑋𝑙 = Dense(MultiHeadSelfAttention(Reshape(𝑋𝑙−1)) (4) Here we first reshape the input embedding and then apply Multi- Head Self-Attention Layer followed by a dense layer to project the output to the desired dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4 Polynomial Interaction Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Polynomial Interaction Net- work [35] is designed to capture bounded degree feature interac- tions explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In this paper, we adopt the PIN layer as one of our feature interaction learning experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Given an input of embedding 𝑋𝑙−1, the mathematical representation of the 𝑙-th PIN layer’s output is given by: 𝑋𝑙 = 𝑋𝑙−1 ◦ (𝑊𝑙 · 𝑋0) (5) where ◦ denotes the Hadamard product and𝑊 denotes the kernel weights of the PIN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We omit the residual connection from the original paper in the above equation as the residual connection will be used across the MoE layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='5 Cross Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Deep Cross Network [32] is later proposed to explore the feature interactions in a recursive fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Given an input of embedding 𝑋𝑙−1, the output of embedding 𝑋𝑙 is obtained from: 𝑋𝑙 = 𝑋0 ◦ (𝑊𝑙 · 𝑋𝑙−1) + 𝑏𝑙 (6) Where 𝑊 and 𝑏 denote the weight matrix and bias vector in the 𝑙-th DCN layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We also omit the residual connection of the original implementation in the above equation, as we will use the residual connection across the MoE layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3 Sparse Mixture-of-Experts Layer The Sparse Mixture-of-Experts layer ensembles aforementioned heterogeneous feature interaction experts and consists of several other essential parts to make the overall model can be stably trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA Yachen and Liubo Sparse MoE Layer Expert 1 Expert 2 Expert 3 Expert n-1 Expert n Input Embedding Output Embedding Gating Network Sparse Dispatcher non-zero index non-zero value .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='. Figure 2: The architecture of Sparse Mixture-of-Experts Layer Input Embedding Feed-Forward Network e1 e2 e3 e4 Expert Embedding Cosine Similarity Learnable Temperature Re-Scaling Top-K L2 Normalization L2 Normalization Routing Score Softmax Noise Injection Figure 3: The Noisy Gating Network within Sparse Mixture- of-Experts Layer 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Noisy Gating Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The gating network essentially com- putes the gating value for selecting and weighting the output em- bedding of each expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For the input embedding of gating network 𝑋0, it firstly pro- cessed by the gating network: a two-layer feed-forward network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' a dimension reduction layer with reduction ratio 𝑟 [10], a non- linear activation function and then a dense layer projecting to hidden state ℎ ∈ 𝑅𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Additionally, we applied multiplicative jitter noise for introducing exploration and promoting load balancing between different experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' ℎ = FFN(𝑋0 ◦ RandomUniform(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='0 − eps, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='0 + eps)) (7) After projecting the input embedding to hidden state ℎ ∈ 𝑅𝑑, we apply the 𝐿2 normalization to both hidden state ℎ ∈ 𝑅𝑑 and learnable expert embeddings 𝑒𝑗 ∈ 𝑅𝑑, where 𝑗 is the index of expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Then, we compute the cosine similarity between the hidden state and expert embedding as the initial routing score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Here we encourage the uniformity of representations to avoid dominated experts issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 𝑠𝑗 = ℎ · 𝑒𝑗 ∥ℎ∥∥𝑒𝑗 ∥ (8) Finally, we use a learnable temperature scalar 𝜏 to re-scale the routing scores to the range [−1, +1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 𝑔𝑗 = 𝑠𝑗/𝜏 (9) For the computed routing score 𝑔, we only keep the top k values and set the rest to −∞, resulting in the corresponding softmax gating values equal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The 𝑖-th element of the output of the gating network is 𝐸𝑥𝑝𝑒𝑟𝑡𝐺(𝑥)𝑖 = exp � TopK(𝑔,𝑘)𝑖 � �𝑁 𝑗=1 exp � TopK(𝑔,𝑘)𝑗 � , (10) where TopK(𝑔,𝑘)𝑗 = � 𝑔𝑗 if 𝑔𝑗 is in the top 𝑘 elements of 𝑔 −∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' (11) These gating values will be used by the sparse dispatcher for routing examples to different experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This is the essential step for achieving sparsity of our Sparse Mixture-of-Experts layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Note that the 𝐺(𝑥) is differentiable regardless the value of 𝑘[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Annealing Top-K Gating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We also introduce annealing mech- anism to the Top-K operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We starts with 𝑘 value equal to the number of experts, which means that we starts as a fully dense gate that routes examples to all experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Then we gradually decrease the 𝑘 and route examples to fewer experts, to adaptively make the structure sparser and continuously improving the computation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' By annealing of the 𝑘 value, we start to train our architecture with a dense structure which allows us to thoroughly learn all ex- perts and adjust the gating network in the correct direction at the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Therefore, we can control the sparsity of our architec- ture while training to not only accelerate the convergence of the gating network but also benefit the experts’ specialty for learning particular types of feature interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3 Sparse Dispatcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The sparse dispatcher takes the examples gating values and experts as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' It firstly dispatches the examples to the experts corresponding to the non-zero gating values, and lets experts generate the output embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The output𝑦 of the Sparse Mixture-of-Experts layer is the linearly weighted combination of expert output embeddings by the non-zero gating values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Research Track Paper Conference’17, July 2017, Washington, DC, USA 𝑦 = ∑︁ 𝑗 ∈𝜙 𝐸𝑥𝑝𝑒𝑟𝑡𝐺𝑗 (𝑥)𝐸𝑗 (𝑥) (12) Where 𝜙 denotes the selected non-zero indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We save com- putation based on the sparsity of 𝐺(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Wherever 𝐺(𝑥)𝑗 = 0, we don’t pass the expert to the corresponding expert and do not need to compute expert embedding 𝐸𝑗 (𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4 Load Distribution Regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' As stated in the previous research [5, 7, 27, 40], the gating network tends to select only a few experts if no regularization is applied, especially when certain experts are easier to train than other experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This phenomenon is self-reinforcing, since the selected experts are trained more and will be selected more frequently by the gating network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Therefore, the load balancing loss is applied to enforce the uniform expert routing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 𝐿balance = 𝜆 · 𝑁 · 𝑁 ∑︁ 𝑗=1 𝑓𝑗 · 𝑃𝑗 (13) where 𝑁 is the number of experts, 𝑓𝑗 is the fraction of examples dispatched to expert j, 𝑃𝑗 is the average of the router probability allocated for expert j, and 𝜆 is the coefficient for the regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 𝑓𝑗 = 1 𝐵 ∑︁ 𝑥 ∈B 1{argmax 𝑝(𝑥) = 𝑗} (14) 𝑃𝑗 = 1 𝐵 ∑︁ 𝑥 ∈B 𝑝𝑗 (𝑥) (15) While the default load balancing loss is applicable and effective when experts are of the same type, AdaEnsemble is using hetero- geneous feature interaction experts, and the optimal load for each expert is not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Therefore, we apply the below load distribu- tion regularization to encourage the expected load distribution of heterogeneous experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 𝐿distribution = 𝜆 · 𝑁 ∑︁ 𝑗=1 𝑓𝑗 · 𝑃𝑗 𝑤𝑗 (16) where 𝑤𝑗 is the expected load fraction of examples dispatched to expert j, and naturally �𝑁 𝑗=1 𝑤𝑗 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In practice, the 𝜆 should be sufficiently large to prevent expert selection self-reinforcing phenomenon at the initial training stage while not overwhelming the primary LogLoss objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4 Estimator Layer The output of the Sparse Mixture-of-Experts layer is a feature map that consists of feature interactions of different degrees and types, including raw input feature map reserved by residual connections and higher-order feature interactions jointly learned by experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For the final prediction, we merely use the formula as follows: ˆ𝑦 = 𝜎(𝑊𝑙𝑋𝑙 + 𝑏𝑙) (17) where 𝜎 is the sigmoid function, 𝑊𝑙 ∈ 𝑅1×𝐹 is a feature map aggre- gation vector that linearly combines all the learned feature interac- tions in the feature map, 𝑏 ∈ 𝑅 is the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='5 Depth Selecting Controller 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Depth Selecting Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The Depth Selecting Network is essentially the same configuration as the aforementioned Noisy Gat- ing Network for SparseMoE layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We denote it by 𝐷𝑒𝑝𝑡ℎ𝐺(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The outputs of 𝐷𝑒𝑝𝑡ℎ𝐺(𝑥) are [𝑔𝑑𝑒𝑝𝑡ℎ 1 ,𝑔𝑑𝑒𝑝𝑡ℎ 2 , · · · ,𝑔𝑑𝑒𝑝𝑡ℎ 𝐿 ], indicating each example’s optimal forward propagation depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The 𝑙-th unit denotes the probability of selecting the 𝑙-th MoE layer to exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The optimal depth is automatically selected as the one corresponding to the largest probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In contrast to the expert selection, when choosing the optimal depth of each example for the dynamic infer- ence, we only keep the top-1 depth index from the output units of the Depth Selecting Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Note that we can also apply the load distribution regularization to encourage the examples’ propagation depth distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Dynamic Propagation Mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' With the depth gates𝑔𝑑𝑒𝑝𝑡ℎ 𝑙 ∈ [0, 1] computed by Depth Selecting Network, we obtain the opti- mal depth for each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' If 𝑔𝑑𝑒𝑝𝑡ℎ 𝑙 = 0, we recursively forward propagate examples through MoE layers and compute deeper repre- sentation until 𝑔𝑑𝑒𝑝𝑡ℎ 𝑙 = 1 or reaching the final layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' If 𝑔𝑑𝑒𝑝𝑡ℎ 𝑙 = 1, the forward propagation will be stopped and the corresponding 𝑙-th estimator will compute the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' To efficiently process a batch of examples with different optimal propagation depths, we utilize algorithm 1 for dynamic forward propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Algorithm 1 Dynamic Propagation 1: DepthGates ← DepthSelectingNetwork(x) 2: �𝑦 ← DynamicPropagation(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' DepthGates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' depth=0) 3: return �𝑦 4: 5: function DynamicPropagation(Inputs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Gates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Depth) 6: Outputs = MoE(Inputs) 7: Depth += 1 8: if Depth == Number of Layer then 9: �𝑦 = Estimator(Outputs) 10: else 11: g = Gates[:,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Depth] 12: Outputskeep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Outputsexit = Dispatch(Outputs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' g) 13: Gateskeep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' _ = Dispatch(Gates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' g) 14: 15: �𝑦keep = DynamicPropagation(Outputskeep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Gateskeep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Depth) 16: �𝑦exit = Estimator(Outputsexit) 17: �𝑦 = Combine(�𝑦keep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' �𝑦exit) 18: end if 19: return �𝑦 20: end function 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='6 Training 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Training Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The loss function we use a linearly weighted combination of the Log Loss and the auxiliary load distribution regularization, 𝐿𝑜𝑠𝑠 = 𝐿LogLoss + 𝜆1𝐿expert distribution + 𝜆2𝐿depth distribution (18) where 𝜆1 and 𝜆2 are the coefficients for weighting the load distri- bution regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA Yachen and Liubo 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Bi-Level Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The optimization task for training the AdaEnsemble is to jointly optimize the parameters𝑊 , which stands for the expert layers and estimator layers, and 𝛼, which represents the expert gating network and depth selecting network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Inspired by the DARTS [17], we apply bi-level optimization algorithm for training our model, where 𝛼 is the upper-level parameters and 𝑊 is the lower-level parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We apply algorithm 2 to optimize 𝑊 and 𝛼 alternatively and iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Algorithm 2 Bi-Level Optimization for AdaEnsemble Input: training examples with corresponding labels, step size 𝑡 Output: well-learned parameters W∗ and 𝛼∗ 1: while not converged do 2: Sample a mini-batch of validation data 3: Updating 𝛼 by descending ∇𝛼 L𝑣𝑎𝑙 �W − 𝜉 ∇WL𝑡𝑟𝑎𝑖𝑛 (W, 𝛼), 𝛼� 4: (𝜉 = 0 for first-order approximation) 5: for 𝑖 ← 1,𝑡 do 6: Sample a mini-batch of training data 7: Update W by descending ∇WL𝑡𝑟𝑎𝑖𝑛 (W, 𝛼) 8: end for 9: end while 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='7 Discussion on AdaEnsemble The combination of sparse experts routing at each SparseMoE layer and the depth selecting controller brings two merits to the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' On one hand, the stacked sparseMoE layers allow the pro- posed model to leverage the exponential combinations of sparsely gated experts, which brings in more predicting power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' On the other hand, the depth selecting controller enables the proposed model to learn the instance-ware model depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' It improves the efficiency during model serving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In the next section, we will illustrate the effectiveness of the proposed model through some experimental studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4 EXPERIMENTS In this section, we focus on evaluating the effectiveness of our proposed models and seeking answers to the following research questions:: Q1: How does our proposed AdaEnsemble perform compared to each baseline in the CTR prediction problem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Q2: How does the SparseMoE layer perform compared to Dense- MoE, which utilizes all feature interaction experts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Does the cascade of SparseMoE layers effectively capture different types of feature interactions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Q3: How does the depth selecting controller perform compared to a full-depth network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Does the early exiting mechanism achieve both effectiveness and efficiency?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Q4: How do different hyper-parameter settings influence the performance of AdaEnsemble?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Experiment Setup 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We evaluate our proposed model on three public real-world datasets widely used for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Criteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Criteo dataset is from Kaggle competition in 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Criteo AI Lab officially released this dataset after, for academic use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This dataset contains 13 numerical features and 26 categor- ical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We discretize all the numerical features to integers by transformation function ⌊𝐿𝑜𝑔 �𝑉 2�⌋ and treat them as categor- ical features, which is conducted by the winning team of Criteo competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Avazu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Avazu dataset is from Kaggle competition in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Avazu provided 10 days of click-through data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We use 21 features in total for modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' All the features in this dataset are categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' iPinYou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3 iPinYou dataset is from iPinYou Global RTB(Real- Time Bidding) Bidding Algorithm Competition in 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We follow the data processing steps of [38] and consider all 16 categorical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For all the datasets, we randomly split the examples into three parts: 70% is for training, 10% is for validation, and 20% is for test- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We also remove each categorical features’ infrequent levels appearing less than 20 times to reduce sparsity issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Note that we want to compare the effectiveness and efficiency on learning higher-order feature interactions automatically, so we do not do any feature engineering but only feature transformation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=', numerical feature bucketing and categorical feature frequency thresholding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We use AUC and LogLoss to evaluate the performance of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' LogLoss LogLoss is both our loss function and evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' It measures the average distance between predicted probability and true label of all the examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' AUC Area Under the ROC Curve (AUC) measures the probabil- ity that a randomly chosen positive example ranked higher by the model than a randomly chosen negative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' AUC only con- siders the relative order between positive and negative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' A higher AUC indicates better ranking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3 Competing Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We compare AdaEnsemble with follow- ing models: LR (Logistic Regression) [19, 20], FM (Factorization Ma- chine) [24], DNN (Multilayer Perceptron), Wide & Deep [4], Deep- Crossing [26], DCN (Deep & Cross Network) [31], PNN (with both inner product layer and outer product layer) [22, 23], DeepFM [8], xDeepFM [16], AutoInt [28], FiBiNET [11], xDeepInt[35] and DCN V2 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Some of the models are state-of-the-art models for CTR prediction problem and are widely used in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4 Reproducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We implement all the models using Tensor- flow [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The mini-batch size is 4096, and the embedding dimension is 16 for all the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For optimization, we employ Adam [13] with learning rate is tuned from 10−4 to 10−3 for all the neural network models, and we apply FTRL [19, 20] with learning rate tuned from 10−2 to 10−1 for both LR and FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For regularization, we choose L2 regularization with 𝜆 ranging from 10−4 to 10−3 for dense layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Grid-search for each competing model’s hyper-parameters is conducted on the validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The number of dense or interaction layers is from 1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The number of neurons ranges 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='com/c/criteo-display-ad-challenge 2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='com/c/avazu-ctr-prediction 3http://contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='ipinyou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='com/ Research Track Paper Conference’17, July 2017, Washington, DC, USA from 128 to 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' All the models are trained with early stopping and are evaluated every 2000 training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The setup is as follows for the hyper-parameters search of AdaEnsem- ble: The number of recursive feature interaction layers 𝑙 is searched from 1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For the number of selected experts 𝑘 per SparseMoE layer, the searched values are from 1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For the reduction ratio for both the expert gating network and depth selecting network, we search from 4 to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We use G-FTRL optimizer for embedding table and Adam for the model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' For AdaEnsemble, as the performance will be generally better when using more experts or layers, we only report the one with fewer experts or layers used if its AUC difference is within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='02% compared to the ones using one more expert or layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4.' metadata={'source': 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+page_content='8086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='7662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='7765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='005593 AdaEnsemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='7687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='7807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='005550 The overall performance of different model architectures is listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We have the following observations in terms of model effectiveness: FM brings the most significant relative boost in performance while we increase model complexity compared to LR baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This reveals the importance of learning feature interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Models with more than two feature interaction modules gen- erally perform better than models with only a single feature interaction module, indicating the importance of jointly learned feature interaction representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The optimal feature interaction depth varies by feature inter- action module type and when combined with different module types, indicating the necessity for dynamically combining differ- ent feature interactions on different interaction depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' AdaEnsemble achieves the best prediction performance among all models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Our model’s superior performance could be attrib- uted to the fact that AdaEnsemble jointly model various types of feature interactions by adaptively selecting the feature inter- action experts combination and determining the optimal feature interaction depth by the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='3 Feature Interaction Expert Selection Analysis (Q2) We compare the model performance and FLOPs between the Dense- MoE and SparseMoE layers in AdaEnsemble architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We also include the performance of different multi-layer single expert mod- els and their ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' All the performance of above methods are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We also draw the alluvial diagram Figure 4 to illus- trate the dependency of each SparseMoE layer’s expert selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The color of the flow is clustered by the frequency of the expert com- bination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Based on the above observations, we developed following understandings: Utilizing different feature interaction experts result in better per- formance than single expert models in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' SparseMoE layer achieves a better tradeoff between accuracy and computation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Only utilizing one expert per SparseMoE layer generally hurts the model performance as the model cannot ensemble different types of feature interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' When utilizing more than one expert per SparseMoE layer, even though only a subset of feature interaction experts are selected, SparseMoE can still effectively capture the most significant fea- ture interactions of different depths and maintain similar perfor- mance as the DenseMoE layer, while including more experts can also result in more computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Figure 4 shows that the SparseMoE layers dynamically utilize a different combination of experts across different layers to capture the complex feature interactions effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' That also explains why fusing different feature interactions is crucial for prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Table 2: Performance Comparison of SparseMoE and Dense- MoE on Criteo Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' AUC LogLoss FLOPs SparseMoE(k=1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4423 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='26M SparseMoE(k=2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4400 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='14M SparseMoE(k=3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4394 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='02M SparseMoE(k=4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4393 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='09M DenseMoE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4392 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='78M Ensemble 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4401 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='15M Dense Expert Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4463 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='71M Cross Expert Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4433 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='36M Polynomial Expert Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4408 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='32M CNN Expert Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4501 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='11M MHSA Expert Only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4465 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='17M 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4 Depth Selection Analysis (Q3) We compare the model performance between the AdaEnsemble with and without depth selecting controller to investigate whether the model achieves the harmony between prediction accuracy and inference efficiency with respect to depth selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The perfor- mance of the different types of MoE layers and ensemble result is listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Conference’17, July 2017, Washington, DC, USA Yachen and Liubo CNN Cross Dense MHSA Poly CNN Cross Dense MHSA Poly CNN Cross Dense MHSA Poly CNN Cross Dense MHSA Poly Layer1 Layer2 Layer3 Layer4 Figure 4: The Alluvial diagram for illustrating the depen- dency of each SparseMoE layer’s expert selection Each vertical axis represents a SparseMoE layer and the proportion of an expert being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The horizontal flows indicate the dependency and relation of each SparseMoE layer’s expert selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The proportion of the expert combination was represented by the width of the flows and further clustered to different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' With the incorporation of the depth selecting controller, we can observe that our model can significantly improve training complex- ity and inference efficiency (measured in FLOPs) while achieving slightly better performance than the full-depth model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We think the full-depth model is easier to overfit compared to AdaEnsem- ble, thus resulting in slightly worse accuracy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The AdaEnsemble with depth selecting controller adaptively selects feature interaction depth per example basis, thus achieving better trade-offs between prediction accuracy and inference efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The distribution of per example forward propagation depth is listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Table 3: Performance Comparison of AdaEnsemble with and without controller on Criteo Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' AUC LogLoss FLOPs w/ controller 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4394 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='02M w/o controller 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='8128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='4396 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='58M Table 4: AdaEnsemble Propagation Depth on Criteo Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Layer 1 Layer 2 Layer 3 Layer 4 Fraction 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='53% 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='36% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='43% 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='68% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='5 Hyper-Parameter Study (4) In order to have deeper insights into the proposed model, we con- duct experiments on the Criteo dataset and compare model perfor- mance on different hyper-parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This section evaluates the model performance change with respect to hyper-parameters that include: 1) depth of SparseMoE layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 2) number of selected experts in SparseMoE layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='442 1 2 3 4 5 LogLoss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='813 1 2 3 4 5 Depth AUC (a) Layer Depth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='440 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='442 1 2 3 4 LogLoss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='813 1 2 3 4 Number of Experts AUC (b) Number of Experts Figure 5: Logloss and AUC v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' feature interaction depth and number of experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='1 Depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The depth of SparsMoE layers 𝑙 determines the max- imum order of feature interactions learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In this experiment, we set the number of selected experts 𝑘 as 3, which is generally a good choice for the Criteo dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Figure 5a shows the performance v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' the depth𝑙 of the AdaEnsem- ble on Criteo dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We observe that the performance keeps in- creasing until we increase the depth up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This aligns with our understanding of the performance v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Note that we still let the controller determine the interaction depth per ex- ample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' the depth here is to control the maximum depth and model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='2 Number of Experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' The number of selected experts of SparsMoE layers 𝑘 determines the number of selected feature interactions ex- perts per SparseMoE layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In this experiment, we set the depth of AdaEnsemble 𝑙 as 4, which is best for the Criteo dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Figure 5b shows the performance v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' the number of experts 𝑘 for AdaEnsemble on Criteo dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We observe that the performance keeps increasing until𝑘 equals 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' This indicates that the incremental gain diminishes while we increase the number of experts selected in SparseMoE layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' 5 CONCLUSION In this paper, we proposed a new CTR model which ensembles the different interaction learning experts using the Sparse-Gated Mixture-of-Experts (SparseMoE) hierarchical architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' We also introduce the Depth Selecting Controller for selecting the optimal depth for each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Based on these two conditional computa- tion mechanisms, our model will select a subset of experts and an optimal depth for each example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' It enlarged the model capacity ex- ponentially without increasing inference cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' Our comprehensive experiments have demonstrated the effectiveness and efficiency of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' In further work, We would like to study how to effectively ex- tend our approach to user behavior sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9E_T4oBgHgl3EQf4hzI/content/2301.08353v1.pdf'} +page_content=' While learning the sparse ensemble of different models, we expect our approach can dynamically select the optimal expert for different behaviors in the user behavior sequence data.' 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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf,len=605 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='04775v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='SY] 12 Jan 2023 On Phase Change Rate Maximization with Practical Applications ⋆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Kao ∗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Hara ∗∗ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Hori ∗∗∗ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Iwasaki ∗∗∗∗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Khong † ∗ Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' of Electrical Engineering, National Sun Yat-Sen University, Taiwan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (e-mail: cykao@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='nsysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='tw) ∗∗ Global Scientific Information and Computing Center, Tokyo Institute of Technology, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (e-mail: shinji hara@ipc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='jp) ∗∗∗ Applied Physics and Physico-Informatics, Keio University, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (e-mail: yhori@appi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='keio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='jp) ∗∗∗∗ Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' of Mechanical and Aerospace Engineering, University of California at Los Angeles, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (e-mail: tiwasaki@ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='edu) † Independent Researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (email: szkhongwork@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='com) Abstract: We recapitulate the notion of phase change rate maximization and demonstrate the usefulness of its solution on analyzing the robust instability of a cyclic network of multi- agent systems subject to a homogenous multiplicative perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Subsequently, we apply the phase change rate maximization result to two practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The first is a magnetic levitation system, while the second is a repressilator with time-delay in synthetic biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' We also state results on robust instability analysis of digital control systems by making use of the bilinear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Keywords: phase change rate maximization, instability analysis, strong stabilization 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' INTRODUCTION Robustness against model uncertainties for feedback sys- tems has been recognized as one of the important issues in control theory from the practical application viewpoint over forty years since the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The most typical and successful theory is the H∞ control which includes robust stability and robust stabilization against norm-bounded dynamic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', (Zhou, 1996) and the ref- erences therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' A counterpart of the robust stability analysis is the so- called “robust instability analysis” for nominally unstable feedback systems, and the problem is to find a stable per- turbation with the smallest H∞-norm which stabilizes the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' A practical motivation of the analysis is maintain- ing nonlinear oscillations caused by instability of an equi- librium point for dynamical systems arising in neuro- science and synthetic biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' See (Hara, 2020) and (Hara, 2021) for applications to the FitzHugh-Nagumo neuron model and repressilator model, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The instability analysis problem is closely related to the strong stabilization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', stabilization by a stable con- troller (Youla, 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Zeren, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Ohta, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Actually, it is equivalent to strong stabilization by a minimum- norm controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The problem is extremely difficult due to the following two reasons: (i) non-convexity nature of minimum-norm controller synthesis and (ii) no upper bound on the order of stable stabilizing controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In other words, the robust instability analysis is similar to ⋆ This work was supported in part by the National Science and Tech- nology Council of Taiwan, under grant MOST 110-2221-E-110-047-MY3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This work has been submitted to IFAC for possible publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' the robust stability analysis in terms of the problem for- mulation, but it is quite different technically and much more challenging as optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Recently, the authors proposed a new optimization prob- lem, which we call the “Phase Change Rate Maximization Problem” in order to provide an almost complete solu- tion to the robust instability analysis for some classes of systems with one or two unstable poles (Hara, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The problem is to find a real-rational transfer function such that its peak gain occurs at a given frequency ωp with a prescribed phase value, and the phase change rate (PCR) at ωp is the maximum among those satisfying the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The essential idea behind is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' One of the key factors for the difficulty of robust in- stability analysis is that we cannot detect the transition from instability to stability by the presence of a pole on the imaginary axis (which successfully characterizes the transition in the opposite direction, making the robust stability analysis tractable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Hence we need an additional criterion for the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' It turned out, roughly speak- ing, that the positivity of the PCR of the loop transfer function at the peak gain frequency is an indication of the instability-to-stability transition for certain systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The aforementioned paper showed that the maximum PCR is attained by a first-order all-pass function and derived conditions under which the exact robust instability anal- ysis is possible in terms of the PCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The purpose of this paper is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The first purpose is to supplement the theoretical results in (Hara, 2022) by a more comprehensive example than those in the reference and illustrate how the PCR plays an important role for the exact robust instability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The class of systems is given as cyclic networks of homogeneous agents, where by changing the number of agents we can treat a variety of situations with respect to the location of stable and unstable complex poles with relatively small dampings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' We focus especially on the relationship between the sign of the PCR and the stable/unstable poles which are fairly close to the imaginary axis and represent under what situation we can get the exact result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The second purpose is to show that the PCR condition derived in (Hara, 2022) works well for two practical applications, namely (i) a minimum-norm strong stabilization for magnetic levita- tion systems and (ii) an exact robust instability analysis for the repressilator with time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The target systems of the former and the latter cases are in G0 1 (one unstable pole with the peak gain attained at zero frequency) and G# 2 (two unstable poles with the peak gain attained at non- zero frequency) , respectively, for which we can get the exact results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This means that the theoretical foundation in (Hara, 2022) can be practically useful although the class of applicable systems may appear restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' An applica- tion of strong stabilization in digital control setting is also presented to show the effectiveness in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Sec- tion 2 is devoted to a brief summary of the PCR maxi- mization problem presented in (Hara, 2022) and an illus- trative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Section 3 provides two practical applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' An extension to strong stabilization in the digital control setting with application to the magnetic levitation system is discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Section 5 summarizes the contributions of this paper and addresses some future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Notation and Terminology: The set of real numbers is denoted by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' ℜ(s) and ℑ(s) denote the real and imag- inary parts of a complex number s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The set of proper real rational functions of one complex variable s is denoted by Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Let L∞ denote the set of functions that are bounded on the imaginary axis jR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The subset of L∞ which consists of real rational functions bounded on jR is denoted by RL∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The stable subsets of L∞ and RL∞ are denoted by H∞ and RH∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The norms in L∞ and H∞ are denoted by ∥ · ∥L∞ and ∥ · ∥H∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The open (closed) left and right half complex planes are abbreviated as OLHP (CLHP) and ORHP (CRHP), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The following terminology will be used for a rational function h ∈ Rp throughout the paper: h is called “stable” (or “exponentially stable”) if all the poles of h are in the OLHP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' “marginally stable” if all the poles of h are in the CLHP and any pole of h on the imaginary axis is simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' “unstable” (or “exponentially unstable”) if at least one of the poles of h is in the ORHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' PHASE CHANGE RATE MAXIMIZATION In this section, we introduce the PCR maximization prob- lem, and motivate the problem by instability analysis and strong stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='1 Problem Formulation Given ωp > 0 and θp ∈ [0, 2π), we consider the following “phase change rate” maximization problem sup f∈RH∞ θ′ f(ωp) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' ∥f∥H∞ = |f(jωp)|, θf(ωp) = θp, (1) where θf(ω) denotes the phase angle of f(jω), and θ′ f(ω) is its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In other words, we seek a function f from RH∞, whose H∞-norm occurs at frequency ωp and phase at ωp is constrained to be θp, and has the maximal “phase change rate” among all functions which satisfy the same constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Such problem arises from robust instability analysis and minimum-norm strong stabilization as ex- plained below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Consider a class of unstable systems defined by G := {g ∈ RL∞ | g is strictly proper and unstable}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (2) The robust instability radius (RIR) for g ∈ G, denoted by ρ∗(g) ∈ R, with respect to real rational dynamic perturba- tion δ ∈ RH∞, is defined as the smallest magnitude of the perturbation that internally stabilizes the system: ρ∗(g) := inf δ∈S(g) ∥δ∥H∞, (3) where S(g) is the set of real-rational, proper, stable trans- fer functions internally stabilizing g, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', S(g) := {δ ∈ RH∞ : δ(s)g(s) = 1 ⇒ ℜ(s) < 0, δ(s) = 0, ℜ(s) > 0 ⇒ |g(s)| < ∞ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (4) The optimization problem stated in (3) is identical to the so-called “minimum-norm strong stabilization” problem for a given (unstable) plant g, where the minimum-norm controller sought is required to be stable itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' It is no- ticed from the well known result on strong stabilizability in (Youla, 1974) that ρ∗(g) is finite if and only if the Parity Interlacing Property (PIP) is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', the number of unstable real poles of g between any pair of real zeros in the closed right half complex plane (including zero at ∞) is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Consequently, the class of systems of our interest is defined as Gn := {g ∈ G | g has n unstable poles and satisfies the PIP condition}, (5) where n is a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Let g ∈ G be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' We have the following lower bound for ρ∗(g) (see (Hara, 2021)) ρ∗(g) ≥ 1/∥g∥L∞, ∥g∥L∞ := sup ω∈R |g(jω)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (6) When ρ∗(g) is exactly equal to its lower bound 1/∥g∥L∞, we say g has the exact RIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' It has been shown in (Hara, 2021) that, if f with ∥f∥H∞ = 1/∥g∥L∞ marginally stabi- lizes g with a single pair of poles on the imaginary axis, then g has the exact RIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Moreover, based on an extended version of the Nyquist criteria, necessary and sufficient conditions were derived in (Hara, 2022) for marginal sta- bilization of g, which in turn are sufficient conditions for obtaining the exact RIR of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' As a part of the necessary and sufficient condition for f with ∥f∥H∞ = 1/∥g∥L∞ to marginally stabilize g, the open-loop transfer function gf must satisfy the following loop-gain and PCR conditions: g(jωp)f(jωp) = 1, θ′ gf(ωp) = θ′ g(ωp) + θ′ f(ωp) > 0, where ωp is the frequency where the L∞-gain of g oc- curs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Searching for such an f boils down to solving a PCR optimization problem of the form described in (1), where the phase θf(ωp) is constrained to −θg(ωp) (and the magnitude of f at ωp is irrelevant to PCR optimiza- tion, as positive scaling of f will not change its phase or phase change rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The solution of the problem provides a tight condition for g to be marginally stabilizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In the next subsection, we summarize the theoretical foun- dation in (Hara, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 The Solution and its Application to Instability Analysis The PCR optimization in (1) can be solved by first nar- rowing down the feasible set using the following sets of functions: RFωp,θp := {f ∈ RH∞ : 1 = ∥f∥H∞ = |f(ωp)|, (7) θf(ωp) = θp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Oωp,θp := {f ∈ RH∞ : f is minimum phase, (8) |f(jωp)| = ∥f∥H∞, and θf(ωp) = θp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' AP ωp,θp := {f ∈ RH∞ : |f(jω)| = 1, ∀ω, (9) |f(jωp)| = ∥f∥H∞, and θf(ωp) = θp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Note that the constraint on the magnitude of the H∞- norm of functions in RF•,• and AP •,• bears no signifi- cance as explained previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The constraint is placed for convenience only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The first result gives an upper bound on the PCR for functions in Oωp,θp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Let θp ∈ (−π, π] and f ∈ Oωp,θp be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' If ωp ̸= 0, then θ′ f(ωp) ≤ − |θp/ωp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Moreover, if ωp = 0, then θ′ f(ωp) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 1 establishes that, for a stable minimum- phase system, its PCR at the peak-frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', where the H∞-norm occurs) is always non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Since any RH∞ function can be factorized as multiplication of an all-pass function and a minimum-phase function, Propo- sition 1 suggests that the PCR maximization problem over the set RF•,• boils down to the problem over the set AP •,•.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This is indeed the case, as the following propo- sition states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Given ωp ̸= 0 and θp ∈ (−π, π] (mod 2π), we have sup f∈RFωp,θp θ′ f(ωp) = sup f∈AP ωp,θp θ′ f(ωp) = − ���� sin(θp) ωp ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (10) Moreover, when θp ̸∈ {0, π}, the supremum is attained by the first-order all-pass function of the form f(s) = a−s a+s or f(s) = s−a a+s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' When θp ∈ {0, π}, the supremum is attained by a zeroth-order all-pass functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', f(s) = 1 or f(s) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For ωp = 0, the only feasible phase angles are θp ∈ {0, π} (mod 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In this case, sup f∈RF0,θp θ′ f(0) = sup f∈AP 0,θp θ′ f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The supremum is attained by f(s) = 1 or f(s) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Using the solutions stated in Proposition 2, the following results were derived for two subclasses of Gn defined by G0 n := {g ∈ Gn | ∥g∥L∞ = |g(0)| > |g(jω)| ∀ω ̸= 0}, (11) G# n := {g ∈ Gn | ∃ ωp > 0 such that ∥g∥L∞ = |g(jωp)| > |g(jω)| ∀ω ̸= ±ωp} (12) based on an extended Nyquist criterion (Hara, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (I) Given g ∈ G0 n, g can be marginally stabilized by a stable system f with ∥f∥H∞ = 1/∥g∥L∞ = 1/|g(0)| if and only if n = 1 and θ′ g(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (13) (II) Given g ∈ G# n for which the peak gain occurs at ωp, g can be marginally stabilized by a stable system f with ∥f∥H∞ = 1/∥g∥L∞ = 1/|g(jωp)| if and only if n = 2 and θ′ g(ωp) > ���� sin(θg(ωp)) ωp ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (14) Note that the marginally stabilizing controllers for cases (I) and (II) can be taken as the zeroth-order and the first- order all-pass functions, respectively, as suggested by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' As marginal stabilization of a system guarantees the exact RIR for the system, Theorem 1 immediately leads to sufficient conditions for attaining the exact RIR of systems in G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Furthermore, necessary conditions can also be derived based on the following result, which gives a PCR condition on the loop-transfer function at the peak frequency when the closed-loop system has all its pole in the closed left half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (Hara, 2022, Lemma 5) Given ωc ≥ 0, an integer n ≥ 1, and a transfer function L ∈ Gn, consider the positive feedback system with loop transfer function L satisfying the following condition 1 = |L(jωp)| = ∥L∥L∞, |L(jω)| < |L(jωp)|, ∀ω ̸= ±ωp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (15) If the feedback system has all its poles in the CLHP, then θ′ L(ωp) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Based on Theorem 1 and Lemma 1, we have necessary conditions and sufficient conditions for the exact RIR as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Let g ∈ G be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Suppose g(jω) takes the peak gain at ωp and consider the exact RIR condition ρ∗(g) = 1/∥g∥L∞ = 1/|g(jωp)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (16) (I) Suppose g ∈ G0 1 and ωp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Then θ′ g(ωp) > 0 ⇒ (16) ⇒ θ′ g(ωp) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (II) Suppose g ∈ G# 2 and ωp > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Then θ′ g(ωp) > ̺(ωp) ⇒ (16) ⇒ θ′ g(ωp) ≥ ̺(ωp), where ̺(ω) := ���� sin(θg(ω)) ω ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (III) For any g ∈ G# 1 , we have ρ∗(g) > 1/∥g∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For the proofs of these results, readers are referred to Section 4 of (Hara, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Also note that, the necessary conditions in statements (I) and (II) hold in fact for sys- tems in G0 n and G# n , respectively, for any n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='3 An Illustrative Example In this subsection we illustrate, by a numerical example, how the PCR condition effectively works for the robust instability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Consider a class of positive feedback systems of which the loop transfer functions are repre- sented by h(s) = −k (s + 1)2m+1 , m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', (17) where we assume that the loop-gain k > 0 is large enough so that the closed-loop system is exponentially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Our interest here is to assess robust instability against a ball type multiplicative stable perturbation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' in other words, the perturbed system ˜h has the form ˜h(s) = (1 + δ(s))h(s), δ(s) ∈ RH∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (18) Such a setting may arise when one considers a cyclic network with 2m + 1 identical agents with a multiplica- tive uncertainty present for the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The corresponding characteristic equation of the closed-loop system is given by 1 − gm(s)δ(s) = 0, where gm(s) := h(s) 1 − h(s) = −k (s + 1)2m+1 + k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (19) For k = 20, we observe that gm ∈ G# 2 for 1 ≤ m ≤ 7, and gm ∈ G# 4 when 8 ≤ m ≤ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The unstable poles of gm increases further when m becomes bigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Table 1 summarizes the findings for m = 1 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For 1 ≤ m ≤ 4, gm has one peak gain, while g5 has two peak gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In all these cases, the PCR condition stated in Theorem 1 holds at the global peak frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 1(a) and 1(b) for an illustration of the magni- tude profiles of g4 and g5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For g5, applying Proposition 2 we obtain the first-order all-pass function of the form δgl,5(s) = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='0896 � s−24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='426 s+24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='426 � , which marginally stabilizes g5 and the closed-loop system has a pair of poles at ±jωp = ±j(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='322).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In this case, we conclude that g5 has the exact RIR equal to 1/|g5(j(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='322))| = 1/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='0896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For m = 6, 7, the PCR condition fails at the global peak frequencies for gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' However for each case, there is a local peak frequency where the PCR holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 1(c) for an illustration of the magnitude profile of g6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Further examination reveals that the global peak-gain is due to a pair of dominating stable poles, while the local peak-gain is the result of a pair of unstable poles which is further away from the imaginary axis compared to the dominat- ing stable poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Take g6 for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Applying Proposi- tion 2 at the global and local peak frequencies, we obtain first-order all-pass functions δgl,6(s) = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='3976 � −s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2522 s+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2522 � and δlc,6(s) = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='0811 � s−18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='02 s+18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='02 � , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The closed- loop system with δgl,6 is exponentially unstable, which has two unstable poles and two imaginary-axis poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' It appears that δgl,6 pushes the dominating stable poles to the imaginary axis while leaving the unstable poles in the ORHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' On the other hand, the closed-loop system with δlc,6 is marginally stable with a pair of poles at ±jωp = ±j(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='276).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In this case, g6 does not have exact RIR, and ρ∗(g6) ∈ (1/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='3976, 1/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='0811].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Note that ρ∗(g6) is strictly larger than 1/∥g6∥L∞ = 1/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='3976, as the necessary condition stated in statement (II) of Theorem 2 is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For 8 ≤ m ≤ 13, gm has two peak-gains and both are caused by unstable poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The PCR condition holds at both peak frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For 14 ≤ m ≤ 16, a third peak is formed, which is caused by a pair of stable poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The PCR of gm is negative at this peak (let’s call it a “stable peak”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For 17 ≤ m ≤ 20, the stable peak over- takes the other two peaks and becomes the global peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 1(d) to 1(f) for an illustration of the magnitude profiles of g8, g16 and g17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Now consider g8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The first- order all-pass functions obtained by the global and lo- cal peak frequencies are δgl,8(s) = 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4116 � s−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='749 s+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='749 � and δlc,8(s) = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='073 � s−29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='498 s+29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='498 � , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The closed-loop system with δgl,8 is exponentially unstable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' apparently δgl,8 pushes a pair of unstable poles to the imaginary axis while leaving the other pair in the ORHP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Similar to g6, δlc,8 is able to marginally stabilize g8, and therefore we have ρ∗(g8) ∈ [1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4116, 1/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='073].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Note that we cannot yet exclude the possibility that ρ∗(g8) = 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4116 since no necessary condition is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For g9 to g16, we have similar results, where the inverse of the L∞-gain of gm gives a lower bound and the second peak-gain of gm gives an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For g17 to g20, the situation is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For those systems, their PCRs at the global peak frequencies violate the necessary condition for having exact RIR’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Therefore, we know that ρ∗(gm) is strictly larger than 1/∥gm∥L∞, for m = 17, · · · , 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For each of these system, an upper bound for ρ∗ is obtained using their respective third peak-gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' PRACTICAL APPLICATIONS In this section, we apply our main results to analyze (in)stability properties of system models that are derived from real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='1 we consider linearized models for magnetic levitation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' These models belong to the class G0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 we consider linearized models for a certain gene regulatory network called “repressilator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' These models belong to the class G# 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The goal is to illustrate that our results are applicable to real applications to provide useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='1 Strong Stabilization for Magnetic Levitation Systems A typical linearized model for the magnetic levitation system (Namerikawa, 2001) at an equilibrium is a third- order system of the following form g(s) = k (−s2 + p2)(τs + 1), (20) where the pair of poles at ±p is due to the mechanical aspect of the system while the stable pole at −τ −1 comes from the electrical part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Typically, we have τ−1 ≫ p and therefore it is generally reasonable to neglect the factor (τs + 1) from the dynamical model for control design purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Here we will show that, for the purpose of minimum-norm strong stabilization, ignoring the (τs+1) factor will lead to a wrong conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For the reduced second-order model: Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Summary of the numbers of peak-gains, satisfaction of the PCR conditions, whether exact RIR occurs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' among different cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' m 1 − 4 5 6 − 7 8 − 13 14 − 16 17 − 20 # of unstable poles 2 2 2 4 4 4 # of peak-gains 1 2 2 2 3 3 # of unstable peak-gains 1 1 1 2 2 2 # of stable peak-gains 0 1 1 0 1 1 global peak-gain is (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='/us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' )?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' us us s us us s PCR holds at global peak?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' y y n y y n PCR holds at a local peak?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' n/a n y y y y RIR = 1/∥gm∥L∞ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' y y n inc inc n RIR > 1/∥gm∥L∞ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' n n y inc inc y Abbreviation: ’s.’ – stable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' ’us.’ – unstable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' ’y’ – yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' ’n’ – no;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' ’n/a’ – not applicable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' ’inc’ – inconclusive Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Magnitude profile of gm for m = 4, 5, 6, 8, 16, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For m = 4 to 6, gm has one pair of unstable poles, while it has two pairs for the other three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The red color indicates the frequency ranges where the PCR condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' A gain-peak where the PCR condition does not hold appears to be caused by a pair of stable poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' gr(s) = k (−s2 + p2), (21) we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Consider gr(s) in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Then inf c∈S(gr) ∥c∥H∞ = 1 |gr(0)| = 1 ∥gr∥L∞ = p2 k , (22) where the infimum is obtained by the sequence of stabi- lizing controllers cǫ, where cǫ(s) = p2 k + ǫs + z s + d, 0 < z < d with ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Here z and d can be any positive real numbers, as long as z < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' It can be readily verified that the poles of the closed- loop system [gr, cǫ] is governed by the characteristic equa- tion s3 + ds2 + kǫs + kǫz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Applying the Routh- Hurwitz stability criterion, we conclude that the closed- loop system is stable for any ǫ > 0, d > z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Clearly, ∥cǫ∥H∞ = p2/k + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Since 1/∥gr∥L∞ is a lower bound for the infimum, we have p2 k = 1 ∥gr∥L∞ ≤ inf c∈S(gr) ∥c∥H∞ ≤ ∥cǫ∥H∞ = p2 k + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Since ∥cǫ∥H∞ → p2/k as ǫ → 0, we conclude that the infimum is equal to p2/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Relating to Theorem 2, note that gr ∈ G1 0 with θ′ gr(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This is a critical case, as gr does not violate the necessary condition for acquiring the exact RIR, but does not satisfy the sufficient PCR condition, either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Nonetheless, here we are able to show that, the exact RIR of gr is achievable, and the infimum of the strong stabilization problem is obtained by a constant feedback equal to 1/∥gr∥L∞ = 1/|gr(0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For the third-order model g, the outcome of minimum- norm strong stabilization is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Based on The- orem 2, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Consider g(s) in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Then inf c∈S(g) ∥c∥H∞ > p2 k = 1 |g(0)| = 1 ∥g∥L∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (23) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' We note that the third-order model g also belongs to G0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Furthermore, it can be readily verified that the PCR of g at the zero frequency is −τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Since having a non- Figure (a) Figure (b) Figure (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='5 m=4 m=5 9=w Magitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 Frequency (rad/sec) Frequency (rad/sec) Frequency (rad/sec) Figure (d) Figure (e) Figure (f) 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='5 m=8 m=16 m=17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='5 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 1 Frequency (rad/sec) Frequency (rad/sec) Frequency (rad/sec)negative PCR is a necessary condition for systems in G0 1 to have the exact RIR by Theorem 2, the infimum of the minimum-norm strong stabilization for g is strictly larger than 1/∥g∥L∞ = p2/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For obtaining an upper bound of the infimum, let us introduce a phase-lead compensator to raise the PCR of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Consider f(s) = (τc+τ)s+1 τcs+1 and gc(s) = g(s)f(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The compensated plant gc has 0 phase change rate at the zero frequency for any τc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This can be readily verified by checking the imaginary part of d dω log(gc(jω)) at the zero frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Furthermore, one can also verify that gc ∈ G0 1 if and only if τc ≤ 1/(p2τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This can be shown by computing the real part of d dω log(gc(jω)), which reveals that Real � d dω log(gc(jω))|ω=0 � = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' when τc ≤ 1/(p2τ), d dω log |gc(jω)| < 0 for any ω > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' when τc > 1/(p2τ), d dω log |gc(jω)| > 0 for ω → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' and hence the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Setting τc = 1/(p2τ), we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The compensated plant gc satisfies inf c∈S(gc) ∥c∥H∞ = 1 |gc(0)| = 1 ∥gc∥L∞ = p2 k , (24) where the infimum is obtained by the sequence of stabi- lizing controllers cǫ, cǫ(s) = p2 k + ǫ s + ǫ2 s + q/(τc + τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Here ǫ is a sufficiently (in fact, arbitrarily) small positive number and q > 0 is chosen sufficiently large for given ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The infimum in (24) in turn implies inf c∈S(g) ∥c∥H∞ ≤ p2(1 + p2τ 2)/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' One can verify that the characteristic equation of the closed-loop system [gc, cǫ] has the form s5 + [(q + 1)d]s4 + [qd2]s3 + [kǫd]s2 + [kǫ( ˆd + ǫ2d)]s + [kǫ3 ˆd], where d := (τ+τc)/(ττc) = τ −1+p2τ, and ˆd := 1/(ττc) = p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The goal here is to select parameters ǫ > 0 and q > 0 such that the roots of the polynomial are all in the open left-half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Applying the Routh-Hurwitz stability criterion, one concludes that it is so when ǫ is sufficiently small and, corresponding to an ǫ, q is chosen sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The infimum in (24) is obtained by taking ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Furthermore, the analysis implies that fcǫ is a stabilizing controller for g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Since ∥fcǫ∥H∞ → p2(1 + p2τ 2)/k as ǫ → 0, it implies p2(1 + p2τ 2)/k is an upper bound for infc∈S(g) ∥c∥H∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Since τ −1 ≫ p, we have 1 + p2τ 2 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' That is, the upper bound on the norm of the minimum-norm strong stabilizing controller is very close to the lower bound p2/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 Robust Instability Analysis for Repressilator Consider a biological network oscillator called the repres- silator with three dynamical units in a cyclic loop (Elowitz, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Its linearized model is given by ξ = he(s)ξ, he(s) = −ke (s + α1)(s + α2)(s + α3), where ξ is a variable, and ke > 0 depends on the equilib- rium state of the original nonlinear system (Hara, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The nominal system with the characteristic equation 1 = he(s) is exponentially unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The subscript •e is used to indicate that the quantity • depends on the equilibrium state, which in turn depends on the perturbation of the DC-gain of the system, denoted by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For more details about the repressilator model, see (Hara, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Here we are interested in assessing robust instability against a ball type multiplicative stable perturbation when the nominal dynamics are further complicated by time-delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' We use the fifth-order Pad´e approximation for the time-delay in order to keep the model rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Let Dτ(s) denote the Pad´e approximation of the time-delay transfer function e−τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The corresponding characteristic equation is 1 − δ(s)ge(s) = 0, where ge(s) = he(s)Dτ(s) 1 − he(s)Dτ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Using the parameters α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4621, α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='5545, α3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='3697, we investigate the case where e = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', no perturbation on the DC-gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In this case, we have k0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='216 and the exact RIR of g0 are confirmed when τ = 0, see Hara (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In what follows, we examine the effect of the time-delay on the exact RIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Numerical computations show that g0 ∈ G# 2 for τ ∈ [0, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='771].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The PCR condition holds at the peak-gain fre- quency of g0 up to τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='481, and ceases to hold when τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Thus, g0 has exact RIR for τ ∈ [0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='481].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Furthermore, one can verify that when τ is large enough, a pair of stable poles of g0 creates a gain-peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' When τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='482, this “stable peak” becomes dominant and the PCR condition ceases to hold at the global peak frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' However, the condition holds at the local (second) peak frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 2 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' More specifically, when τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='482, ∥g0∥L∞ = |g(j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='5009)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='10273, while a local peak occurs at ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='396 rad/sec with |g(j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='396)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='10268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The first-order all-pass function 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='10268 � s−18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8246 s+18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8246 � , obtained by applying Proposition 2 to the local peak frequency, marginally stabilizes g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Thus, we conclude that 1/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='10273 < ρ∗(g0) ≤ 1/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='10268 when τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For τ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4, a marginally stabilizing perturbation with norm equal to 1/∥g0∥L∞ is 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='1044 � s−18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4747 s+18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4747 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This per- turbation is further multiplied by a high-pass filter to make the DC-gain of δ(s) equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Specifically, δ(s) is defined by δ(s) = s + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='01γ s + ξ (1 + ǫ) 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='1044 �s − 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4747 s + 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4747 � , (25) where γ = −1/(1 + ǫ) with a small non-negative number ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The closed-loop systems of g0 is marginally stabilized with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The nonlinear repressilator models with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='95 and ǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='05 were simulated, and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 3 (left and right figures, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Clearly, δ(s) with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='95 is not able to stabilize g0 and the closed-loop system exhibits oscillatory behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' On the other hand, δ(s) with ǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='05 stabilizes g0 and the oscillatory behavior ceases to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='coli cells, the delay factor mainly repre- sents the protein maturation time, which is usually 6 to 60 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For the repressilator model presented in this sec- tion, the unit of time is “hour”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' therefore, the delay time τ of the range [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='1, 1] corresponds to realistic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Our analysis shows that the L∞-norm of g0 gives the exact RIR for τ ∈ [0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='481] , which indicates that it is a useful metric for determining the instability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', oscillation) of practical repressilators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Magnitude profile of g0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The red color indicates the frequency range where the PCR condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Time-course simulations of the closed-loop sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Left: g0 and δ(s) with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Right: g0 and δ(s) with ǫ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' EXTENSIONS TO DIGITAL CONTROL SYSTEMS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='1 Robust Instability Analysis of Discrete-Time Systems via Bilinear Transformation to Continuous-Time Systems In this section, we propose a procedure for robust insta- bility analysis and finding minimum-norm strongly stabi- lizing controller for discrete-time LTI (unstable) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For a discrete-time LTI system with transfer function g(z), its stability can be assessed by the location of its poles inside, on, and/or outside the unit circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Let the open unit disk be denoted by Ω, the unit circle by ∂Ω, and the outside of the closed unit disk by Ωc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For a discrete-time transfer function, the “stable” region is Ω, “unstable region” is Ωc, and “stability boundary” is ∂Ω, which corresponds to the OLHP, the ORHP, and jR, respectively, for its continuous-time counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' It is well-known that the bilinear transformation z = 1 + s 1 − s ⇔ s = z − 1 z + 1 is a one-to-one mapping between the regions in each aforementioned pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' that is, s ∈ OLHP ⇔ z ∈ Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' s ∈ ORHP ⇔ z ∈ Ωc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' s ∈ jR ⇔ z ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Therefore, when we apply the bilinear transformation to a discrete-time transfer function, its stability property is preserved by the resulting continuous-time representative, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Moreover, it is straightforward to see that the norm of the transfer function is also preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Observing these, we propose the following procedure for applying our results in Theorems 1 and 2 to discrete-time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Given a discrete-time transfer function gd(z), we verify whether infc∈S(gz) ∥c∥H∞ = 1/∥gz∥L∞ (exact RIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Step 1: Apply bilinear transformation z = 1+s 1−s to find the continuous-time representative gd,c of gd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=', gd,c(s) := gd � 1+s 1−s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Step 2(a): If gd,c ∈ G0 1 and θ′ gd,c(0) > 0, or gd,c ∈ G# n and θ′ gd,c(ωp) > | sin(θgd,c(ωp))|/|ωp|, then gd has exact RIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proceed to Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Step 2(b): If gd,c ∈ G0 n and θ′ gd,c(0) < 0, or gd,c ∈ G# n and θ′ gd,c(ωp) < | sin(θgd,c(ωp))|/|ωp|, then gd does not have exact RIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' infc∈S(gz) ∥c∥H∞ is strictly larger than 1/∥gz∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The analysis is completed and stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Step 2(c): If gd,c ∈ G# 1 , then gd does not have exact RIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' infc∈S(gz) ∥c∥H∞ is strictly larger than 1/∥gz∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The analysis is completed and stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Step 2(d): If gd,c does not belong to one of the sce- narios described in 2(a) to 2(c), then the analysis is inconclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Step 3: Obtain the all-pass function c∗(s) which marginally stabilizes gd,c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Then c∗,d(z) := c∗ � z−1 z+1 � is the all-pass function that marginally stabilizes gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The analysis is completed and stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 Application to the Magnetic Levitation System In this section, we apply the procedure outlined in the previous section to a strong stabilization problem of the magnetic levitation system presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='1 in the digital control setting, and discuss the sampling affect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For simplicity, we consider the second-order reduced model gr(s) in (21) of the magnetic levitation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' For the digital control setting, assume an ideal sampler and a synchronized zeroth-order holder are placed around the continuous-time plant, which leads to the following time- discretized model gd(z) = κ z + 1 (z − e−pT )(z − epT ), (26) where κ := k(1 − epT )(1 − e−pT )/(2p2) and T is the sampling period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Applying the bilinear transformation z ← (1 + s)/(1 − s), the continuous-time representative of gd(z) has the following form gd,c(s) = kc (1 − s) (s − q)(s + q), where kc = 2κ/((1 + epT )(1 + e−pT )) and q = (1 − e−pT )/(1 + e−pT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Note that kc can also be expressed as kc = −kq2/p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Therefore, we have qd,c(0) = k/p2 = gr(0), which is also equal to |gd(0)| = ∥gd∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Consider the continuous-time representa- tion gd,c of the discrete-time plant gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' We have inf c∈S(gd,c) ∥c∥H∞ > p2 k = 1 |gd,c(0)| = 1 ∥gd,c∥L∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' (27) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 T=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='482 Megnitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content='8 2 Freguency(rad/sec)This in turn implies the discrete-time plant gd can not be stabilized by a stable controller with norm arbitrarily close to 1/|gd(1)| = 1/∥gd∥L∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 6 follows from the facts that gd,c ∈ G0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This is verified by d dω log |gd,c(jω)| = ω(−ω2 − 2ω2 + q2(q2 − 2)) (ω2 + 1)(ω2 + q2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Since q ∈ (0, 1), the derivative is 0 when ω = 0, and strictly negative when ω ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The PCR of gd,c at zero frequency is −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This is verified by the imaginary part of d dω log(gd,c(jω)), which is equal to −j 1 − jω − j jω − q − j jω + q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' When ω = 0, the imaginary part is equal to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Thus, gd,c violates the necessary condition for having exact RIR stated in statement (I) of Theorem 2, and hence comes inequality (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This in turn implies that the discrete-time model gd can not be stabilized by a stable controller with norm arbitrarily close to the inverse of the L∞-norm of gd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 6 is in stark contrast to Proposition 3, where the continuous-time plant gr is shown to have strongly stabilizing controller with its norm arbitrarily close to 1/|gr(0)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' It appears that the very act of sampling and the introduction of the zeroth-order-hold discretization decimate this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' To obtain an upper bound of the minimum-norm strongly stabilizing controller of gd, we apply a lead compensator to gd,c, similar to that intro- duced for the third-order continuous-time plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' One can readily verify that, with a lead compensator, ˜gd,c(s) = gd,c(s) (τ+1)s+1 τs+1 belongs to G0 1 with zero PCR at the zero frequency for any 0 < τ ≤ 1/q2 − 1 = 4 e−pT /(1 − e−pT )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Setting τ to be 1/q2 − 1, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Let τ = 1/q2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The plant ˜gd,c satisfies inf c∈S(˜gd,c) ∥c∥H∞ = 1 |˜gd,c(0)| = 1 ∥˜gd,c∥L∞ = p2 k , (28) where the infimum is obtained by the sequence of stabi- lizing controllers cǫ, cǫ(s) = p2 k + ǫp2τ kq2 � s + ǫ2 τ 1+τ s + (α + ǫ)(1 + τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Here ǫ is a sufficiently (in fact, arbitrarily) small positive number, and α > 0 is chosen sufficiently large for given ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' This in turn implies infc∈S(gd,c) ∥c∥H∞ ≤ p2/(k(1 − q2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' It can be verified that the characteristic equation of the closed-loop system with the controller cǫ has the form s4 + α(1 + τ)s3 + τǫ(1 − ǫ2)s2 + ǫ � 1 + ǫ2 τ 2 1 + τ � s + ǫ3 τ 1 + τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Let ai, i = 0, · · · , 3, denote the coefficients of si, i = 0, · · · , 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Applying the Routh-Hurwitz sta- bility criterion, the closed-loop system is stable if and only if a1a2a3 −a2 1 −a0a2 3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Now one can readily verify that, for every small enough ǫ, there is a corresponding large enough α such that the inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The infi- mum in (28) is obtained by taking ǫ to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' By Propositions 6 and 7, we have p2/k < infc∈S(gd,c) ∥c∥H∞ ≤ p2/(k(1 − q2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The upper bound has a factor 1/(1 − q2) = (1 + e−pT )2/(4 e−pT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Note that this factor approaches 1 as e−pT approaches 1, or as pT approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Thus, when T approaches 0, the result matches that of the continuous-time 2nd-order reduced model gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' The closed-loop system of gd,c(s) with c∗(s) = p2 k �(1 + τ)s + 1 τs + 1 � has three poles at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' Applying the bilinear trans- formation s ← z−1 z+1, we obtain the discrete-time controller cd,∗(z) := c∗( z−1 z+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' One can verify that the closed-loop system of gd(z) with cd,∗(z) has three poles at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' CONCLUDING REMARKS We recalled the phase change rate maximization problem and solution from Hara (2022) and illustrated the latter’s utility in the robust instability analysis of a cyclic net- work of homogenous multi-agent systems subject to an identical multiplicative stable perturbation on each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' We also applied the result to two practical applications — magnetic levitation systems and repressilators with time-delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' In addition, robust instability of digital con- trol 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} +page_content=' 1080–1087, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQf4wuZ/content/2301.04775v1.pdf'} diff --git a/itE3T4oBgHgl3EQfIwnu/content/tmp_files/2301.04338v1.pdf.txt b/itE3T4oBgHgl3EQfIwnu/content/tmp_files/2301.04338v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6007c3d28356cc1db34d9d1a6b2a212bfb1902b6 --- /dev/null +++ b/itE3T4oBgHgl3EQfIwnu/content/tmp_files/2301.04338v1.pdf.txt @@ -0,0 +1,1281 @@ +Synthetic data generation method for data-free knowledge +distillation in regression neural networks +Tianxun Zhou1 and Keng-Hwee Chiam1 +1Bioinformatics Institute, Singapore +Abstract +Knowledge distillation is the technique of compressing a larger neural network, known as the +teacher, into a smaller neural network, known as the student, while still trying to maintain the +performance of the larger neural network as much as possible. Existing methods of knowledge +distillation are mostly applicable for classification tasks. +Many of them also require access to +the data used to train the teacher model. To address the problem of knowledge distillation for +regression tasks under the absence of original training data, previous work has proposed a data-free +knowledge distillation method where synthetic data are generated using a generator model trained +adversarially against the student model. These synthetic data and their labels predicted by the +teacher model are then used to train the student model. In this study, we investigate the behavior +of various synthetic data generation methods and propose a new synthetic data generation strategy +that directly optimizes for a large but bounded difference between the student and teacher model. +Our results on benchmark and case study experiments demonstrate that the proposed strategy +allows the student model to learn better and emulate the performance of the teacher model more +closely. +1 +Introduction +In the recent decade, advances in algorithms, computational hardware and data availability have +enabled significant developments in artificial neural networks and deep learning [Lecun et al., 2015]. +Neural networks models are now state-of-the-art in many fields of application including computer +vision ([O’Mahony et al., 2020], natural language processing [Otter et al., 2021], and signal processing +[Purwins et al., 2019]. However, as models become increasingly larger in size measured by number +of parameters, they too become computationally expensive to store and perform inference on. Large +neural networks can be unusable for real world deployment scenarios where hardware may be limited, +such as on mobile devices or microcontrollers, or when deployed for as a service to a large number of +users such as web applications [Cheng et al., 2018, Deng et al., 2020]. +Knowledge distillation is a class of method to address this problem by distilling the predictive ca- +pabilities of a larger neural network into a smaller neural network, allowing for faster inference and +lower memory requirements [Gou et al., 2021]. There have been several knowledge distillation meth- +ods proposed in the past, typically requiring the original data that was used to train the teacher +model. However, in many real-world applications, the original data may not be available for perform- +ing knowledge distillation to student models due to reasons such as data size and data privacy [Chen +et al., 2019, Gou et al., 2021]. +To deal with such situations, data-free knowledge distillation methods have been proposed to allow +distillation of knowledge without the original training data [Hu et al., 2020, Lopes et al., 2017, Micaelli +and Storkey, 2019, Ye et al., 2020, Yoo et al., 2019]. Data-free knowledge distillation works by gener- +ating synthetic data and training the student model with these data and their teacher model predicted +labels. +Much of the existing research for knowledge distillation has been focused on classification tasks. How- +ever, regression tasks are common in many engineering applications [Guo et al., 2021, Schweidtmann +1 +arXiv:2301.04338v1 [cs.LG] 11 Jan 2023 + +et al., 2021] and there are limited methods available on knowledge distillation for regression neural +networks. Recently, [Kang and Kang, 2021] proposed the first data-free knowledge distillation method +for regression where a generator model was trained in an adversarial manner to generate synthetic +data. Motivated by the need for data-free model distillation on regression models in real world ap- +plications, in this work we investigate the behaviors of several synthetic data generation methods +including random sampling and adversarial generator. +Based on the insights gained from this investigation, we propose an improved method to generate +synthetic data for data-free knowledge distillation of regression neural networks by optimizing for a +loss function defined using the student and teacher model predictions directly rather than implicitly +through an additional generator model. +Compared to existing methods, synthetic data generated +through this process can provide large difference in prediction between the student and teacher model +while mimicking real data better. We demonstrate that this method for synthetic data generation +can provide better performance than existing methods through experiments in 7 standard regression +datasets, as well as on the MNIST handwritten digit dataset adapted for regression, and a real-world +bioinformatics case study of protein solubility prediction. +2 +Related work +2.1 +Knowledge distillation +As neural networks become increasingly large in number of parameters, the deployment of such models +faces a difficult challenge for applications such as mobile devices and embedded systems due to limita- +tions in computational resources and memory [Cheng et al., 2018, Deng et al., 2020]. To address such +problems, model compression through knowledge distillation has become an active area of research +in recent years. Knowledge distillation is the technique where knowledge learned by a larger teacher +model is transferred to a smaller student model [Gou et al., 2021, Hinton et al., 2015]. The main idea is +that the student model mimics the teacher model to achieve a similar or even a superior performance. +Various methods of knowledge distillation define and focus on different forms of knowledge. Following +the nomenclature in [Gou et al., 2021], these can be largely grouped as response-based knowledge, +feature-based knowledge, and relation-based knowledge. For response-based knowledge, outputs of the +teacher model are used to supervise the training of the student model. For example, [Hinton et al., +2015] uses soft targets from the logits output of the teacher model to train the student. For feature- +based knowledge, outputs of intermediate layers, or feature maps learned by the teacher model can be +to supervise the training of the student model. For example, [Romero et al., 2014] trains the student +model to match the feature activations of the teacher model. For relationship-based knowledge, the +relationships between different layers or data samples are used. For example, [Yim et al., 2017] uses +the inner products between features from two layers to represent the relationship between different +layers, while [Chen et al., 2021] trains the student model to learn to preserve the similarity of samples’ +feature embeddings in the intermediate layers of the teacher models. +2.2 +Data-free knowledge distillation +In some situations, access to the original data used to train the teacher model is not available due to +issues such as privacy and legal reasons. Data-free knowledge distillation methods have been proposed +to allow model distillation in the absence of original training data. This is achieved by generating +synthetic data for training. +Many methods achieve this by using generative adversarial networks +(GAN) [Chen et al., 2019, Hu et al., 2020, Micaelli and Storkey, 2019, Ye et al., 2020, Yoo et al., 2019]. +For example, [Micaelli and Storkey, 2019] train a generator model to generate synthetic images that +maximizes the difference in prediction (measured by KL divergence) between the teacher and student +models. The student model is then trained to minimize the difference on these synthetic images. +Other methods such as [Lopes et al., 2017] make use of metadata collected during training of the +teacher model, in the form of the layer activation records of the teacher model to reconstruct dataset +for training the student model. +2 + +Figure 1: Generic data-free knowledge distillation method +2.3 +Knowledge distillation for regression +Most of the methods currently existing in knowledge distillation literature deal with classification +problems. These methods generally are not immediately applicable to regression problems where the +predictions are unbounded real values. For regression problems, [Chen et al., 2017] uses a teacher +bounded regression loss where the teacher’s predictions serve as an upper bound for the student model +instead of using it directly as a target. [Takamoto et al., 2020] uses a teacher outlier rejection loss, +that rejects outliers in training samples based on the teacher model predictions. +[Kang and Kang, 2021] introduced the first work that addresses data-free knowledge distillation for +regression, by using a generator model that generates synthetic datapoints that is trained adversarially +together with the student model. +3 +Material and methods +3.1 +Overview of methods +Given a trained teacher model T, and a student model Sθ parameterized by θ, we generate synthetic +data x via some data generation method. The student model is trained by minimizing the student loss +LS(x) defined in equation 1 using gradient descent. This generic method is illustrated in Figure 1. +LS(x) = (T(x) − Sθ(x))2 +(1) +The performance of the student model in mimicking the performance of the teacher is dependent on the +representation strength θ of the student model, and the data x used to train it, and the optimization +process of minimizing student loss. Hence for a fixed student model architecture and training process, +the synthetic data generation process plays the key role in determining the performance of the student +model. +3.2 +Synthetic data generation methods +Three types of synthetic data generation methods are investigated in this study: random sampling, +generative model, and direct optimization. +3.2.1 +Random sampling +Synthetic data are generated by sampling randomly from an input distribution. Assuming the input +has been standardized, random samples can be drawn from a Gaussian distribution ∼ N(0, I). Random +sampling can also be drawn through quasi-Monte Carlo method such as Latin Hypercube sampling +and Halton sequences which are designed to evenly cover the input space. Input space bounds may be +defined using the maximum and minimum values of an available validation or test set, or based upon +some prior knowledge. +3 + +Teacher Model, T +T(Xg) +Synthetic Data +Generation +Student Model, S +→ S(Xg)Figure 2: Data-free distillation with generator method +3.2.2 +Generator model +Generator model for generating synthetic data was proposed for data-free knowledge distillation for +regression tasks by [Kang and Kang, 2021], follows similar methods in classification tasks [Micaelli and +Storkey, 2019]. In this method, a generator model Gφ parameterized by φ is trained to output samples +that would result in a large difference between the student and teacher model’s predictions. This +generator model is trained in an adversarial manner against the student model during the distillation +process by optimizing the generator loss function in equation 2. +LG(z) = Exg Gφ(z)[−(T(xg) − Sθ(xg))2] +(2) +The student is trained using the student loss to minimize the difference between teacher and its own +predictions, the two opposing learning objectives are trained in a sequential adversarial manner, and +the student model is able to learn to match the predictions of the teacher model as training continues. +This process is illustrated in Figure 2. +In practice, regularization terms may be added to the generator loss to prevent complete deviation +from underlying data distribution, for e.g. by adding the square of L2-norm of xg and Sθ(xg), yielding: +LG(z) = Exg Gφ(z)[−(T(xg) − Sθ(xg))2 + β∥xg∥2 + γSθ(xg)2] +(3) +3.2.3 +Direct optimization from random samples +The generator model approach attempts to train the generative model Gθ to approximate the inverse +function of the student loss implicitly, where the generative model predicts x given the objective of +high student loss. It is not immediately clear whether the generative model is able to learn this inverse +function easily. +Since the goal of the generative model approach is to generate samples that maximize the student loss, +it is more straightforward to maximize the student loss directly, as formulated below. +max +xg +(T(xg) − Sθ(xg))2 +Or following conventions: +min +xg +−(T(xg) − Sθ(xg))2 +(4) +In practice, following the generator method, we may add regularization terms as well, such as in +equation 5. +4 + +Teacher Model, T +T(Xa +Generator Network, +Z ~N(0, D - +LG +G +Student Model, S +S(Xg) +Updateparameters ofGFigure 3: Data-free model distillation with direct optimization method +min +xg −(T(xg) − Sθ(xg))2 + β∥xg∥2 + γ(xg)2 +(5) +It is later shown in 3.5 and 3.6 that the methodology is very flexible, and any arbitrary loss function +may be used to incorporate loss terms designed to capture important properties of the data. +This minimization can be done through various optimization algorithms. If both the student and +teacher models are differentiable, gradient descent can be used. Black box metaheuristic optimization +methods such as genetic algorithms and simulated annealing may also be used, especially if the teacher +model gradients are unavailable. The method is illustrated in Figure 3. +When using direct optimization of the student loss with gradient descent, it is possible to derive theo- +retical guarantees for (a) generating samples that are better than random sample and (b) generating +samples that are bounded in their deviation away from underlying distribution. +The gradient descent updates as such: +xg,t+1 = xg,t + η +∂ +∂xg,t +[T(xg) − Sθ(xg)]2 +(6) +Assuming the neural networks are locally smooth (Lipschitz continuous), given some sufficiently small +learning rate η, xg,t+1 always improves upon xg,t fulfilling guarantee (a). Given some learning rate η +and number of gradient descent steps tmax, xg,t+1 deviates from xg,0 randomly sampled from underlying +distribution by an arbitrary bound, fulfilling guarantee (b). Proof for guarantee (a) is provided in [Boyd +and Vandenberghe, 2009] p.466 and proof for guarantee (b) is provided in the supplementary materials. +It is not obvious to us that the generator model method can fulfil guarantee (a) because xg is generated +from Gaussian noise z of an arbitrary dimension and is not related to random samples in input space; +and to fulfil guarantee (b), a bound on the deviation of xg from 0 exist only if a regularization term is +applied to xg. The proof for bound on magnitude of xg for generator method with L2 regularization +is provided in the supplementary materials. +3.2.4 +Proposed method for knowledge distillation +The proposed data-free knowledge distillation method generates training data xg through direct opti- +mization of student loss with gradient descent. In the synthetic data generation step, assuming inputs +are standardized, a batch of random samples are drawn from a Gaussian distribution ∼ N(0, 1). Gra- +dient descent is used to perturb these random samples to the direction of maximizing their student +5 + +Teacher Model, T +T(Xg) +Xg,0 ~ N(0, I) +Xg +Student Model, S +→ S(Xg) +Update Xg directlyloss values, obtaining xg. In the student training step, the student weights are updated to minimize +the student loss with respect to the synthetic data xg. +Following the methods proposed in [Kang and Kang, 2021], generated data is also supplemented with +random samples xp drawn from Gaussian distribution ∼ N(0, 1). The sample weights for the generated +samples xg and random samples xp are controlled by a factor α, which can be a fixed value or follow +a schedule based on the training epoch. +LS = αLS(xg) + (1 − α)LS(xp) +(7) +Setting α to 0 is equivalent to the random sampling strategy. Setting α to 1 is a pure generative +sampling strategy. Note that for both edge cases, since the loss of only 1 set of samples contributes to +the training, the number of training samples in each epoch needs to be doubled for a fair comparison +with cases where α is between 0 and 1. We investigate a decreasing alpha schedule as well as a pure +xg training strategy in the experiments. +The training process is provided in algorithm 1 & 2. +In the main procedure Data-free model +distillation where the data distillation training happens, the number of training epochs for the +student model is defined as tmax, and the number of batches per epoch is defined as ns. In the sub- +procedure Optimize, where direct optimization to generate synthetic data is done via gradient descent, +the number of gradient descent steps is defined as τmax. +Algorithm 1: Main procedure: Data-free model distillation +Input: teacher model, T +Output: student model, Sθ +1 for t=1 to tmax do +2 +for 1 to ns do +3 +z ∼ N(0, I) +4 +xg ← Optimize(z) +5 +xp ∼ N(0, I) +6 +L ← αLS(xg) + (1 − α)LS(xp) +7 +Update Sθ with gradient descent w.r.t. L +Algorithm 2: Sub-procedure: Optimize +Input: z +Output: xg +1 xg ← z +2 for τ=1 to τmax do +3 +LS ← −(T(xg) − Sθ(xg))2 + β∥xg∥2 + γ(xg)2 +4 +xg ← xg − η +∂ +∂xg LS +3.3 +Regression datasets for experiments +To facilitate comparison with the previous work by [Kang and Kang, 2021], the experiments were +conducted on the same datasets. These 7 datasets are regression problem sets available from UCI +machine learning repository [Dheeru and Casey, 2019] and KEEL dataset repository [Alcala-Fdez +et al., 2010]. ‘longitude’ was selected as the output variable for Indoorloc. Details of the datasets are +provided in the Table 1. +The data are split into training and test set. The training set consists of 5000 samples for each dataset. +10% of the remainder samples are placed into the validation set, and the remaining 90% is the test +set. The validation set is used to periodically evaluate the training of the student model. +6 + +Table 1: List of regression datasets +Dataset +Number of features +Number of samples +Compactiv +21 +8192 +Cpusmall +12 +8192 +CTScan +384 +53500 +Indoorloc +520 +19337 +Mv +10 +40768 +Pole +26 +14998 +Puma32h +32 +8192 +For data processing step, all values were standardized to a mean of 0 and a standard deviation of 1. +Two processing workflows were tested where the scaling factors were calculated for the training set +only and then applied to the test set, and where the scaling was done on the whole dataset prior to +splitting of training and testing data. No significant differences were observed for both workflows, and +the second workflow was used for the results for simplicity. +3.4 +Experiment setup for regression datasets +To facilitate comparison, we used the same experiment setup for the neural networks as was used in +[Kang and Kang, 2021]. The teacher model is a fully connected feed forward network containing 1 +hidden layer of 500 units with Tanh activation function. The student model is also a fully connected +feed forward network containing 1 hidden layer of either 25, or 50 units with Tanh activation function. +The teacher model is trained with the training data, while student models are trained without access +to any real data from the training set. RMSProp optimizer is used for gradient descent, with a learning +rate of 10−3 and weight decay regularization of 10−5. Batch size m is set to be 50, and the number +of batches in each epoch, ns is set to be 10. β and γ are selected to be 10−5. The number of epochs +is selected as 2000. Models that performed the best on the validation loss was used to evaluate on +the test set. For the direct optimization method to generate synthetic data, RMSProp optimizer with +a learning rate of 10−1, and 2 epochs were used, how these two hyperparameters were selected are +elaborated in the results section 4.1. +3.5 +Experiments on MNIST dataset +To further test the applicability of our method on different types of inputs, and on deeper and more +complex neural network architectures, we designed an experiment for data-free knowledge distillation +for regression on the MNIST handwritten digits dataset. +The MNIST dataset is originally intended to be used for classification task, following the method +presented in [Wang et al., 2020], we adapt it for regression task by making the neural network to +predict a continuous number that represent the class value of the digit label of the input image. The +performance of the model is measured in mean absolute error (MAE) between the predicted value and +the actual value of the digit. For e.g. for perfect performance, the model should predict a value of 3.0 +for an image with the handwritten digit 3. A prediction of 2.9 will result in a MAE of 0.1. +The input image in MNIST is a single channel image of size 28 by 28 pixels, each pixel taking a value +between 0 – 1. The mean µ and standard deviation σ of each pixel position is calculated for the entire +dataset and is used to generate random datapoints with a normal distribution N(µ, σ) clipped between +0 – 1. +As proposed by [Wang et al., 2020], we used a multi-layer convolutional neural network with the +architecture specified in Table 2. +The teacher and student network follow the same architecture, +except that the number of filters, f for each convolutional layer in the teacher network is higher than +that in the student network. f is chosen to be 10 for the teacher network and 5 for the student network. +Log hyperbolic cosine (Log-Cosh) loss was used instead of mean squared error as the loss function to +improve training [Wang et al., 2020]. +7 + +Table 2: Architecture of neural network for MNIST regression +Name +Filters/units +Activation function +Conv2D-1 +3 x 3 x f +ReLU +Conv2D-2 +3 x 3 x 2f +softplus +Maxpool2D-1 +2 x 2 +Conv2D-3 +3 x 3 x 4f +softplus +Maxpool2D-2 +2 x 2 +Flatten +Fully connected-1 +500 +softplus +Dropout-1 (0.5) +Fully connected-1 +100 +softplus +Dropout-2 (0.25) +Fully connected-1 +20 +softplus +Fully connected-1 +1 +softplus +Due to the different nature of the input, which are images rather than standardized tabular data in +the regression datasets, and the output which are natural number, we designed a different loss function +for generating synthetic data. This loss function differs from equations 3 and 5 by replacing the mean- +squared error loss with Log-Cosh loss and by changing the regularization terms to better capture the +distribution of real data. Firstly, instead of penalizing the L2 norm of xg, we penalize the L1 norm +of xg because the handwritten digits image tends to be sparse. Secondly, instead of penalizing the +student prediction on xg, we randomly sample a whole number from 0 – 9 and penalize the distance +of the teacher’s prediction to the random whole number. The purpose of this penalty is to allow the +synthetically generated sample xg to match more closely with the actual data distribution, as real data +should generally not be predicted too far away from whole number by the teacher model for this task. +yrand ∼ {n ∈ Z : 0 ≤ n ≤ 9} +LG(xg) = −ϵ log[cosh(T(xg) − Sθ(xg))] + β|xg|+γ(T(xg) − yrand)2 +(8) +To train the student model, RMSProp optimizer was used for gradient descent, with a learning rate of +10−3 and weight decay regularization of 10−5. Batch size m is set to be 50, and the number of batches +in each epoch, ns is set to be 10. The number of epochs is selected as 1000. +For the direct optimization method to generate synthetic data, RMSProp optimizer with a learning +rate of 10−3, and 20 epochs were used. For the generator network, the number of rounds for training +the generator per epoch was also set to 20. +3.6 +Case study on protein solubility prediction +A bioinformatics problem, predicting continuous protein solubility value with the constituent amino +acids [Han et al., 2019], was used as a case study to test the effectiveness of data-free knowledge +distillation for regression on a real-world scientific problem. Predicting continuous solubility value is +useful for in-silico screening and design of proteins for industrial applications. +We also want to test how the method can be used when the gradients of the teacher model are not +available. For example, many bioinformatics tools such as protein solubility prediction are hosted on +servers that allow users to query proteins and obtain predictions. However, both the model and data +used to train the model are not available to the user. To recreate the model, data-free knowledge +distillation without gradient access to the teacher model is required. If gradient information of the +teacher model is unavailable, it is not possible to train the generative network as described in 3.2.2 +directly. However, for direct optimization, it is possible to use metaheuristics optimization that does +not require gradients instead of gradient descent. +8 + +The dataset used contains 3148 proteins with solubility represented as a continuous value between +0 – 1 from the eSol database [Niwa et al., 2009]. The input features are the proportion of each of +the 20 amino acids within the protein sequence. 2500 proteins are selected for the training set, and +the remaining as test set. The teacher model used is a support vector machine, which represents the +black-box teacher model that contains no gradient information and only output prediction value is +available. +As in the MNIST example, we introduce diversity in the predicted value by the teacher model on xg +with a penalty term on distance away from a random y value sampled for every batch. +yrand ∼ {n ∈ R+ : 0 ≤ n ≤ 1} +LG(xg) = −ϵ (T(xg) − Sθ(xg))2 + (1 − ϵ)(T(xg) − yrand)2 +(9) +The student model is made up of a fully connected Gaussian kernel radial basis function layer with +output size of 100, followed by a fully connected linear layer that outputs the prediction. For training +the student models in both baseline and direct optimization method, RMSProp optimizer is used for +gradient descent, with a learning rate of 10−3 and weight decay regularization of 10−6. Batch size m +is set to be 50 with decreasing α schedule. +Random sampling was used for training baseline model and providing initial points for direct opti- +mization method. The mean µ and standard deviation σ of each amino acid feature is calculated from +the training dataset and is used to generate random datapoints with a normal distribution N(µ, σ) +clipped between 0 – 1. The feature values are then normalized such that the value sums to 1. This is +done as the features which are proportion of each of the 20 amino acids within the protein sequence +must sum to 1. For the direct optimization method to generate synthetic data, differential evolution +algorithm [Storn and Price, 1997] with 25 iterations of best2bin strategy was used, with initial points +generated with the random sampling just described. +4 +Results +4.1 +Properties of synthetic data generated +We first investigate the properties of the synthetic data generated by various methods, namely the +student loss value of the synthetic data, and the distribution of the synthetic data. +4.1.1 +Student loss value of synthetic data +Intuitively, the goal of the synthetic data generation process is to generate data that gives large +differences in student and teacher prediction (i.e. student loss LS in equation 1) in the hope that by +learning to correct these large mistakes, the student model is able to learn faster and better mimic the +outputs of the teacher model. +To verify the actual behavior of the various methods at achieving this goal, we compare the student +loss values of synthetic data generated by the various methods at different stages of training a student +model with random samples: when student is first randomly initialized at 0th epoch, during the middle +stage of training at the 50th and 100th epoch, and when the student model has converged at the 500th +epoch. The results shown in Figure 4 are for Indoorloc dataset. +As expected, it can be observed that the synthetic data generated by the generator method and the +direct optimization method have higher student loss than random Gaussian samples at all stages of +training. Compared to the direct optimization method, the generator method tends to generate data +with smaller loss at the early stages of training, and larger loss at later stages of training. +9 + +Directly optimizing with metaheuristics algorithms, in this case differential evolution, appears to also +produce synthetic data with high loss. However, the running speed of metaheuristics algorithms is much +slower than gradient descent and is not ideal practically unless gradient information is unavailable. +Figure 4: Boxplot of student loss of synthetic data at different epochs +4.1.2 +Distribution of synthetic data +Synthetic data generated should reasonably overlap with the underlying distribution. Out of distri- +bution data generated may either be not useful or even detrimental to model performance on test +data. Ideally the synthetic data generated should also be well spread out from each other rather than +clustered closely together to allow for better coverage of the data distribution. +To verify the actual behavior of the generator and direct optimization methods at achieving this +goal, we visualize the distribution of synthetic datapoints generated by the generator method and +direct optimization (gradient descent) method at different stages of training using plots of 2D UMAP +(Uniform Manifold Approximation and Projections) shown in Figure 5. +It is observed that the synthetic data generated by the generator approach tends to converge around +one or two tight clusters, leaving the rest of the input space untouched. +Even though the direct +optimization approach also tends to have some datapoints concentrated at a few clusters, the rest of +the datapoints tends to be much better spread out in the input space, while still maintaining similarity +with real data. This suggests greater diversity of synthetic data generated with direct optimization +should be helpful for training the student model. +It is also observed that at the later stage of training, many more datapoints generated by the generator +method cluster at regions where there are no real datapoints compared to datapoints generated by the +direct optimization method. This may explain the larger student loss for the generator method than +10 + +(a) Oth epoch +(b) 5oth epoch +14 - +random sample (gaussian) ++ +1.2 - +random sample (latin hypercube) +generator method +12 - +1.0 - +direct optimization (gradient descent) +direct optimization (metaheuristics) +10 - +0.8 +8. +student +6. +0.4 - +4 - +0.2 +2 - +0.0 +0 +2 +3 +5 +2 +4 +5 +4 +method +method +(c) 100th epoch +(d) 500th epoch +0.30 +0.8 +0.25 +0.6 +0.20 +0.10 +0.2 +0.05 +0.0 +0.00 +2 +3 +5 +4 +5 +method +methodFigure 5: Distribution of synthetic datapoints at different epochs +direct optimization method at later stage of training. This suggests that the decreasing schedule for +sample weights parameter α which controls how much the generated data, xg influence the training +loss compared to random samples xp, would likely play a much more important role when using the +generator method. Because at later stage of training, xg generated by the generator method will likely +deviate more from the underlying distribution and may lead to negative learning, which necessitates +a smaller weight α. +We have experimented and found that direct optimizing for 2 steps with a step size of 10−1 leads +to generating synthetic data that do not deviate much from the underlying distribution while still +providing a substantially higher student loss than random samples. Hence these two hyperparameters +were selected for the direct optimization method. +4.2 +Comparison of different methods for data-free distillation on regression +datasets +Table 3 and Table 4 shows the comparison of root mean squared error (RMSE) for 5 methods of +data-free distillation using student size of 25 and 50 respectively. For the generator method and direct +optimization method, both a decreasing α schedule and α value of 1 are tested. The α value of 1 +means that the training uses the generated synthetic data xg entirely without any randomly sampled +datapoints. +Comparing the results for student model size of 25 and 50 hidden units, it is observed that with an +increase in student model size, the RMSE is lower for all datasets due to the greater representation +power of the student model. For most of the datasets tested, the direct optimization method achieves +the lowest RMSE and most closely matches the performance of the teacher model. Compared against +11 + +(b) 50th epoch +(a) Oth +epoch +14 +real data +generator method +direct optimization (gradient descent) +12 - +10 +4 +": +: +3 +7 +2 +5 +6 +8 +0 +3 +9 +(c) 100th epoch +(d) 500th epoch +19 +18 - +17 +16 - +2 +15 - +14 - +1 +13 +0 +12 - +6 +10 +8 +10 +11 +12 +13 +8 +UMAPrandom sampling, direct optimization with decreasing alpha achieves lower RMSE on 6 out of 7 +datasets. Compared against generator method with decreasing α, direct optimization with decreasing +α achieves lower RMSE on 5 out of 7 datasets. +Table 3: RMSE results achieved with different methods for student model size of 25 +Dataset +Teacher +Model +Random +Sampling +Generator; +decreasing +α +Generator; +α = 1 +Direct +op- +timizer; de- +creasing α +Direct +op- +timizer; +α += 1 +Compactv 0.1441 +± 0.0039 +0.1588 +± 0.0050 +0.1606 +± 0.0061 +0.1693 +± 0.0069 +0.1562 +± 0.0043 +0.1599 +± 0.0067 +Cpusmall +0.1672 +± 0.0031 +0.1840 +± 0.0065 +0.1875 +± 0.0070 +0.1918 +± 0.0101 +0.1817 +± 0.0042 +0.1822 +± 0.0048 +CTScan +0.1058 +± 0.0060 +0.2248 +± 0.0170 +0.1601 +± 0.0044 +0.2091 +± 0.0090 +0.1649 +± 0.0058 +0.1593 +± 0.0054 +Indoorloc +0.0847 +± 0.0018 +0.105 +± 0.0051 +0.1034 +± 0.0034 +0.1629 +± 0.0134 +0.0944 +± 0.0015 +0.0957 +± 0.0035 +Mv +0.0236 +± 0.0022 +0.0250 +± 0.0019 +0.0255 +± 0.0016 +0.0428 +± 0.0045 +0.0252 +± 0.0016 +0.0284 +± 0.0017 +Pole +0.1549 +± 0.0064 +0.2893 +± 0.0141 +0.2748 +± 0.0161 +0.3484 +± 0.0304 +0.2836 +± 0.0198 +0.3523 +± 0.0206 +Puma32h +0.2589 +± 0.0055 +0.2474 +± 0.0043 +0.2499 +± 0.0035 +0.2686 +± 0.0091 +0.2464 +± 0.0034 +0.2460 +± 0.0034 +Table 4: RMSE results achieved with different methods for student model size of 50 +Dataset +Teacher +Model +Random +Sampling +Generator; +decreasing +α +Generator; +α = 1 +Direct +op- +timizer; de- +creasing α +Direct +op- +timizer; +α += 1 +Compactv 0.1450 +± 0.0062 +0.15534 +± 0.0077 +0.1551 +± 0.0060 +0.1837 +± 0.0124 +0.1514 +± 0.0068 +0.1531 +± 0.0066 +Cpusmall +0.1663 +± 0.0037 +0.1760 +± 0.0043 +0.1744 +± 0.0040 +0.1842 +± 0.0079 +0.1737 +± 0.0027 +0.1737 +± 0.0049 +CTScan +0.1032 +± 0.0048 +0.1980 +± 0.0111 +0.1458 +± 0.0058 +0.2165 +± 0.0092 +0.1320 +± 0.0047 +0.1316 +± 0.0050 +Indoorloc +0.0844 +± 0.0039 +0.0965 +± 0.0043 +0.1020 +± 0.0035 +0.1549 +± 0.0076 +0.0890 +± 0.0021 +0.0913 +± 0.0027 +Mv +0.0226 +± 0.0027 +0.0237 +± 0.0023 +0.0235 +± 0.0019 +0.0441 +± 0.0059 +0.0235 +± 0.0021 +0.0271 +± 0.0022 +Pole +0.1539 +± 0.0055 +0.2163 +± 0.0143 +0.1964 +± 0.0074 +0.2094 +± 0.0059 +0.2092 +± 0.0114 +0.2324 +± 0.0165 +Puma32h +0.2625 +± 0.0057 +0.2521 +± 0.0035 +0.2554 +± 0.0036 +0.2639 +± 0.0123 +0.2518 +± 0.0057 +0.2495 +± 0.0045 +When α is set to 1, we observe a substantial increase in RMSE for the generator method. However, +12 + +for the direct optimization method, setting α to 1 generally does not lead to much worse performance. +This matches our hypothesis that a decreasing α schedule is much more important for the generator +method as the synthetic datapoint generated tends to deviate more from the underlying distribution +at a later stage of training. Compared with generator method with decreasing α, direct optimization +α = 1 (i.e. training on xg only) achieves lower RMSE on 5 out of 7 datasets. +Figure 6 shows the RMSE on the validation set over the course of training of the student model of +size 50. The direct optimization method shows a faster decrease in RMSE and a generally more stable +learning behavior than the generator method. (Plot values have been smoothed with a Savitzky–Golay +filter of window size of 15 epochs to reduce noise for better visualization) +Figure 6: RMSE on the validation set against training epochs +We also examine the student loss on xg for the two models where α = 1. +As seen in Figure 7, +the generator method often produce unexpectedly large losses during training that could results in +negative learning for the model, whereas the direct optimization method generally produces a stable +and consistently decreasing loss. +Figure 7: Student loss on synthetic data xg against training epochs +4.3 +Comparison of different methods for data-free distillation on MNIST +We experimented with different settings of β, γ and ϵ weights for the various components in the loss +function (equation 8) and found that a low β value (10−6), high γ value (set to 1), low ϵ value (set +to 10−6) and provides good regularization that encourages synthetic data generated to be diverse and +resemble real data distribution more closely. +Table 5 below shows the comparison of mean absolute error (MAE) achieved by training the student +13 + +(a)compactiv +(b)cpusmall +(c) CTScan +(d) Indoorloc +0.300 +0.30 +0.7 - +0.35 +0.275 +0.28 +0.6 - +0.30 +0.250 +0.26 +0.5 - +0.25 +0.24 +MSE +MSE +M0.200 +0.20 +0.175 +0.15 - +0.2 +0.18 +0.150- +0.10 +0.16 - +0.1 +(e) mv +(f) pole +(g)puma32h +0.10 - +0.7 +0.30 +Teacher model +0.6 +0.29 +Random sample +0.08- +Generatormethod +0.5 +0.28 +Direct optimizer method + 0.06 +0.04 - +0.3 +0.26 +0.2 +0.25 +0.02 - +0 +500 +1000 +1500 +2000 +0 +500 +1000 +1500 +2000 +500 +1000 +1500 +2000 +Epochs +Epochs +Epochs(a)compactiv +(b) cpusmall +0.40 +(c) CTScan +0.25 +(d) Indoorloc +0.5 - +0.5 - +0.35 +0.4 +Student loss on xg +0.4 +0.20 +0.30 + 0.25 +0.3 +0.15 +0.2 +60.10 +0.1 - +0.1 - +0.05 +0.05 +0.0 - +0.0 1 +0.00 +0.00 +(e) mv +(f) pole +(g)puma32h +0.30 +0.8 +1.2 +Generatormethod +0.7 +0.25 +1.0 +Direct optimizer method +0 0.6 + 0.20 . + 0.5 +0 0.8. +ent +号 0.10 + 0.2 +0.05 +0.1 - +0.2 +0.00 +0.0 - +0.0 +0 +500 +1000 +1500 +2000 +0 +500 +1000 +1500 +2000 +500 +1000 +1500 +2000 +Epochs +Epochs +Epochsmodel with synthetic data sampled randomly, generated by the generator method and by the direct +optimization method. Results are averaged over 5 runs. Note that the best performing random model +that outputs a constant value of 4.5 would give a MAE of approximately 2.5 for a class balanced test +set. +Table 5: MAE results achieved with different methods on MNIST regression +Teacher Model +Random Sampling +Generator +Direct optimizer +0.157 +2.872 +± 0.052 +2.422 +± 0.027 +1.179 +± 0.132 +We can observe that random sampling synthetic data trains a student model that performs worse than +a random prediction while the generator method is only slightly better than random prediction. The +direct optimization method was able to provide a substantial improvement in performance compared +to the other methods. We examine samples of the synthetic data generated by each method in Figure +8. It is observed that the direct optimization method generates synthetic data closer to what appears +to be handwritten digits compared to the other methods. We also examine the histograms of predicted +values by teacher networks on a batch of 50 synthetically generated data in Figure 9. It is observed that +direct optimization method generates samples with the most diversity while maintaining closeness to +integer values. The closer resemblance to real data distribution is likely the reason the student model +trained on those synthetic data distills more useful knowledge from the teacher model and outperforms +the other methods. +Figure 8: Samples of a synthetic image generated by different methods +This experiment demonstrates that the direct optimization method can be easily and effectively +adapted for data-free knowledge distillation for regression tasks on image inputs, different types of +student or generator loss functions and for multilayer networks with non-MLP architectures which po- +tentially addresses the limitation raised in [Kang and Kang, 2021] on poor applicability of the generator +method for data-free knowledge distillation of multilayer networks for regression. +4.4 +Case study of data-free distillation for protein solubility predictions +We experimented with different settings of ϵ value and found that a value of 0.05 encourages synthetic +data generated to be diverse and provides the best training results. +Table 6 shows the comparison of root mean squared error (RMSE) for the teacher model. The perfor- +mance obtained for the teacher model is comparable with those obtained in the original study [Han +et al., 2019] on regression predictive model for protein solubility. Results are averaged over 5 runs. +Note that a random model that outputs values uniformly drawn from 0 – 1 will give a RMSE of +approximately 0.43. +It is observed that direct optimization method with differential evolution outperforms random sampling +significantly and approaches the RMSE of the teacher model. This case study demonstrated that direct +14 + +5 - +10 - +15 - +15 +J2 +20 +os +20 +25 +2s +25. +15 +JOFigure 9: Histograms of predicted values by teacher networks for synthetic data generated by different +methods +Table 6: RMSE results achieved with different methods on protein solubility prediction +Teacher Model (SVM) +Random Sampling +Direct optimizer +0.250 +0.287 +± 0.001 +0.267 +± 0.005 +optimization can be easily and effectively applied to cases where gradient information from teacher +model is not available, or if the teacher model is not a differentiable neural network at all, such as the +support vector machine teacher model in this case. This is achieved by simply swapping the gradient +descent with a metaheuristics algorithm for direct optimization. Whereas, using a conventional neural +network generative model for synthetic data generation is not possible as the training for the generative +model relies on gradients of both the teacher and student model. +The limitation however is that metaheuristics optimization methods tend to be much slower than +gradient based optimization and thus incur and significant increase in runtime over the baseline method +during training. This may be improved by using a faster metaheuristics algorithm, or one that has +been optimized to run on GPU, but that is beyond the scope of this paper. +5 +Conclusion +In this study, we investigated the behavior of various synthetic data generation methods including ran- +dom sampling and using an adversarially trained generator. We propose a straightforward synthetic +data generation strategy that optimizes the difference between the student and teacher model pre- +dictions directly, with additional flexibility to incorporate arbitrary regularization terms that capture +15 + +Synthetic data generated by direct optimizer +10 +5 +0 +Number of samples +Synthetic data generated by generator model +20 +10 +0 +Random samples +20 - +10 - +0 +-1 +0 +1 +2 +3 +4. +5 +6 +7 +8 +9 +10 +Prediction by teacher modelproperties of the data. We show that synthetic data generated by an adversarially trained generator +tends not to represent underlying data distribution well, requiring the need to supplement training +with random samples and balancing the loss contributions. +Our results demonstrate that the proposed strategy of direct optimization generates synthetic data with +higher loss than random samples while deviating less from underlying distribution than the generator +method. This allows the student model to learn better and emulate the performance of the teacher +model more closely. In the experiments, the proposed method achieves lower RMSE than baseline +and generator method for most regression datasets tested. We also demonstrate the applicability and +flexibility of the method applied to image inputs and deeper convolutional networks on the MNIST +dataset, as well as performing distillation on a non-differentiable model in the case study for predicting +protein solubility. +We hope that this study furthers the understanding of data-free distillation for regression and highlights +the key role of the synthetic data generation process in allowing the student model to effectively distill +the teacher model. +All codes and data used in this study are available at https://github.com/ +zhoutianxun/data_free_KD_regression. +References +[Alcala-Fdez et al., 2010] Alcala-Fdez, J., Fern´andez, A., Luengo, J., Derrac, J., Garcia, S., Sanchez, +L., and Herrera, F. (2010). Keel data-mining software tool: Data set repository, integration of algo- +rithms and experimental analysis framework. 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Advances in Neural Information Processing Systems, 32. +Supplementary Materials +Guarantee (b) for direct optimization method with fixed steps: +Assuming a d-dimensional K-Lipschitz continuous loss function, direct optimization of x with gradient +descent in fixed number of steps t on the loss function results in xt that is bounded in distance away +from x. +Proof: +Given a d-dimensional K-Lipschitz function, by definition, for all real x1 and x2: +|f(x1) − f(x2)|≤ K|x1 − x2| +Let x1 = x2 + δx and taking the limits of δx → 0: +lim +δx→0 +���� +f(x2 + δx) − f(x2) +δx +���� ≤ K +|∇f|≤ K +For gradient descent, the update formula is: +xt+1 = xt − η∇f(xt) +The squared distance moved in the first step of gradient descent is bounded: +∥x1 − x0∥2 = (η∥∇f(x0)∥)2 +∥x1 − x0∥2 ≤ η2dK2 +Therefore, the distance moved in t steps of gradient descent where t ≥ 1 on a d-dimensional K-Lipschitz +function is bounded by: +∥xt − x0∥ ≤ ηt +√ +dK +Guarantee (b) for generator method with L2 regularization: +Assuming a d-dimensional K-Lipschitz continuous loss function, generating synthetic data with a +generator network trained with the L2 regularized loss function [equation 3] will result in xg that is +bounded in distance away from 0, i.e. L2 norm of xg. +Proof: Given a d-dimensional K-Lipschitz continuous loss function, the gradient at any point is +bounded by K: +−K ≤ ∇f ≤ K +The generator network is trained to map random noise vector z to xg to minimize the loss function +with a L2 regularization on xg, i.e +LG(z) = Exg∼Gφ(z)[f(xg) + β∥xg∥2] +As we are minimizing the loss function, we only focus on the lower bound on f(xg) +−K ≤ ∇f +18 + +Then assuming sufficiently small learning rate, the solution xg will converge upon a point where +gradient of the loss function is 0: +∂ +∂xg +(f(xg) + β∥xg∥2) = 0 +∂ +∂xg +(f(xg) = −2βxg +−K ≤ 2βxg +xg ≤ K +2β +Therefore, the bound for the L2 norm of xg is: +∥xg∥ ≤ +√ +dK +2β +19 + diff --git a/itE3T4oBgHgl3EQfIwnu/content/tmp_files/load_file.txt b/itE3T4oBgHgl3EQfIwnu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7fea6c8f56c15864ce2fcb50771af06724587ed3 --- /dev/null +++ b/itE3T4oBgHgl3EQfIwnu/content/tmp_files/load_file.txt @@ -0,0 +1,870 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf,len=869 +page_content='Synthetic data generation method for data-free knowledge distillation in regression neural networks Tianxun Zhou1 and Keng-Hwee Chiam1 1Bioinformatics Institute, Singapore Abstract Knowledge distillation is the technique of compressing a larger neural network, known as the teacher, into a smaller neural network, known as the student, while still trying to maintain the performance of the larger neural network as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Existing methods of knowledge distillation are mostly applicable for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Many of them also require access to the data used to train the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' To address the problem of knowledge distillation for regression tasks under the absence of original training data, previous work has proposed a data-free knowledge distillation method where synthetic data are generated using a generator model trained adversarially against the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' These synthetic data and their labels predicted by the teacher model are then used to train the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' In this study, we investigate the behavior of various synthetic data generation methods and propose a new synthetic data generation strategy that directly optimizes for a large but bounded difference between the student and teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Our results on benchmark and case study experiments demonstrate that the proposed strategy allows the student model to learn better and emulate the performance of the teacher model more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 1 Introduction In the recent decade, advances in algorithms, computational hardware and data availability have enabled significant developments in artificial neural networks and deep learning [Lecun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Neural networks models are now state-of-the-art in many fields of application including computer vision ([O’Mahony et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020], natural language processing [Otter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2021], and signal processing [Purwins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' However, as models become increasingly larger in size measured by number of parameters, they too become computationally expensive to store and perform inference on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Large neural networks can be unusable for real world deployment scenarios where hardware may be limited, such as on mobile devices or microcontrollers, or when deployed for as a service to a large number of users such as web applications [Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2018, Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Knowledge distillation is a class of method to address this problem by distilling the predictive ca- pabilities of a larger neural network into a smaller neural network, allowing for faster inference and lower memory requirements [Gou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' There have been several knowledge distillation meth- ods proposed in the past, typically requiring the original data that was used to train the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' However, in many real-world applications, the original data may not be available for perform- ing knowledge distillation to student models due to reasons such as data size and data privacy [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2019, Gou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' To deal with such situations, data-free knowledge distillation methods have been proposed to allow distillation of knowledge without the original training data [Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020, Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2017, Micaelli and Storkey, 2019, Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020, Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Data-free knowledge distillation works by gener- ating synthetic data and training the student model with these data and their teacher model predicted labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Much of the existing research for knowledge distillation has been focused on classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' How- ever, regression tasks are common in many engineering applications [Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2021, Schweidtmann 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='04338v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='LG] 11 Jan 2023 et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2021] and there are limited methods available on knowledge distillation for regression neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Recently, [Kang and Kang, 2021] proposed the first data-free knowledge distillation method for regression where a generator model was trained in an adversarial manner to generate synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Motivated by the need for data-free model distillation on regression models in real world ap- plications, in this work we investigate the behaviors of several synthetic data generation methods including random sampling and adversarial generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Based on the insights gained from this investigation, we propose an improved method to generate synthetic data for data-free knowledge distillation of regression neural networks by optimizing for a loss function defined using the student and teacher model predictions directly rather than implicitly through an additional generator model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Compared to existing methods, synthetic data generated through this process can provide large difference in prediction between the student and teacher model while mimicking real data better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We demonstrate that this method for synthetic data generation can provide better performance than existing methods through experiments in 7 standard regression datasets, as well as on the MNIST handwritten digit dataset adapted for regression, and a real-world bioinformatics case study of protein solubility prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 2 Related work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 Knowledge distillation As neural networks become increasingly large in number of parameters, the deployment of such models faces a difficult challenge for applications such as mobile devices and embedded systems due to limita- tions in computational resources and memory [Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2018, Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' To address such problems, model compression through knowledge distillation has become an active area of research in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Knowledge distillation is the technique where knowledge learned by a larger teacher model is transferred to a smaller student model [Gou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2021, Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The main idea is that the student model mimics the teacher model to achieve a similar or even a superior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Various methods of knowledge distillation define and focus on different forms of knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Following the nomenclature in [Gou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2021], these can be largely grouped as response-based knowledge, feature-based knowledge, and relation-based knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For response-based knowledge, outputs of the teacher model are used to supervise the training of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For example, [Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2015] uses soft targets from the logits output of the teacher model to train the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For feature- based knowledge, outputs of intermediate layers, or feature maps learned by the teacher model can be to supervise the training of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For example, [Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2014] trains the student model to match the feature activations of the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For relationship-based knowledge, the relationships between different layers or data samples are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For example, [Yim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2017] uses the inner products between features from two layers to represent the relationship between different layers, while [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2021] trains the student model to learn to preserve the similarity of samples’ feature embeddings in the intermediate layers of the teacher models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 Data-free knowledge distillation In some situations, access to the original data used to train the teacher model is not available due to issues such as privacy and legal reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Data-free knowledge distillation methods have been proposed to allow model distillation in the absence of original training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This is achieved by generating synthetic data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Many methods achieve this by using generative adversarial networks (GAN) [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2019, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020, Micaelli and Storkey, 2019, Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020, Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For example, [Micaelli and Storkey, 2019] train a generator model to generate synthetic images that maximizes the difference in prediction (measured by KL divergence) between the teacher and student models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The student model is then trained to minimize the difference on these synthetic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Other methods such as [Lopes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2017] make use of metadata collected during training of the teacher model, in the form of the layer activation records of the teacher model to reconstruct dataset for training the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 2 Figure 1: Generic data-free knowledge distillation method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='3 Knowledge distillation for regression Most of the methods currently existing in knowledge distillation literature deal with classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' These methods generally are not immediately applicable to regression problems where the predictions are unbounded real values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For regression problems, [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2017] uses a teacher bounded regression loss where the teacher’s predictions serve as an upper bound for the student model instead of using it directly as a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' [Takamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020] uses a teacher outlier rejection loss, that rejects outliers in training samples based on the teacher model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' [Kang and Kang, 2021] introduced the first work that addresses data-free knowledge distillation for regression, by using a generator model that generates synthetic datapoints that is trained adversarially together with the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 3 Material and methods 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 Overview of methods Given a trained teacher model T, and a student model Sθ parameterized by θ, we generate synthetic data x via some data generation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The student model is trained by minimizing the student loss LS(x) defined in equation 1 using gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This generic method is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' LS(x) = (T(x) − Sθ(x))2 (1) The performance of the student model in mimicking the performance of the teacher is dependent on the representation strength θ of the student model, and the data x used to train it, and the optimization process of minimizing student loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Hence for a fixed student model architecture and training process, the synthetic data generation process plays the key role in determining the performance of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 Synthetic data generation methods Three types of synthetic data generation methods are investigated in this study: random sampling, generative model, and direct optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 Random sampling Synthetic data are generated by sampling randomly from an input distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Assuming the input has been standardized, random samples can be drawn from a Gaussian distribution ∼ N(0, I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Random sampling can also be drawn through quasi-Monte Carlo method such as Latin Hypercube sampling and Halton sequences which are designed to evenly cover the input space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Input space bounds may be defined using the maximum and minimum values of an available validation or test set, or based upon some prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 3 Teacher Model, T T(Xg) Synthetic Data Generation Student Model, S → S(Xg)Figure 2: Data-free distillation with generator method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 Generator model Generator model for generating synthetic data was proposed for data-free knowledge distillation for regression tasks by [Kang and Kang, 2021], follows similar methods in classification tasks [Micaelli and Storkey, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' In this method, a generator model Gφ parameterized by φ is trained to output samples that would result in a large difference between the student and teacher model’s predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This generator model is trained in an adversarial manner against the student model during the distillation process by optimizing the generator loss function in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' LG(z) = Exg Gφ(z)[−(T(xg) − Sθ(xg))2] (2) The student is trained using the student loss to minimize the difference between teacher and its own predictions, the two opposing learning objectives are trained in a sequential adversarial manner, and the student model is able to learn to match the predictions of the teacher model as training continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This process is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' In practice, regularization terms may be added to the generator loss to prevent complete deviation from underlying data distribution, for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' by adding the square of L2-norm of xg and Sθ(xg), yielding: LG(z) = Exg Gφ(z)[−(T(xg) − Sθ(xg))2 + β∥xg∥2 + γSθ(xg)2] (3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='3 Direct optimization from random samples The generator model approach attempts to train the generative model Gθ to approximate the inverse function of the student loss implicitly, where the generative model predicts x given the objective of high student loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' It is not immediately clear whether the generative model is able to learn this inverse function easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Since the goal of the generative model approach is to generate samples that maximize the student loss, it is more straightforward to maximize the student loss directly, as formulated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' max xg (T(xg) − Sθ(xg))2 Or following conventions: min xg −(T(xg) − Sθ(xg))2 (4) In practice, following the generator method, we may add regularization terms as well, such as in equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 4 Teacher Model, T T(Xa Generator Network, Z ~N(0, D - LG G Student Model, S S(Xg) Updateparameters ofGFigure 3: Data-free model distillation with direct optimization method min xg −(T(xg) − Sθ(xg))2 + β∥xg∥2 + γ(xg)2 (5) It is later shown in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='6 that the methodology is very flexible, and any arbitrary loss function may be used to incorporate loss terms designed to capture important properties of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This minimization can be done through various optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' If both the student and teacher models are differentiable, gradient descent can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Black box metaheuristic optimization methods such as genetic algorithms and simulated annealing may also be used, especially if the teacher model gradients are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The method is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' When using direct optimization of the student loss with gradient descent, it is possible to derive theo- retical guarantees for (a) generating samples that are better than random sample and (b) generating samples that are bounded in their deviation away from underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The gradient descent updates as such: xg,t+1 = xg,t + η ∂ ∂xg,t [T(xg) − Sθ(xg)]2 (6) Assuming the neural networks are locally smooth (Lipschitz continuous), given some sufficiently small learning rate η, xg,t+1 always improves upon xg,t fulfilling guarantee (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Given some learning rate η and number of gradient descent steps tmax, xg,t+1 deviates from xg,0 randomly sampled from underlying distribution by an arbitrary bound, fulfilling guarantee (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Proof for guarantee (a) is provided in [Boyd and Vandenberghe, 2009] p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='466 and proof for guarantee (b) is provided in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' It is not obvious to us that the generator model method can fulfil guarantee (a) because xg is generated from Gaussian noise z of an arbitrary dimension and is not related to random samples in input space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' and to fulfil guarantee (b), a bound on the deviation of xg from 0 exist only if a regularization term is applied to xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The proof for bound on magnitude of xg for generator method with L2 regularization is provided in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='4 Proposed method for knowledge distillation The proposed data-free knowledge distillation method generates training data xg through direct opti- mization of student loss with gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' In the synthetic data generation step, assuming inputs are standardized, a batch of random samples are drawn from a Gaussian distribution ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Gra- dient descent is used to perturb these random samples to the direction of maximizing their student 5 Teacher Model, T T(Xg) Xg,0 ~ N(0, I) Xg Student Model, S → S(Xg) Update Xg directlyloss values, obtaining xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' In the student training step, the student weights are updated to minimize the student loss with respect to the synthetic data xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Following the methods proposed in [Kang and Kang, 2021], generated data is also supplemented with random samples xp drawn from Gaussian distribution ∼ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The sample weights for the generated samples xg and random samples xp are controlled by a factor α, which can be a fixed value or follow a schedule based on the training epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' LS = αLS(xg) + (1 − α)LS(xp) (7) Setting α to 0 is equivalent to the random sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Setting α to 1 is a pure generative sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Note that for both edge cases, since the loss of only 1 set of samples contributes to the training, the number of training samples in each epoch needs to be doubled for a fair comparison with cases where α is between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We investigate a decreasing alpha schedule as well as a pure xg training strategy in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The training process is provided in algorithm 1 & 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' In the main procedure Data-free model distillation where the data distillation training happens, the number of training epochs for the student model is defined as tmax, and the number of batches per epoch is defined as ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' In the sub- procedure Optimize, where direct optimization to generate synthetic data is done via gradient descent, the number of gradient descent steps is defined as τmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Algorithm 1: Main procedure: Data-free model distillation Input: teacher model, T Output: student model, Sθ 1 for t=1 to tmax do 2 for 1 to ns do 3 z ∼ N(0, I) 4 xg ← Optimize(z) 5 xp ∼ N(0, I) 6 L ← αLS(xg) + (1 − α)LS(xp) 7 Update Sθ with gradient descent w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' L Algorithm 2: Sub-procedure: Optimize Input: z Output: xg 1 xg ← z 2 for τ=1 to τmax do 3 LS ← −(T(xg) − Sθ(xg))2 + β∥xg∥2 + γ(xg)2 4 xg ← xg − η ∂ ∂xg LS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='3 Regression datasets for experiments To facilitate comparison with the previous work by [Kang and Kang, 2021], the experiments were conducted on the same datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' These 7 datasets are regression problem sets available from UCI machine learning repository [Dheeru and Casey, 2019] and KEEL dataset repository [Alcala-Fdez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' ‘longitude’ was selected as the output variable for Indoorloc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Details of the datasets are provided in the Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The data are split into training and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The training set consists of 5000 samples for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 10% of the remainder samples are placed into the validation set, and the remaining 90% is the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The validation set is used to periodically evaluate the training of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 6 Table 1: List of regression datasets Dataset Number of features Number of samples Compactiv 21 8192 Cpusmall 12 8192 CTScan 384 53500 Indoorloc 520 19337 Mv 10 40768 Pole 26 14998 Puma32h 32 8192 For data processing step, all values were standardized to a mean of 0 and a standard deviation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Two processing workflows were tested where the scaling factors were calculated for the training set only and then applied to the test set, and where the scaling was done on the whole dataset prior to splitting of training and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' No significant differences were observed for both workflows, and the second workflow was used for the results for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='4 Experiment setup for regression datasets To facilitate comparison, we used the same experiment setup for the neural networks as was used in [Kang and Kang, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The teacher model is a fully connected feed forward network containing 1 hidden layer of 500 units with Tanh activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The student model is also a fully connected feed forward network containing 1 hidden layer of either 25, or 50 units with Tanh activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The teacher model is trained with the training data, while student models are trained without access to any real data from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' RMSProp optimizer is used for gradient descent, with a learning rate of 10−3 and weight decay regularization of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Batch size m is set to be 50, and the number of batches in each epoch, ns is set to be 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' β and γ are selected to be 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The number of epochs is selected as 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Models that performed the best on the validation loss was used to evaluate on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For the direct optimization method to generate synthetic data, RMSProp optimizer with a learning rate of 10−1, and 2 epochs were used, how these two hyperparameters were selected are elaborated in the results section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 Experiments on MNIST dataset To further test the applicability of our method on different types of inputs, and on deeper and more complex neural network architectures, we designed an experiment for data-free knowledge distillation for regression on the MNIST handwritten digits dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The MNIST dataset is originally intended to be used for classification task, following the method presented in [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020], we adapt it for regression task by making the neural network to predict a continuous number that represent the class value of the digit label of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The performance of the model is measured in mean absolute error (MAE) between the predicted value and the actual value of the digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' for perfect performance, the model should predict a value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 for an image with the handwritten digit 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' A prediction of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='9 will result in a MAE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The input image in MNIST is a single channel image of size 28 by 28 pixels, each pixel taking a value between 0 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The mean µ and standard deviation σ of each pixel position is calculated for the entire dataset and is used to generate random datapoints with a normal distribution N(µ, σ) clipped between 0 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' As proposed by [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020], we used a multi-layer convolutional neural network with the architecture specified in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The teacher and student network follow the same architecture, except that the number of filters, f for each convolutional layer in the teacher network is higher than that in the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' f is chosen to be 10 for the teacher network and 5 for the student network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Log hyperbolic cosine (Log-Cosh) loss was used instead of mean squared error as the loss function to improve training [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 7 Table 2: Architecture of neural network for MNIST regression Name Filters/units Activation function Conv2D-1 3 x 3 x f ReLU Conv2D-2 3 x 3 x 2f softplus Maxpool2D-1 2 x 2 Conv2D-3 3 x 3 x 4f softplus Maxpool2D-2 2 x 2 Flatten Fully connected-1 500 softplus Dropout-1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5) Fully connected-1 100 softplus Dropout-2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='25) Fully connected-1 20 softplus Fully connected-1 1 softplus Due to the different nature of the input, which are images rather than standardized tabular data in the regression datasets, and the output which are natural number, we designed a different loss function for generating synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This loss function differs from equations 3 and 5 by replacing the mean- squared error loss with Log-Cosh loss and by changing the regularization terms to better capture the distribution of real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Firstly, instead of penalizing the L2 norm of xg, we penalize the L1 norm of xg because the handwritten digits image tends to be sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Secondly, instead of penalizing the student prediction on xg, we randomly sample a whole number from 0 – 9 and penalize the distance of the teacher’s prediction to the random whole number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The purpose of this penalty is to allow the synthetically generated sample xg to match more closely with the actual data distribution, as real data should generally not be predicted too far away from whole number by the teacher model for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' yrand ∼ {n ∈ Z : 0 ≤ n ≤ 9} LG(xg) = −ϵ log[cosh(T(xg) − Sθ(xg))] + β|xg|+γ(T(xg) − yrand)2 (8) To train the student model, RMSProp optimizer was used for gradient descent, with a learning rate of 10−3 and weight decay regularization of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Batch size m is set to be 50, and the number of batches in each epoch, ns is set to be 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The number of epochs is selected as 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For the direct optimization method to generate synthetic data, RMSProp optimizer with a learning rate of 10−3, and 20 epochs were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For the generator network, the number of rounds for training the generator per epoch was also set to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='6 Case study on protein solubility prediction A bioinformatics problem, predicting continuous protein solubility value with the constituent amino acids [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2019], was used as a case study to test the effectiveness of data-free knowledge distillation for regression on a real-world scientific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Predicting continuous solubility value is useful for in-silico screening and design of proteins for industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We also want to test how the method can be used when the gradients of the teacher model are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For example, many bioinformatics tools such as protein solubility prediction are hosted on servers that allow users to query proteins and obtain predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' However, both the model and data used to train the model are not available to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' To recreate the model, data-free knowledge distillation without gradient access to the teacher model is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' If gradient information of the teacher model is unavailable, it is not possible to train the generative network as described in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' However, for direct optimization, it is possible to use metaheuristics optimization that does not require gradients instead of gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 8 The dataset used contains 3148 proteins with solubility represented as a continuous value between 0 – 1 from the eSol database [Niwa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The input features are the proportion of each of the 20 amino acids within the protein sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 2500 proteins are selected for the training set, and the remaining as test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The teacher model used is a support vector machine, which represents the black-box teacher model that contains no gradient information and only output prediction value is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' As in the MNIST example, we introduce diversity in the predicted value by the teacher model on xg with a penalty term on distance away from a random y value sampled for every batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' yrand ∼ {n ∈ R+ : 0 ≤ n ≤ 1} LG(xg) = −ϵ (T(xg) − Sθ(xg))2 + (1 − ϵ)(T(xg) − yrand)2 (9) The student model is made up of a fully connected Gaussian kernel radial basis function layer with output size of 100, followed by a fully connected linear layer that outputs the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For training the student models in both baseline and direct optimization method, RMSProp optimizer is used for gradient descent, with a learning rate of 10−3 and weight decay regularization of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Batch size m is set to be 50 with decreasing α schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Random sampling was used for training baseline model and providing initial points for direct opti- mization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The mean µ and standard deviation σ of each amino acid feature is calculated from the training dataset and is used to generate random datapoints with a normal distribution N(µ, σ) clipped between 0 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The feature values are then normalized such that the value sums to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This is done as the features which are proportion of each of the 20 amino acids within the protein sequence must sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For the direct optimization method to generate synthetic data, differential evolution algorithm [Storn and Price, 1997] with 25 iterations of best2bin strategy was used, with initial points generated with the random sampling just described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 4 Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 Properties of synthetic data generated We first investigate the properties of the synthetic data generated by various methods, namely the student loss value of the synthetic data, and the distribution of the synthetic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 Student loss value of synthetic data Intuitively, the goal of the synthetic data generation process is to generate data that gives large differences in student and teacher prediction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' student loss LS in equation 1) in the hope that by learning to correct these large mistakes, the student model is able to learn faster and better mimic the outputs of the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' To verify the actual behavior of the various methods at achieving this goal, we compare the student loss values of synthetic data generated by the various methods at different stages of training a student model with random samples: when student is first randomly initialized at 0th epoch, during the middle stage of training at the 50th and 100th epoch, and when the student model has converged at the 500th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The results shown in Figure 4 are for Indoorloc dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' As expected, it can be observed that the synthetic data generated by the generator method and the direct optimization method have higher student loss than random Gaussian samples at all stages of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Compared to the direct optimization method, the generator method tends to generate data with smaller loss at the early stages of training, and larger loss at later stages of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 9 Directly optimizing with metaheuristics algorithms, in this case differential evolution, appears to also produce synthetic data with high loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' However, the running speed of metaheuristics algorithms is much slower than gradient descent and is not ideal practically unless gradient information is unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Figure 4: Boxplot of student loss of synthetic data at different epochs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 Distribution of synthetic data Synthetic data generated should reasonably overlap with the underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Out of distri- bution data generated may either be not useful or even detrimental to model performance on test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Ideally the synthetic data generated should also be well spread out from each other rather than clustered closely together to allow for better coverage of the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' To verify the actual behavior of the generator and direct optimization methods at achieving this goal, we visualize the distribution of synthetic datapoints generated by the generator method and direct optimization (gradient descent) method at different stages of training using plots of 2D UMAP (Uniform Manifold Approximation and Projections) shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' It is observed that the synthetic data generated by the generator approach tends to converge around one or two tight clusters, leaving the rest of the input space untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Even though the direct optimization approach also tends to have some datapoints concentrated at a few clusters, the rest of the datapoints tends to be much better spread out in the input space, while still maintaining similarity with real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This suggests greater diversity of synthetic data generated with direct optimization should be helpful for training the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' It is also observed that at the later stage of training, many more datapoints generated by the generator method cluster at regions where there are no real datapoints compared to datapoints generated by the direct optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This may explain the larger student loss for the generator method than 10 (a) Oth epoch (b) 5oth epoch 14 - random sample (gaussian) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 - random sample (latin hypercube) generator method 12 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 - direct optimization (gradient descent) direct optimization (metaheuristics) 10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' student 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='4 - 4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 0 2 3 5 2 4 5 4 method method (c) 100th epoch (d) 500th epoch 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='00 2 3 5 4 5 method methodFigure 5: Distribution of synthetic datapoints at different epochs direct optimization method at later stage of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This suggests that the decreasing schedule for sample weights parameter α which controls how much the generated data, xg influence the training loss compared to random samples xp, would likely play a much more important role when using the generator method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Because at later stage of training, xg generated by the generator method will likely deviate more from the underlying distribution and may lead to negative learning, which necessitates a smaller weight α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We have experimented and found that direct optimizing for 2 steps with a step size of 10−1 leads to generating synthetic data that do not deviate much from the underlying distribution while still providing a substantially higher student loss than random samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Hence these two hyperparameters were selected for the direct optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 Comparison of different methods for data-free distillation on regression datasets Table 3 and Table 4 shows the comparison of root mean squared error (RMSE) for 5 methods of data-free distillation using student size of 25 and 50 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For the generator method and direct optimization method, both a decreasing α schedule and α value of 1 are tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The α value of 1 means that the training uses the generated synthetic data xg entirely without any randomly sampled datapoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Comparing the results for student model size of 25 and 50 hidden units, it is observed that with an increase in student model size, the RMSE is lower for all datasets due to the greater representation power of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' For most of the datasets tested, the direct optimization method achieves the lowest RMSE and most closely matches the performance of the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Compared against 11 (b) 50th epoch (a) Oth epoch 14 real data generator method direct optimization (gradient descent) 12 - 10 4 ": : 3 7 2 5 6 8 0 3 9 (c) 100th epoch (d) 500th epoch 19 18 - 17 16 - 2 15 - 14 - 1 13 0 12 - 6 10 8 10 11 12 13 8 UMAPrandom sampling, direct optimization with decreasing alpha achieves lower RMSE on 6 out of 7 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Compared against generator method with decreasing α, direct optimization with decreasing α achieves lower RMSE on 5 out of 7 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Table 3: RMSE results achieved with different methods for student model size of 25 Dataset Teacher Model Random Sampling Generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' decreasing α Generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' α = 1 Direct op- timizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' de- creasing α Direct op- timizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' α = 1 Compactv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1441 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1588 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1606 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1693 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1562 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1599 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0067 Cpusmall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1672 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1840 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1875 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1918 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1817 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1822 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0048 CTScan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1058 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2248 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0170 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1601 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2091 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1649 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1593 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0054 Indoorloc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0847 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='105 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1034 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1629 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0944 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0957 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0035 Mv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0236 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0250 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0428 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0252 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0284 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0017 Pole 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1549 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2893 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2748 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='3484 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2836 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='3523 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0206 Puma32h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2589 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2474 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2499 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2686 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2464 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2460 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0034 Table 4: RMSE results achieved with different methods for student model size of 50 Dataset Teacher Model Random Sampling Generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' decreasing α Generator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' α = 1 Direct op- timizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' de- creasing α Direct op- timizer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' α = 1 Compactv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1450 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='15534 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1551 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1837 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1514 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1531 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0066 Cpusmall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1663 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1760 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1744 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1842 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1737 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1737 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0049 CTScan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1032 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1980 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1458 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2165 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1320 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1316 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0050 Indoorloc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0844 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0965 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1020 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1549 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0890 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0913 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0027 Mv 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0226 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0237 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0235 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0441 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0235 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0271 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0022 Pole 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1539 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2163 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1964 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2094 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2092 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2324 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0165 Puma32h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2625 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2521 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2639 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2518 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2495 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0045 When α is set to 1, we observe a substantial increase in RMSE for the generator method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' However, 12 for the direct optimization method, setting α to 1 generally does not lead to much worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This matches our hypothesis that a decreasing α schedule is much more important for the generator method as the synthetic datapoint generated tends to deviate more from the underlying distribution at a later stage of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Compared with generator method with decreasing α, direct optimization α = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' training on xg only) achieves lower RMSE on 5 out of 7 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Figure 6 shows the RMSE on the validation set over the course of training of the student model of size 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The direct optimization method shows a faster decrease in RMSE and a generally more stable learning behavior than the generator method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' (Plot values have been smoothed with a Savitzky–Golay filter of window size of 15 epochs to reduce noise for better visualization) Figure 6: RMSE on the validation set against training epochs We also examine the student loss on xg for the two models where α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' As seen in Figure 7, the generator method often produce unexpectedly large losses during training that could results in negative learning for the model, whereas the direct optimization method generally produces a stable and consistently decreasing loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Figure 7: Student loss on synthetic data xg against training epochs 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='3 Comparison of different methods for data-free distillation on MNIST We experimented with different settings of β, γ and ϵ weights for the various components in the loss function (equation 8) and found that a low β value (10−6), high γ value (set to 1), low ϵ value (set to 10−6) and provides good regularization that encourages synthetic data generated to be diverse and resemble real data distribution more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Table 5 below shows the comparison of mean absolute error (MAE) achieved by training the student 13 (a)compactiv (b)cpusmall (c) CTScan (d) Indoorloc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='7 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='24 MSE MSE M0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='15 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='150- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='16 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 (e) mv (f) pole (g)puma32h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='10 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='30 Teacher model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='29 Random sample 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='08- Generatormethod 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='28 Direct optimizer method 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='04 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='02 - 0 500 1000 1500 2000 0 500 1000 1500 2000 500 1000 1500 2000 Epochs Epochs Epochs(a)compactiv (b) cpusmall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='40 (c) CTScan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='25 (d) Indoorloc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='4 Student loss on xg 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='00 (e) mv (f) pole (g)puma32h 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 Generatormethod 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 Direct optimizer method 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' ent 号 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='0 0 500 1000 1500 2000 0 500 1000 1500 2000 500 1000 1500 2000 Epochs Epochs Epochsmodel with synthetic data sampled randomly, generated by the generator method and by the direct optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Results are averaged over 5 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Note that the best performing random model that outputs a constant value of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 would give a MAE of approximately 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='5 for a class balanced test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Table 5: MAE results achieved with different methods on MNIST regression Teacher Model Random Sampling Generator Direct optimizer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='157 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='872 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='052 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='422 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='027 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='179 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='132 We can observe that random sampling synthetic data trains a student model that performs worse than a random prediction while the generator method is only slightly better than random prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The direct optimization method was able to provide a substantial improvement in performance compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We examine samples of the synthetic data generated by each method in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' It is observed that the direct optimization method generates synthetic data closer to what appears to be handwritten digits compared to the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We also examine the histograms of predicted values by teacher networks on a batch of 50 synthetically generated data in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' It is observed that direct optimization method generates samples with the most diversity while maintaining closeness to integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The closer resemblance to real data distribution is likely the reason the student model trained on those synthetic data distills more useful knowledge from the teacher model and outperforms the other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Figure 8: Samples of a synthetic image generated by different methods This experiment demonstrates that the direct optimization method can be easily and effectively adapted for data-free knowledge distillation for regression tasks on image inputs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' different types of student or generator loss functions and for multilayer networks with non-MLP architectures which po- tentially addresses the limitation raised in [Kang and Kang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 2021] on poor applicability of the generator method for data-free knowledge distillation of multilayer networks for regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='4 Case study of data-free distillation for protein solubility predictions We experimented with different settings of ϵ value and found that a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='05 encourages synthetic data generated to be diverse and provides the best training results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Table 6 shows the comparison of root mean squared error (RMSE) for the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The perfor- mance obtained for the teacher model is comparable with those obtained in the original study [Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2019] on regression predictive model for protein solubility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Results are averaged over 5 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Note that a random model that outputs values uniformly drawn from 0 – 1 will give a RMSE of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' It is observed that direct optimization method with differential evolution outperforms random sampling significantly and approaches the RMSE of the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This case study demonstrated that direct 14 5 - 10 - 15 - 15 J2 20 os 20 25 2s 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 15 JOFigure 9: Histograms of predicted values by teacher networks for synthetic data generated by different methods Table 6: RMSE results achieved with different methods on protein solubility prediction Teacher Model (SVM) Random Sampling Direct optimizer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='287 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='267 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='005 optimization can be easily and effectively applied to cases where gradient information from teacher model is not available, or if the teacher model is not a differentiable neural network at all, such as the support vector machine teacher model in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This is achieved by simply swapping the gradient descent with a metaheuristics algorithm for direct optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Whereas, using a conventional neural network generative model for synthetic data generation is not possible as the training for the generative model relies on gradients of both the teacher and student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' The limitation however is that metaheuristics optimization methods tend to be much slower than gradient based optimization and thus incur and significant increase in runtime over the baseline method during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This may be improved by using a faster metaheuristics algorithm, or one that has been optimized to run on GPU, but that is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 5 Conclusion In this study, we investigated the behavior of various synthetic data generation methods including ran- dom sampling and using an adversarially trained generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We propose a straightforward synthetic data generation strategy that optimizes the difference between the student and teacher model pre- dictions directly, with additional flexibility to incorporate arbitrary regularization terms that capture 15 Synthetic data generated by direct optimizer 10 5 0 Number of samples Synthetic data generated by generator model 20 10 0 Random samples 20 - 10 - 0 1 0 1 2 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' 5 6 7 8 9 10 Prediction by teacher modelproperties of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We show that synthetic data generated by an adversarially trained generator tends not to represent underlying data distribution well, requiring the need to supplement training with random samples and balancing the loss contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Our results demonstrate that the proposed strategy of direct optimization generates synthetic data with higher loss than random samples while deviating less from underlying distribution than the generator method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' This allows the student model to learn better and emulate the performance of the teacher model more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' In the experiments, the proposed method achieves lower RMSE than baseline and generator method for most regression datasets tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We also demonstrate the applicability and flexibility of the method applied to image inputs and deeper convolutional networks on the MNIST dataset, as well as performing distillation on a non-differentiable model in the case study for predicting protein solubility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' We hope that this study furthers the understanding of data-free distillation for regression and highlights the key role of the synthetic data generation process in allowing the student model to effectively distill the teacher model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' All codes and data used in this study are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='com/ zhoutianxun/data_free_KD_regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' References [Alcala-Fdez et al.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', and Song, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Data-free knowledge amalga- mation via group-stack dual-gan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' [Yim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2017] Yim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', Joo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', Bae, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', and Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' A gift from knowledge distillation: Fast optimization, network minimization and transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' [Yoo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', 2019] Yoo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', Cho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', Kim, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=', and Kang, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Knowledge extraction with no observable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Supplementary Materials Guarantee (b) for direct optimization method with fixed steps: Assuming a d-dimensional K-Lipschitz continuous loss function, direct optimization of x with gradient descent in fixed number of steps t on the loss function results in xt that is bounded in distance away from x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Proof: Given a d-dimensional K-Lipschitz function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' by definition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' for all real x1 and x2: |f(x1) − f(x2)|≤ K|x1 − x2| Let x1 = x2 + δx and taking the limits of δx → 0: lim δx→0 ���� f(x2 + δx) − f(x2) δx ���� ≤ K |∇f|≤ K For gradient descent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' the update formula is: xt+1 = xt − η∇f(xt) The squared distance moved in the first step of gradient descent is bounded: ∥x1 − x0∥2 = (η∥∇f(x0)∥)2 ∥x1 − x0∥2 ≤ η2dK2 Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' the distance moved in t steps of gradient descent where t ≥ 1 on a d-dimensional K-Lipschitz function is bounded by: ∥xt − x0∥ ≤ ηt √ dK Guarantee (b) for generator method with L2 regularization: Assuming a d-dimensional K-Lipschitz continuous loss function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' generating synthetic data with a generator network trained with the L2 regularized loss function [equation 3] will result in xg that is bounded in distance away from 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' L2 norm of xg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content=' Proof: Given a d-dimensional K-Lipschitz continuous loss function, the gradient at any point is bounded by K: −K ≤ ∇f ≤ K The generator network is trained to map random noise vector z to xg to minimize the loss function with a L2 regularization on xg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} +page_content='e LG(z) = Exg∼Gφ(z)[f(xg) + β∥xg∥2] As we are minimizing the loss function, we only focus on the lower bound on f(xg) −K ≤ ∇f 18 Then assuming sufficiently small learning rate, the solution xg will converge upon a point where gradient of the loss function is 0: ∂ ∂xg (f(xg) + β∥xg∥2) = 0 ∂ ∂xg (f(xg) = −2βxg −K ≤ 2βxg xg ≤ K 2β Therefore, the bound for the L2 norm of xg is: ∥xg∥ ≤ √ dK 2β 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfIwnu/content/2301.04338v1.pdf'} diff --git a/jdAyT4oBgHgl3EQfkfjo/vector_store/index.faiss b/jdAyT4oBgHgl3EQfkfjo/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..25dd16cbb6a67be405bdbc58a2c5a2627bae33dc --- /dev/null +++ b/jdAyT4oBgHgl3EQfkfjo/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:97e5e83be6ddb394ea4f307faebe6e936147f8d6de9b6e3c6c2b943cdfe82b7e +size 4456493 diff --git 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S.A.It. Vol. 75, 282 +© SAIt 2022 +Memorie della +Virtual reality for the analysis and visualization +of scientific numerical models +S. Orlando1, M. Miceli2,1, U. Lo Cicero1, and S. Ustamujic1 +1 INAF-Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134, Palermo, +Italy e-mail: salvatore.orlando@inaf.it +2 Dipartimento di Fisica e Chimica E. Segr`e, Universit`a degli Studi di Palermo, Via +Archirafi, 36, 90123, Palermo, Italy +Received: Day Month Year; Accepted: Day Month Year +Abstract. The complexity of the data generated by (magneto)-hydrodynamic (HD/MHD) +simulations requires advanced tools for their analysis and visualization. The dramatic im- +provements in virtual reality (VR) technologies have inspired us to seek the long-term goal +of creating VR tools for scientific model analysis and visualization that would allow re- +searchers to study and perform data analysis on their models within an immersive environ- +ment. Here, we report the results obtained at INAF-Osservatorio Astronomico di Palermo in +the development of these tools, which would allow for the exploration of 3D models inter- +actively, resulting in highly detailed analysis that cannot be performed with traditional data +visualization and analysis platforms. Additionally, these VR-based tools offer the ability to +produce high-impact VR content for efficient audience engagement and awareness. +1. Introduction +The development of (magneto)-hydrodynamic +(HD/MHD) models of astronomical objects +and phenomena can be crucial for our under- +standing of the structure, dynamics and ener- +getics of these phenomena and for the anal- +ysis and interpretation of astronomical obser- +vations. These models are described by set +of HD/MHD equations which can be solved +through sophisticated parallel numerical codes +as, for instance, the codes for astrophysical +plasmas FLASH (Fryxell et al. 2000) and +PLUTO (Mignone et al. 2012). Running these +codes requires high-performance parallel com- +puting systems (using thousands of proces- +sors in parallel) and a significant number of +numerical resources (in terms of CPU hours +and storage memory). The codes are optimized +to reduce the computational cost and to im- +prove the parallel efficiency when using several +thousands of processors. Typical 3D HD/MHD +simulations are executed using of the order of +10 thousands CPUs on parallel supercomput- +ers and require millions of computer hours for +the whole computation. +Over the years, the HD/MHD models of as- +tronomical phenomena have become increas- +ingly accurate and complex, also thanks to +the improvement of the algorithms for solving +the system of HD/MHD equations and to the +greater amount of numerical resources avail- +able for scientific research. Today, fully three- +dimensional (3D) HD/MHD simulations of as- +trophysical phenomena contain a great wealth +of scientific information, which can be diffi- +arXiv:2301.11334v1 [cs.HC] 26 Jan 2023 + +S. Orlando: Virtual reality for analysis and visualization of scientific models +283 +cult to unravel and extract, and produce large +amounts of data. For these reasons, modern +3D HD/MHD simulations pose a challenge for +their analysis and standard data visualization +for scientific purposes. +In the last decade, the potential of virtual +reality (VR) hardware and software has started +to be exploited in different fields for public +outreach and education with excellent results. +For example, online digital media stores and +education and public outreach websites (such +as those promoted by NASA1 and eduINAF2) +offer, among other things, high-impact VR +content in Astrophysics and Space Sciences. +Unfortunately, the routine use of VR environ- +ments for analyzing and visualizing models for +scientific research is still in its infancy. +Driven by the need for efficient tools +for analyzing and viewing 3D HD/MHD +models, in 2019 we launched 3DMAP-VR +(3-Dimensional Modeling of Astrophysical +Phenomena in Virtual Reality; see Sect. 2), a +laboratory for the development and testing of +VR-based tools for the analysis and visualiza- +tion of numerical scientific models (Orlando +et al. 2019b) and, more recently, StarBlast (see +Sect. 3) a standalone app available for free +on Steam that offers a VR tour of the re- +sults of stellar explosions accessible even to +the general public of non-experts. In 2022 we +started the development of a VR-based tool for +the analysis of numerical scientific simulations +(see Sect. 4). In the next sections, we introduce +these projects in more details. +2. The 3DMAP-VR Project +The project aims to provide an environment +for developing tools that exploit VR technolo- +gies for visualizing scientific models. In fact, +due to the complexity of modern 3D HD/MHD +simulations full of details that characterize the +structure and evolution of the simulated astro- +nomical objects, untangling the different pro- +cesses that govern the object under study and +extracting relevant information from the model +can be a difficult task that cannot be easily re- +1 https://chandra.si.edu/vr/ +2 https://edu.inaf.it +solved by traditional 2D representations of the +models as, for instance, in screen displays. For +example, the stellar debris ejected after the ex- +plosion of a supernova (SN) can be charac- +terized by rather complex structures in which +the distributions of the different species of el- +ements intertwine in a chaotic and, apparently, +disordered way. In these cases, traditional 2D +representations fail to describe inherently 3D +structures, making it difficult to identify and +follow the evolution of specific features and +phenomena of interest. +Conversely, exploring models immersively +through VR equipment allows scientists to +navigate and interact with their MHD mod- +els in a more natural and intuitive way. The +3DMAP-VR project aims at providing a VR- +based enviromnent for the exploration and vi- +sualization of the models. More specifically, +in the 3DMAP-VR environment, the workflow +for creating VR visualizations of models con- +sists in three steps. +i) Accurate 3D HD/MHD simulations per- +formed for scientific purposes. First, the 3D +HD/MHD models are produced through nu- +merical simulations that are executed on par- +allel supercomputers. The simulations account +for all the relevant physical processes in astro- +physical phenomena: gravity, magnetic-field- +oriented thermal conduction, energy losses due +to radiation, gas viscosity, deviations from +proton-electron temperature equilibration, de- +viations from the ionization equilibrium, cos- +mic rays acceleration, etc.. +ii) Tools for model analysis and for mesh- +ing and texture production of the different +model components. As a second step, interac- +tive and navigable 3D graphics of the astro- +physical simulations are realized by exploiting +the tools commonly used by the scientific com- +munity for the analysis of data, e.g., Interactive +Data Language, YT project, ParaView, Visit, +MeshLab, MeshMixer. A mixed technique +consisting of multilayer isodensity surfaces +with different opacities is used to realize the +3D graphics. +iii) Tools for the creation of objects that can +be explored in VR. Finally, a VR representation +of the astrophysical models is realized by up- + +284 +S. Orlando: Virtual reality for analysis and visualization of scientific models +Fig. 1. Galleries of models available on Artstation: https://www.artstation.com/saorlando4. +loading the 3D graphics to Sketchfab3, one of +the largest open access platforms for publish- +ing and sharing 3D VR and augmented real- +ity content. Once the 3D graphics have been +loaded, the VR representations of the model +are created through a friendly environment +provided by Sketchfab for defining the prop- +erties (e.g. opacity, textures, luminosity, etc.) +of the objects that make up the model and for +the post-processing of the model to improve +the rendering and highlight specific features of +interest. +In the framework of 3DMAP-VR, we have +realized several galleries of models, that can be +also explored in VR, on the Sketchfab4 site and +on Artstation5, a leading showcase platform +for art and design (see examples in Fig. 1). +Through these galleries, we promote wide dis- +semination of astrophysical model results for +both scientific and public outreach purposes. +More specifically, in the Sketchfab plat- +form we realized the gallery “Universe in +3 https://sketchfab.com +4 https://sketchfab.com/sorlando +5 https://www.artstation.com +hands6” dedicated to 3D HD/MHD models de- +veloped by our team for scientific purposes +and published in international scientific jour- +nals. For instance, thanks to these models, we +have investigated (see Fig. 2): accretion phe- +nomena in young stellar objects (e.g., Orlando +et al. 2011; Colombo et al. 2019); protostellar +jets (e.g., Bonito et al. 2011; Ustamujic et al. +2016); nova outbursts (e.g., Drake & Orlando +2010; Orlando & Drake 2012); the outcome of +SN explosions (e.g., Ono et al. 2020; Orlando +et al. 2020). +The assets derived from scientific simula- +tions have been very successful during pub- +lic outreach events and among non-experts. +This positive reception encouraged us to go +beyond numerical simulations. So we started +creating new models not based on numerical +simulations but illustrating astrophysical ob- +jects on the base of our current knowledge +of these phenomena. The first collection of +this new class of resources was “The art of +Astrophysical Phenomena7” where it is possi- +6 https://skfb.ly/oxHxq +7 https://skfb.ly/oxHx9 + +All +33 +Cosmic Explosions +13 +Stars +Planets +10 +Training +The science of science fictionS. Orlando: Virtual reality for analysis and visualization of scientific models +285 +Fig. 2. Examples of scientific models available on the Sketchfab platform: a young accreting star (up- +per left panel; e.g., Colombo et al. 2019), a protostellar jet (upper right; e.g., Ustamujic et al. 2016), +the 2010 outburst of nova U Scorpii (lower left; e.g., Drake & Orlando 2010), and the remnant of SN +1987A (lower right; e.g., Orlando et al. 2020). The corresponding interactive graphics can be visited at +the following links: https://skfb.ly/6RPnX, https://skfb.ly/6Rq69, https://skfb.ly/6Rn8u, +(https://skfb.ly/6YNNE. +ble to visit and explore artists’ views of astro- +physical phenomena and objects. Then we cre- +ated two additional collections: “Anatomy of +Astrophysical Objects8” and “The Science of +Science Fiction9”. The first collection reports +schematic representations of the structure of +astrophysical objects based on our knowledge. +The assets of the second collection spot famous +science fiction movies to highlight whether and +in which parts they get the science right (thus +providing accurate and plausible science). +The resources produced in the framework +of the 3DMAP-VR project were also used for +dissemination purposes by other international +scientific institutes. For example, a few as- +sets belonging to the galleries mentioned above +were used by NASA to make a series of 3D vi- +sualizations of astronomical objects observed +with X-ray observatories10 and 3D print kits +8 https://skfb.ly/oxHxz +9 https://skfb.ly/oxHxv +10 https://youtu.be/wIMB-D9l3I0 +have been produced for people with visual +impairments (e.g., Arcand et al. 2019, 2020). +Other models illustrating five of the most +popular supernova remnants (SNRs) in our +Galaxy were used in the standalone application +“StarBlast” (see Sect. 3) which exploits the +power of VR for the dissemination and educa- +tional projects within the scientific activities of +the international PHAROS project. The assets +were also used to create a series of videos de- +scribing astrophysical objects and phenomena, +available in Italian (SocialMente: condividi- +AMO l’universo11) and in English (Universe in +hands12). Finally, other assets have been used +to create exhibitions in public outreach events +and others have been requested for planetarium +exhibitions. +11 https://youtube.com/playlist?list= +PLITL-h-D-WYQQ7UDMP3CNhp_eYMN94UPU +12 https://youtube.com/playlist?list= +PLITL-h-D-WYRfehOoiTRx_bZ5YFI-bZ8w + +286 +S. Orlando: Virtual reality for analysis and visualization of scientific models +3. StarBlast: a VR tour of the +outcome of stellar explosions +As mentioned before, the remnants of SN ex- +plosions are characterised by a strong com- +plexity in the distribution of their physical and +chemical properties. The adiabatic cooling of +the expanding ejecta results in very cold, and +tenuous, metal-rich material, which is com- +monly considered as an important “dust fac- +tory” for our Galaxy. At the same time, the +supersonic expansion drives strong (highly su- +personic and super-Alfvenic) shocks, which +heat the interstellar material and the ejecta +themselves up to X-ray emitting temperature. +The presence of such a multi-phase material +is further complicated by intrinsic anisotropies +in the ejecta properties (velocity field, den- +sity, chemical composition, pressure, etc) and +by the knotty and complex interstellar envi- +ronment where the remnants evolve (e.g., Vink +2020). +Deciphering the structure of SNRs is cru- +cial for our understanding of the physics gov- +erning their formation and evolution. At the +same time, the physical modeling of these +objects relies on multi-dimensional HD/MHD +simulations, whose outputs pose a serious chal- +lenge for standard data visualization tools. The +intrinsic complexity of the system under exam, +indeed requires a novel approach for its proper +visualization and analysis. VR provides an im- +mersive experience which enhances the capa- +bility of grasping details of the simulations and +extract information on SNRs from the numeri- +cal models. +To this end, we developed “StarBlast: a +VR tour of the outcome of stellar explo- +sions” (hereafter StarBlast), a standalone app +which exploits the power of VR to offer an +immersive experience inside 3D simulations +and actual observations of SNRs and pulsar +wind nebulae (PWNe). StarBlast took advan- +tage of the successful results obtained within +the 3DMAP-VR project (see Sect. 2). The de- +velopment of the app was supported by the +COST Action PHAROS13, thanks to the pro- +posal “Real power of virtual reality: develop- +ing a standalone app for outreach and teach- +13 http://www.pharos.ice.csic.es/ +ing activities” (PI M. Miceli), and by INAF- +Osservatorio Astronomico G. S. Vaiana of +Palermo and the University of Palermo. +StarBlast was carefully designed to meet +the needs of different typologies of user, so +as to be adopted for outreach, teaching and +research projects. The app is easily accessi- +ble to the general public. A voice over (avail- +able in English, Italian and Spanish) offers a +simple but comprehensive description of the +objects during the navigation. Moreover, the +very smooth rendering of the astrophysical ob- +jects, and the user-friendly control system al- +low the users to “naturally” interact with SNRs +and PWNe, by literally using the hands (see +Fig. 3). On the other hand, StarBlast offers +also a deeper level of fruition, specifically con- +ceived for students, who can get a clear view +of the system and a more intuitive grasp on the +physics governing it. We successfully adopted +StarBlast in many public outreach events and +as a support for lectures in the courses of +“Astrophysics” and “Stellar Evolution” at the +University of Palermo (Italy). Finally, the app +offers a substantial support for researchers, +who can access state of the art MHD mod- +els and observations (together with the corre- +sponding references, which are provided for all +objects), to get a deeper level of diagnostics. +The objects currently included in the app +are: SN 1987a (based on the results presented +in Orlando et al. 2016, 2019a; Miceli et al. +2019) with its putative PWN (Greco et al. +2021, 2022), SN 1006 (Bocchino et al. 2011; +Orlando et al. 2012; Miceli et al. 2016), the +Crab Nebula (Olmi et al. 2016), Cassiopeia A +(Orlando et al. 2016, 2021, 2022) and IC 443 +(Ustamujic et al. 2021). However, the library of +astrophysical sources in StarBlast can be easily +enhanced with new objects. +StarBlast works with the most diffused VR +headsets (provided that SteamVR is installed) +and is freely available (download link in the +footnote14). +14 https://axt.oapa.inaf.it/starblast/ + +S. Orlando: Virtual reality for analysis and visualization of scientific models +287 +Fig. 3. Two examples of scenes in the StarBlast app showing the Crab nebula (on the left; Olmi et al. 2016) +and SN 1987A (on the right; Orlando et al. 2020). +4. A VR-based tool for the analysis of +numerical scientific simulations +In addition to the immersive visualization of +3D HD/MHD models, VR technologies can be +exploited for a deep analysis of these models. +Analysis tools based on VR would enable the +exploration of the 3D models in an interactive +way, achieving a highly detailed analysis that +cannot be performed with classical data visual- +ization and analysis platforms. However, VR- +based tools specifically tailored for the analysis +of numerical simulations are not routinely used +in the scientific community and it is necessary +to develop ad-hoc techniques and methods. +As a first step, we started exploring soft- +ware frameworks that could help develop this +kind of tools, probing their capabilities in terms +of dealing with the data we are interested +in and implementing analysis functions. The +high-interactivity and the powerful visualiza- +tion capabilities required, lead us to consider +the use of a game engine, a software frame- +work oriented to the development of video- +games. Usually, a modern game engine in- +cludes features like a level/map editor, a ren- +derer, a physics engine, a collision manager, +a scripting system, etc. The use of such a +framework allows to accelerate game develop- +ment, since developers can make use of the +mechanics embedded in the engine and fo- +cus on the game specific contents and rules. +Moreover, engines often provide platform ab- +straction, meaning that a developed game can +seamlessly be deployed on multiple platforms +(Windows, Mac, console, smartphone, etc.). +Now some game engines have evolved for all- +purpose 3D creation, including film-making, +architecture, and “serious game” simulations. +For our purpose, we selected Unreal Engine +(UE; Epic Games 2023), v.5, for its power- +ful graphics, for having a powerful scripting +language (C++), and for its licensing policy +allowing the use of full-featured software for +free with no royalty due for developed products +that generate a lifetime revenue < 1 M. Other +works already proposed the use of UE for sci- +entific data visualization and analysis (Friese +et al. 2008; Marsden & Shankar 2020; Smith & +Greenwood 2020; Mayerich et al. 2020) but, to +our knowledge, no VR tool has yet been devel- +oped for interactively performing data analysis +on models described by data-cubes, such as the +results of HD/MHD simulations or similar 3D +data (e.g. computer tomography scans). +We tested the feasibility of two operations, +that we consider the first steps for developing +VR analysis tools: volumetric rendering visu- +alization of our models, and performing inter- +active data processing leveraging an external +analysis software from within the UE gener- +ated VR environment. +Volumetric rendering is computationally +expensive, since it implies a full volumetric +ray-tracing: the interaction of light with every +volume element (voxel) has to be dynamically +calculated in real-time in order to reconstruct a +correct visual representation. A real-time volu- +metric rendering of a science simulation output +data-cube, in VR, is a task that many 3D visu- + +Torus +Jet288 +S. Orlando: Virtual reality for analysis and visualization of scientific models +alization software and platforms are not able to +perform. UE is highly optimized for rendering, +and we verified that it is indeed able to render +a VR volumetric model. Moreover, it is pos- +sible to change model visualization parameters +in real-time, allowing a high level of user inter- +action. The first step to achieve the volumetric +rendering is to use the simulation output data, +combining its variables, to generate 3D arrays +containing representations of the model (e.g., +the total mass density, the mass density of spe- +cific components, the temperature, the X-ray +emission, etc.). We used a model of the SNR +IC 443 (Ustamujic et al. 2021) to create two +3D arrays of 5123 elements that represent the +mass density of the SN ejecta, and the mass +density of an interacting molecular cloud. Each +3D array is then sliced along one dimension +obtaining a sequence of tiled slices. A 2D im- +age is generated, containing in its red and blue +channels the tiled slices from the two 3D ar- +rays. This procedure, that we perform using a +Python script, allows us to convert the visu- +alization data to an image, a format that UE +can import as 2D Texture. Since an image has +four channels (red, green, blue, transparency), +it is possible to embed the data of up to four +visual representations. Within UE, we convert +the data to a Volumetric Texture, that in turn is +used as a base to create a Volumetric Material. +A Material, assigned to an object within a vir- +tual environment, describes how it will be dis- +played, defining its color, transparency, emis- +sivity, etc. Assigning the Volumetric Material +to a base object placed in the VR environment, +in our case a sphere, allows UE to perform +the volumetric rendering of the 3D model, as +shown in Fig. 4. We also implemented a con- +trol panel, visible in the virtual environment as +floating over the user wrist, featuring sliders +that can control in real-time some properties of +the rendered model. We tested, as a proof-of- +concept, the variation of the emission intensity +of the model and the intensity balance between +the two components of the model (see Fig. 4). +The result is very promising, indicating that it +is possible to modify visualizations properties +on the fly while keeping a fluid rendering in +VR. We also implemented the possibility to in- +teract with the model, grabbing it and moving +Fig. 4. Volumetric rendering of an SNR IC 443 +model (Ustamujic et al. 2021) within a UE gener- +ated VR environment. The model was produced by +a HD simulation and contains two components, the +ejecta density on the red channel and the molecu- +lar cloud density in the blue channel. The floating +interface allows to change in real-time the emissive +intensity of the model and the intensity balance be- +tween the two components. +and rotating it as if it were held in the user +hand, and to enlarge it or scale it down using +intuitive hand gestures. +As a second feasibility test we checked if +it is possible to use UE in synergy with an ex- +ternal data analysis software. Building a com- +plete analysis suite fully within UE, while tech- +nically feasible, would require a huge effort, a +lot of manpower with high programming skills +and deep understanding of data manipulation +algorithms and methods. We investigated the +use of ParaView (Kitware 2022), widely used +advanced software for analysis of 3D data- +cubes, from within UE. ParaView exposes C +APIs that could be exploited from UE for direct +integration. For a preliminary concept demon- +strator, however, we adopted an indirect and +simpler approach. From the VR environment, +built with UE, we seamlessly call a Python +script that uses ParaView to load a 3D model +(HD/MHD simulation output), applies an anal- +ysis pipeline (i.e. extracts a iso-density surface, +a contour), and save the resulting 3D object to +the disk. The newly created 3D asset is then +loaded and rendered within the VR environ- + +Vol. model intens +Molec. cloud <-> EjectaS. Orlando: Virtual reality for analysis and visualization of scientific models +289 +Fig. 5. Iso-density contour of SNR IC 443 ejecta +(see Ustamujic et al. 2021) generated with ParaView +and rendered in VR within UE. The floating inter- +face allows to change the density value of the con- +tour. The ”Set contour value” button executes an +analysis pipeline, leveraging ParaView to create an +updated contour that is then rendered in the environ- +ment. +ment (see the contour of SNR IC 443 in Fig. 5). +We can pass parameters to the script to mod- +ify the analysis pipeline, e.g. changing the den- +sity used to extract the iso-surface using the +bottom slider shown in Fig. 5. The delay be- +tween the trigger in the VR world and the visu- +alization of the model resulting from the anal- +ysis depends on the size of the data-cube, on +the computer performances, and on the anal- +ysis pipeline; for extracting a contour from a +2563 elements data-cube we found a delay of +about 4 s. +We thus demonstrated the feasibility of +two fundamental tasks with UE, the volumet- +ric rendering of a model created for scientific +purposes, and interacting with an external anal- +ysis software to perform data processing. The +next steps for developing VR analysis tools are +the C++ implementation, within UE, of some +analysis operations (e.g. slicing), and the inte- +gration of ParaView, through its APIs, to sen- +sibly speed up the analysis process. +5. Conclusions +In this contribution we reported the recent +achievements obtained at INAF-Osservatorio +Astronomico di Palermo in the development of +VR-based tools aimed at the analysis and vi- +sualization of 3D scientific models describing +astronomical objects and phenomena. These +tools allow to explore the models interactively +in an immersive fashion, resulting in highly de- +tailed analysis that cannot be performed with +traditional data visualization (e.g., based on +screen displays) and analysis platforms. In ad- +dition our tools have been proved to be very +powerful in the production of high-impact VR +content for efficient audience engagement and +outreach. +Acknowledgements. We thank the anonymous ref- +eree for the careful reading of the paper. S.O. and +M.M. acknowledge financial contribution from the +PRIN INAF 2019 grant “From massive stars to su- +pernovae and supernova remnants: driving mass, en- +ergy and cosmic rays in our Galaxy” and the INAF +mainstream program “Understanding particle accel- +eration in galactic sources in the CTA era”. The de- +velopment of the app StarBlast was supported by +the COST Action PHAROS, thanks to the proposal +“Real power of virtual reality: developing a stan- +dalone app for outreach and teaching activities”. +We acknowledge the high performance comput- +ing (HPC) facility at CINECA through the ISCRA +programme and the SCAN (Sistema di Calcolo +per l’Astrofisica Numerica) HPC facility at INAF- +Osservatorio Astronomico di Palermo for the avail- +ability of HPC resources and support. +References +Arcand, K., Jubett, A., Watzke, M., et al. 2019, +JCOM Journal of Science Communication, +18, 18040201 +Arcand, K. K., Price, S. 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T., +et al. 2021, A&A, 645, A66 +Smith, +M. +& +Greenwood, +M. +2020, +in +Transactions +of +the +American +Nuclear +Society - Volume 123 (AMNS) +Ustamujic, S., Orlando, S., Bonito, R., et al. +2016, A&A, 596, A99 +Ustamujic, S., Orlando, S., Greco, E., et al. +2021, A&A, 649, A14 +Vink, J. 2020, Physics and Evolution of +Supernova Remnants + diff --git a/ldFIT4oBgHgl3EQfsCtp/content/tmp_files/load_file.txt b/ldFIT4oBgHgl3EQfsCtp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..397b9416f022f99f4c20e310a2e01202e4969d65 --- /dev/null +++ b/ldFIT4oBgHgl3EQfsCtp/content/tmp_files/load_file.txt @@ -0,0 +1,441 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf,len=440 +page_content='Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 75, 282 © SAIt 2022 Memorie della Virtual reality for the analysis and visualization of scientific numerical models S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Miceli2,1, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Lo Cicero1, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Ustamujic1 1 INAF-Osservatorio Astronomico di Palermo, Piazza del Parlamento 1, 90134, Palermo, Italy e-mail: salvatore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='orlando@inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='it 2 Dipartimento di Fisica e Chimica E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Segr`e, Universit`a degli Studi di Palermo, Via Archirafi, 36, 90123, Palermo, Italy Received: Day Month Year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Accepted: Day Month Year Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The complexity of the data generated by (magneto)-hydrodynamic (HD/MHD) simulations requires advanced tools for their analysis and visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The dramatic im- provements in virtual reality (VR) technologies have inspired us to seek the long-term goal of creating VR tools for scientific model analysis and visualization that would allow re- searchers to study and perform data analysis on their models within an immersive environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Here, we report the results obtained at INAF-Osservatorio Astronomico di Palermo in the development of these tools, which would allow for the exploration of 3D models inter- actively, resulting in highly detailed analysis that cannot be performed with traditional data visualization and analysis platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Additionally, these VR-based tools offer the ability to produce high-impact VR content for efficient audience engagement and awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Introduction The development of (magneto)-hydrodynamic (HD/MHD) models of astronomical objects and phenomena can be crucial for our under- standing of the structure, dynamics and ener- getics of these phenomena and for the anal- ysis and interpretation of astronomical obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' These models are described by set of HD/MHD equations which can be solved through sophisticated parallel numerical codes as, for instance, the codes for astrophysical plasmas FLASH (Fryxell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2000) and PLUTO (Mignone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Running these codes requires high-performance parallel com- puting systems (using thousands of proces- sors in parallel) and a significant number of numerical resources (in terms of CPU hours and storage memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The codes are optimized to reduce the computational cost and to im- prove the parallel efficiency when using several thousands of processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Typical 3D HD/MHD simulations are executed using of the order of 10 thousands CPUs on parallel supercomput- ers and require millions of computer hours for the whole computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Over the years, the HD/MHD models of as- tronomical phenomena have become increas- ingly accurate and complex, also thanks to the improvement of the algorithms for solving the system of HD/MHD equations and to the greater amount of numerical resources avail- able for scientific research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Today, fully three- dimensional (3D) HD/MHD simulations of as- trophysical phenomena contain a great wealth of scientific information, which can be diffi- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='11334v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='HC] 26 Jan 2023 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando: Virtual reality for analysis and visualization of scientific models 283 cult to unravel and extract, and produce large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' For these reasons, modern 3D HD/MHD simulations pose a challenge for their analysis and standard data visualization for scientific purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' In the last decade, the potential of virtual reality (VR) hardware and software has started to be exploited in different fields for public outreach and education with excellent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' For example, online digital media stores and education and public outreach websites (such as those promoted by NASA1 and eduINAF2) offer, among other things, high-impact VR content in Astrophysics and Space Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Unfortunately, the routine use of VR environ- ments for analyzing and visualizing models for scientific research is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Driven by the need for efficient tools for analyzing and viewing 3D HD/MHD models, in 2019 we launched 3DMAP-VR (3-Dimensional Modeling of Astrophysical Phenomena in Virtual Reality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2), a laboratory for the development and testing of VR-based tools for the analysis and visualiza- tion of numerical scientific models (Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2019b) and, more recently, StarBlast (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 3) a standalone app available for free on Steam that offers a VR tour of the re- sults of stellar explosions accessible even to the general public of non-experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' In 2022 we started the development of a VR-based tool for the analysis of numerical scientific simulations (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' In the next sections, we introduce these projects in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The 3DMAP-VR Project The project aims to provide an environment for developing tools that exploit VR technolo- gies for visualizing scientific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' In fact, due to the complexity of modern 3D HD/MHD simulations full of details that characterize the structure and evolution of the simulated astro- nomical objects, untangling the different pro- cesses that govern the object under study and extracting relevant information from the model can be a difficult task that cannot be easily re- 1 https://chandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='edu/vr/ 2 https://edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='it solved by traditional 2D representations of the models as, for instance, in screen displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' For example, the stellar debris ejected after the ex- plosion of a supernova (SN) can be charac- terized by rather complex structures in which the distributions of the different species of el- ements intertwine in a chaotic and, apparently, disordered way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' In these cases, traditional 2D representations fail to describe inherently 3D structures, making it difficult to identify and follow the evolution of specific features and phenomena of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Conversely, exploring models immersively through VR equipment allows scientists to navigate and interact with their MHD mod- els in a more natural and intuitive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The 3DMAP-VR project aims at providing a VR- based enviromnent for the exploration and vi- sualization of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' More specifically, in the 3DMAP-VR environment, the workflow for creating VR visualizations of models con- sists in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' i) Accurate 3D HD/MHD simulations per- formed for scientific purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' First, the 3D HD/MHD models are produced through nu- merical simulations that are executed on par- allel supercomputers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The simulations account for all the relevant physical processes in astro- physical phenomena: gravity, magnetic-field- oriented thermal conduction, energy losses due to radiation, gas viscosity, deviations from proton-electron temperature equilibration, de- viations from the ionization equilibrium, cos- mic rays acceleration, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='. ii) Tools for model analysis and for mesh- ing and texture production of the different model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' As a second step, interac- tive and navigable 3D graphics of the astro- physical simulations are realized by exploiting the tools commonly used by the scientific com- munity for the analysis of data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Interactive Data Language, YT project, ParaView, Visit, MeshLab, MeshMixer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' A mixed technique consisting of multilayer isodensity surfaces with different opacities is used to realize the 3D graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' iii) Tools for the creation of objects that can be explored in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Finally, a VR representation of the astrophysical models is realized by up- 284 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando: Virtual reality for analysis and visualization of scientific models Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Galleries of models available on Artstation: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='artstation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='com/saorlando4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' loading the 3D graphics to Sketchfab3, one of the largest open access platforms for publish- ing and sharing 3D VR and augmented real- ity content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Once the 3D graphics have been loaded, the VR representations of the model are created through a friendly environment provided by Sketchfab for defining the prop- erties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' opacity, textures, luminosity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=') of the objects that make up the model and for the post-processing of the model to improve the rendering and highlight specific features of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' In the framework of 3DMAP-VR, we have realized several galleries of models, that can be also explored in VR, on the Sketchfab4 site and on Artstation5, a leading showcase platform for art and design (see examples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Through these galleries, we promote wide dis- semination of astrophysical model results for both scientific and public outreach purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' More specifically, in the Sketchfab plat- form we realized the gallery “Universe in 3 https://sketchfab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='com 4 https://sketchfab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='com/sorlando 5 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='artstation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='com hands6” dedicated to 3D HD/MHD models de- veloped by our team for scientific purposes and published in international scientific jour- nals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' For instance, thanks to these models, we have investigated (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2): accretion phe- nomena in young stellar objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' protostellar jets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Bonito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Ustamujic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' nova outbursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Drake & Orlando 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando & Drake 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' the outcome of SN explosions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Ono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The assets derived from scientific simula- tions have been very successful during pub- lic outreach events and among non-experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' This positive reception encouraged us to go beyond numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' So we started creating new models not based on numerical simulations but illustrating astrophysical ob- jects on the base of our current knowledge of these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The first collection of this new class of resources was “The art of Astrophysical Phenomena7” where it is possi- 6 https://skfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ly/oxHxq 7 https://skfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ly/oxHx9 All 33 Cosmic Explosions 13 Stars Planets 10 Training The science of science fictionS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando: Virtual reality for analysis and visualization of scientific models 285 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Examples of scientific models available on the Sketchfab platform: a young accreting star (up- per left panel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Colombo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2019), a protostellar jet (upper right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Ustamujic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016), the 2010 outburst of nova U Scorpii (lower left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Drake & Orlando 2010), and the remnant of SN 1987A (lower right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The corresponding interactive graphics can be visited at the following links: https://skfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ly/6RPnX, https://skfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ly/6Rq69, https://skfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ly/6Rn8u, (https://skfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ly/6YNNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' ble to visit and explore artists’ views of astro- physical phenomena and objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Then we cre- ated two additional collections: “Anatomy of Astrophysical Objects8” and “The Science of Science Fiction9”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The first collection reports schematic representations of the structure of astrophysical objects based on our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The assets of the second collection spot famous science fiction movies to highlight whether and in which parts they get the science right (thus providing accurate and plausible science).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The resources produced in the framework of the 3DMAP-VR project were also used for dissemination purposes by other international scientific institutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' For example, a few as- sets belonging to the galleries mentioned above were used by NASA to make a series of 3D vi- sualizations of astronomical objects observed with X-ray observatories10 and 3D print kits 8 https://skfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ly/oxHxz 9 https://skfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ly/oxHxv 10 https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='be/wIMB-D9l3I0 have been produced for people with visual impairments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Arcand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Other models illustrating five of the most popular supernova remnants (SNRs) in our Galaxy were used in the standalone application “StarBlast” (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 3) which exploits the power of VR for the dissemination and educa- tional projects within the scientific activities of the international PHAROS project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The assets were also used to create a series of videos de- scribing astrophysical objects and phenomena, available in Italian (SocialMente: condividi- AMO l’universo11) and in English (Universe in hands12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Finally, other assets have been used to create exhibitions in public outreach events and others have been requested for planetarium exhibitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 11 https://youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='com/playlist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='list= PLITL-h-D-WYQQ7UDMP3CNhp_eYMN94UPU 12 https://youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='com/playlist?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='list= PLITL-h-D-WYRfehOoiTRx_bZ5YFI-bZ8w 286 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando: Virtual reality for analysis and visualization of scientific models 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' StarBlast: a VR tour of the outcome of stellar explosions As mentioned before, the remnants of SN ex- plosions are characterised by a strong com- plexity in the distribution of their physical and chemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The adiabatic cooling of the expanding ejecta results in very cold, and tenuous, metal-rich material, which is com- monly considered as an important “dust fac- tory” for our Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' At the same time, the supersonic expansion drives strong (highly su- personic and super-Alfvenic) shocks, which heat the interstellar material and the ejecta themselves up to X-ray emitting temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The presence of such a multi-phase material is further complicated by intrinsic anisotropies in the ejecta properties (velocity field, den- sity, chemical composition, pressure, etc) and by the knotty and complex interstellar envi- ronment where the remnants evolve (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Vink 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Deciphering the structure of SNRs is cru- cial for our understanding of the physics gov- erning their formation and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' At the same time, the physical modeling of these objects relies on multi-dimensional HD/MHD simulations, whose outputs pose a serious chal- lenge for standard data visualization tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The intrinsic complexity of the system under exam, indeed requires a novel approach for its proper visualization and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' VR provides an im- mersive experience which enhances the capa- bility of grasping details of the simulations and extract information on SNRs from the numeri- cal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' To this end, we developed “StarBlast: a VR tour of the outcome of stellar explo- sions” (hereafter StarBlast), a standalone app which exploits the power of VR to offer an immersive experience inside 3D simulations and actual observations of SNRs and pulsar wind nebulae (PWNe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' StarBlast took advan- tage of the successful results obtained within the 3DMAP-VR project (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The de- velopment of the app was supported by the COST Action PHAROS13, thanks to the pro- posal “Real power of virtual reality: develop- ing a standalone app for outreach and teach- 13 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='pharos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='csic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='es/ ing activities” (PI M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Miceli), and by INAF- Osservatorio Astronomico G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Vaiana of Palermo and the University of Palermo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' StarBlast was carefully designed to meet the needs of different typologies of user, so as to be adopted for outreach, teaching and research projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The app is easily accessi- ble to the general public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' A voice over (avail- able in English, Italian and Spanish) offers a simple but comprehensive description of the objects during the navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Moreover, the very smooth rendering of the astrophysical ob- jects, and the user-friendly control system al- low the users to “naturally” interact with SNRs and PWNe, by literally using the hands (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' On the other hand, StarBlast offers also a deeper level of fruition, specifically con- ceived for students, who can get a clear view of the system and a more intuitive grasp on the physics governing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We successfully adopted StarBlast in many public outreach events and as a support for lectures in the courses of “Astrophysics” and “Stellar Evolution” at the University of Palermo (Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Finally, the app offers a substantial support for researchers, who can access state of the art MHD mod- els and observations (together with the corre- sponding references, which are provided for all objects), to get a deeper level of diagnostics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The objects currently included in the app are: SN 1987a (based on the results presented in Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Miceli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2019) with its putative PWN (Greco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2021, 2022), SN 1006 (Bocchino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Miceli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016), the Crab Nebula (Olmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016), Cassiopeia A (Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016, 2021, 2022) and IC 443 (Ustamujic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' However, the library of astrophysical sources in StarBlast can be easily enhanced with new objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' StarBlast works with the most diffused VR headsets (provided that SteamVR is installed) and is freely available (download link in the footnote14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 14 https://axt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='oapa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='it/starblast/ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando: Virtual reality for analysis and visualization of scientific models 287 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Two examples of scenes in the StarBlast app showing the Crab nebula (on the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Olmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016) and SN 1987A (on the right;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' A VR-based tool for the analysis of numerical scientific simulations In addition to the immersive visualization of 3D HD/MHD models, VR technologies can be exploited for a deep analysis of these models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Analysis tools based on VR would enable the exploration of the 3D models in an interactive way, achieving a highly detailed analysis that cannot be performed with classical data visual- ization and analysis platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' However, VR- based tools specifically tailored for the analysis of numerical simulations are not routinely used in the scientific community and it is necessary to develop ad-hoc techniques and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' As a first step, we started exploring soft- ware frameworks that could help develop this kind of tools, probing their capabilities in terms of dealing with the data we are interested in and implementing analysis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The high-interactivity and the powerful visualiza- tion capabilities required, lead us to consider the use of a game engine, a software frame- work oriented to the development of video- games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Usually, a modern game engine in- cludes features like a level/map editor, a ren- derer, a physics engine, a collision manager, a scripting system, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The use of such a framework allows to accelerate game develop- ment, since developers can make use of the mechanics embedded in the engine and fo- cus on the game specific contents and rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Moreover, engines often provide platform ab- straction, meaning that a developed game can seamlessly be deployed on multiple platforms (Windows, Mac, console, smartphone, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Now some game engines have evolved for all- purpose 3D creation, including film-making, architecture, and “serious game” simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' For our purpose, we selected Unreal Engine (UE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Epic Games 2023), v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='5, for its power- ful graphics, for having a powerful scripting language (C++), and for its licensing policy allowing the use of full-featured software for free with no royalty due for developed products that generate a lifetime revenue < 1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Other works already proposed the use of UE for sci- entific data visualization and analysis (Friese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Marsden & Shankar 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Smith & Greenwood 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Mayerich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2020) but, to our knowledge, no VR tool has yet been devel- oped for interactively performing data analysis on models described by data-cubes, such as the results of HD/MHD simulations or similar 3D data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' computer tomography scans).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We tested the feasibility of two operations, that we consider the first steps for developing VR analysis tools: volumetric rendering visu- alization of our models, and performing inter- active data processing leveraging an external analysis software from within the UE gener- ated VR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Volumetric rendering is computationally expensive, since it implies a full volumetric ray-tracing: the interaction of light with every volume element (voxel) has to be dynamically calculated in real-time in order to reconstruct a correct visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' A real-time volu- metric rendering of a science simulation output data-cube, in VR, is a task that many 3D visu- Torus Jet288 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando: Virtual reality for analysis and visualization of scientific models alization software and platforms are not able to perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' UE is highly optimized for rendering, and we verified that it is indeed able to render a VR volumetric model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Moreover, it is pos- sible to change model visualization parameters in real-time, allowing a high level of user inter- action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The first step to achieve the volumetric rendering is to use the simulation output data, combining its variables, to generate 3D arrays containing representations of the model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', the total mass density, the mass density of spe- cific components, the temperature, the X-ray emission, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We used a model of the SNR IC 443 (Ustamujic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2021) to create two 3D arrays of 5123 elements that represent the mass density of the SN ejecta, and the mass density of an interacting molecular cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Each 3D array is then sliced along one dimension obtaining a sequence of tiled slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' A 2D im- age is generated, containing in its red and blue channels the tiled slices from the two 3D ar- rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' This procedure, that we perform using a Python script, allows us to convert the visu- alization data to an image, a format that UE can import as 2D Texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Since an image has four channels (red, green, blue, transparency), it is possible to embed the data of up to four visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Within UE, we convert the data to a Volumetric Texture, that in turn is used as a base to create a Volumetric Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' A Material, assigned to an object within a vir- tual environment, describes how it will be dis- played, defining its color, transparency, emis- sivity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Assigning the Volumetric Material to a base object placed in the VR environment, in our case a sphere, allows UE to perform the volumetric rendering of the 3D model, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We also implemented a con- trol panel, visible in the virtual environment as floating over the user wrist, featuring sliders that can control in real-time some properties of the rendered model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We tested, as a proof-of- concept, the variation of the emission intensity of the model and the intensity balance between the two components of the model (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The result is very promising, indicating that it is possible to modify visualizations properties on the fly while keeping a fluid rendering in VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We also implemented the possibility to in- teract with the model, grabbing it and moving Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Volumetric rendering of an SNR IC 443 model (Ustamujic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2021) within a UE gener- ated VR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The model was produced by a HD simulation and contains two components, the ejecta density on the red channel and the molecu- lar cloud density in the blue channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The floating interface allows to change in real-time the emissive intensity of the model and the intensity balance be- tween the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' and rotating it as if it were held in the user hand, and to enlarge it or scale it down using intuitive hand gestures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' As a second feasibility test we checked if it is possible to use UE in synergy with an ex- ternal data analysis software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Building a com- plete analysis suite fully within UE, while tech- nically feasible, would require a huge effort, a lot of manpower with high programming skills and deep understanding of data manipulation algorithms and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We investigated the use of ParaView (Kitware 2022), widely used advanced software for analysis of 3D data- cubes, from within UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' ParaView exposes C APIs that could be exploited from UE for direct integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' For a preliminary concept demon- strator, however, we adopted an indirect and simpler approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' From the VR environment, built with UE, we seamlessly call a Python script that uses ParaView to load a 3D model (HD/MHD simulation output), applies an anal- ysis pipeline (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' extracts a iso-density surface, a contour), and save the resulting 3D object to the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The newly created 3D asset is then loaded and rendered within the VR environ- Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' model intens Molec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' cloud <-> EjectaS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Orlando: Virtual reality for analysis and visualization of scientific models 289 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Iso-density contour of SNR IC 443 ejecta (see Ustamujic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2021) generated with ParaView and rendered in VR within UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The floating inter- face allows to change the density value of the con- tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The ”Set contour value” button executes an analysis pipeline, leveraging ParaView to create an updated contour that is then rendered in the environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' ment (see the contour of SNR IC 443 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We can pass parameters to the script to mod- ify the analysis pipeline, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' changing the den- sity used to extract the iso-surface using the bottom slider shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The delay be- tween the trigger in the VR world and the visu- alization of the model resulting from the anal- ysis depends on the size of the data-cube, on the computer performances, and on the anal- ysis pipeline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' for extracting a contour from a 2563 elements data-cube we found a delay of about 4 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We thus demonstrated the feasibility of two fundamental tasks with UE, the volumet- ric rendering of a model created for scientific purposes, and interacting with an external anal- ysis software to perform data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The next steps for developing VR analysis tools are the C++ implementation, within UE, of some analysis operations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' slicing), and the inte- gration of ParaView, through its APIs, to sen- sibly speed up the analysis process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Conclusions In this contribution we reported the recent achievements obtained at INAF-Osservatorio Astronomico di Palermo in the development of VR-based tools aimed at the analysis and vi- sualization of 3D scientific models describing astronomical objects and phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' These tools allow to explore the models interactively in an immersive fashion, resulting in highly de- tailed analysis that cannot be performed with traditional data visualization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', based on screen displays) and analysis platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' In ad- dition our tools have been proved to be very powerful in the production of high-impact VR content for efficient audience engagement and outreach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We thank the anonymous ref- eree for the careful reading of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' acknowledge financial contribution from the PRIN INAF 2019 grant “From massive stars to su- pernovae and supernova remnants: driving mass, en- ergy and cosmic rays in our Galaxy” and the INAF mainstream program “Understanding particle accel- eration in galactic sources in the CTA era”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' The de- velopment of the app StarBlast was supported by the COST Action PHAROS, thanks to the proposal “Real power of virtual reality: developing a stan- dalone app for outreach and teaching activities”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' We acknowledge the high performance comput- ing (HPC) facility at CINECA through the ISCRA programme and the SCAN (Sistema di Calcolo per l’Astrofisica Numerica) HPC facility at INAF- Osservatorio Astronomico di Palermo for the avail- ability of HPC resources and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' References Arcand, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Jubett, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Watzke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2019, JCOM Journal of Science Communication, 18, 18040201 Arcand, K.' metadata={'source': 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2010, ApJ, 720, L195 Epic Games, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2023, Unreal Engine Friese, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Herrlich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', & Wolter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2008, in New Frontiers for Entertainment Computing, ed.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Miceli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Petruk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', & Pumo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2012, ApJ, 749, 156 Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' & Drake, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2012, MNRAS, 419, 2329 Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Miceli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Petruk, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2019a, A&A, 622, A73 Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Miceli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Pumo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', & Bocchino, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016, ApJ, 822, 22 Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Ono, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Nagataki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2020, A&A, 636, A22 Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Pillitteri, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Bocchino, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Daricello, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', & Leonardi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2019b, Research Notes of the American Astronomical Society, 3, 176 Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Reale, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Peres, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', & Mignone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2011, MNRAS, 415, 3380 Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Wongwathanarat, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Janka, H.' metadata={'source': 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al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2021, A&A, 645, A66 Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' & Greenwood, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2020, in Transactions of the American Nuclear Society - Volume 123 (AMNS) Ustamujic, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Bonito, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2016, A&A, 596, A99 Ustamujic, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Orlando, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', Greco, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2021, A&A, 649, A14 Vink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} +page_content=' 2020, Physics and Evolution of Supernova Remnants' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFIT4oBgHgl3EQfsCtp/content/2301.11334v1.pdf'} diff --git a/ltFPT4oBgHgl3EQfHzTL/content/tmp_files/2301.13009v1.pdf.txt b/ltFPT4oBgHgl3EQfHzTL/content/tmp_files/2301.13009v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6c46d622c7fe8f2d46c38d9ebf91c94b32c5a0de --- /dev/null +++ b/ltFPT4oBgHgl3EQfHzTL/content/tmp_files/2301.13009v1.pdf.txt @@ -0,0 +1,2324 @@ +DEFI: DATA-DRIVEN CHARACTERISATION OF UNISWAP V3 +ECOSYSTEM & AN IDEAL CRYPTO LAW FOR LIQUIDITY POOLS +Deborah Miori +Mathematical Institute +University of Oxford +deborah.miori@maths.ox.ac.uk +Mihai Cucuringu +Department of Statistics & Mathematical Institute +University of Oxford +The Alan Turing Institute, London, UK +mihai.cucuringu@stats.ox.ac.uk +January 31, 2023 +ABSTRACT +The Uniswap v3 ecosystem is built upon liquidity pools, where pairs of tokens are exchanged subject +to a fee. We propose a systematic workflow to extract a meaningful but tractable sub-universe out of +the current > 6,000 pools. We filter by imposing minimum levels on individual pool features, e.g. +liquidity locked and agents’ activity, but also maximising the interconnection between the chosen +pools to support broader dynamics. Then, we investigate liquidity consumption behaviour on the most +relevant pools for Jan-June 2022. We propose to describe each liquidity taker by a transaction graph, +which is a complete graph where nodes are transactions on pools and edges have weights from the +time elapsed between pairs of transactions. Each graph is embedded into a vector by our own variant +of the NLP rooted graph2vec algorithm. Thus, we are able to investigate the structural equivalence of +liquidity takers behaviour and extract seven clusters with interpretable features. Finally, we introduce +an ideal crypto law inspired from the ideal gas law of thermodynamics. Our model tests a relationship +between variables that govern the mechanisms of each pool, i.e. liquidity provision, consumption, +and price variation. If the law is satisfied, we say the pool has high cryptoness and demonstrate that it +constitutes a better venue for the activity of market participants. Our metric could be employed by +regulators and practitioners for developing pool health monitoring tools and establishing minimum +levels of requirements. +Keywords Decentralised Finance · Uniswap v3 · Network Analysis · NLP · Clustering · Ideal gas law +1 +Introduction +A blockchain is a type of distributed ledger technology (DLT) that stores users transactions on an increasingly long +sequence of blocks of data. This ledger is duplicated and distributed across the entire network of computer systems on +the blockchain to allow the validation of new transactions by the peer-to-peer (P2P) computer network and subsequent +addition of blocks. During 2007 and 2008, the person(s) known via the pseudonymous Satoshi Nakamoto designed +the Bitcoin blockchain and described it in the whitepaper [1]. The project was released as an open source software +in 2009 and from that moment onwards, Bitcoin started slowly acquiring increasing value and seeing higher trading +volumes. However, Bitcoin is not Turing complete and indeed it lacks the capacity for logical loops and conditionals. +This limitation fueled the rise of the Ethereum blockchain, which was first described in the 2013 whitepaper [2]. +Ethereum supports smart contract functionality and, due to this, it is able to offer financial instruments that do not rely +on intermediaries such as brokerages, exchanges or banks. Thus, Ethereum built the foundations for Decentralised +Finance (DeFi). Within DeFi, individuals can lend, trade and borrow using software that automatically broadcasts +their intentions for P2P verification, and records valid financial actions on a blockchain. Decentralised Exchanges +(DEXs) are a direct result of this setup, and started being designed and implemented mainly from 2017. They differ +from the usual centralised exchanges, since they are non-custodial and leverage the self-execution of smart contracts +for P2P trading, allowing users to retain control of their private keys and funds. One of the first and most established +arXiv:2301.13009v1 [q-fin.TR] 20 Dec 2022 + +JANUARY 31, 2023 +DEXs at the time of writing is Uniswap, built indeed on Ethereum and launched in November 2018. There exist three +versions of Uniswap (namely v1, v2, v3, see the whitepapers https://hackmd.io/@HaydenAdams/HJ9jLsfTz, [3], +[4] respectively) that update its design and evolve its functionalities. +Since we focus on Uniswap v3 data in this research, and specifically investigate its ecosystem, we consider essential to +first highlight some of its core aspects. Uniswap is an automated market maker (AMM), and in particular, a constant +function market maker (CFMM). This implies that digital assets are traded without centralised permission and the +pricing occurs following a mathematical formula, rather than relying on an order book as in traditional exchanges. +Uniswap smart contracts hold liquidity reserves of various tokens, and trades initiated by liquidity takers (LTs) are +executed directly against these reserves. Reserves are pooled between a network of liquidity providers (LPs) who supply +the system with tokens in exchange for a proportional share of transaction fees. A standard Uniswap liquidity pool +allows the exchange, or swap, of two assets via the constant product market maker mechanism +(x − ∆x) × +� +y + (1 − γ +106 )∆y +� += x × y = k, +(1) +where x, y are the current pool reserves of tokens X, Y respectively. Then, k tracks the evolution of liquidity of the +pool, and γ ∈ {100, 500, 3000, 10000} (i.e. {1, 5, 30, 100} basis points) denotes the feeTier characteristic of the pool. +Here, we are assuming that a LT sells an amount ∆y of token Y to the pool and receives ∆x of token X back. The +exchange rate Z between the two digital assets is given by the proportion of respective reserves in the pool, which +changes following the trades of LTs. A proportion γ of each swap is kept by the pool to reward LPs for their service. +Swaps do not change k, while this invariant does vary if new liquidity is minted (added) or burned (destroyed) in the +pool by LPs. Of course, higher liquidity assures less price slippage for LTs and is thus preferred. LPs profit an amount +proportional to their involvement into the whole liquidity of the pool for each trade occurred. However, they also incur +impermanent loss due to the need to stake both tokens to provide liquidity, while bearing the risk of varying exchange +rates. +In Uniswap v3, it is important to notice that concentrated liquidity is implemented. This means that LPs can choose +the range of prices and proportions over which they stake their tokens, and they will collect LTs’ fees when the +exchange rate of executed trades lies between two ticks over which they are indeed actively providing liquidity. For +completeness, it is also worth mentioning that every action (i.e. creation of a pool, swap, mint or burn operation...) that +occurs on Uniswap, or in the general DeFi universe, must be validated and registered on the blockchain before being +considered executed. This introduces a further cost for the initiator of the action, who needs to pay non-negligible gas +fees [5] to miners to compensate them for the computational power they consume. This is especially significant for +blockchains that use a Proof-of-Work consensus protocol, such as Ethereum until September 2022. Indeed, Ethereum +then transitioned to Proof-of-Stake via the upgrade named “The Merge”, allowing validation of transactions not to only +rely on computational power, and opening the opportunity to have lower gas fees and enhanced users participation in +DeFi (despite not yet reality). +Many further interesting new and old finance concepts live within DeFi and beyond DEXs. One such concept worth +mentioning first is that of stablecoins. Stablecoins are digital assets that are pegged to the value of a fiat currency, and +can be useful to exit risky positions while remaining inside the crypto ecosystem. Some stablecoins are fiat-backed +(e.g. USDC, Tether), while others are backed by an over-collateralised pool of cryptocurrencies (e.g. DAI). There +also exist algorithmic coins (e.g. UST), which closely resemble traditional pegged exchange rates and are indeed also +vulnerable to speculative attacks, i.e. as it happened with the Terra-Luna crash in May 2022. Apart from stablecoins, +DeFi provides several lending protocols (e.g. Aave, Compound, Instadapp, Maker), protocols for derivatives trading +(e.g. dYdX, Futureswap, Nexus), and DEX aggregators (e.g. 1inch) that optimise routing to take advantage of the best +exchange rates across multiple other exchanges. In [6], we find an interesting study of the interactions between different +blockchain protocols. +While DeFi is fascinating, it is also the stage of many scams, speculative high-risk investments, direct blockchain +attacks, and money laundering events. On top of that, its complexity and atomicity might disadvantage small users, +whose transactions can e.g. be re-ordered before execution by the validators for their own profit, known here as miner +extractable value (MEV). Despite the current effort of regulators to penetrate the crypto world and establish some +equilibrium between centralisation and decentralisation, the current situation and possible upcoming developments are +still highly confusing, especially for outsiders or newcomers. Interesting overviews and critical thoughts are presented +in [7] and [8], where the latter work especially discusses enforcing tax compliance, anti-money laundering laws and +how to possibly prevent financial malfeasance. [9] further studies different layers of DeFi and the related risks involved, +i.e. at the blockchain, protocol, pool and token level. They focus on Uniswap and propose a related risk parity approach +for portfolio construction. The current academic research is also at its very early stages in terms of understanding the +inner dynamics of DEXs and external relationships with the well-known traditional stock market, especially from an +empirical and data-driven point of view. In [10], the authors investigate how promoting a greater diversity of price-space +partitions in Uniswap v3 can simultaneously benefit both liquidity providers and takers. [11] studies whether AMM +2 + +JANUARY 31, 2023 +protocols, such as Uniswap, can sustainably retain a portion of their trading fees for the protocol and the expected +outflow of traders to competitor venues. Inefficiencies between Uniswap and SushiSwap are investigated in [12], where +sub-optimal trade routing is addressed. However, [13] shows that constant product markets should closely track the +reference market price. Flows of liquidity between Uniswap v2 pools are studied in [14], while [15] and [16] show the +difficulty of earning significant returns by providing liquidity in Uniswap v3. +Main Contributions. +We divert from the available literature in many ways. To the best of our knowledge, this is the +first study to methodically define a set of pools that are necessary to be considered for a full view of Uniswap dynamics. +We assess both the pools’ inner features and their interconnectedness, and deliver a workflow for extracting significant +sub-universes of pools in time, which can be completely reproduced by the reader. Then, we leverage on this first point +to cluster and characterise the broad behaviour of LTs. Due to the eclectic set of pools that we consider, our view should +approximate well the overall patterns of the entire ecosystem. Our final contribution is to propose an ideal crypto law +for liquidity pools, inspired by the ideal gas law from thermodynamics. We provide motivation for it and show that +pools with high cryptoness, i.e. strongly adhering to our law, are healthier crypto environments on which to trade. The +level of cryptoness of a pool can evolve in time and hence it is important to track it, along with various metrics that +quantify the risks associated to the respective pool. +Structure of the Paper. +In Section 2, we identify the most important and interconnected liquidity pools for different +time windows within 2022. Next, we cluster LTs according to their behaviour on the relevant sub-universes in Section 3. +Due to the complexity of this ecosystem, we draw on intuition from Natural Language Processing (NLP) and graph +embedding techniques to assess structural equivalence of trading behaviour in a novel way. Section 4 expands our +investigations by proposing an ideal crypto law to simultaneously model LTs, LPs and price dynamics for each pool +under consideration. Finally, we summarise our thoughts and discuss future research directions in Section 5. +2 +Systematic selection of Uniswap v3 pools of interest +2.1 +Empirical introduction to the ecosystem +At the time of writing, Uniswap v3 is the latest implementation of this DEX. It launched in May 2021, introduced the +concept of concentrated liquidity and allowed multiple feeTiers. For each Uniswap version N = 1, 2, 3, the addresses +of related liquidity pool smart contracts are stored in the respective “UniswapVNFactory” contracts. We access them on +Etherscan1 by querying for transactions related to UniswapVNFactory addresses234 and filtering for methods “Create +Exchange”, “Create Pair” or “Create Pool”, for the three versions respectively. We find total numbers of 3, 857 and 992 +and 40 associated calls until 15 November 2022, which we agglomerate at daily level and plot in Fig. 1. The dates of +transition from Uniswap v1 to v2, and v2 to v3, are also depicted. It is interesting to notice that the previous protocols +remain active after the transitions, but their liquidity can be easily moved to the new Uniswap versions via “Migrator” +contracts. In terms of pools, the main differences between v1, v2 and v3 are that the first protocol allows only pools +where one token is ETH and feeTier γ = 3000, while the second one introduces the ability to create a pool between +1https://etherscan.io/ +2UniswapV1Factory: 0xc0a47dFe034B400B47bDaD5FecDa2621de6c4d95 +3UniswapV2Factory: 0x5C69bEe701ef814a2B6a3EDD4B1652CB9cc5aA6f +4UniswapV3Factory: 0x1f98431c8ad98523631ae4a59f267346ea31f984 +Figure 1: Daily count of new pools created via UniswapV1Factory, UniswapV2Factory and UniswapV3Factory smart +contracts. The two vertical orange lines depict the dates of official transition from Uniswap v1 to v2, and from v2 to v3. +3 + +Uniswap newpools created in time +35 +v1tov2 +30 +V2toV3 +v1 +25 +V2 +Count +20 +V3 +15 +10 +5 +0 +2019-012019-072020-012020-072021-012021-072022-012022-072023-01JANUARY 31, 2023 +Figure 2: Evolution in time of the TVL in USD on Uniswap main Ethereum chain. The higher the TVL, the more liquid +the ecosystem is considered to be. In orange, we show the dates of transition from Uniswap v1 to v2, and from v2 to v3. +any two tokens. Then, v3 expands pools to possibly have also feeTier 100, 500 or 10000. Although there is a total +of 4, 889 pools directly created with UniswapVNFactory contracts, the majority of them is the result of the 2020-21 +cryptomania and inflated creation of new tokens. This translates to many pools not containing any relevant amount of +liquidity locked, but which do not disappear due to the immutability of the blockchain. On the other hand, we also +expect wrapped calls to the Factory contracts and thus refer to the above as a lower bound to the number of pools +created. +As a last note, we shown the evolution of Uniswap liquidity on its main Ethereum chain in Fig. 2. The data are +downloaded via the Defi Llama5 API and we proxy liquidity by the total amount of USD locked on the protocol, i.e. +Total Value Locked (TVL). +2.2 +Data download and coarse refinement of pools +Each version of Uniswap has its own dedicated subgraph, which has a precise endpoint for querying data and a schema +to expose the available fields. Our terminology follows the Uniswap v3 schema6 and we download pools data via the +related subgraph. Our first aim is to identify the pools most representative of the Uniswap ecosystem, which we interpret +as having significant liquidity consumption and provision events, but also showing high interconnectedness. To this end, +we download the latest summary data of all possible pools, full historical record of liquidity consumption operations, and +full record of liquidity provision actions. Then, we develop a systematic approach that aims at increasingly discarding +layers of pools with weakest features first and then weakest dynamics too, respectively in this subsection and in the next +one. The final data set will be our starting point for the subsequent analyses, but aims at being useful to a wider group +of researchers that desire to empirically investigate Uniswap v3. +Download summary data of pools. +As of 15 November 2022, we download the latest “Pool” data as described in +Uniswap v3 subgraph schema. We download pools in descending transaction count (txnCount) order, since this variable +is strictly increasing in time, while e.g. TVL does not need to be. This allows us to select a universe of pools which +have had a minimum number of transactions thus far. We apply this weak initial filtering on the first 6, 000 pools by +txnCount, and find that only 1, 344 pools report at least 1, 000 transactions by 15 November 2022. Then, we also +restrict ourselves to pools where both exchanged tokens are traded in at least 3 pools (e.g. token T is traded against a +stablecoin, against ETH and against ETH with different feeTier), in order to focus on interesting dynamics of the full +ecosystem. The result is that we subset to a universe of 696 pools to consider in our study. +Download LP data. +We download liquidity provision data for these 696 pools and find non-empty entries for 629 of +them. Liquidity provision data are downloaded to have a historical record of all liquidity mint and burn operations on +each pool, with related USD value. By computing the total cumulative sum of LPs activity, we proxy the TVL in USD +that each pool contains at every moment in time and denote it as “proxyTVL”. Unfortunately, we cannot simply rely on +the “PoolDayData” values provided by the subgraph due to incoherences found when cross-checking with Ethereum +blochckain data on Etherscan. +Uniswap v3 data start on 6 May 2021, when the transition from the previous version of the protocol successfully +completed. While for the first months only the 500, 3000 and 10000 feeTiers were implemented, in November 2021 a +5https://defillama.com/ +6https://github.com/Uniswap/v3-subgraph/blob/main/schema.graphql#L1 +4 + +le10 +TVL in time +1.2 +v1 to v2 +v2 to v3 +1.0 +ETH chain +0.8 +USD +0.6 +0.4 +0.2 +0.0 +2019-012019-07 +2020-012020-07 +72021-01 2021-072022-01 2022-072023-01JANUARY 31, 2023 +fourth feeTier γ = 100 was activated. This generated structural flows, noise and adjustments that we prefer to exclude +from our analyses. On top of that, we recognise that the transition of Uniswap’s foundation blockchain (i.e. Ethereum) +from Proof-of-Work to Proof-of-Stake in September 2022 could have triggered turbulences on the ecosystem too. Thus, +we decide to focus our analyses on the six-months period from January 2022 to the end of June 2022, which we consider +as the most representative of the actual DEX dynamics. We check for every pool if our proxyTVL passes the threshold +of 1, 000, 000 USD (one million dollars) at any point before the end of June 2022. This is motivated by the aim to find +pools that were liquid enough at some point in our time window to show interesting behaviour of LTs and LPs. We find +282 pools that satisfy this further requirement. Of these, 210 had that much TVL already at some point before January +2022, and 261 at some point before April 2022. While some pools acquire relevance as time passes, other ones can +also lose liquidity, as in the extreme case of pools related to the Terra-Luna crash of May 2022. As a final detail, we +highlight that we additionally check whether a pool has at least one million USD in TVL for two consecutive points in +time in order to avoid pools where a substantial amount of liquidity is minted and immediately burned by an agent, to +likely take advantage of specific external information. +Download LT data. +For the above 282 pools, we download related liquidity consumption data and find all non-empty +data sets. Thus, we are left with a final set of 282 liquidity pools, for which we have a summary file, a LP database and +a LT database each. This completes our coarse filtering of pools, which implements the least invasive possible initial +requirements, while still filtering down the universe of Uniswap v3 liquidity pools to a tractable number of instances. +The diagram in Fig. 3 summarises the steps completed. +Figure 3: Summary diagram of the filtration steps pursued during our coarse refinement of pools. Stronger con- +straints on the TVL and txnCount of pools will follow in the next subsection, such as an attention to maximise the +interconnectedness of the final sub-universe of pools. +2.3 +Final refinements +Stronger TVL and txnCount constraints. +From our first coarse filtration, we recover 282 pools to consider further +in Uniswap v3 analyses. However, we shall be stricter about the minimum number of transactions taking place on a +pool and its TVL in time, in order to lower the noise-to-signal ratio in the data. As already motivated, we want to focus +on the six-months window [January, July) 2022, which we denote as our case A. We also consider five sub-ranges, +namely the two three-months windows [January, April), [April, July) that we denote as cases B1/B2, and the three +two-months windows [January, March), [March, May), [May, July), that we call cases C1/C2/C3. For each +case and related time window [start, end), we extract the pools with at least 1, 000 transactions before start (where the +number of transactions in time is calculated via the cumulative sum of both swap events and mint or burn operations) and +that also had at least 1, 000, 000 USD in proxyTVL both at the start and end of the interval. Considering sub-ranges +allows us to further account for the appearance of new pools that became significantly liquid or active after January +2022, or pools that lost the majority of their liquidity before July 2022. For cases A/B1/B2/C1/C2/C3 in order, we +find respectively 113/126/148/131/146/155 pools that satisfy the above requirements, for which we save the related +addresses and information. Taking the union of these sets of pools, we notice that we are considering 177 different +pools overall. Of these, five pools belong to the 100 feeTier, 28 pools to the 500 feeTier, 84 pools to the 3000 feeTier +and 60 pools to the 10000 feeTier. +To gain a brief insight into the most liquid and active venues, we consider the pools extracted for case A and plot in Fig. +4a the 10 pools with highest proxyTVL at the end of June 2022, and in Fig. 4b the 10 pools with highest total number +of transactions over the six months of relevance. For the first pool in the ranking of both measures, we plot the related +evolution of liquidity and daily number of transactions in Fig. 5. As a convention, we refer to pools with the format +“SYMBOL1-SYMBOL2/feeTier”, where we deploy the trading symbols of the two tokens exchanged by the pool. +Stablecoins, wrapped Ether (WETH) and wrapped Bitcoin (WBTC) dominate the landscape of tokens swapped in the +most liquid and active venues, which is expected since they are the oldest, most established, or safest cryptocurrencies +that agents can trade and develop strategies onto. Then, it is interesting to observe how the DAI-USDC/100 pool is +5 + +Summary data +ofUniswapv3 +pools as of +TVLand txnCount constraints +15Nov2022 +Interconnectedness +>6,000 +1,344 +629 +282 +pools +pools +pools +poolsJANUARY 31, 2023 +(a) Pools with highest liquidity at the end of June 2022. +(b) Pools with highest total number of transactions during case A. +Figure 4: The 10 most liquid and active pools for case A, i.e. over the time window between January and June 2022. +much younger than the USDC-WETH/500 one, but quickly gained strong liquidity levels due to its tokens being both +stablecoins. +(a) Full history of TVL for the pool with highest final TVL. +(b) Daily txns count for the pool with highest final TVL. +(c) Full history of TVL for the pool with highest total txnCount. +(d) Daily txns count for the pool with highest total txnCount. +Figure 5: Evolution of liquidity, i.e. proxyTVL, and daily number of transactions for the pool with highest TVL at the +end of June 2022 (DAI-USDC/100), and for the one with largest total number of transactions during the full six-months +window of case A (USDC-WETH/500). +Filtering of pools by interconnectedness. +Considering the full list of data sets of liquidity provision and consumption +actions to pursue investigations of more than 100 pools becomes highly computationally expensive very soon. Thus, +we further subset our pools of interest by requiring minimum levels of interconnection between them. This is also +functional in assuring a focus on the deepest dynamics that characterise the Uniswap ecosystem as a whole. +For each one of our cases A/B1/B2/C1/C2/C3, we build a weighted graph G = (P, E). The set of nodes P denotes +relevant pools, and edges (p, q) ∈ E with p, q ∈ P have weights wpq that encode some measure of similarity defined +below. We start by considering two possible different measures of connection between pools: +1. Number of common LTs (or LPs) active on both pools, which are identified by the entry “origin” in the +Uniswap data. +2. Number of common smart contracts, i.e. “senders” in the Uniswap data, called by origins to execute swap +transactions (or to execute liquidity provision operations). +6 + +1eB +Total value locked at the end of june 2o22 +proxyTVL +6- +5 +3 - +2 - +1Total number of txns between January and June 2o22 +100 +tot txns +10leB +DAI-USDC/10D +proxyTVL +6 +5 - +4 +3 +2 - +1 +0 - +Dec +Jan +Feb +Mar +Apr +222 +JunDAI-USDC/10D +1750 +dailyTxns +1500 +1250 +750 +500 +250 +Jen +Feb +Mar +Apr +2422 +Jun +imestampleB +USDC-WETH/SO0 +proxyTVL +t +3 +02 +1 +0 - +Jul +Crt +Jan +Apr +2422USDC-WETH/SO0 +000 +- +dailyTxns +16000 +14000 +12000 +Co +DO +8 +DO+9 +4000 +Jean +Feb +Mar +Apr +Mey +Jun +2422 +timestampJANUARY 31, 2023 +Figure 6: The intersection of origins and senders is always zero since the former are wallets of users and the latter +smart contracts. Recipients can instead be both, hinting to more complex patterns in the execution of transactions. +To be precise, here we are specifically considering the sub-universe of pools relevant for case A at the end of all our +refinements. +We separate between a focus on liquidity consumption or provision in the above measures, since the two dynamics +differ substantially, and one might prefer to enhance the sub-universe under consideration to be more representative of +one or the other. Of course, the intersection or union of the results can be then used to pursue broader analyses. To +clarify some Uniswap terminology, every “swap action” is initiated by an origin O, then it calls a smart contract referred +to as sender S, and ends to the recipient R. In liquidity provision, only the origin and sender of operations are relevant. +Figure 6 shows the distribution of number of origins, senders and recipients in each pool for both LT and LP data over +the time window of case A. We also show the distribution of the intersection between origins, senders and recipients’ +addresses. +We now proceed to studying the relationship between the size of each graph’s giant component and a minimum threshold +on the value of the measure used to create the link between each pair of pools. After fixing a threshold, we consider +the pools in the related giant component as our relevant interconnected sub-universe. Figure 7 shows the variation in +size of the giant component for case A, when modifying the minimum number of common origins or senders for LT +and LP data. We aim at considering the tails of the distributions for each case (i.e. time interval), which amounts to +∼ 20/30 pools in each instance, to retain the most significant connections and possible dynamics of the ecosystem. +For case A, this results in the choice of thresholds 2, 000 and 100 for minimum common origins and common senders +respectively, on the LT data. Similarly, we choose thresholds 30 and 3 for minimum common origins and common +senders respectively, on the LP data. Finally, we consider the intersection of survival pools for the two graphs generated +by LT data, and find 27 common pools (out of the 34 and 36 pools, respectively in each graph). For LP data, we find a +number of 19 final relevant pools (from the intersection of 25 and 30 pools). The full pipeline is repeated for cases +B1/B2/C1/C2/C3. +(a) LT data. +(b) LP data. +Figure 7: Evolution of the size of the giant component for graphs of pools in case A, when varying the threshold of +common origins and senders for (a) swap transactions, (b) liquidity provision operations. +7 + +SWAP data +LP Data +104 +105 +000 +0 +o +103 +104 +Count +Count +103 +102 +0 +00 +102 +101, +no. 0 +s'ou +no. R +O&S +O&R +oou +s'ou +O&5size giant component +swap +100 +commonorigins +75 +50 +25 +0 +0 +500 +QQOT +1500 +2000 +2500 +3000 +3500 +minimum threshold +size giant component +swap +100 +commonsenders +75 +50 +25 +0 +0 +50 +100 +150 +200 +250 +minimumthresholdcomponent +LP +100 +common origins +75 +size giant +50 +25 +0 +0 +20 +40 +60 +Bo +minimum threshold +size giant component +LP +100 +commonsenders +75 +50 +25 +0 +0 +2 +4 +6 +B +10 +12 +14 +minimum thresholdJANUARY 31, 2023 +For the interested reader, Fig. 8 depicts the distribution of origins, also called Externally Owned Accounts (EOAs), +that are both LTs and LPs on each same pool for case A before filtering by interconnectedness. This quantity is +shown both as a ratio of the total number of LTs and of LPs on each pool. We witness an extreme case for pool +“WETH-sETH2/3000”, for which more than 20% of the total amount of LTs are also LPs. However, the number of +LTs that also act as liquidity providers is a negligible minority. If we further count the total number of LTs, LPs +and LTs acting also as LPs regardless of the pool, we find 479, 161 and 23, 952 and 13, 640 such market participants, +respectively. Approximately half of LPs also act as LTs, but LTs acting also as LPs are still a small minority. +Figure 8: Distribution of origins that act both as LTs and LPs on the same pool. +Final enhancement on pools for liquidity consumption analysis. +Ideally, we should also consider the flow of funds +across pools and find the related most interconnected graph. However, this is intractable if using only Uniswap data and +not the full list of Ethereum blockchain transactions. Indeed, LTs are active across different DeFi protocols and can +easily move liquidity from one venue to another and back. We propose an approximation to the problem by taking +advantage of the fact that each trader’s transaction can include more actions, which happen “instantaneously but in +order” when the full transaction is validated. Thus, if a LT executes two swaps of the form X → Y, Y → Z for tokens +X, Y, Z in one same transaction, then we interpret Y as a bridge between the action of selling X to buy Z. We view +this as an indication of the flow of (smart) money between pools and of possible arbitrage opportunities, relevant to the +LT sub-universe. In summary, we consider the following steps: +1. Merge all LT data before the interconnectedness analyses, e.g. data for the 113 pools of case A. +2. Keep all the transactions for which there are at least two inner actions, i.e. same “transaction id” but different +“logIndex” in Uniswap terminology. +3. For each resulting transaction: +(a) For each token that appears in the transaction actions, keep a flow list of related buying (−1) or selling +(+1) trades in all the related pools by looking at the sign of the amount swapped by the pool. +(b) For each token, consider its flow list and find all the occasions when a −1 is immediately followed by a ++1 (i.e. the token was first bought in a pool and then sold in another pool, acting as one of our bridges). +(c) Save this occurrence of a flow between pools as a bridge transaction, +where we are approximating only jumps of length one. As an example, for a flow list of the form [−1, +1, +1], we only +consider the flow as from the first pool to the second one. For more specific analyses, one could consider the specific +amounts traded and check the relative proportions exchanged from the first pool to the second and third ones, but this is +outside the scope of our current investigation. +We extract all bridge transactions between pools and create a directed graph for each one of our temporal cases. Nodes +are pools as usual, and edges are built for each pair of pools that have at least some number of bridge transactions +between them. Of course, each pair of pools can have up to two edges between them according to the direction of related +bridge transactions. Then, we keep the largest connected component from the undirected version of the graph and add +the resultant set of nodes to the LTs pools saved from the previous interconnectedness analyses. For case A, we require +at least 800 bridge transactions between two pools to create the related edges. The resultant giant component (see Fig. +9a for a visualisation) has 22 nodes, seven of which were not included in our LT set of pools from the previous analyses +and are thus added. Figure 9b further highlights the nodes with highest eigenvector centrality in the graph, where we +can especially notice how several pools of WETH against a stablecoin are proposed. This is intuitively sensible, since +LTs can take advantage of routing to complete specific re-balancing of tokens via more liquid and favourable pools, +which tend to have stablecoins, WETH and WBTC as their tokens, as shown in the earlier analyses. +The final list of 34 pools that we propose to consider for LTs analyses is: DAI-WETH/3000, CEL-WETH/3000, +USDC-UOS/10000, DAI-USDC/100, SPELL-WETH/3000, WETH-CRV/10000, USDC-USDT/500, DAI-FRAX/500, +8 + +No. of EOAs that are both LTs and LPs on the same pool, wrt total LTs +No. of EOAs that are both LTs and LPs on the same pool, wrt total LPs +101 , +. of pools +No. +2 + oOT +100 +0.00 +0.05 +0.10 +0.15 +0.20 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Fraction +FractionJANUARY 31, 2023 +(a) The resulting giant component, with edge weights reported in both directions. +(b) Pools with highest eigenvector centralities. +Figure 9: Results from our bridges investigation for case A, which covers the six-months window from January to June +2022. If a LT executes two swaps X → Y, Y → Z one after the other (for tokens X, Y, Z), then we interpret Y as a +bridge between the action of selling X to buy Z. We save all pairs of pools for which there is a common token that acts +as a bridge, with the related number of occurrences of bridge transactions. Then, we create a directed graph where +nodes are pools and edges are built for each pair of pools that have at least 800 bridge transactions between them. +WETH-BTRFLY/10000, GALA-WETH/3000, WETH-USDT/3000, WBTC-USDC/3000, DAI-USDT/500, UNI- +WETH/3000, WETH-ENS/3000, DAI-USDC/500, WBTC-WETH/500, MATIC-WETH/3000, DAI-WETH/500, +WETH-USDT/500, USDC-WETH/500, LINK-WETH/3000, WBTC-WETH/3000, FXS-WETH/10000, FRAX- +USDC/500, USDC-WETH/3000, USDC-WETH/10000, LUSD-USDC/500, HEX-USDC/3000, USDC-NCR/500, +SHIB-WETH/3000, DYDX-WETH/3000, USDC-USDT/100, HEX-WETH/3000. +Regarding LP pools, we have instead the following 19 pools: WETH-CRV/10000, MKR-WETH/3000, WETH- +USDT/3000, WBTC-USDC/3000, UNI-WETH/3000, WETH-ENS/3000, WBTC-WETH/500, MATIC-WETH/3000, +DAI-WETH/500, WETH-USDT/500, USDC-WETH/500, LINK-WETH/3000, WBTC-WETH/3000, USDC- +WETH/3000, SHIB-WETH/3000, WBTC-USDT/3000, USDC-USDT/100, USDC-USDT/500, SHIB-WETH/10000. +The final results for cases B1/B2/C1/C2/C3 are then listed in Appendix A, for the benefit of the reader that can +use these sub-universes of pools as starting point for their own investigations on Uniswap v3. In our next steps, we +specifically focus on the pools extracted for longest cases A/B1/B2. +3 +Structural Investigation of the Uniswap v3 Ecosystem +3.1 +Clustering of Liquidity Takers +3.1.1 +Overview and pre-processing +The DeFi ecosystem has grown increasingly complex in the recent years. The first step to shed more light on its +intrinsic features and dynamics is to better understand its own components, which is what motivates the following +empirical investigation of LTs trading behaviour on Uniswap v3. This is a non-trivial task, for a number of reasons. +First of all, agents can easily generate numerous crypto wallets, and hence in some sense, “multiply” their identities +to hide or obfuscate their full behaviour. Their actions are then generally spread over a broad set of possible pools, +vary significantly in size both within and across different types of pools, and also happen with evolving frequencies +over time. Applying the usual initial clustering methodologies would indeed be difficult (i.e. defining a set of features +that characterise pools to then perform dimensionality reduction, and finally compute similarity measures), due to the +9 + +WETH-BT +Y/10000 +1051 +.034 +WETH +HEx- +3000 +1316 +803 +WSD +5/500 +1413936 +WETH +T/3000 +1208 +USDE +T100 +2985 +6467 +1753 +HEx-0 +000 +USDC- +1/3000 +984 +2012 +676 +USDC +T/500 +WBTC +H/500 +USDg +/500 +2041 +DAI-L +/100 +USDC-I +DOOQT +WBTC- +C/3000 +1541 +911 +1007 +2743 +DDAI-W +000E +9 +WBTC- +H/3000 +DAI-M +/500 +944 +10211254 +1461 +666 +DAI-F +y500 +DAI-U +1500 +DAI-U +/500Eigenvector Centrality ? 0 +eig-centrality +0.5 +0.4 +0.3 +0.2 +0.1 +0.0 +////////JANUARY 31, 2023 +Figure 10: Distribution of total number of transactions (txns) performed by LTs during case A. We show the full +distribution, the result after requiring a minimum of at least 25 transactions, and the distribution after applying +thresholds of minimum 60 and maximum 15, 000 transactions. The latter scenario results in our final set for case A, +which comprises 3, 415 LTs. A small cluster of LTs much more active than others is already discernible. +complexity of the ecosystem. Thus, we propose a novel method to express and cluster structural trading equivalence +of agents on multiple environments by leveraging both network analysis and NLP techniques. A sample of possibly +external features is then used to judge and characterise the groups unravelled and extract insights on the main species of +agents present in the ecosystem. +We focus on the LT data for our three longest periods A/B1/B2, which we defined and described in the previous section. +For each case, we first look at the distribution of the total number of transactions performed by the different LTs over +each full time window. We require a minimum number of transactions completed by each LT, since considering only a +very small sample of trades per agent would not provide meaningful structural information. We impose a lower bound +of at least 10 transactions per month on average over the time window of each case, and define maximum thresholds by +considering the respective own distributions and removing only extreme singular outliers for computational purposes. +We show the initial total distribution for case A in Fig. 10, where we also highlight how it changes when requiring +a minimum number of transactions equal to 25, and when we require our final minimum and maximum thresholds +of 60 and 15, 000 total number of transactions, respectively. For cases B1/B2, we require the range 30 to 5, 000 +transactions for the former, and 30 to 11, 000 transactions for the latter. Overall, we find a number of LTs approximately +between 3, 500 and 5, 000 for all our periods A/B1/B2. This altogether defines the final sets of LTs along with their +transactions. Next, we proceed to computing their embeddings, which are subsequently used for the final clustering +stage. +3.1.2 +Methodology +NLP background and graph2vec. +The field of Natural Language Processing (NLP) studies the development of +algorithms for processing, analysing, and extracting meaningful insights from large amounts of natural language +data. Examples of its myriad applications include sentiment analysis of news articles, text summary generation, +topic extraction and speech recognition. One of the turning points in NLP was the development of the word2vec +word embedding technique [17], which considers sentences as directed subgraphs with nodes as words, and uses a +shallow two-layer neural network to map each word to a unique vector. The learned word representations capture +meaningful syntactic and semantic regularities, and if pairs of words share a particular relation then they are related +by the same constant offset in the embedding space. As an example, the authors observe that the singular/plural +relation is captured, e.g. xapple − xapples ≈ xcar − xcars, where we denote the vector for word i as xi. Words +sharing a common context in the corpus of sentences also lie closer to each other, and therefore, relationships such as +xking − xman + xwoman ≈ xqueen are satisfied with the analogies indeed predicted by the model. +Taking inspiration from this idea of preserving knowledge of the context window of a word in its embedding, the +node2vec algorithm [18] learns a mapping of nodes in a graph to a low-dimensional space of features by maximising +the likelihood of preserving network neighbourhoods of nodes. The optimisation problem is given by +max +f +� +s∈S +log Pr(NL(s)|f(s)), +(2) +where G = (S, T) is a graph with nodes S and edges T, f is the mapping function for nodes to n-dimensional vectors +that we aim to learn, and NL(s) ⊂ S is the network neighbourhood of node s generated with sampling strategy L. The +10 + +Case A +all +105 +min 25 +60-15,000 +104 +S17 +102 +101 +100 +0 +5000 +10000 +15000 +20000 +Total no. txns over the periodJANUARY 31, 2023 +latter is designed by the authors of node2vec as a biased random walk procedure, which can be tuned to either focus on +sampling a broader set of immediate neighbours, or a sequence of deeper nodes at increasing distances. Then, Problem +(2) is solved for f by simulating several random walks from each node and applying stochastic gradient descent (SGD) +and backpropagation. +By taking a further step towards general language representations, [19] proposes the unsupervised algorithm Paragraph +Vector (also known as doc2vec), which learns continuous fixed-length vector embeddings from variable-length pieces +of text, i.e. sentences, paragraphs and documents. The vector representation is trained to predict the next word of a +paragraph from a sample of the previous couple of sentences. Both word vectors and paragraph vectors need to be +trained, which is again performed via SGD and backpropagation. +As doc2vec extends word2vec, graph2vec [20] is a neural embedding framework that aims to learn data-driven +distributed representations of an ensemble of arbitrary sized graphs. The authors propose to view an entire graph as a +document, and to consider the rooted subgraphs around every node in the graph as words that compose the document, +in order to finally apply doc2vec. This approach is able to consider non-linear substructures and has thus the advantage +to preserve and capture structural equivalences. One necessary requirement to pursue this analogy is for nodes to have +labels, since differently labelled nodes can be then considered as different words. These labels can be decided by the +user, or can be simply initiated with the degree of each node. Thus, doc2vec considers a set of graphs G = {G1, G2...}, +where the nodes S of each graph G = (S, T, λ) can be labelled via the mapping function λ : S → L to the alphabet L. +The algorithm begins by randomly initialising the embeddings for all graphs in the set G, then proceeds with extracting +rooted subgraphs around every node in each one of the graphs, and finally iteratively refines the corresponding graph +embedding in several epochs via SGD and backpropagation, in the spirit of doc2vec. The rooted subgraphs act as +the context words, which are used to train the paragraph (i.e. graph) vector representations. Subgraphs are extracted +following the Weisfeiler-Lehman (WL) relabeling process [21]. The intuition is that, for each node in a graph, all its +(breadth-first) neighbours are extracted up to some depth d. Labels are then propagated from the furthest nodes to the +root one, and concatenated at each step. In this way, a unique identifier for each node is identified from its “context” +and the full set can be used to train an embedding for the graph. The optimisation problem thus becomes +max +f ′ +� +G∈G +� +s∈S +log Pr(gd +W L(s)|f ′(G)), +(3) +where the aim is to maximise the probability of the WL subgraphs given the current vector representation of the graph. +Here, f ′ is a mapping function of graphs to n-dimensional representations, and gd +W L are WL subgraphs with depth d. +A modification of graph2vec for LTs embedding. +For each one of our cases A/B1/B2, we consider all the related +LTs and their full set of transactions on the sub-universe of LTs’ pools of relevance. We then introduce the concept of a +transaction graph Gtxn, which we use to represent the behaviour of each active agent. +Definition 3.1 (Transaction graph). A transaction graph Gtxn = (S, T, W) is the complete weighted graph where +nodes S are the swap actions that the LT under consideration has completed, and edges (s, r) ∈ T with s, r ∈ S +are built between every pair of nodes. Each edge has a weight wsr ∈ W, which encodes the amount of time ∆t (in +seconds) elapsed between the two transactions s, r. Each node s ∈ S has a label ls from the alphabet L , which uniquely +identifies the pool that the swap was executed into. Importantly, L is shared among the full set of LTs and related +transaction graphs. +Labels in the alphabet L differentiate between swaps executed on different pools, i.e. pools with unique combination of +tokens exchanged and feeTier implemented. This implies that the algorithm receives as input only general identifiers of +pools , while we can consider intuitive differences (e.g. expected volatility of the exchange rate on pools of stablecoins +versus on pools of more exotic tokens) only afterwards, when assessing and investigating the meaningfulness and +interpretability of the extracted clusters. +We now have a set of graphs representing LTs, and our aim is to find a n-dimensional vector representation of each +one of its elements. We cannot plainly apply the graph2vec algorithm, since the concept of neighbours of a node is +irrelevant in a complete graph. Thus, we modify its mechanism to take advantage of the weight that the different links +between nodes have, while maintaining the overall intuition. For each node s ∈ S of a graph Gtxn, we sample a set of +neighbours Ntxn(s) by generating random numbers from a uniform distribution between [0, 1] and comparing them +to the cut-value of the edges between the node and possible neighbours. If the value is below the cut-value, then the +link is kept and the associated node added to Ntxn(s). In this way, the probability of an edge to be chosen is inversely +proportional to its weight ∆t, and the sub-structures kept represent clustered activity in time. +11 + +JANUARY 31, 2023 +Definition 3.2 (Cut-value). The cut-value C(wsr) of an edge (s, r) ∈ T with weight wsr ∈ W in graph Gtxn = +(S, T, W) is computed as +C(wsr) = +H(f scal(wsr)) +H(f scal(min W)), +with +H(wsr) = +� +2 +π exp −w2 +sr +2 +, wsr ≥ 0, +f scal(wsr) = wsr − min W +(max W)/|S| , +(4) +where we are using a half-norm that is shifted and scaled to adapt to each LT’s extreme features, i.e. min W and +max W. The final cut-value is also normalised to impose a value of C(min W) = 1, meaning that the shortest link(s) +in the graph is chosen with probability 1 (of course only if it is involved in the current node under consideration). +After having generated the set of Ntxn(s), ∀s ∈ S, we perform WL relabeling and proceed as in the vanilla version of +the graph2vec algorithm. We set all the hyperparameters to their default values, i.e. number of workers = 4, number of +epochs = 10, minimal structural feature count = 5, initial learning rate = 0.025, and down sampling rate of features = +0.0001. The only exception is the number of WL iterations, which in our case must be set to 1 instead of 2. The result +is an embedding for each graph in our set of transaction graphs, which becomes a set that we can subsequently cluster +via the popular k-means++ methodology. Importantly, we want to underline that our embeddings and clusters do not +depend on the real magnitudes of weights ∆t, since the sampling is adjusted on that. In addition, they also have no +notion of the amount of USD traded, thus being agnostic to the transaction value. As a final note, we refer the reader to +[22] for a version of graph2vec that uses edge labels. However, the algorithm creates the dual version of the graph and +would not be effective in our case, thus providing ground for our proposed extension. +An illustrative example. +To increase clarity on our approach, we briefly describe a simple example. Consider an +agent that performs 20 transactions. She performs the first 10 transactions shortly clustered in time, waiting only 60 +seconds one after the other. Then, she waits 42 minutes to action on the final 10 transactions with, again, a frequency of +one minute. Her behaviour is plotted in the transaction graph Gtxn of Fig. 11a, where we assume for simplicity that +each transaction is performed on the same pool and thus colour-code nodes all the same. We also number nodes to +show the order in which the related transactions are executed. We do not draw all the edges of this complete graph +for cleanness of the diagram and ease of visualisation, but hint with the green dashed lines that indeed there are more +connections to be remembered. In this example, the minimum time between transactions is 60 seconds (light blue edges) +and the maximum one is one hour, i.e. 3, 600 seconds (light grey edge). Some intermediate times are depicted as edges +with the same colour for the same weight. The resultant cut-value function C(w) that defines our sampling probabilities +to choose edges is shown in Fig. 11b. As intuitively desired, we aim at always keeping the shortest edges and indeed +these have probability 1. Then, we also aim to keep the most clustered “communities”, and indeed, we observe from the +plot that transactions five minutes away are still chosen with 40% probability, but longer times are very easily dropped. +(a) Transaction graph Gtxn = (S, T, W). +(b) Cut-value C(w). +Figure 11: For our illustrative example, we show in (a) a simplified representation of the LT’s transaction graph, and in +(b) the cut-value that defines probabilities of keeping edges as neighbours. +12 + +1.0 +/tokeep +0.6 +Probability +0.4 +0.2 +0.0 +0 +100 +200 +O0E +400 +500 +600 +700 +800 +edge weight (△t in seconds)JANUARY 31, 2023 +3.1.3 +Discussion of results +For each case A/B1/B2, we study the structural equivalence of LTs’ trading activity by clustering the representations +generated via our modified graph2vec algorithm. Focusing first on case A, we compute embeddings for dimensions +n ∈ {8, 16, 32, 64}, and confirm with Principal Component Analysis (PCA) that the proportions of data’s variance +captured by different dimensions are well-distributed. For each n-dimensional set of vectors, we then group LTs by +performing a series of k-means++ clusterings with different number of desired groups, and choosing the result lying at +the elbow of the related inertia plot, i.e. applying the elbow method. The similarity between optimal clusterings for +different dimensions is then computed, in order to investigate the stability of results across representations of increasing +dimensionality. We achieve this by computing the Adjusted Rank Index (ARI) [23], which is a measure of similarity +between two data clusterings in the range [−1, 1], adjusted for the chance of grouping of elements. We find ARIs for +clusterings on 8-vs-{16, 32, 64} dimensional data around 0.75, while clusterings on 16-vs-32, 16-vs-64 and 32-vs-64 +dimensional data reach approximately the value of 0.90. Therefore, we conclude that there is a high stability of results +when our data are embedded at least in 16 dimensions, and use the related 16-dimensional vector representations for +our final analyses. The related optimal number of clusters of LTs for case A is seven. Similar results arise for cases +B1/B2 too, and the related optimal numbers of clusters of LTs are six and seven, respectively. +Each extracted clustering is based on the structural similarity of LTs’ trading behaviour. To judge the goodness of our +modified algorithm and assess the results, we investigate whether there are specific features or trends that are highly +representative of only some of the groups. Thus, we proceed to build a set of characteristics to compute for each LT and +calculate the average of these results over the LTs belonging to each different group. The features that we consider are: +• average and median USD traded, +• average and median time ∆t in seconds between transactions, +• proportion of transactions done in “SS”, “EXOTIC” or “ECOSYS” pools, and related entropy, +• proportion of transactions done in pools with a specific feeTier, and related entropy, +• proportions of trades on days when the SP LargeCap Crypto Index7 increased or decreased in value, or when +the market was closed, due to weekends and bank holidays. +The distinction between “SS”, “EXOTIC” or “ECOSYS” pools is inspired by the classification in [14], where the +authors introduce a notion of normal pools, stable pools and exotic pools. For them, stable pools exchange tokens that +are both stablecoins. Normal pools trade instead tokens that are both recognised in the crypto ecosystem, while exotic +pools deal with at least one token that is extremely volatile in price (e.g. YAM, MOON and KIMCHI). We slightly +divert from this classification and define “SS” pools as pools whose tokens are both stablecoins, “ECOSYS” pools as +pools that exchange only tokens that are either stablecoins or pegged to the most established BTC and ETH coins, and +“EXOTIC” pools as the remaining ones. ECOSYS pools can be seen as the venues carrying the “safest” opportunity +for profit for a novice crypto investor, since they trade volatile tokens though directly related to the most established +blockchains that are the true foundations of the whole DeFi environment. +The average magnitude of features computed over the LTs belonging to each different cluster for case A, i.e. over +Jan-June 2022, is reported in Fig. 12a. We focus on the groups found specifically for this period because it is the +longest one and thus, it provides us the most general results and insights. Cases B1/B2 will be later described +too, in order to assess the overall stability of recovered species of LTs and highlight any specific variations due to +different sub-periods in time and related pools of relevance considered. The seven clusters of LTs found have sizes of +304/142/512/978/379/186/914 agents respectively, which means that we are able to find a well-balanced distribution +of cluster sizes without any dominant clusters in terms of size. Thanks to the heatmap, we also easily confirm that our +methodology is able to extract different groups of LTs that have significant variation of behaviour with respect to the +outer features defined. However, a few columns had to be dropped due to non-significance of their results. Importantly, +we also remind that inner biases on ratios are present (e.g. when considering that our sub-universe does not have +a uniform distribution of numbers of pools with specific feeTier), and thus we can expect more/less transactions of +some type on average. For visualisation purposes, we also embed the 16-dimensional representations of LTs into a +2-dimensional view via t-SNE, and plot them with perplexity = 15 in Fig. 12b. LTs are colour-coded according to the +cluster they belong to, and we indeed observe that different groups lie on different parts of the plane. +Focusing on Fig. 12a, one can immediately draw the following high-level remarks. +• Groups 0 and 1 have a strong focus on trading exotic cryptocurrencies. The former set of LTs mainly uses +feeTier 3000 for the purpose, and shows slightly higher than average tendency to trade when the market is +7https://www.spglobal.com/spdji/en/indices/digital-assets/sp-cryptocurrency-largecap-index/ +#overview +13 + +JANUARY 31, 2023 +closed. The latter group uses significantly both the 3000 and 10000 feeTiers, meaning that the related LTs +are willing to accept also extremely high transaction costs. This behaviour could indicate that they have high +confidence on their intentions and possibly urgency. +• On the other hand, groups 2 and 3 trade stablecoins more than usual. The former cluster could point to +an enhanced use of SS pools to take advantage of optimised routing, while the latter has a non-negligible +proportion of trades in exotic pools with feeTier 10000. Likely, group 3 isolates a set of LTs that are interested +in niche exotic tokens, which are only proposed in pools against stablecoins that do not overlap. Diverting +funds between two of these exotic tokens requires an exchange between the two related stablecoins too, which +motivates the recovered statistics. We also witness strong usage of the feeTier 100, which hints to traders +trying to compensate the high costs suffered in pools with feeTier 10000 by paying the lowest possible fees on +the SS pools. +• Groups 4 and 6 are more active than average on ECOSYS pools. The two groups differ noticeably from their +opposite relative strength of USD traded and time between operations. Overall, group 6 trades less money and +waits longer, mainly using pools with low feeTier 500. These features can be interpreted as characteristics of +cautious retail traders that invest in less risky and highly well-known crypto possibilities. And indeed, we also +find that this group is one of the largest in size. Then, group 4 also relates to ECOSYS pools. However, these +users tend to trade more USD with higher frequency, and this is also the cluster with much higher than average +proportion of LTs that also act as LPs (∼ 16%) . Therefore, we identify here a group of more professional +investors. +• Finally, group 5 shows a significant usage of all the three types of liquidity pools, but trades are concentrated +in pools with cheap feeTier 500. These agents trade often, and indeed show the smallest median time between +transactions. These eclectic, active and thrifty LTs are probably our group of smartest investors. +Our results confirm that the proposed algorithm is able to recognise variance in the data, and we also manage to extract +interesting insights into the species of LTs’ behaviour. We highlight the recovered importance of different types of pools, +despite no full notion of tokens and feeTier is used in the generation of the embeddings. Indeed, only a unique label per +pool is applied, e.g. USDC-WETH/500 could be pool “P1”, USDC-WETH/3000 pool “P2” and FXS-WETH/10000 +pool “P3”, and these would be considered equally different if no structural pattern was inherently characteristic of the +first two and recognised by the methodology. +However, we have mainly focused on the liquidity consumption component of the crypto ecosystem thus far. In the next +step of our investigation, we shift the focus from LTs to pools. We first aim to perform a clustering of pools based on +features built from simple statistics that consider both liquidity consumption and liquidity provision. This will allow us +to assess whether the SS, ECOSYS and EXOTIC classification really describes the crypto ecosystem or is only useful +for LTs characterisation. Before proceeding to this task, we first briefly report on a stability analysis component. +Stability analyses. +As already motivated, we now pursue the same analyses described above but for cases B1/B2. +We cluster the n-dimensional embeddings for n ∈ {8, 16, 32, 64} and compute the ARIs between each pair of resultant +sets of LT groups. We confirm that at least a 16-dimensional embedding is required in order to have a stability of +clusters in case B1, while only eight dimensions suffice for the case B2. For simplicity, we use the 16-dimensional +representations consistently in all cases. We recover six groups of LTs in case B1, and seven in case B2. In both cases, +we find two clusters with same characteristics as groups 4 and 6 of case A, i.e. traders mainly active on ECOSYS +pools. We also recover the eclectic traders of group 5. Therefore, we observe several stable and persistent types of LTs. +Small perturbations happen instead on the groups trading on SS or EXOTIC pools, as one could expect from the mere +evolution of time and external market conditions, and consequently generation of different behaviours. In particular, all +case A species, except group 1, are also found in case B1. On the other hand, case B2 shows less intensity on group +3, probably due to investors diversifying more during the crypto turmoils of the second quarter of 2022. Overall, we +observe general agreement on the groups and main features recovered during cases A/B1/B2, and we can thus rely on +our species of LTs found for the longest duration case A as descriptors of the ecosystem. +The above findings are of interest in themselves, first of all, since central banks started hiking interest rates in March +2022. This consequently stopped a strong influx of liquidity into the crypto ecosystem and accentuated a period of +significant underperformance, that could have weakened the stability of results. On top of that, the Terra-Luna crash +happened in May 2022 and it could have in theory enhanced noise and instabilities especially in the structural clustering +on case B2. As a very last remark, we notice that only ∼ 20% addresses are present in all cases A/B1/B2. Therefore, +we are either recovering similar behaviour but for different people, or in some cases it could be the same person simply +employing a new wallet to hide their trading behaviour better. +14 + +JANUARY 31, 2023 +(a) Average features for LTs in case A. +(b) t-SNE embedding visualisation of case A. +Figure 12: Clustering of LTs for case A, i.e. over the six-months time window between January and June 2022. In (a), +each row represents one of the recovered clusters and columns are the different features computed to characterise species +of LTs. The color-code employed applies to each column separately to be able to quickly identify the related smallest +and biggest values in magnitude, and judge the general distribution. It is essential to always check the magnitudes +of cells per se too, due to highly variable variance between columns. In (b), the t-SNE plot of embeddings of LTs is +reported with perplexity = 15 and points are color-coded according to their cluster of membership. +3.2 +Clustering of pools +3.2.1 +Motivation and Methodology +The above analyses revealed a characterisation of the main types of LTs structural trading behaviour. While the +importance of different types of pools in the ecosystem seems to be also clear, we stress that a full understanding +of liquidity pools goes beyond the mere liquidity consumption mechanism (i.e. it needs to further account for both +liquidity provision and price evolution). Thus, we now pursue an intuitive initial investigation of the similarity of pools +themselves, in order to gain additional insights on the entire ecosystem. +We focus on case A, as it covers the longest period in time. We consider the intersection of pools relevant for both LTs +and LPs to properly account for both mechanisms, and find a resulting set of 16 pools. For each pool, we compute the +following 13 features: +• average daily number of active LTs/LPs - “SdailyLT” and “LdailyLP” respectively, +• volatility of the execution price of the pool - “SstdP”, +• average size of swap/mint/burn operations in dollars - “SavgUSD”, “LavgUSDmint” and “LavgUSDburn”, +15 + +A +1.0 +2.4e+04 +1.6e+04 +1.2e+05 +1.8e+04 +0.066 +0.59 +0.34 +0.74 +0.054 +0.37 +0.55 +0.022 +0.7 +0.28 +7e+04 +4.4e+04 +6e+04 +1.7e+04 +0.012 +0.55 +0.44 +0.52 +0.011 +0.33 +0.61 +0.046 +0.59 +0.25 +0.8 +1 +2 +5.1e+04 +3.9e+04 +9.3e+04 +1.9e+04 +0.15 +0.043 +0.81 +0.46 +0.11 +0.76 +0.11 +0.022 +0.61 +0.25 +0.6 +m-1.2e+05 +57.1e+04 +1.2e+05 +1.8e+04 +0.19 +0.1 +0.71 +0.67 +0.14 +0.65 +0.15 +0.054 +0.84 +0.23 +0.4 +4 +2.6e+05 +1.3e+05 +3e+04 +8.1e+03 +0.0017 +0.074 +0.92 +0.26 +0.0014 +0.63 +0.34 +0.02 +0.71 +0.24 +5 +7.6e+04 +3.5e+04 +1.8e+04 +4.2e±03 +0.1 +0.11 +0.79 +0.61 +0.059 +0.79 +0.12 +0.026 +0.65 +0.25 +0.2 +6e+04 +1.2e+05 +2.3e+04 +0.061 +0.034 +0.9 +0.3 +0.052 +0.78 +0.16 +0.012 +0.59 +0.23 +_ECOSYS +0.0 +_USD +asn +_100 +500 +fees +avg_c +close +avg. +ratio_1 +ratio_5 +"suxA +60 +40 +20 +0 +0 +-20 +1 +7 +-40 +3 +4 +-09- +5 +6 +-80 +-75 +-50 +-25 +0 +25 +50 +75JANUARY 31, 2023 +Figure 13: Spearman correlation between the computed features for pools, with the addition of feeTier, for our case A. +• average daily amount of dollars used in swap/mint/burn operations, i.e. volume - “SdailyVol”, “LdailyVolMint” +and “LdailyVolBurn”, +• average daily number of LTs/LPs transactions - “SdailyTxn” and “LdailyTxn”, +• average daily number of different senders, i.e. smart contracts, called within swap transactions - “SdailyS”, +• number of agents with only one transaction normalised by the number of days considered - “Sdaily1txn”. +This measure is computed to gauge the tendency of external smart investors to hide their behavior by creating +several different wallets on the pool. +For the above features, we create related labels for ease of reference, which start with letter “S” if the quantity is +computed from swap operations, or letter “L” if the quantity is computed from liquidity provision operations. In Fig. +13, we show the heatmap of Spearman correlations between the above attributes plus feeTier (“SfeeTier”) for our pools. +There are significant positive correlations, especially among features developed from LT data and LP data, respectively. +Thus, we standardise entries and employ linear PCA and kernel PCA (with both “rbf” and “cosine” kernel in the latter) +to reduce the dimensionality of our data. The eigenvalue decay for all three mentioned cases is shown in Fig. 14, where +only the first seven eigenvalues are depicted for clarity of visualisation. The cosine kernel PCA is seen to capture more +variance in fewer dimensions, and thus we embed the data by projecting on its related first three components. The +resulting 3D embedding is shown in Fig. 15 from three different angles, where we color-code pools according to their +feeTier. In particular, green relates to feeTier 100, blue to 500, orange to 3000, and red to 10000. +3.2.2 +Discussion +From the projections shown in Fig. 15 and initial trials of clustering, it is clear that the division between SS, ECOSYS +and EXOTIC pools does not hold when considering the full set of dynamics on pools (while it is indeed suitable in +(a) Linear PCA. +(b) rbf kernel PCA. +(c) Cosine kernel PCA. +Figure 14: Decay of eigenvalues for the first seven out of 13 eigenvalues for different PCA kernels. +16 + +A +-0.096-0.34 +-0.57-0.66 +-0.15 +-0.28 +0.12 +0.15 +-0.086-0.023 + 1.0 +SfeeTier - +1 +-0.65-0.64 +-0.6 +Sstdp +-0.096 +1 +-0.19 +0.11 +0.056 +-0.038 +0.16 +0.18 +-0.16 +-0.05 +-0.035 +-0.074 +-0.091 +-0.11 +- 0.8 +SavgUSD +-0.34 +-0.19 +1 +0.38 +0.37 +0.79 +0.35 +0.26 +0.41 +0.59 +0.6 +0.59 +0.81 +0.79 +SdailyLT +-0.65 +0.11 +8E0 +1 +86'0 +0.78 +0.92 +0.96 +-0.035 +0.19 +0.3 +EEO +0.27 +0.31 + 0.6 +SdailyTxn +-0.64 +0.056 +0.37 +86°0 +1 +0.8 +0.94 +0.91 +-0.059 +0.17 +0.36 +0.38 +0.27 +EEO +SdailyVol +-0.6 +-0.038 +0.79 +0.78 +0.8 +1 +0.78 +69°0 +0.2 +0.45 +0.67 +0.67 +0.7 +0.71 + 0.4 +SdailyS +-0.57 +0.16 +0.35 +0.92 +0.94 +0.78 +1 +0.84 +-0.15 +0.094 +0.39 +0.4 +0.27 +0.32 + 0.2 +Sdaily1txn +-0.66 +0.18 +0.26 +0.96 +0.91 +0.69 +0.84 +1 +-0.076 +0.15 +0.17 +0.21 +0.16 +0.19 +LavgUSDmint +-0.15 +-0.16 +0.41 +-0.035 +-0.059 +0.2 +-0.15 +-0.076 +1 +E60 +0.091 +0.097 +0.64 +0.59 + 0.0 +LavgUSDburn +-0.28 +-0.05 +0.59 +0.19 +0.17 +0.45 +0.094 +0.15 +E6'0 +1 +0.25 +0.26 +0.78 +0.74 +LdailyLP +0.12 +0.035 +0.6 +0.3 +0.36 +0.67 +0.39 +0.17 +0.091 +0.25 +1 +86'0 +0.76 +0.81 +0.2 +LdailyTxn +0.15 +-0.074 +0.59 +EEO +8E0 +0.67 +0.4 +0.21 +0.097 +0.26 +86'0 +1 +0.76 +0.82 +0.4 +LdailyVolMint +-0.086 +-0.091 +0.81 +0.27 +0.27 +0.7 +0.27 +0.16 +0.64 +0.78 +0.76 +0.76 +1 +86'0 +LdailyVolBurn +-0.023 +-0.11 +0.79 +0.31 +EEO +0.71 +0.32 +0.19 +0.59 +0.74 +0.81 +0.82 +0.98 +1 +0.6 +SfeeTier +asnGAes +SdailyLT +SdailyTxn +SdailyVol , +SdailyS +Sdaily1txn +LavgUSDmint +LavgUSDburn +ilyLP +LdailyTxn +LdailyVoIMint - +LdailyVolBurn +1 +lep7Linear PCA, Eigenvalue Decay +0.6 +0.5 +0.4 +0.2 +0.1 +0'0 +0 +1 +2 +E +4 +5 +6 +EigenvalueKernel PCA - rbf, Eigenvalue Decay +0.25 +0.20 +0.15 +0.10 +0.05 +0.00 +0 +1 +2 +E +4 +5 +6 +EigenvalueKernel PCA - cosine, Eigenvalue Decay +0.40 +0.35 +OE'0 +0.25 +0.20 +0.15 +0.10 +0.05 +0.00 +0 +1 +2 +E +4 +5 +6 +EigenvalueJANUARY 31, 2023 +(a) Angle = 45 degrees. +(b) Angle = 135 degrees. +(c) Angle = 315 degrees. +Figure 15: Projection on a 3D space of the vectors encoding different features of pools, from the application of PCA with +cosine kernel. Views from different angles are reported for a better judgment of the results, and pools are color-coded +according to their feeTier (green relates to feeTier 100, blue to 500, orange to 3000 and red to 10000). +connection to specifically LTs’ behaviour). Similarly, we do not witness strong proximity of pools with same feeTier. +Liquidity consumption, provision and price evolution are all essential mechanisms to consider for a full description +of the Uniswap ecosystem, and our intuition is that certain combinations of tokens and feeTiers are more similar and +suitable for trading at different moments in time. LPs are more incentivised to enhance liquidity on pools with strong +LTs activity, low volatility of the exchange rate to avoid impermanent loss, and possibly high feeTier from which they +indeed mainly profit. In parallel, LTs are more interested in pools with low fees but some volatility of the price of tokens, +and high liquidity to diminish the market impact of their trades. Thus, different adjustments of these mechanisms can +result in the proximity or not of our projections of pools. In the next Section of our work, we propose a model to judge +the health of each pool’s combination of mechanisms and characterise the best venues for market participants (i.e. both +LTs and LPs) to be active on. +4 +The ideal crypto law and a cryptoness measure of a liquidity pool +4.1 +Model: from the ideal gas law of thermodynamics to the ideal crypto law for pools +In physics, an ideal gas is a theoretical gas composed of many randomly moving particles with negligible volume that +are not subject to interparticle interactions. On the other hand, real gases occupy space and molecules interact between +themselves. Ideal gases obey the ideal gas law, which says that +PV = nRT, +(5) +where P is the pressure of the gas, V its volume and T its temperature. Furthermore, n denotes the constant number of +moles of particles in the considered closed system, and R is the gas constant. Despite being very simple and elegant, +this law describes very well complex dynamics. +Our intuition finds resemblances between the law in Eq. (5) and the crypto ecosystem, and we consider an analogy +in which each pool is a gas. To define the similitude between variables that is summarised in Table 1, we follow how +the ideal gas law was discovered and reason about both the possible meaning of variables and expected relationships. +First of all, P can be intuitively compared to the USD volume traded by active LTs over e.g. a day. If everything else +but T is kept constant, then we expect T to increase with higher P. Therefore, we can interpret T as the liquidity +of the pool, i.e. the value of our proxyTVL in USD for the pool at that date. Indeed, the evolution of liquidity of a +Ideal gas law +Ideal crypto law +Symbol +Meaning +Symbol +Meaning +P +pressure +Pvol +daily USD volume traded by LTs +V +volume +Vstab = STD(Z)−1 +daily stability of the exchange rate Z +n +moles of particles +nfee = feeTier−1 +stimulus to LTs’ activation +R +gas constant +Rpool +pool crypto constant +T +temperature +Tliq +daily liquidity, i.e. proxyTVL value +Table 1: Parallelism drawn between the ideal gas law and our ideal crypto law. +17 + +WBTC-WETH/3000 +0.8 +WETH-USDT/3000 +0.6 +JSDC-WETH/3000 +0.4 +UNI-WETH/3000 +3 +0.2 +WBTC-WETH/500 +LINK-WETWEUENS/3000 +0.0 +0.2 +SHIB-WETH/3000 +JUSDC-USDT/100 +DAI-WETH/500 +0.4 +JSDC-WETH/500 +WETH-USDT/500 +0.25 +0.00 +0.25 +0.25 +0.00 ~0.25~0.50~0.75 +0.50 +0.75 +0.50 +1 +0.75 +1.00WBTC-WETH/3000 +0.8 +USDC-WETH/3000 +WETH-USDT/3000 +0.6 +WBTC-WETH/500 +0.4 +0.2 +USDC-USDT/100 +0.0 +MANETWERVSCOO00 +SHIB-WETH/3000 +0.2 +JSDC-WETH/500 +DAI-WETH/500 +0.4 +WETH-USDT/500 +1.00 +0.75 +0.50 +0.25 +0.00 +~0.25 ~0.50~0.75 +1 +00'0 +0.25 0.50 +0.50 +0.25 +2 +0.75 +0.75WBTC-WETH/3000 +WETH-USDT/3000 +0.8 +USDC-WETH/3000 +0.6 +0.4 +UNI-WETH/2A993000 +WBTC-WETH/500 +0.2 +LINK-WETH/3000 +3 +MATIC-WETH/3000 +0.0 +DAI-WETH/500 +WETH-CRV/10000 +SHIB-WETH/3000 +0.2 +USDC-USDT/500 +WBTC-USDC/3000 +WETH-USDT/500 +JUSDC-USDT/100 +JSDC-WETH/50D +0.4 +0.50 +0.75 +0.00 +0.25 +0.00 +0.25 +1 +0.50 +0.75 +2 +1.00JANUARY 31, 2023 +pool accounts for the behaviour of LPs, and more LPs should execute mint operations when there are more LTs active, +in order to collect higher profit from the fees that the latter pay for each swap transaction. Clearly, we also expect a +stronger overall volume traded by LTs with more liquidity, due to more convenient, smaller price impact. Variable V +is then the volume of the gas in the thermodynamics interpretation, i.e. ideal gas law. Despite being a more subtle +relationship, we find that it is reasonable to consider V as the stability of the exchange rate between the two tokens in +the pool, i.e. STD(Z)−1, where Z is the exchange rate. Keeping everything else constant, one would expect that, with +higher liquidity, the price is more stable. Similarly, a compressed gas with small volume will be less stable than an +expanded gas. In addition to that, the concentrated liquidity mechanism of Uniswap v3 implies that LPs encounter +the risk of no gains from LTs’ fees if the exchange rate moves outside the range of prices over which they are actively +providing liquidity. Thus, a more stable Z for the same USD volume traded by LTs (i.e. P) is likely to attract more +minting operations, especially close to the current rate. Finally, a higher P with the same level of liquidity is indeed +likely to cause a less stable relative price of the tokens, due to the impact of surplus swap operations. +Keeping the above in mind, Eq. (5) thus becomes +Pvol × Vstab = nfee × Rpool × Tliq +⇒ Pvol × STD(Z)−1 = feeTier−1 × Rpool × Tliq, +(6) +allowing us to bring together into a single formula, all the variables that govern the three mechanisms of liquidity +consumption, provision and exchange rate evolution. Constant Rpool is instead the invariant characteristic of each pool, +which we aim at regressing with some level of significance. Here, n is the fixed number of moles (molecules), thus a +constant. A higher n means more interactions in the physical description, which we relate to lower feeTier and stronger +activity of the LTs, that we know dominate LPs. Thus, we express n = feeTier−1, which is indeed constant too. +Real gases and van der Waals forces. +Our above model is based on the thermodynamics law for ideal gases. +However, it is worth mentioning that there also exists a law for real gases that interact via van der Waals forces. This is +governed by the Van der Waals equation +� +P + a n2 +V 2 +� +(V − nb) = nRT, +(7) +where the variables have same meaning as before. In addition, a is a constant whose value depends on the gas and +represents intermolecular forces, while b is the volume occupied by the molecules of one mole of the gas. Based on a +preliminary analysis, we believe it could be of interest to expand the ideal crypto law in this direction; however, this is +beyond the scope of our current work, and we leave this for future investigation. +4.2 +Regression analysis and interpretation of results +As a first step, we test for empirical instances that support the validity of our ideal crypto law given by Eq. +(6). +We focus on case A and consider the intersection of pools that are relevant for both liquidity provision +and liquidity consumption. +This results in a set of 16 pools, namely USDC-USDT/100, WBTC-WETH/500, +WETH-USDT/3000, WETH-ENS/3000, MATIC-WETH/3000, WBTC-USDC/3000, WBTC-WETH/3000, DAI- +WETH/500, UNI-WETH/3000, SHIB-WETH/3000, USDC-USDT/500, USDC-WETH/500, WETH-CRV/10000, +LINK-WETH/3000, USDC-WETH/3000, WETH-USDT/500. +For each individual pool, we compute the quantities Pvol, Vstab, Tliq reported in Table 1 at daily scale for the full +six-months time window between January and June 2022. We suggest not to use higher frequencies due to significant +general low rate of activity of LPs. We then scatter-plot these daily realisations for each combination of pairs of +variables and critically analyse the results. Observations with z-score > 3 in absolute value are considered outliers and +discarded. A few representative examples of the results are shown in Fig. 16, where we observe hints that the ansatz +relationships are indeed satisfied, apart for the case of pools whose tokens are both stablecoins. The latter dissimilarity +is perhaps not surprising, since our ideal crypto law aims at encompassing the full ensemble of crypto mechanisms, +while the behaviour of LTs on pools of stablecoins has lower relevance. +We broaden our sub-universe of venues of interest by taking the union of pools significant to LTs’ and LPs’ behaviour +in case A. After filtering for relevance of the samples in the daily frequency, we are left with a set of 32 pools. For +each pool, we perform a linear regression over the available six months of daily values, and thus estimate Rpool by +rearranging Eq. (6) to +Pvol = Rpool × +�nfee × Tliq +Vstab +� +⇒ ypool = Rpool × xpool, +(8) +18 + +JANUARY 31, 2023 +(a) USDC-WETH/3000 pool. +(b) UNI-WETH/3000 pool. +(c) USDC-USDT/100 pool. +Figure 16: Visualisation of pair relationships from our ideal crypto law for a sample of pools. From left to right for +each case, we plot first Vstab vs. Pvol, then Tliq vs. Pvol, and finally Tliq vs. Vstab. +where the intercept is zero. We compute the coefficient of determination R2 of the regression, which we refer to as the +cryptoness ξ. Thus, pools with high cryptoness are meant to adhere well to our proposed ideal crypto law model. We +compute the average ξSS over the seven pools found to exchange only stablecoins in our sub-universe, and compare +it to the average ξnotSS of all other pools. We find values of ξSS = −0.44 and ξnotSS = −0.21, meaning that SS +pools should not be considered in our model, as expected and already motivated. For the remaining pools, we show +in Fig. 17a the ones that have ξ > 0. We notice that different feeTiers appear to be relevant and that there is one +interesting occurrence of two pools, both with high ξ, that exchange the same tokens, i.e. WBTC-WETH/3000 and +WBTC-WETH/500. This latter result raises the question of whether these two pools are generally adhering to our ideal +crypto law, or if they might follow it at disjoint periods of time that however influence the overall cryptoness values and +render them both significant. Before investigating this idea further, it is worth highlighting the fact that several pools +with high ξ are seen to exchange exotic tokens. Examples are pools trading CRV and UOS, which are respectively the +tokens of the well-know DEX Curve.fi, and of the blockchain-based gaming company Ultra. The general pattern that +we read is that pools exchanging digital assets linked to a tangible and established business idea might adhere better to +19 + +USDC-WETH/3000 +le10 +le10 +3.0 +3.0 +0.08 +2.8 +2.8 +2.6 +2.6 +0.06 +Vstab +2.4 +2.4 +0.04 - +2.2 +2.2 +0.02 +2.0 +2.0 +XX +1.8 +1.8 +X +0.00 ++ +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.00 +0.02 +0.04 +0.06 +0.08 +Pvol +1e8 +Pvol +1e8 +VstabUNI-WETH/3000 +1e7 +le7 +0.8 +7.8 +7.8 +0.7 +7.7 +7.7 +0.6 +0.5 +7.6 +7.6 +0.4 +7.5 +7.5 +0.3 +7.4 +7.4 +0.2 +X +X +X +X +7.3 +7.3 +0.1 +7.2 +0 +6 +0 +、 +5 +6 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Pvol +1e6 +Pvol +1e6 +VstabUSDC-USDT/1OO +1e8 +1e8 +10000 +2.2 +2.2 - +X +X +9000 + 0. +(b) The xpool, ypool values for WETH-CRV/10000 pool, which has +highest ξ. Dots are colour-coded to encode the temporal evolution, +with lighter (resp. darker) ones denoting earlier (resp. more recent) +observations. The straight line denotes the linear regression line. +Figure 17: For each pool under analysis, the linear regression between xpool, ypool is computed for daily values over the +six months of case A. Each related coefficient of determination provides the cryptoness ξ of the pool. +our ideal crypto law, hinting at the idea that they are interpreted as equity investments. Thus, we can think of such pools +as being strongly rooted in the crypto ecosystem, at least over the time window of reference in our analyses. +For the sake of clarity, we explicitly depict in Fig. 17b the linear regression pursued for the pool with highest cryptoness, +i.e. WETH-CRV/10000, where occurrences that happened more recently in time are color-coded darker. We also +mention that pools might exhibit a less good fit to our ideal crypto law due to noise introduced by the characteristic +frequency of executed swaps, mint and burn operations on each pool. Liquidity provision events are generally rare, as +reported in Table 2. Thus, it might become necessary to define ad-hoc frequencies, possibly dynamic, to be used in the +regression for each pool, in order to better investigate the related evolution of behaviours. +Liquidity consumption +Liquidity provision +Pool +Daily +Swap +Pool +Daily +Mint +Pool +Daily +Burn +USDC-WETH/500 +6181 +USDC-WETH/3000 +95 +USDC-WETH/3000 +70 +WETH-USDT/500 +2395 +USDC-WETH/500 +65 +USDC-WETH/500 +70 +DAI-WETH/500 +817 +WBTC-WETH/3000 +19 +WBTC-WETH/3000 +26 +WETH-CRV/10000 +55 +DYDX-WETH/3000 +1.0 +FXS-WETH/10000 +1.4 +USDC-UOS/10000 +39 +USDC-UOS/10000 +0.6 +USDC-UOS/10000 +1.0 +SHIB-WETH/10000 +32 +CEL-WETH/3000 +0.4 +CEL-WETH/3000 +0.8 +Table 2: Average daily frequency of operations on different pools for the time window of case A. We report the first and +last three values when sorting by magnitude. +Time-evolving analyses. +We now analyse the evolution of our proposed cryptoness metric for the set of pools, over +the six-months time window of case A, i.e. Jan-June 2022. We again compute the regression in Eq. (8) and record the +related coefficient of determination as the cryptoness value ξ, but we now use the observations contained in a 30-day +window in time, sliding every day. This allows us to recognise pools that adhere more or less to our ideal crypto law at +different points in time, and investigate the patterns generated. Interesting results for specific subsets of pools are shown +in Fig. 18, where we threshold all the irrelevant ξ < 0 to ξ = 0 for ease of visualisation. In reference to our ranking +of Fig. 17a and related discussion, we consider in Fig. 18a the evolution of ξ for a subset of pools exchanging exotic +tokens, which relate to well-established blockchain-based companies. As expected, we observe ξs that are strongly +significant for almost the entire six-months time range under consideration. Thus, we confirm that the exotic tokens +depicted are deemed by our model to have solid associated companies by the dynamics of market participants and price, +and we construe the related pools as healthy venues for trading. +Our intuition then suggests that there should exist only one ideal feeTier per point in time, for the exchange of two +same tokens. This feeTier would be the key that balances the set of mechanisms we are modeling and the beliefs of the +market participants. As a first related example, Fig. 18b shows some association between the drops in cryptoness of the +20 + +Pools with > 0 +0.5 +0.4 +m +0.3 +0.2 +0.1 +0.0 +WETH-CRV/10000 +WBTC-WETH/3000 +USDC-UOS/10000 +WETH-ENS/3000 +WBTC-WETH/500 +WBTC-USDT/3000 +LINK-WETH/3000 +SHIB-WETH/10000 +MKR-WETH/3000 +WBTC-USDC/3000 +USDC-WETH/3000 +DAI-WETH/3000le6 +WETH-CRV/10000 +1.0 +7 +6 - + 0.8 +5 - +0.6 +3 + 0.4 +1 +0.2 +0 - +0.0 +0.01 +0.02 +EO'0 +0.04 +Xpoo/JANUARY 31, 2023 +(a) Evolution of ξ for a sample of pools exchanging exotic tokens. +(b) Evolution of ξ for two pools exchanging SHIB-WETH. +(c) Evolution of ξ for a sample of pools exchanging WETH vs. stablecoin. The dominance of +different pools over time can lead to hypothesise cryptoness as a measure of the varying health +of pools and indicator of where market participants should prefer to be active on. +Figure 18: Evolution of cryptoness ξ for different subsets of pools, where ξ is now repeatedly computed from the +regression over a 30-days window of instances, sliding one day each time. +liquidity pool SHIB-WETH/10000, and the peaks in cryptoness of SHIB-WETH/3000, allowing us to consider our +cryptoness measure as an indicator of the described tendency. With a similar view, we also expect the redundancy of +pools exchanging the same token against different stablecoins to reduce the overall health of single pools, due to the +related fragmentation of market participants’ dynamics. Thus, we believe that a natural stabilisation of each token to +one specific stablecoin per period in time should also be reached with the stronger establishment of crypto ecosystems. +The results of this section provide supporting evidence towards the above concepts. In particular, we show that our +cryptoness measure signals the healthiest venues (i.e. liquidity pools) where agents should be active on, by comparing +simultaneous changes in ξ, versus variations in agents’ activity and TVL. This is performed for the set of relevant +pools that exchange WETH against stablecoins, i.e. DAI-WETH, WETH-USDT and USDC-WETH, also with different +feeTiers, i.e. 500 and 3000. Figure 18c shows the related evolutions of the cryptoness measure, where we can clearly +see that feeTier 3000 is the relevant one for each pair DAI-WETH, WETH-USDT and USDC-WETH for the entire time +window, except during April 2022. Interestingly, over this month, no liquidity pools exhibit strong cryptoness, except +for DAI-WETH/500, which we thus claim to be the only healthy venue for trading at that point in time. To support our +claim, we compute the average daily number of swap actions, mint operations and burn operations, for each one of our +six pools under analysis, as related measures of activity and good usage. In particular, we compute average quantities +over the month of April, and then over the time window Jan-June 2022 but excluding April, and then calculate the +percentage change. This results in the set of values +opChange = opApril − opNotApril +opNotApril +, +(9) +where “op” indicates an operation between swap, mint and burn. The actual computations are reported in Fig. 19a, +where we further include avgChange as the average of swapChange, mintChange and burnChange. In addition, we also +plot in Fig. 19b the evolution of TVL in USD over the pools of interest, in order to be able to compare occurrences +of drops in liquidity over time. Figure 19a reveals that DAI-WETH/500 is indeed the liquidity pool with the least +damage of activity during April 2022. While there exists another pool which performs better than others, namely +USDC-WETH/500, it suffered from a significant drop in liquidity in April 2022. On the other hand, DAI-WETH/500 +had constant TVL during this month, as clearly shown in Fig. 19b. We conclude that the only pool with a significant +cryptoness ξ score during April 2022 is indeed the healthiest and preferred trading venue during the month of relevance. +This provides further empirical motivation and utility to our proposed ideal crypto law. +21 + + evolution - 30-days window, sliding 1 day +0.9 +LINK-WETH/3000 +0.8 +MKR-WETH/3000 +WETH-CRV/10000 +0.7 +WETH-ENS/3000 +0.6 +W +0.5 +0.4 +E'O +0.2 +0.1 +0.0 +2022-02 +2022-03 +2022-04 +2022-05 +2022-06 +2022-07E evolution - 30-days window, sliding 1 day +0.7 +SHIB-WETH/10000 +SHIB-WETH/3000 +0.6 +0.5 +0.4 +E'0 +0.2 +0.1 +0.0 +2022-02 +2022-03 +2022-04 +2022-05 +2022-06 +2022-07E evolution - 30-days window, sliding 1 day +0.8 +DAI-WETH/3000 +0.7 +DAI-WETH/500 +WETH-USDT/3000 +0.6 +WETH-USDT/500 +USDC-WETH/3000 +0.5 +USDC-WETH/500 +w 0.4 +E'0 +0.2 +0.1 +0.0 +2022-02 +2022-03 +2022-04 +2022-05 +2022-06 +2022-07JANUARY 31, 2023 +Our measure of cryptoness of liquidity pools aims at becoming a useful tool for market participants, allowing them to +assess the health of trading venues and decide the best environments on which to be active. Ideally, we would like to +witness smooth dynamics of the evolution of cryptoness in the future, with new pools adhering better to the ideal crypto +law once they become well-rooted components of the crypto ecosystem. From a regulatory point of view, dynamic +requirements could then be imposed, such as less over-collateral required for trades on pools with cryptoness above a +certain threshold. This metric could thus be employed both by regulators and practitioners for developing pool health +monitoring tools, and establishing minimum levels of requirement. Finally, we believe that it would also be interesting +to investigate and interpret variations in the characteristic constant Rpool for specific pools, after drops in cryptoness. +As a motivating example, Fig. 20 shows two regressions over the first and second three months of case A, where clearly +the slope of the recovered lines varies significantly. +(a) Variation in activity, i.e. in the frequency of the +different transaction types, over April. +(b) Our proxyTVL in USD. +Figure 19: Comparison of activity variation and liquidity evolution between our six pools exchanging WETH against +a stablecoin, over the six months of case A. We discuss changes in average activity over April 2022 versus on the +remaining five months of case A, and similarly compare disequilibria of minting versus burning operations in USD +traded. Pool DAI-WETH/500 is the only venue of the subset with significant cryptoness in April 2022 and indeed, it +reports both the least damage in activity and no drop in proxyTVL over that month. +Figure 20: A pool that shows the evolution in time of its characteristic coefficient Rpool. The yellow and green lines are +generated from the regressions over data from the first three months, or last three months respectively, for case study A. +5 +Conclusions +Blockchain, DeFi and DEXs are recent concepts that just started taking roots in the common language and knowledge +of both practitioners and academics. However, a real comprehension of the characteristic dynamics of these protocols is +still far away. Similarly, academic research is at its dawn on the topic, despite having strongly accelerated in the past year. +To this end, our investigations aim at being a stepping stone towards a deeper understanding of the crypto ecosystem, +22 + +0.1 +DAI-WETH/3000 +-0.42 +-0.28 +-0.42 +-0.37 +0.0 +DAI-WETH/500 +-0.025 +0.026 +-0.054 +-0.018 +WETH-USDT/3000 +-0.25 +-0.37 +-0.36 +-0.32 +-0.1 +WETH-USDT/500 +-0.013 +-0.59 +-0.43 +-0.34 +0.2 +USDC-WETH/3000 +-0.4 +-0.49 +-0.29 +-0.39 +0.3 +USDC-WETH/500 +-0.18 +0.082 +-0.026 +-0.041 +-0.4 +swapChange mintChange burnChange avgChange1e8 +le10 +5 +DAI-WETH/3000 +USDC-WETH/3000 +3.0 +DAI-WETH/500 +WETH-USDT/3000 +2.5 +4 +WETH-USDT/500 +USDC-WETH/500 +USD proxyTVL +2.0 +proxyTVL +m +1.5 +2 +USD +1.0 +1 +0.5 +0.0 +2021-05 +2021-07 +2021-09 +2021-11 +2022-01 +2022-03 +2022-05 +2022-071e6 +SHIB-WETH/10000 +1.0 +2.0 +0.8 +1.5 +0.6 +/ood,^ +1.0 + 0.4 +0.5 + 0.2 +0.0 + 0.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Xpoo/ +le10JANUARY 31, 2023 +and we achieve this task by empirically studying and characterising Uniswap v3 DEX. We build a workflow to define the +most relevant liquidity pools over time by assessing the inner features of pools along with their interconnectedness, and +provide related lists of liquidity pools significant for six different windows in time, i.e. cases A/B1/B2/C1/C2/C3, +that can be directly used for future research studies. We then focus on LTs and show the existence of seven “species” of +traders with interpretable features. These clusters are recovered by assessing the equivalence of LTs structural trading +behaviour, and suggest a connection between patterns in swap transactions and specific types of pools on which these +operations are indeed executed. Finally, we also propose a novel metric that could aid practitioners and regulators in the +challenge of assessing the “health” of different trading venues, i.e. liquidity pools, by proposing an ideal crypto law and +proving the efficacy of the related cryptoness goodness measure. +Future work. +Regarding future directions of research, there are three main threads we aim to pursue. The first one is +a detailed investigation into the behaviour of LPs, along with the corresponding clustering of species, also leveraging on +a strongly data-driven approach. Secondly, we are considering the development of a broader integrated crypto market +indicator. This would be based on an aggregated cryptoness measure over a set of pools, with different associated +weights quantifying the contribution to the index, proportional to the evolving total value locked in each liquidity +pool. And as a last point, we plan to leverage the entire data on the Ethereum blockchain to track the flow of funds +over multiple DEXs and active protocols. This will enable us to gain a better understanding of LTs, as we will then +be able to approximate their profit and loss (PnL). Indeed, the crypto ecosystem is highly interconnected, and agents +easily trade between different exchanges on the same blockchain, also with the possibility to enhance their positions +via borrowing. In parallel, one could also investigate the “optimal routing problem” [24] on the Ethereum blockchain, +which is formulated as the problem of optimally executing an order involving multiple crypto assets on a network of +multiple constant function market makers. +Acknowledgements +We are grateful to Álvaro Cartea, Fayçal Drissi and Marcello Monga for insightful discussions. Deborah Miori acknowl- +edges financial support from the EPSRC CDT in Mathematics of Random Systems (EPSRC Grant EP/S023925/1). +References +[1] Satoshi Nakamoto. Bitcoin: A Peer-to-Peer Electronic Cash System. May 2009. +[2] Vitalik Buterin. Ethereum White Paper: A Next Generation Smart Contract & Decentralized Application Platform. +2013. +[3] Hayden Adams, Noah Zinsmeister, and Dan Robinson. Uniswap v2 Core. 2020. +[4] Hayden Adams. Uniswap v3 Core. 2021. +[5] Giuseppe Antonio Pierro and Henrique Rocha. The Influence Factors on Ethereum Transaction Fees. In 2019 +IEEE/ACM 2nd International Workshop on Emerging Trends in Software Engineering for Blockchain (WETSEB), +pages 24–31, 2019. +[6] Stefan Kitzler, Friedhelm Victor, Pietro Saggese, and Bernhard Haslhofer. Disentangling Decentralized Finance +(DeFi) Compositions, November 2021. +[7] Fabian Schär. Decentralized Finance: On Blockchain- and Smart Contract-based Financial Markets. 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Behavior of Liquidity Providers in Decentralized Exchanges, +2021. +23 + +JANUARY 31, 2023 +[15] Lioba Heimbach, Eric Schertenleib, and Roger Wattenhofer. Risks and Returns of Uniswap V3 Liquidity Providers. +arXiv preprint arXiv:2205.08904, 2022. +[16] Alvaro Cartea, Faycal Drissi, and Marcello Monga. Decentralised Finance and Automated Market Making: +Permanent Loss and Optimal Liquidity Provision. November 2022. +[17] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient Estimation of Word Representations in +Vector Space, 2013. +[18] Aditya Grover and Jure Leskovec. node2vec: Scalable Feature Learning for Networks, 2016. +[19] Quoc V. Le and Tomas Mikolov. Distributed Representations of Sentences and Documents, 2014. +[20] Annamalai Narayanan, Mahinthan Chandramohan, Rajasekar Venkatesan, Lihui Chen, Yang Liu, and Shantanu +Jaiswal. graph2vec: Learning Distributed Representations of Graphs, 2017. +[21] Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M. Borgwardt. +Weisfeiler-Lehman Graph Kernels. Journal of Machine Learning Research, 12(77):2539–2561, 2011. +[22] Hong Chen and Hisashi Koga. GL2vec: Graph Embedding Enriched by Line Graphs with Edge Features. In +International Conference on Neural Information Processing, 2019. +[23] Lawrence J. Hubert and Phipps Arabie. Comparing partitions. Journal of Classification, 2:193–218, 1985. +[24] Guillermo Angeris, Alex Evans, Tarun Chitra, and Stephen Boyd. Optimal Routing for Constant Function Market +Makers. In Proceedings of the 23rd ACM Conference on Economics and Computation, EC ’22, page 115–128, +New York, NY, USA, 2022. Association for Computing Machinery. +A +Sub-universes of pools for cases B1/B2/C1/C2/C3 +We report here the pools that our methodology finds most significant for cases B1/B2/C1/C2/C3. For each one of +these time windows, we list separately the pools relevant for liquidity consumption (LT data) and liquidity provision (LP +data) investigations. We also provide the thresholds chosen to define the giant components in our interconnectedness +analyses, always with reference to the workflow of Section 2. +Case B1. +LT data: thresholds for common origins, senders and bridges are 1, 500 and 60 and 600 respectively. +We recover a final set of 32 pools, which are: DAI-WETH/500, FRAX-USDC/500, SPELL-WETH/3000, agEUR- +USDC/500, USDC-USDT/100, LUSD-USDC/500, USDC-UST/100, WBTC-WETH/3000, DAI-USDC/100, USDC- +NCR/500, WETH-USDT/500, XSGD-USDC/500, LINK-WETH/3000, USDC-WETH/3000, DAI-WETH/3000, +WETH-BTRFLY/10000, FEI-USDC/500, HEX-USDC/3000, WETH-ENS/3000, MATIC-WETH/3000, USDC- +USDT/500, XSGD-WETH/500, WBTC-WETH/500, FXS-WETH/10000, GALA-WETH/3000, WBTC-USDC/3000, +WETH-CRV/10000, WETH-USDT/3000, USDC-UOS/10000, HEX-WETH/3000, USDC-WETH/500, USDC- +GF/3000. +LP data: thresholds for common origins and senders are 20 and 3 respectively. We recover a final set of 16 pools, which +are: DAI-WETH/500, WBTC-USDC/3000, USDC-WETH/3000, LINK-WETH/3000, WETH-CRV/10000, WBTC- +WETH/3000, MATIC-WETH/3000, WETH-ENS/3000, USDC-USDT/100, UNI-WETH/3000, SHIB-WETH/3000, +USDC-WETH/500, WETH-USDT/500, MKR-WETH/3000, SHIB-WETH/10000, WBTC-WETH/500. +Case B2. +LT data: thresholds for common origins, senders and bridges are 1, 500 and 80 and 600 respectively. +We recover a final set of 33 pools, which are: DAI-WETH/500, FRAX-USDC/500, FEI-USDC/100, USDC- +USDT/100, UST-WETH/3000, HDRN-USDC/10000, USDC-STG/3000, CEL-WETH/3000, DAI-USDC/500, WBTC- +WETH/3000, DAI-USDC/100, WETH-USDT/500, APE-WETH/3000, LINK-WETH/3000, USDC-WETH/3000, DAI- +WETH/3000, USDC-WETH/10000, HEX-USDC/3000, WETH-ENS/3000, MATIC-WETH/3000, USDC-USDT/500, +APE-USDC/3000, WBTC-WETH/500, WBTC-USDC/3000, WETH-USDT/3000, WETH-LUNA/10000, USDC- +UOS/10000, BUSD-USDC/500, HEX-WETH/3000, WETH-LOOKS/3000, SHIB-WETH/3000, USDC-WETH/500, +DAI-FRAX/500. +LP data: thresholds for common origins and senders are 15 and 3 respectively. We recover a final set of 16 pools, which +are: WBTC-USDT/3000, DAI-WETH/500, WBTC-USDC/3000, LINK-WETH/3000, USDC-WETH/3000, WETH- +USDT/3000, WETH-LUNA/10000, HEX-USDC/3000, WBTC-WETH/3000, MATIC-WETH/3000, USDC-USDT/500, +HEX-WETH/3000, UNI-WETH/3000, USDC-WETH/500, WETH-USDT/500, SHIB-WETH/10000. +24 + +JANUARY 31, 2023 +Case C1. +LT data: thresholds for common origins, senders and bridges are 1, 000 and 50 and 400 respectively. +We recover a final set of 29 pools, which are: DAI-WETH/500, FRAX-USDC/500, SPELL-WETH/3000, agEUR- +USDC/500, USDC-USDT/100, USDC-UST/100, WBTC-WETH/3000, DAI-USDC/100, USDC-NCR/500, WETH- +USDT/500, XSGD-USDC/500, LINK-WETH/3000, USDC-WETH/3000, DAI-WETH/3000, WETH-BTRFLY/10000, +FEI-USDC/500, HEX-USDC/3000, WETH-ENS/3000, MATIC-WETH/3000, SOS-WETH/10000, XSGD-WETH/500, +WBTC-WETH/500, FXS-WETH/10000, GALA-WETH/3000, WBTC-USDC/3000, WETH-CRV/10000, WETH- +USDT/3000, HEX-WETH/3000, USDC-WETH/500. +LP data: thresholds for common origins and senders are 10 and 3 respectively. We recover a final set of 15 pools, +which are: DAI-WETH/500, WBTC-USDC/3000, LINK-WETH/3000, USDC-WETH/3000, WETH-CRV/10000, +WETH-BTRFLY/10000, WBTC-WETH/3000, WETH-ENS/3000, MATIC-WETH/3000, UNI-WETH/3000, USDC- +WETH/500, SHIB-WETH/3000, WETH-USDT/500, MKR-WETH/3000, SHIB-WETH/10000. +Case C2. +LT data: thresholds for common origins, senders and bridges are 1, 000 and 50 and 500 respectively. We re- +cover a final set of 30 pools, which are: DAI-WETH/500, FRAX-USDC/500, WETH-WRLD/10000, USDC-USDT/100, +DAI-USDC/500, USDC-UST/100, WBTC-WETH/3000, DAI-USDC/100, USDC-NCR/500, WETH-USDT/500, +XSGD-USDC/500, LINK-WETH/3000, USDC-WETH/3000, DAI-WETH/3000, WETH-BTRFLY/10000, HEX- +USDC/3000, WETH-ENS/3000, MATIC-WETH/3000, XSGD-WETH/500, WBTC-WETH/500, FXS-WETH/10000, +GALA-WETH/3000, WBTC-USDC/3000, WETH-USDT/3000, HEX-WETH/3000, USDC-RSS3/3000, WETH- +LOOKS/3000, SHIB-WETH/3000, USDC-WETH/500, DAI-FRAX/500. +LP data: thresholds for common origins and senders are 10 and 3 respectively. We recover a final set of 16 pools, +which are: DAI-WETH/500, WBTC-USDC/3000, LINK-WETH/3000, USDC-WETH/3000, WETH-USDT/3000, +WETH-LUNA/10000, WETH-WRLD/10000, WBTC-WETH/3000, MATIC-WETH/3000, USDC-USDT/100, UNI- +WETH/3000, SHIB-WETH/3000, USDC-WETH/500, WETH-USDT/500, MKR-WETH/3000, SHIB-WETH/10000. +Case C3. +LT data: thresholds for common origins, senders and bridges are 1, 000 and 75 and 500 respectively. We re- +cover a final set of 30 pools, which are: DAI-WETH/500, FRAX-USDC/500, FEI-USDC/100, USDC-USDT/100, UST- +WETH/3000, HDRN-USDC/10000, CEL-WETH/3000, WBTC-WETH/3000, DAI-USDC/100, WETH-USDT/500, +UNI-WETH/3000, APE-WETH/3000, LINK-WETH/3000, USDC-WETH/3000, DAI-WETH/3000, HEX-USDC/3000, +WETH-ENS/3000, MATIC-WETH/3000, USDC-USDT/500, APE-USDC/3000, WBTC-WETH/500, WBTC- +USDC/3000, WETH-CRV/10000, WETH-USDT/3000, WETH-LUNA/10000, BUSD-USDC/500, HEX-WETH/3000, +WETH-LOOKS/3000, SHIB-WETH/3000, USDC-WETH/500. +LP data: thresholds for common origins and senders are 10 and 3 respectively. We recover a final set of 11 pools, which +are: WBTC-USDT/3000, DAI-WETH/500, WBTC-USDC/3000, USDC-WETH/3000, UNI-USDC/3000, WETH- +USDT/3000, WETH-LUNA/10000, HEX-USDC/3000, HEX-WETH/3000, WETH-USDT/500, USDC-WETH/500. +25 + diff --git a/ltFPT4oBgHgl3EQfHzTL/content/tmp_files/load_file.txt b/ltFPT4oBgHgl3EQfHzTL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c33871342b33d12f8993ee170e48885674a8543d --- /dev/null +++ b/ltFPT4oBgHgl3EQfHzTL/content/tmp_files/load_file.txt @@ -0,0 +1,1683 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf,len=1682 +page_content='DEFI: DATA-DRIVEN CHARACTERISATION OF UNISWAP V3 ECOSYSTEM & AN IDEAL CRYPTO LAW FOR LIQUIDITY POOLS Deborah Miori Mathematical Institute University of Oxford deborah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='miori@maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='uk Mihai Cucuringu Department of Statistics & Mathematical Institute University of Oxford The Alan Turing Institute, London, UK mihai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='cucuringu@stats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='ox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='uk January 31, 2023 ABSTRACT The Uniswap v3 ecosystem is built upon liquidity pools, where pairs of tokens are exchanged subject to a fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We propose a systematic workflow to extract a meaningful but tractable sub-universe out of the current > 6,000 pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We filter by imposing minimum levels on individual pool features, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' liquidity locked and agents’ activity, but also maximising the interconnection between the chosen pools to support broader dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, we investigate liquidity consumption behaviour on the most relevant pools for Jan-June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We propose to describe each liquidity taker by a transaction graph, which is a complete graph where nodes are transactions on pools and edges have weights from the time elapsed between pairs of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Each graph is embedded into a vector by our own variant of the NLP rooted graph2vec algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we are able to investigate the structural equivalence of liquidity takers behaviour and extract seven clusters with interpretable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Finally, we introduce an ideal crypto law inspired from the ideal gas law of thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Our model tests a relationship between variables that govern the mechanisms of each pool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' liquidity provision, consumption, and price variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' If the law is satisfied, we say the pool has high cryptoness and demonstrate that it constitutes a better venue for the activity of market participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Our metric could be employed by regulators and practitioners for developing pool health monitoring tools and establishing minimum levels of requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Keywords Decentralised Finance · Uniswap v3 · Network Analysis · NLP · Clustering · Ideal gas law 1 Introduction A blockchain is a type of distributed ledger technology (DLT) that stores users transactions on an increasingly long sequence of blocks of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This ledger is duplicated and distributed across the entire network of computer systems on the blockchain to allow the validation of new transactions by the peer-to-peer (P2P) computer network and subsequent addition of blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' During 2007 and 2008, the person(s) known via the pseudonymous Satoshi Nakamoto designed the Bitcoin blockchain and described it in the whitepaper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The project was released as an open source software in 2009 and from that moment onwards, Bitcoin started slowly acquiring increasing value and seeing higher trading volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, Bitcoin is not Turing complete and indeed it lacks the capacity for logical loops and conditionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This limitation fueled the rise of the Ethereum blockchain, which was first described in the 2013 whitepaper [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Ethereum supports smart contract functionality and, due to this, it is able to offer financial instruments that do not rely on intermediaries such as brokerages, exchanges or banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, Ethereum built the foundations for Decentralised Finance (DeFi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Within DeFi, individuals can lend, trade and borrow using software that automatically broadcasts their intentions for P2P verification, and records valid financial actions on a blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Decentralised Exchanges (DEXs) are a direct result of this setup, and started being designed and implemented mainly from 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' They differ from the usual centralised exchanges, since they are non-custodial and leverage the self-execution of smart contracts for P2P trading, allowing users to retain control of their private keys and funds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' One of the first and most established arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='13009v1 [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='TR] 20 Dec 2022 JANUARY 31, 2023 DEXs at the time of writing is Uniswap, built indeed on Ethereum and launched in November 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' There exist three versions of Uniswap (namely v1, v2, v3, see the whitepapers https://hackmd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='io/@HaydenAdams/HJ9jLsfTz, [3], [4] respectively) that update its design and evolve its functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Since we focus on Uniswap v3 data in this research, and specifically investigate its ecosystem, we consider essential to first highlight some of its core aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Uniswap is an automated market maker (AMM), and in particular, a constant function market maker (CFMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This implies that digital assets are traded without centralised permission and the pricing occurs following a mathematical formula, rather than relying on an order book as in traditional exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Uniswap smart contracts hold liquidity reserves of various tokens, and trades initiated by liquidity takers (LTs) are executed directly against these reserves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Reserves are pooled between a network of liquidity providers (LPs) who supply the system with tokens in exchange for a proportional share of transaction fees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' A standard Uniswap liquidity pool allows the exchange, or swap, of two assets via the constant product market maker mechanism (x − ∆x) × � y + (1 − γ 106 )∆y � = x × y = k, (1) where x, y are the current pool reserves of tokens X, Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, k tracks the evolution of liquidity of the pool, and γ ∈ {100, 500, 3000, 10000} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' {1, 5, 30, 100} basis points) denotes the feeTier characteristic of the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Here, we are assuming that a LT sells an amount ∆y of token Y to the pool and receives ∆x of token X back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The exchange rate Z between the two digital assets is given by the proportion of respective reserves in the pool, which changes following the trades of LTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' A proportion γ of each swap is kept by the pool to reward LPs for their service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Swaps do not change k, while this invariant does vary if new liquidity is minted (added) or burned (destroyed) in the pool by LPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Of course, higher liquidity assures less price slippage for LTs and is thus preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' LPs profit an amount proportional to their involvement into the whole liquidity of the pool for each trade occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, they also incur impermanent loss due to the need to stake both tokens to provide liquidity, while bearing the risk of varying exchange rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In Uniswap v3, it is important to notice that concentrated liquidity is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This means that LPs can choose the range of prices and proportions over which they stake their tokens, and they will collect LTs’ fees when the exchange rate of executed trades lies between two ticks over which they are indeed actively providing liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For completeness, it is also worth mentioning that every action (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' creation of a pool, swap, mint or burn operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=') that occurs on Uniswap, or in the general DeFi universe, must be validated and registered on the blockchain before being considered executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This introduces a further cost for the initiator of the action, who needs to pay non-negligible gas fees [5] to miners to compensate them for the computational power they consume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This is especially significant for blockchains that use a Proof-of-Work consensus protocol, such as Ethereum until September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Indeed, Ethereum then transitioned to Proof-of-Stake via the upgrade named “The Merge”, allowing validation of transactions not to only rely on computational power, and opening the opportunity to have lower gas fees and enhanced users participation in DeFi (despite not yet reality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Many further interesting new and old finance concepts live within DeFi and beyond DEXs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' One such concept worth mentioning first is that of stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Stablecoins are digital assets that are pegged to the value of a fiat currency, and can be useful to exit risky positions while remaining inside the crypto ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Some stablecoins are fiat-backed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' USDC, Tether), while others are backed by an over-collateralised pool of cryptocurrencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' DAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' There also exist algorithmic coins (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' UST), which closely resemble traditional pegged exchange rates and are indeed also vulnerable to speculative attacks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' as it happened with the Terra-Luna crash in May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Apart from stablecoins, DeFi provides several lending protocols (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Aave, Compound, Instadapp, Maker), protocols for derivatives trading (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' dYdX, Futureswap, Nexus), and DEX aggregators (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 1inch) that optimise routing to take advantage of the best exchange rates across multiple other exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In [6], we find an interesting study of the interactions between different blockchain protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' While DeFi is fascinating, it is also the stage of many scams, speculative high-risk investments, direct blockchain attacks, and money laundering events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' On top of that, its complexity and atomicity might disadvantage small users, whose transactions can e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' be re-ordered before execution by the validators for their own profit, known here as miner extractable value (MEV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Despite the current effort of regulators to penetrate the crypto world and establish some equilibrium between centralisation and decentralisation, the current situation and possible upcoming developments are still highly confusing, especially for outsiders or newcomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Interesting overviews and critical thoughts are presented in [7] and [8], where the latter work especially discusses enforcing tax compliance, anti-money laundering laws and how to possibly prevent financial malfeasance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' [9] further studies different layers of DeFi and the related risks involved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' at the blockchain, protocol, pool and token level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' They focus on Uniswap and propose a related risk parity approach for portfolio construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The current academic research is also at its very early stages in terms of understanding the inner dynamics of DEXs and external relationships with the well-known traditional stock market, especially from an empirical and data-driven point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In [10], the authors investigate how promoting a greater diversity of price-space partitions in Uniswap v3 can simultaneously benefit both liquidity providers and takers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' [11] studies whether AMM 2 JANUARY 31, 2023 protocols, such as Uniswap, can sustainably retain a portion of their trading fees for the protocol and the expected outflow of traders to competitor venues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Inefficiencies between Uniswap and SushiSwap are investigated in [12], where sub-optimal trade routing is addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, [13] shows that constant product markets should closely track the reference market price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Flows of liquidity between Uniswap v2 pools are studied in [14], while [15] and [16] show the difficulty of earning significant returns by providing liquidity in Uniswap v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Main Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We divert from the available literature in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' To the best of our knowledge, this is the first study to methodically define a set of pools that are necessary to be considered for a full view of Uniswap dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We assess both the pools’ inner features and their interconnectedness, and deliver a workflow for extracting significant sub-universes of pools in time, which can be completely reproduced by the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, we leverage on this first point to cluster and characterise the broad behaviour of LTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Due to the eclectic set of pools that we consider, our view should approximate well the overall patterns of the entire ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Our final contribution is to propose an ideal crypto law for liquidity pools, inspired by the ideal gas law from thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We provide motivation for it and show that pools with high cryptoness, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' strongly adhering to our law, are healthier crypto environments on which to trade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The level of cryptoness of a pool can evolve in time and hence it is important to track it, along with various metrics that quantify the risks associated to the respective pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Structure of the Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In Section 2, we identify the most important and interconnected liquidity pools for different time windows within 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Next, we cluster LTs according to their behaviour on the relevant sub-universes in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Due to the complexity of this ecosystem, we draw on intuition from Natural Language Processing (NLP) and graph embedding techniques to assess structural equivalence of trading behaviour in a novel way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Section 4 expands our investigations by proposing an ideal crypto law to simultaneously model LTs, LPs and price dynamics for each pool under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Finally, we summarise our thoughts and discuss future research directions in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2 Systematic selection of Uniswap v3 pools of interest 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 Empirical introduction to the ecosystem At the time of writing, Uniswap v3 is the latest implementation of this DEX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' It launched in May 2021, introduced the concept of concentrated liquidity and allowed multiple feeTiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each Uniswap version N = 1, 2, 3, the addresses of related liquidity pool smart contracts are stored in the respective “UniswapVNFactory” contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We access them on Etherscan1 by querying for transactions related to UniswapVNFactory addresses234 and filtering for methods “Create Exchange”, “Create Pair” or “Create Pool”, for the three versions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We find total numbers of 3, 857 and 992 and 40 associated calls until 15 November 2022, which we agglomerate at daily level and plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The dates of transition from Uniswap v1 to v2, and v2 to v3, are also depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' It is interesting to notice that the previous protocols remain active after the transitions, but their liquidity can be easily moved to the new Uniswap versions via “Migrator” contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In terms of pools, the main differences between v1, v2 and v3 are that the first protocol allows only pools where one token is ETH and feeTier γ = 3000, while the second one introduces the ability to create a pool between 1https://etherscan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='io/ 2UniswapV1Factory: 0xc0a47dFe034B400B47bDaD5FecDa2621de6c4d95 3UniswapV2Factory: 0x5C69bEe701ef814a2B6a3EDD4B1652CB9cc5aA6f 4UniswapV3Factory: 0x1f98431c8ad98523631ae4a59f267346ea31f984 Figure 1: Daily count of new pools created via UniswapV1Factory, UniswapV2Factory and UniswapV3Factory smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The two vertical orange lines depict the dates of official transition from Uniswap v1 to v2, and from v2 to v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 3 Uniswap newpools created in time 35 v1tov2 30 V2toV3 v1 25 V2 Count 20 V3 15 10 5 0 2019-012019-072020-012020-072021-012021-072022-012022-072023-01JANUARY 31, 2023 Figure 2: Evolution in time of the TVL in USD on Uniswap main Ethereum chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The higher the TVL, the more liquid the ecosystem is considered to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In orange, we show the dates of transition from Uniswap v1 to v2, and from v2 to v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' any two tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, v3 expands pools to possibly have also feeTier 100, 500 or 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Although there is a total of 4, 889 pools directly created with UniswapVNFactory contracts, the majority of them is the result of the 2020-21 cryptomania and inflated creation of new tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This translates to many pools not containing any relevant amount of liquidity locked, but which do not disappear due to the immutability of the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' On the other hand, we also expect wrapped calls to the Factory contracts and thus refer to the above as a lower bound to the number of pools created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As a last note, we shown the evolution of Uniswap liquidity on its main Ethereum chain in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The data are downloaded via the Defi Llama5 API and we proxy liquidity by the total amount of USD locked on the protocol, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Total Value Locked (TVL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 Data download and coarse refinement of pools Each version of Uniswap has its own dedicated subgraph, which has a precise endpoint for querying data and a schema to expose the available fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Our terminology follows the Uniswap v3 schema6 and we download pools data via the related subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Our first aim is to identify the pools most representative of the Uniswap ecosystem, which we interpret as having significant liquidity consumption and provision events, but also showing high interconnectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' To this end, we download the latest summary data of all possible pools, full historical record of liquidity consumption operations, and full record of liquidity provision actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, we develop a systematic approach that aims at increasingly discarding layers of pools with weakest features first and then weakest dynamics too, respectively in this subsection and in the next one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The final data set will be our starting point for the subsequent analyses, but aims at being useful to a wider group of researchers that desire to empirically investigate Uniswap v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Download summary data of pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As of 15 November 2022, we download the latest “Pool” data as described in Uniswap v3 subgraph schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We download pools in descending transaction count (txnCount) order, since this variable is strictly increasing in time, while e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' TVL does not need to be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This allows us to select a universe of pools which have had a minimum number of transactions thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We apply this weak initial filtering on the first 6, 000 pools by txnCount, and find that only 1, 344 pools report at least 1, 000 transactions by 15 November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, we also restrict ourselves to pools where both exchanged tokens are traded in at least 3 pools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' token T is traded against a stablecoin, against ETH and against ETH with different feeTier), in order to focus on interesting dynamics of the full ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The result is that we subset to a universe of 696 pools to consider in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Download LP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We download liquidity provision data for these 696 pools and find non-empty entries for 629 of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Liquidity provision data are downloaded to have a historical record of all liquidity mint and burn operations on each pool, with related USD value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' By computing the total cumulative sum of LPs activity, we proxy the TVL in USD that each pool contains at every moment in time and denote it as “proxyTVL”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Unfortunately, we cannot simply rely on the “PoolDayData” values provided by the subgraph due to incoherences found when cross-checking with Ethereum blochckain data on Etherscan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Uniswap v3 data start on 6 May 2021, when the transition from the previous version of the protocol successfully completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' While for the first months only the 500, 3000 and 10000 feeTiers were implemented, in November 2021 a 5https://defillama.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='com/ 6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='com/Uniswap/v3-subgraph/blob/main/schema.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='graphql#L1 4 le10 TVL in time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 v1 to v2 v2 to v3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 ETH chain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8 USD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 2019-012019-07 2020-012020-07 72021-01 2021-072022-01 2022-072023-01JANUARY 31, 2023 fourth feeTier γ = 100 was activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This generated structural flows, noise and adjustments that we prefer to exclude from our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' On top of that, we recognise that the transition of Uniswap’s foundation blockchain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Ethereum) from Proof-of-Work to Proof-of-Stake in September 2022 could have triggered turbulences on the ecosystem too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we decide to focus our analyses on the six-months period from January 2022 to the end of June 2022, which we consider as the most representative of the actual DEX dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We check for every pool if our proxyTVL passes the threshold of 1, 000, 000 USD (one million dollars) at any point before the end of June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This is motivated by the aim to find pools that were liquid enough at some point in our time window to show interesting behaviour of LTs and LPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We find 282 pools that satisfy this further requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Of these, 210 had that much TVL already at some point before January 2022, and 261 at some point before April 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' While some pools acquire relevance as time passes, other ones can also lose liquidity, as in the extreme case of pools related to the Terra-Luna crash of May 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As a final detail, we highlight that we additionally check whether a pool has at least one million USD in TVL for two consecutive points in time in order to avoid pools where a substantial amount of liquidity is minted and immediately burned by an agent, to likely take advantage of specific external information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Download LT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For the above 282 pools, we download related liquidity consumption data and find all non-empty data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we are left with a final set of 282 liquidity pools, for which we have a summary file, a LP database and a LT database each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This completes our coarse filtering of pools, which implements the least invasive possible initial requirements, while still filtering down the universe of Uniswap v3 liquidity pools to a tractable number of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 3 summarises the steps completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 3: Summary diagram of the filtration steps pursued during our coarse refinement of pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Stronger con- straints on the TVL and txnCount of pools will follow in the next subsection, such as an attention to maximise the interconnectedness of the final sub-universe of pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='3 Final refinements Stronger TVL and txnCount constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' From our first coarse filtration, we recover 282 pools to consider further in Uniswap v3 analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, we shall be stricter about the minimum number of transactions taking place on a pool and its TVL in time, in order to lower the noise-to-signal ratio in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As already motivated, we want to focus on the six-months window [January, July) 2022, which we denote as our case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We also consider five sub-ranges, namely the two three-months windows [January, April), [April, July) that we denote as cases B1/B2, and the three two-months windows [January, March), [March, May), [May, July), that we call cases C1/C2/C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each case and related time window [start, end), we extract the pools with at least 1, 000 transactions before start (where the number of transactions in time is calculated via the cumulative sum of both swap events and mint or burn operations) and that also had at least 1, 000, 000 USD in proxyTVL both at the start and end of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Considering sub-ranges allows us to further account for the appearance of new pools that became significantly liquid or active after January 2022, or pools that lost the majority of their liquidity before July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For cases A/B1/B2/C1/C2/C3 in order, we find respectively 113/126/148/131/146/155 pools that satisfy the above requirements, for which we save the related addresses and information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Taking the union of these sets of pools, we notice that we are considering 177 different pools overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Of these, five pools belong to the 100 feeTier, 28 pools to the 500 feeTier, 84 pools to the 3000 feeTier and 60 pools to the 10000 feeTier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' To gain a brief insight into the most liquid and active venues, we consider the pools extracted for case A and plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 4a the 10 pools with highest proxyTVL at the end of June 2022, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 4b the 10 pools with highest total number of transactions over the six months of relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For the first pool in the ranking of both measures, we plot the related evolution of liquidity and daily number of transactions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As a convention, we refer to pools with the format “SYMBOL1-SYMBOL2/feeTier”, where we deploy the trading symbols of the two tokens exchanged by the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Stablecoins, wrapped Ether (WETH) and wrapped Bitcoin (WBTC) dominate the landscape of tokens swapped in the most liquid and active venues, which is expected since they are the oldest, most established, or safest cryptocurrencies that agents can trade and develop strategies onto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, it is interesting to observe how the DAI-USDC/100 pool is 5 Summary data ofUniswapv3 pools as of TVLand txnCount constraints 15Nov2022 Interconnectedness >6,000 1,344 629 282 pools pools pools poolsJANUARY 31, 2023 (a) Pools with highest liquidity at the end of June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) Pools with highest total number of transactions during case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 4: The 10 most liquid and active pools for case A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' over the time window between January and June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' much younger than the USDC-WETH/500 one, but quickly gained strong liquidity levels due to its tokens being both stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (a) Full history of TVL for the pool with highest final TVL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) Daily txns count for the pool with highest final TVL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (c) Full history of TVL for the pool with highest total txnCount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (d) Daily txns count for the pool with highest total txnCount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 5: Evolution of liquidity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' proxyTVL, and daily number of transactions for the pool with highest TVL at the end of June 2022 (DAI-USDC/100), and for the one with largest total number of transactions during the full six-months window of case A (USDC-WETH/500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Filtering of pools by interconnectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Considering the full list of data sets of liquidity provision and consumption actions to pursue investigations of more than 100 pools becomes highly computationally expensive very soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we further subset our pools of interest by requiring minimum levels of interconnection between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This is also functional in assuring a focus on the deepest dynamics that characterise the Uniswap ecosystem as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each one of our cases A/B1/B2/C1/C2/C3, we build a weighted graph G = (P, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The set of nodes P denotes relevant pools, and edges (p, q) ∈ E with p, q ∈ P have weights wpq that encode some measure of similarity defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We start by considering two possible different measures of connection between pools: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Number of common LTs (or LPs) active on both pools, which are identified by the entry “origin” in the Uniswap data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Number of common smart contracts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' “senders” in the Uniswap data, called by origins to execute swap transactions (or to execute liquidity provision operations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1eB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Total value locked at the end of june 2o22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='proxyTVL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='3 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1Total number of txns between January and June 2o22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='tot txns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='10leB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='DAI-USDC/10D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='proxyTVL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='5 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Dec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Jan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Feb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Mar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Apr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='222 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='JunDAI-USDC/10D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='dailyTxns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Jen ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Feb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Mar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Apr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2422 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Jun ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='imestampleB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='USDC-WETH/SO0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='proxyTVL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Jul ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Crt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Jan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Apr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2422USDC-WETH/SO0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='000 dailyTxns ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='16000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='14000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='12000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Co ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='DO +8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='DO+9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Jean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Feb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Mar ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Apr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Mey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='Jun ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2422 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='timestampJANUARY 31,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2023 Figure 6: The intersection of origins and senders is always zero since the former are wallets of users and the latter smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Recipients can instead be both, hinting to more complex patterns in the execution of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' To be precise, here we are specifically considering the sub-universe of pools relevant for case A at the end of all our refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We separate between a focus on liquidity consumption or provision in the above measures, since the two dynamics differ substantially, and one might prefer to enhance the sub-universe under consideration to be more representative of one or the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Of course, the intersection or union of the results can be then used to pursue broader analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' To clarify some Uniswap terminology, every “swap action” is initiated by an origin O, then it calls a smart contract referred to as sender S, and ends to the recipient R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In liquidity provision, only the origin and sender of operations are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 6 shows the distribution of number of origins, senders and recipients in each pool for both LT and LP data over the time window of case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We also show the distribution of the intersection between origins, senders and recipients’ addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We now proceed to studying the relationship between the size of each graph’s giant component and a minimum threshold on the value of the measure used to create the link between each pair of pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' After fixing a threshold, we consider the pools in the related giant component as our relevant interconnected sub-universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 7 shows the variation in size of the giant component for case A, when modifying the minimum number of common origins or senders for LT and LP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We aim at considering the tails of the distributions for each case (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' time interval), which amounts to ∼ 20/30 pools in each instance, to retain the most significant connections and possible dynamics of the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For case A, this results in the choice of thresholds 2, 000 and 100 for minimum common origins and common senders respectively, on the LT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Similarly, we choose thresholds 30 and 3 for minimum common origins and common senders respectively, on the LP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Finally, we consider the intersection of survival pools for the two graphs generated by LT data, and find 27 common pools (out of the 34 and 36 pools, respectively in each graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For LP data, we find a number of 19 final relevant pools (from the intersection of 25 and 30 pools).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The full pipeline is repeated for cases B1/B2/C1/C2/C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (a) LT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) LP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 7: Evolution of the size of the giant component for graphs of pools in case A, when varying the threshold of common origins and senders for (a) swap transactions, (b) liquidity provision operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 7 SWAP data LP Data 104 105 000 0 o 103 104 Count Count 103 102 0 00 102 101, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=" 0 s'ou no." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=" R O&S O&R oou s'ou O&5size giant component swap 100 commonorigins 75 50 25 0 0 500 QQOT 1500 2000 2500 3000 3500 minimum threshold size giant component swap 100 commonsenders 75 50 25 0 0 50 100 150 200 250 minimumthresholdcomponent LP 100 common origins 75 size giant 50 25 0 0 20 40 60 Bo minimum threshold size giant component LP 100 commonsenders 75 50 25 0 0 2 4 6 B 10 12 14 minimum thresholdJANUARY 31," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2023 For the interested reader,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 8 depicts the distribution of origins, also called Externally Owned Accounts (EOAs), that are both LTs and LPs on each same pool for case A before filtering by interconnectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This quantity is shown both as a ratio of the total number of LTs and of LPs on each pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We witness an extreme case for pool “WETH-sETH2/3000”, for which more than 20% of the total amount of LTs are also LPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, the number of LTs that also act as liquidity providers is a negligible minority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' If we further count the total number of LTs, LPs and LTs acting also as LPs regardless of the pool, we find 479, 161 and 23, 952 and 13, 640 such market participants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Approximately half of LPs also act as LTs, but LTs acting also as LPs are still a small minority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 8: Distribution of origins that act both as LTs and LPs on the same pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Final enhancement on pools for liquidity consumption analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Ideally, we should also consider the flow of funds across pools and find the related most interconnected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, this is intractable if using only Uniswap data and not the full list of Ethereum blockchain transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Indeed, LTs are active across different DeFi protocols and can easily move liquidity from one venue to another and back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We propose an approximation to the problem by taking advantage of the fact that each trader’s transaction can include more actions, which happen “instantaneously but in order” when the full transaction is validated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, if a LT executes two swaps of the form X → Y, Y → Z for tokens X, Y, Z in one same transaction, then we interpret Y as a bridge between the action of selling X to buy Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We view this as an indication of the flow of (smart) money between pools and of possible arbitrage opportunities, relevant to the LT sub-universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In summary, we consider the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Merge all LT data before the interconnectedness analyses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' data for the 113 pools of case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Keep all the transactions for which there are at least two inner actions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' same “transaction id” but different “logIndex” in Uniswap terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each resulting transaction: (a) For each token that appears in the transaction actions, keep a flow list of related buying (−1) or selling (+1) trades in all the related pools by looking at the sign of the amount swapped by the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) For each token, consider its flow list and find all the occasions when a −1 is immediately followed by a +1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' the token was first bought in a pool and then sold in another pool, acting as one of our bridges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (c) Save this occurrence of a flow between pools as a bridge transaction, where we are approximating only jumps of length one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As an example, for a flow list of the form [−1, +1, +1], we only consider the flow as from the first pool to the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For more specific analyses, one could consider the specific amounts traded and check the relative proportions exchanged from the first pool to the second and third ones, but this is outside the scope of our current investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We extract all bridge transactions between pools and create a directed graph for each one of our temporal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Nodes are pools as usual, and edges are built for each pair of pools that have at least some number of bridge transactions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Of course, each pair of pools can have up to two edges between them according to the direction of related bridge transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, we keep the largest connected component from the undirected version of the graph and add the resultant set of nodes to the LTs pools saved from the previous interconnectedness analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For case A, we require at least 800 bridge transactions between two pools to create the related edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The resultant giant component (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 9a for a visualisation) has 22 nodes, seven of which were not included in our LT set of pools from the previous analyses and are thus added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 9b further highlights the nodes with highest eigenvector centrality in the graph, where we can especially notice how several pools of WETH against a stablecoin are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This is intuitively sensible, since LTs can take advantage of routing to complete specific re-balancing of tokens via more liquid and favourable pools, which tend to have stablecoins, WETH and WBTC as their tokens, as shown in the earlier analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The final list of 34 pools that we propose to consider for LTs analyses is: DAI-WETH/3000, CEL-WETH/3000, USDC-UOS/10000, DAI-USDC/100, SPELL-WETH/3000, WETH-CRV/10000, USDC-USDT/500, DAI-FRAX/500, 8 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' of EOAs that are both LTs and LPs on the same pool, wrt total LTs No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' of EOAs that are both LTs and LPs on the same pool, wrt total LPs 101 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' of pools No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 2 oOT 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 Fraction FractionJANUARY 31, 2023 (a) The resulting giant component, with edge weights reported in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) Pools with highest eigenvector centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 9: Results from our bridges investigation for case A, which covers the six-months window from January to June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' If a LT executes two swaps X → Y, Y → Z one after the other (for tokens X, Y, Z), then we interpret Y as a bridge between the action of selling X to buy Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We save all pairs of pools for which there is a common token that acts as a bridge, with the related number of occurrences of bridge transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, we create a directed graph where nodes are pools and edges are built for each pair of pools that have at least 800 bridge transactions between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' WETH-BTRFLY/10000, GALA-WETH/3000, WETH-USDT/3000, WBTC-USDC/3000, DAI-USDT/500, UNI- WETH/3000, WETH-ENS/3000, DAI-USDC/500, WBTC-WETH/500, MATIC-WETH/3000, DAI-WETH/500, WETH-USDT/500, USDC-WETH/500, LINK-WETH/3000, WBTC-WETH/3000, FXS-WETH/10000, FRAX- USDC/500, USDC-WETH/3000, USDC-WETH/10000, LUSD-USDC/500, HEX-USDC/3000, USDC-NCR/500, SHIB-WETH/3000, DYDX-WETH/3000, USDC-USDT/100, HEX-WETH/3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Regarding LP pools, we have instead the following 19 pools: WETH-CRV/10000, MKR-WETH/3000, WETH- USDT/3000, WBTC-USDC/3000, UNI-WETH/3000, WETH-ENS/3000, WBTC-WETH/500, MATIC-WETH/3000, DAI-WETH/500, WETH-USDT/500, USDC-WETH/500, LINK-WETH/3000, WBTC-WETH/3000, USDC- WETH/3000, SHIB-WETH/3000, WBTC-USDT/3000, USDC-USDT/100, USDC-USDT/500, SHIB-WETH/10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The final results for cases B1/B2/C1/C2/C3 are then listed in Appendix A, for the benefit of the reader that can use these sub-universes of pools as starting point for their own investigations on Uniswap v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In our next steps, we specifically focus on the pools extracted for longest cases A/B1/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 3 Structural Investigation of the Uniswap v3 Ecosystem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 Clustering of Liquidity Takers 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 Overview and pre-processing The DeFi ecosystem has grown increasingly complex in the recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The first step to shed more light on its intrinsic features and dynamics is to better understand its own components, which is what motivates the following empirical investigation of LTs trading behaviour on Uniswap v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This is a non-trivial task, for a number of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' First of all, agents can easily generate numerous crypto wallets, and hence in some sense, “multiply” their identities to hide or obfuscate their full behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Their actions are then generally spread over a broad set of possible pools, vary significantly in size both within and across different types of pools, and also happen with evolving frequencies over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Applying the usual initial clustering methodologies would indeed be difficult (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' defining a set of features that characterise pools to then perform dimensionality reduction, and finally compute similarity measures), due to the 9 WETH-BT Y/10000 1051 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='034 WETH HEx- 3000 1316 803 WSD 5/500 1413936 WETH T/3000 1208 USDE T100 2985 6467 1753 HEx-0 000 USDC- 1/3000 984 2012 676 USDC T/500 WBTC H/500 USDg /500 2041 DAI-L /100 USDC-I DOOQT WBTC- C/3000 1541 911 1007 2743 DDAI-W 000E 9 WBTC- H/3000 DAI-M /500 944 10211254 1461 666 DAI-F y500 DAI-U 1500 DAI-U /500Eigenvector Centrality ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 0 eig-centrality 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 ////////JANUARY 31, 2023 Figure 10: Distribution of total number of transactions (txns) performed by LTs during case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We show the full distribution, the result after requiring a minimum of at least 25 transactions, and the distribution after applying thresholds of minimum 60 and maximum 15, 000 transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The latter scenario results in our final set for case A, which comprises 3, 415 LTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' A small cluster of LTs much more active than others is already discernible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' complexity of the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we propose a novel method to express and cluster structural trading equivalence of agents on multiple environments by leveraging both network analysis and NLP techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' A sample of possibly external features is then used to judge and characterise the groups unravelled and extract insights on the main species of agents present in the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We focus on the LT data for our three longest periods A/B1/B2, which we defined and described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each case, we first look at the distribution of the total number of transactions performed by the different LTs over each full time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We require a minimum number of transactions completed by each LT, since considering only a very small sample of trades per agent would not provide meaningful structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We impose a lower bound of at least 10 transactions per month on average over the time window of each case, and define maximum thresholds by considering the respective own distributions and removing only extreme singular outliers for computational purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We show the initial total distribution for case A in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 10, where we also highlight how it changes when requiring a minimum number of transactions equal to 25, and when we require our final minimum and maximum thresholds of 60 and 15, 000 total number of transactions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For cases B1/B2, we require the range 30 to 5, 000 transactions for the former, and 30 to 11, 000 transactions for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Overall, we find a number of LTs approximately between 3, 500 and 5, 000 for all our periods A/B1/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This altogether defines the final sets of LTs along with their transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Next, we proceed to computing their embeddings, which are subsequently used for the final clustering stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 Methodology NLP background and graph2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The field of Natural Language Processing (NLP) studies the development of algorithms for processing, analysing, and extracting meaningful insights from large amounts of natural language data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Examples of its myriad applications include sentiment analysis of news articles, text summary generation, topic extraction and speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' One of the turning points in NLP was the development of the word2vec word embedding technique [17], which considers sentences as directed subgraphs with nodes as words, and uses a shallow two-layer neural network to map each word to a unique vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The learned word representations capture meaningful syntactic and semantic regularities, and if pairs of words share a particular relation then they are related by the same constant offset in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As an example, the authors observe that the singular/plural relation is captured, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' xapple − xapples ≈ xcar − xcars, where we denote the vector for word i as xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Words sharing a common context in the corpus of sentences also lie closer to each other, and therefore, relationships such as xking − xman + xwoman ≈ xqueen are satisfied with the analogies indeed predicted by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Taking inspiration from this idea of preserving knowledge of the context window of a word in its embedding, the node2vec algorithm [18] learns a mapping of nodes in a graph to a low-dimensional space of features by maximising the likelihood of preserving network neighbourhoods of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The optimisation problem is given by max f � s∈S log Pr(NL(s)|f(s)), (2) where G = (S, T) is a graph with nodes S and edges T, f is the mapping function for nodes to n-dimensional vectors that we aim to learn, and NL(s) ⊂ S is the network neighbourhood of node s generated with sampling strategy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The 10 Case A all 105 min 25 60-15,000 104 S17 102 101 100 0 5000 10000 15000 20000 Total no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' txns over the periodJANUARY 31, 2023 latter is designed by the authors of node2vec as a biased random walk procedure, which can be tuned to either focus on sampling a broader set of immediate neighbours, or a sequence of deeper nodes at increasing distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, Problem (2) is solved for f by simulating several random walks from each node and applying stochastic gradient descent (SGD) and backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' By taking a further step towards general language representations, [19] proposes the unsupervised algorithm Paragraph Vector (also known as doc2vec), which learns continuous fixed-length vector embeddings from variable-length pieces of text, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' sentences, paragraphs and documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The vector representation is trained to predict the next word of a paragraph from a sample of the previous couple of sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Both word vectors and paragraph vectors need to be trained, which is again performed via SGD and backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As doc2vec extends word2vec, graph2vec [20] is a neural embedding framework that aims to learn data-driven distributed representations of an ensemble of arbitrary sized graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The authors propose to view an entire graph as a document, and to consider the rooted subgraphs around every node in the graph as words that compose the document, in order to finally apply doc2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This approach is able to consider non-linear substructures and has thus the advantage to preserve and capture structural equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' One necessary requirement to pursue this analogy is for nodes to have labels, since differently labelled nodes can be then considered as different words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' These labels can be decided by the user, or can be simply initiated with the degree of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, doc2vec considers a set of graphs G = {G1, G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='}, where the nodes S of each graph G = (S, T, λ) can be labelled via the mapping function λ : S → L to the alphabet L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The algorithm begins by randomly initialising the embeddings for all graphs in the set G, then proceeds with extracting rooted subgraphs around every node in each one of the graphs, and finally iteratively refines the corresponding graph embedding in several epochs via SGD and backpropagation, in the spirit of doc2vec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The rooted subgraphs act as the context words, which are used to train the paragraph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' graph) vector representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Subgraphs are extracted following the Weisfeiler-Lehman (WL) relabeling process [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The intuition is that, for each node in a graph, all its (breadth-first) neighbours are extracted up to some depth d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Labels are then propagated from the furthest nodes to the root one, and concatenated at each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In this way, a unique identifier for each node is identified from its “context” and the full set can be used to train an embedding for the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The optimisation problem thus becomes max f ′ � G∈G � s∈S log Pr(gd W L(s)|f ′(G)), (3) where the aim is to maximise the probability of the WL subgraphs given the current vector representation of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Here, f ′ is a mapping function of graphs to n-dimensional representations, and gd W L are WL subgraphs with depth d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' A modification of graph2vec for LTs embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each one of our cases A/B1/B2, we consider all the related LTs and their full set of transactions on the sub-universe of LTs’ pools of relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We then introduce the concept of a transaction graph Gtxn, which we use to represent the behaviour of each active agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 (Transaction graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' A transaction graph Gtxn = (S, T, W) is the complete weighted graph where nodes S are the swap actions that the LT under consideration has completed, and edges (s, r) ∈ T with s, r ∈ S are built between every pair of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Each edge has a weight wsr ∈ W, which encodes the amount of time ∆t (in seconds) elapsed between the two transactions s, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Each node s ∈ S has a label ls from the alphabet L , which uniquely identifies the pool that the swap was executed into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Importantly, L is shared among the full set of LTs and related transaction graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Labels in the alphabet L differentiate between swaps executed on different pools, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' pools with unique combination of tokens exchanged and feeTier implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This implies that the algorithm receives as input only general identifiers of pools , while we can consider intuitive differences (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' expected volatility of the exchange rate on pools of stablecoins versus on pools of more exotic tokens) only afterwards, when assessing and investigating the meaningfulness and interpretability of the extracted clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We now have a set of graphs representing LTs, and our aim is to find a n-dimensional vector representation of each one of its elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We cannot plainly apply the graph2vec algorithm, since the concept of neighbours of a node is irrelevant in a complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we modify its mechanism to take advantage of the weight that the different links between nodes have, while maintaining the overall intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each node s ∈ S of a graph Gtxn, we sample a set of neighbours Ntxn(s) by generating random numbers from a uniform distribution between [0, 1] and comparing them to the cut-value of the edges between the node and possible neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' If the value is below the cut-value, then the link is kept and the associated node added to Ntxn(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In this way, the probability of an edge to be chosen is inversely proportional to its weight ∆t, and the sub-structures kept represent clustered activity in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 11 JANUARY 31, 2023 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 (Cut-value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The cut-value C(wsr) of an edge (s, r) ∈ T with weight wsr ∈ W in graph Gtxn = (S, T, W) is computed as C(wsr) = H(f scal(wsr)) H(f scal(min W)), with H(wsr) = � 2 π exp −w2 sr 2 , wsr ≥ 0, f scal(wsr) = wsr − min W (max W)/|S| , (4) where we are using a half-norm that is shifted and scaled to adapt to each LT’s extreme features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' min W and max W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The final cut-value is also normalised to impose a value of C(min W) = 1, meaning that the shortest link(s) in the graph is chosen with probability 1 (of course only if it is involved in the current node under consideration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' After having generated the set of Ntxn(s), ∀s ∈ S, we perform WL relabeling and proceed as in the vanilla version of the graph2vec algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We set all the hyperparameters to their default values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' number of workers = 4, number of epochs = 10, minimal structural feature count = 5, initial learning rate = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='025, and down sampling rate of features = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The only exception is the number of WL iterations, which in our case must be set to 1 instead of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The result is an embedding for each graph in our set of transaction graphs, which becomes a set that we can subsequently cluster via the popular k-means++ methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Importantly, we want to underline that our embeddings and clusters do not depend on the real magnitudes of weights ∆t, since the sampling is adjusted on that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In addition, they also have no notion of the amount of USD traded, thus being agnostic to the transaction value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As a final note, we refer the reader to [22] for a version of graph2vec that uses edge labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, the algorithm creates the dual version of the graph and would not be effective in our case, thus providing ground for our proposed extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' An illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' To increase clarity on our approach, we briefly describe a simple example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Consider an agent that performs 20 transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' She performs the first 10 transactions shortly clustered in time, waiting only 60 seconds one after the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, she waits 42 minutes to action on the final 10 transactions with, again, a frequency of one minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Her behaviour is plotted in the transaction graph Gtxn of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 11a, where we assume for simplicity that each transaction is performed on the same pool and thus colour-code nodes all the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We also number nodes to show the order in which the related transactions are executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We do not draw all the edges of this complete graph for cleanness of the diagram and ease of visualisation, but hint with the green dashed lines that indeed there are more connections to be remembered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In this example, the minimum time between transactions is 60 seconds (light blue edges) and the maximum one is one hour, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 3, 600 seconds (light grey edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Some intermediate times are depicted as edges with the same colour for the same weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The resultant cut-value function C(w) that defines our sampling probabilities to choose edges is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 11b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As intuitively desired, we aim at always keeping the shortest edges and indeed these have probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, we also aim to keep the most clustered “communities”, and indeed, we observe from the plot that transactions five minutes away are still chosen with 40% probability, but longer times are very easily dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (a) Transaction graph Gtxn = (S, T, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) Cut-value C(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 11: For our illustrative example, we show in (a) a simplified representation of the LT’s transaction graph, and in (b) the cut-value that defines probabilities of keeping edges as neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 /tokeep 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 0 100 200 O0E 400 500 600 700 800 edge weight (△t in seconds)JANUARY 31, 2023 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='3 Discussion of results For each case A/B1/B2, we study the structural equivalence of LTs’ trading activity by clustering the representations generated via our modified graph2vec algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Focusing first on case A, we compute embeddings for dimensions n ∈ {8, 16, 32, 64}, and confirm with Principal Component Analysis (PCA) that the proportions of data’s variance captured by different dimensions are well-distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each n-dimensional set of vectors, we then group LTs by performing a series of k-means++ clusterings with different number of desired groups, and choosing the result lying at the elbow of the related inertia plot, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' applying the elbow method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The similarity between optimal clusterings for different dimensions is then computed, in order to investigate the stability of results across representations of increasing dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We achieve this by computing the Adjusted Rank Index (ARI) [23], which is a measure of similarity between two data clusterings in the range [−1, 1], adjusted for the chance of grouping of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We find ARIs for clusterings on 8-vs-{16, 32, 64} dimensional data around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75, while clusterings on 16-vs-32, 16-vs-64 and 32-vs-64 dimensional data reach approximately the value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Therefore, we conclude that there is a high stability of results when our data are embedded at least in 16 dimensions, and use the related 16-dimensional vector representations for our final analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The related optimal number of clusters of LTs for case A is seven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Similar results arise for cases B1/B2 too, and the related optimal numbers of clusters of LTs are six and seven, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Each extracted clustering is based on the structural similarity of LTs’ trading behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' To judge the goodness of our modified algorithm and assess the results, we investigate whether there are specific features or trends that are highly representative of only some of the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we proceed to build a set of characteristics to compute for each LT and calculate the average of these results over the LTs belonging to each different group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The features that we consider are: average and median USD traded, average and median time ∆t in seconds between transactions, proportion of transactions done in “SS”, “EXOTIC” or “ECOSYS” pools, and related entropy, proportion of transactions done in pools with a specific feeTier, and related entropy, proportions of trades on days when the SP LargeCap Crypto Index7 increased or decreased in value, or when the market was closed, due to weekends and bank holidays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The distinction between “SS”, “EXOTIC” or “ECOSYS” pools is inspired by the classification in [14], where the authors introduce a notion of normal pools, stable pools and exotic pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For them, stable pools exchange tokens that are both stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Normal pools trade instead tokens that are both recognised in the crypto ecosystem, while exotic pools deal with at least one token that is extremely volatile in price (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' YAM, MOON and KIMCHI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We slightly divert from this classification and define “SS” pools as pools whose tokens are both stablecoins, “ECOSYS” pools as pools that exchange only tokens that are either stablecoins or pegged to the most established BTC and ETH coins, and “EXOTIC” pools as the remaining ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' ECOSYS pools can be seen as the venues carrying the “safest” opportunity for profit for a novice crypto investor, since they trade volatile tokens though directly related to the most established blockchains that are the true foundations of the whole DeFi environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The average magnitude of features computed over the LTs belonging to each different cluster for case A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' over Jan-June 2022, is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We focus on the groups found specifically for this period because it is the longest one and thus, it provides us the most general results and insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Cases B1/B2 will be later described too, in order to assess the overall stability of recovered species of LTs and highlight any specific variations due to different sub-periods in time and related pools of relevance considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The seven clusters of LTs found have sizes of 304/142/512/978/379/186/914 agents respectively, which means that we are able to find a well-balanced distribution of cluster sizes without any dominant clusters in terms of size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thanks to the heatmap, we also easily confirm that our methodology is able to extract different groups of LTs that have significant variation of behaviour with respect to the outer features defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, a few columns had to be dropped due to non-significance of their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Importantly, we also remind that inner biases on ratios are present (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' when considering that our sub-universe does not have a uniform distribution of numbers of pools with specific feeTier), and thus we can expect more/less transactions of some type on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For visualisation purposes, we also embed the 16-dimensional representations of LTs into a 2-dimensional view via t-SNE, and plot them with perplexity = 15 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 12b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' LTs are colour-coded according to the cluster they belong to, and we indeed observe that different groups lie on different parts of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Focusing on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 12a, one can immediately draw the following high-level remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Groups 0 and 1 have a strong focus on trading exotic cryptocurrencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The former set of LTs mainly uses feeTier 3000 for the purpose, and shows slightly higher than average tendency to trade when the market is 7https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='spglobal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='com/spdji/en/indices/digital-assets/sp-cryptocurrency-largecap-index/ #overview 13 JANUARY 31, 2023 closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The latter group uses significantly both the 3000 and 10000 feeTiers, meaning that the related LTs are willing to accept also extremely high transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This behaviour could indicate that they have high confidence on their intentions and possibly urgency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' On the other hand, groups 2 and 3 trade stablecoins more than usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The former cluster could point to an enhanced use of SS pools to take advantage of optimised routing, while the latter has a non-negligible proportion of trades in exotic pools with feeTier 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Likely, group 3 isolates a set of LTs that are interested in niche exotic tokens, which are only proposed in pools against stablecoins that do not overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Diverting funds between two of these exotic tokens requires an exchange between the two related stablecoins too, which motivates the recovered statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We also witness strong usage of the feeTier 100, which hints to traders trying to compensate the high costs suffered in pools with feeTier 10000 by paying the lowest possible fees on the SS pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Groups 4 and 6 are more active than average on ECOSYS pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The two groups differ noticeably from their opposite relative strength of USD traded and time between operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Overall, group 6 trades less money and waits longer, mainly using pools with low feeTier 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' These features can be interpreted as characteristics of cautious retail traders that invest in less risky and highly well-known crypto possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' And indeed, we also find that this group is one of the largest in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Then, group 4 also relates to ECOSYS pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, these users tend to trade more USD with higher frequency, and this is also the cluster with much higher than average proportion of LTs that also act as LPs (∼ 16%) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Therefore, we identify here a group of more professional investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Finally, group 5 shows a significant usage of all the three types of liquidity pools, but trades are concentrated in pools with cheap feeTier 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' These agents trade often, and indeed show the smallest median time between transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' These eclectic, active and thrifty LTs are probably our group of smartest investors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Our results confirm that the proposed algorithm is able to recognise variance in the data, and we also manage to extract interesting insights into the species of LTs’ behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We highlight the recovered importance of different types of pools, despite no full notion of tokens and feeTier is used in the generation of the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Indeed, only a unique label per pool is applied, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' USDC-WETH/500 could be pool “P1”, USDC-WETH/3000 pool “P2” and FXS-WETH/10000 pool “P3”, and these would be considered equally different if no structural pattern was inherently characteristic of the first two and recognised by the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, we have mainly focused on the liquidity consumption component of the crypto ecosystem thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In the next step of our investigation, we shift the focus from LTs to pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We first aim to perform a clustering of pools based on features built from simple statistics that consider both liquidity consumption and liquidity provision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This will allow us to assess whether the SS, ECOSYS and EXOTIC classification really describes the crypto ecosystem or is only useful for LTs characterisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Before proceeding to this task, we first briefly report on a stability analysis component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Stability analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As already motivated, we now pursue the same analyses described above but for cases B1/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We cluster the n-dimensional embeddings for n ∈ {8, 16, 32, 64} and compute the ARIs between each pair of resultant sets of LT groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We confirm that at least a 16-dimensional embedding is required in order to have a stability of clusters in case B1, while only eight dimensions suffice for the case B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For simplicity, we use the 16-dimensional representations consistently in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We recover six groups of LTs in case B1, and seven in case B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In both cases, we find two clusters with same characteristics as groups 4 and 6 of case A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' traders mainly active on ECOSYS pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We also recover the eclectic traders of group 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Therefore, we observe several stable and persistent types of LTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Small perturbations happen instead on the groups trading on SS or EXOTIC pools, as one could expect from the mere evolution of time and external market conditions, and consequently generation of different behaviours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In particular, all case A species, except group 1, are also found in case B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' On the other hand, case B2 shows less intensity on group 3, probably due to investors diversifying more during the crypto turmoils of the second quarter of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Overall, we observe general agreement on the groups and main features recovered during cases A/B1/B2, and we can thus rely on our species of LTs found for the longest duration case A as descriptors of the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The above findings are of interest in themselves, first of all, since central banks started hiking interest rates in March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This consequently stopped a strong influx of liquidity into the crypto ecosystem and accentuated a period of significant underperformance, that could have weakened the stability of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' On top of that, the Terra-Luna crash happened in May 2022 and it could have in theory enhanced noise and instabilities especially in the structural clustering on case B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' As a very last remark, we notice that only ∼ 20% addresses are present in all cases A/B1/B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Therefore, we are either recovering similar behaviour but for different people, or in some cases it could be the same person simply employing a new wallet to hide their trading behaviour better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 14 JANUARY 31, 2023 (a) Average features for LTs in case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) t-SNE embedding visualisation of case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 12: Clustering of LTs for case A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' over the six-months time window between January and June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In (a), each row represents one of the recovered clusters and columns are the different features computed to characterise species of LTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The color-code employed applies to each column separately to be able to quickly identify the related smallest and biggest values in magnitude, and judge the general distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' It is essential to always check the magnitudes of cells per se too, due to highly variable variance between columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In (b), the t-SNE plot of embeddings of LTs is reported with perplexity = 15 and points are color-coded according to their cluster of membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 Clustering of pools 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 Motivation and Methodology The above analyses revealed a characterisation of the main types of LTs structural trading behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' While the importance of different types of pools in the ecosystem seems to be also clear, we stress that a full understanding of liquidity pools goes beyond the mere liquidity consumption mechanism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' it needs to further account for both liquidity provision and price evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we now pursue an intuitive initial investigation of the similarity of pools themselves, in order to gain additional insights on the entire ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We focus on case A, as it covers the longest period in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We consider the intersection of pools relevant for both LTs and LPs to properly account for both mechanisms, and find a resulting set of 16 pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each pool, we compute the following 13 features: average daily number of active LTs/LPs - “SdailyLT” and “LdailyLP” respectively, volatility of the execution price of the pool - “SstdP”, average size of swap/mint/burn operations in dollars - “SavgUSD”, “LavgUSDmint” and “LavgUSDburn”, 15 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6e+04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2e+05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8e+04 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='23 _ECOSYS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 _USD asn _100 500 fees avg_c close avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' ratio_1 ratio_5 "suxA 60 40 20 0 0 20 1 7 40 3 4 09- 5 6 80 75 50 25 0 25 50 75JANUARY 31, 2023 Figure 13: Spearman correlation between the computed features for pools, with the addition of feeTier, for our case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' average daily amount of dollars used in swap/mint/burn operations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' volume - “SdailyVol”, “LdailyVolMint” and “LdailyVolBurn”, average daily number of LTs/LPs transactions - “SdailyTxn” and “LdailyTxn”, average daily number of different senders, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' smart contracts, called within swap transactions - “SdailyS”, number of agents with only one transaction normalised by the number of days considered - “Sdaily1txn”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This measure is computed to gauge the tendency of external smart investors to hide their behavior by creating several different wallets on the pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For the above features, we create related labels for ease of reference, which start with letter “S” if the quantity is computed from swap operations, or letter “L” if the quantity is computed from liquidity provision operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 13, we show the heatmap of Spearman correlations between the above attributes plus feeTier (“SfeeTier”) for our pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' There are significant positive correlations, especially among features developed from LT data and LP data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we standardise entries and employ linear PCA and kernel PCA (with both “rbf” and “cosine” kernel in the latter) to reduce the dimensionality of our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The eigenvalue decay for all three mentioned cases is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 14, where only the first seven eigenvalues are depicted for clarity of visualisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The cosine kernel PCA is seen to capture more variance in fewer dimensions, and thus we embed the data by projecting on its related first three components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The resulting 3D embedding is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 15 from three different angles, where we color-code pools according to their feeTier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In particular, green relates to feeTier 100, blue to 500, orange to 3000, and red to 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 Discussion From the projections shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 15 and initial trials of clustering, it is clear that the division between SS, ECOSYS and EXOTIC pools does not hold when considering the full set of dynamics on pools (while it is indeed suitable in (a) Linear PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) rbf kernel PCA.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='98 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 SfeeTier asnGAes SdailyLT SdailyTxn SdailyVol , SdailyS Sdaily1txn LavgUSDmint LavgUSDburn ilyLP LdailyTxn LdailyVoIMint - LdailyVolBurn 1 lep7Linear PCA, Eigenvalue Decay 0.' 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+page_content="35 OE'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00 0 1 2 E 4 5 6 EigenvalueJANUARY 31, 2023 (a) Angle = 45 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) Angle = 135 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (c) Angle = 315 degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 15: Projection on a 3D space of the vectors encoding different features of pools, from the application of PCA with cosine kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Views from different angles are reported for a better judgment of the results, and pools are color-coded according to their feeTier (green relates to feeTier 100, blue to 500, orange to 3000 and red to 10000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' connection to specifically LTs’ behaviour).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Similarly, we do not witness strong proximity of pools with same feeTier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Liquidity consumption, provision and price evolution are all essential mechanisms to consider for a full description of the Uniswap ecosystem, and our intuition is that certain combinations of tokens and feeTiers are more similar and suitable for trading at different moments in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' LPs are more incentivised to enhance liquidity on pools with strong LTs activity, low volatility of the exchange rate to avoid impermanent loss, and possibly high feeTier from which they indeed mainly profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In parallel, LTs are more interested in pools with low fees but some volatility of the price of tokens, and high liquidity to diminish the market impact of their trades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, different adjustments of these mechanisms can result in the proximity or not of our projections of pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In the next Section of our work, we propose a model to judge the health of each pool’s combination of mechanisms and characterise the best venues for market participants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' both LTs and LPs) to be active on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 4 The ideal crypto law and a cryptoness measure of a liquidity pool 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='1 Model: from the ideal gas law of thermodynamics to the ideal crypto law for pools In physics, an ideal gas is a theoretical gas composed of many randomly moving particles with negligible volume that are not subject to interparticle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' On the other hand, real gases occupy space and molecules interact between themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Ideal gases obey the ideal gas law, which says that PV = nRT, (5) where P is the pressure of the gas, V its volume and T its temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Furthermore, n denotes the constant number of moles of particles in the considered closed system, and R is the gas constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Despite being very simple and elegant, this law describes very well complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Our intuition finds resemblances between the law in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (5) and the crypto ecosystem, and we consider an analogy in which each pool is a gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' To define the similitude between variables that is summarised in Table 1, we follow how the ideal gas law was discovered and reason about both the possible meaning of variables and expected relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' First of all, P can be intuitively compared to the USD volume traded by active LTs over e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' If everything else but T is kept constant, then we expect T to increase with higher P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Therefore, we can interpret T as the liquidity of the pool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' the value of our proxyTVL in USD for the pool at that date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Indeed, the evolution of liquidity of a Ideal gas law Ideal crypto law Symbol Meaning Symbol Meaning P pressure Pvol daily USD volume traded by LTs V volume Vstab = STD(Z)−1 daily stability of the exchange rate Z n moles of particles nfee = feeTier−1 stimulus to LTs’ activation R gas constant Rpool pool crypto constant T temperature Tliq daily liquidity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' proxyTVL value Table 1: Parallelism drawn between the ideal gas law and our ideal crypto law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 17 WBTC-WETH/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8 WETH-USDT/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 JSDC-WETH/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 UNI-WETH/3000 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 WBTC-WETH/500 LINK-WETWEUENS/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 SHIB-WETH/3000 JUSDC-USDT/100 DAI-WETH/500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 JSDC-WETH/500 WETH-USDT/500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00WBTC-WETH/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8 USDC-WETH/3000 WETH-USDT/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 WBTC-WETH/500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 USDC-USDT/100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 MANETWERVSCOO00 SHIB-WETH/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 JSDC-WETH/500 DAI-WETH/500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 WETH-USDT/500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content="75 1 00'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75WBTC-WETH/3000 WETH-USDT/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8 USDC-WETH/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 UNI-WETH/2A993000 WBTC-WETH/500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 LINK-WETH/3000 3 MATIC-WETH/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 DAI-WETH/500 WETH-CRV/10000 SHIB-WETH/3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 USDC-USDT/500 WBTC-USDC/3000 WETH-USDT/500 JUSDC-USDT/100 JSDC-WETH/50D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='25 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='75 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='00JANUARY 31, 2023 pool accounts for the behaviour of LPs, and more LPs should execute mint operations when there are more LTs active, in order to collect higher profit from the fees that the latter pay for each swap transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Clearly, we also expect a stronger overall volume traded by LTs with more liquidity, due to more convenient, smaller price impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Variable V is then the volume of the gas in the thermodynamics interpretation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' ideal gas law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Despite being a more subtle relationship, we find that it is reasonable to consider V as the stability of the exchange rate between the two tokens in the pool, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' STD(Z)−1, where Z is the exchange rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Keeping everything else constant, one would expect that, with higher liquidity, the price is more stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Similarly, a compressed gas with small volume will be less stable than an expanded gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In addition to that, the concentrated liquidity mechanism of Uniswap v3 implies that LPs encounter the risk of no gains from LTs’ fees if the exchange rate moves outside the range of prices over which they are actively providing liquidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, a more stable Z for the same USD volume traded by LTs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' P) is likely to attract more minting operations, especially close to the current rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Finally, a higher P with the same level of liquidity is indeed likely to cause a less stable relative price of the tokens, due to the impact of surplus swap operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Keeping the above in mind, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (5) thus becomes Pvol × Vstab = nfee × Rpool × Tliq ⇒ Pvol × STD(Z)−1 = feeTier−1 × Rpool × Tliq, (6) allowing us to bring together into a single formula, all the variables that govern the three mechanisms of liquidity consumption, provision and exchange rate evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Constant Rpool is instead the invariant characteristic of each pool, which we aim at regressing with some level of significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Here, n is the fixed number of moles (molecules), thus a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' A higher n means more interactions in the physical description, which we relate to lower feeTier and stronger activity of the LTs, that we know dominate LPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, we express n = feeTier−1, which is indeed constant too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Real gases and van der Waals forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Our above model is based on the thermodynamics law for ideal gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' However, it is worth mentioning that there also exists a law for real gases that interact via van der Waals forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This is governed by the Van der Waals equation � P + a n2 V 2 � (V − nb) = nRT, (7) where the variables have same meaning as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' In addition, a is a constant whose value depends on the gas and represents intermolecular forces, while b is the volume occupied by the molecules of one mole of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Based on a preliminary analysis, we believe it could be of interest to expand the ideal crypto law in this direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' however, this is beyond the scope of our current work, and we leave this for future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 Regression analysis and interpretation of results As a first step, we test for empirical instances that support the validity of our ideal crypto law given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We focus on case A and consider the intersection of pools that are relevant for both liquidity provision and liquidity consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This results in a set of 16 pools, namely USDC-USDT/100, WBTC-WETH/500, WETH-USDT/3000, WETH-ENS/3000, MATIC-WETH/3000, WBTC-USDC/3000, WBTC-WETH/3000, DAI- WETH/500, UNI-WETH/3000, SHIB-WETH/3000, USDC-USDT/500, USDC-WETH/500, WETH-CRV/10000, LINK-WETH/3000, USDC-WETH/3000, WETH-USDT/500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each individual pool, we compute the quantities Pvol, Vstab, Tliq reported in Table 1 at daily scale for the full six-months time window between January and June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We suggest not to use higher frequencies due to significant general low rate of activity of LPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We then scatter-plot these daily realisations for each combination of pairs of variables and critically analyse the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Observations with z-score > 3 in absolute value are considered outliers and discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' A few representative examples of the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 16, where we observe hints that the ansatz relationships are indeed satisfied, apart for the case of pools whose tokens are both stablecoins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The latter dissimilarity is perhaps not surprising, since our ideal crypto law aims at encompassing the full ensemble of crypto mechanisms, while the behaviour of LTs on pools of stablecoins has lower relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We broaden our sub-universe of venues of interest by taking the union of pools significant to LTs’ and LPs’ behaviour in case A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' After filtering for relevance of the samples in the daily frequency, we are left with a set of 32 pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For each pool, we perform a linear regression over the available six months of daily values, and thus estimate Rpool by rearranging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (6) to Pvol = Rpool × �nfee × Tliq Vstab � ⇒ ypool = Rpool × xpool, (8) 18 JANUARY 31, 2023 (a) USDC-WETH/3000 pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (b) UNI-WETH/3000 pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' (c) USDC-USDT/100 pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Figure 16: Visualisation of pair relationships from our ideal crypto law for a sample of pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' From left to right for each case, we plot first Vstab vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Pvol, then Tliq vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Pvol, and finally Tliq vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Vstab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' where the intercept is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We compute the coefficient of determination R2 of the regression, which we refer to as the cryptoness ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Thus, pools with high cryptoness are meant to adhere well to our proposed ideal crypto law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We compute the average ξSS over the seven pools found to exchange only stablecoins in our sub-universe, and compare it to the average ξnotSS of all other pools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We find values of ξSS = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='44 and ξnotSS = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='21, meaning that SS pools should not be considered in our model, as expected and already motivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' For the remaining pools, we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' 17a the ones that have ξ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' We notice that different feeTiers appear to be relevant and that there is one interesting occurrence of two pools, both with high ξ, that exchange the same tokens, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' WBTC-WETH/3000 and WBTC-WETH/500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' This latter result raises the question of whether these two pools are generally adhering to our ideal crypto law, or if they might follow it at disjoint periods of time that however influence the overall cryptoness values and render them both significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Before investigating this idea further, it is worth highlighting the fact that several pools with high ξ are seen to exchange exotic tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' Examples are pools trading CRV and UOS, which are respectively the tokens of the well-know DEX Curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='fi, and of the blockchain-based gaming company Ultra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content=' The general pattern that we read is that pools exchanging digital assets linked to a tangible and established business idea might adhere better to 19 USDC-WETH/3000 le10 le10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='06 Vstab 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='8 Pvol 1e6 Pvol 1e6 VstabUSDC-USDT/1OO 1e8 1e8 10000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltFPT4oBgHgl3EQfHzTL/content/2301.13009v1.pdf'} +page_content='2 - X X 9000 12 +QPU +O(VN) +(0) +01 +3 +02 +I《E[r>|2 +(3) +04 +0(8) +E +(2) +0 +(1) +C(0) +(4) +tr(pRA) +CPU +CPU/QPU2 +12], which employ a digital quantum computer aided +by a classical optimizer. +Although generic VQAs suf- +fer from the well known barren plateau problem [13– +15] which suggests unscalablility in full generality, there +is evidence that VQAs can calculate the ground state +of certain Hamiltonians using only polynomial quan- +tum resources, e.g. by using the Hamiltonian variational +ansatz for the transverse field Ising model [16]. +Re- +cent works have also considered using VQAs to pre- +pare Gibbs states using cost functions such as the rel- +ative entropy or relative free energy between the cur- +rent state and target state [17]; strategies to overcome +the costly evaluation of the entropic term have also +been proposed [18, 19]. Other finite-temperature VQAs +prepare thermofield-double (TFD) states, which require +doubling the number of qubits in the physical system +being simulated—for example the algorithm of Ref. [20] +can prepare the TFD state of free fermions efficiently +at any inverse temperature. Alternative quantum algo- +rithms for preparing thermal states include a quantum +version of the minimally-entangled typical thermal states +algorithm (QMETTS) that involves imaginary time evo- +lution on quantum hardware [21], and an algorithm based +on random quantum circuits [22]. +In this work, we task a VQA with calculating mi- +crocanonical averages of local observables in a one- +dimensional (1D) nonintegrable spin model. Our work is +partially inspired by analog quantum simulation [23] and +classical tensor network [24] algorithms for estimating +microcanonical properties. The algorithm takes advan- +tage of the eigenstate thermalization hypothesis (ETH), +in particular the “diagonal” ETH which states that in +a nonintegrable model the energy-basis diagonal matrix +elements ⟨E|A|E⟩ of an observable A approach a smooth +function A(E) in the thermodynamic limit. +The algorithm, which we call the variational micro- +canonical estimator (VME), works as follows (see Fig. 1). +We initialize the QPU in a random product state [step +(0)] |ψ0 +r⟩, whose energy variance is typically extensive in +N (the number of sites) [24]. Given a target energy λ +and microcanonical window size δ, a classical optimizer +is then tasked with minimizing the cost function +C(θ) = ⟨ψ(θ)| (H − λ)2 |ψ(θ)⟩ +(1.1) +[steps (1) and (2)] originally proposed in [9]. +How- +ever, instead of trying to reach a local or global min- +imum, we stop the optimization as soon as Var(H) = +⟨(H − ⟨H⟩)2⟩ ≤ δ2. This produces states whose energy +support is roughly limited to the microcanonical window +of interest [step (3)], and the resulting variational states +|ψr⟩ are then used to compute the expectation of a local +observable A by averaging ⟨ψr|A|ψr⟩ over R variational +states [step (4)]. The ensemble average in step (4) en- +ables a parametric reduction in the error and is essential +to the algorithm’s performance. +We benchmark the VME algorithm on a nonintegrable +Ising chain by comparing its estimates for local ob- +servables to corresponding Gaussian microcanonical en- +semble predictions obtained from exact diagonalization +(ED). For (i) local operators appearing as terms the +Hamiltonian, which we refer to as local operators cou- +pled to energy density as in Ref. [25], and (ii) for target +energies λ in the bulk of the spectrum we conjecture that +the absolute error in the VME estimate scales as +ϵR ≲ O(Rm) + O(δ/N) + O(D−1/2(λ)) +(1.2) +in the thermodynamic limit and for large R. Here, D(λ) +is the density of states at the target energy λ and m ≈ +−1/2. The last two terms in this formula are predicted +by ETH and the first term we give a heuristic argument +for that we substantiate with numerical evidence. +At +the lowest energy density considered, which is near the +edge of the spectrum, we find instead that m ≤ −0.36 +for A coupled to energy density. For A not coupled to +energy density, the O(Rm) scaling appears to be less well +established but power law fits yield −0.51 ≤ m ≤ −0.1. +We then generalize the problem to the reduced state of +small subsystems of the chain and find numerically that +when choosing R = O(N 2) and for certain λ, the VME +appears to approach the corresponding microcanonical +state in the thermodynamic limit. The states prepared +by the VME are consistent with area law entanglement +for a fixed N, and require roughly linearly deep quantum +circuits to prepare. We find that every random initial +product state is able to converge, which we attribute to +the fact that the algorithm does not seek global minima +of the cost function. An additional distinction from other +current VQAs for preparing mixed states is that we pre- +pare pure states one at a time, thus avoiding storage of +a large ensemble of pure quantum states in a quantum +memory. Should the decrease in the trace distance with +R = O(N 2) continue to hold for larger N than we access +in this work, an interesting consequence of the VME algo- +rithm would be that the microcanonical ensemble, which +involves at least one (via ETH) highly entangled (i.e. vol- +ume law) eigenstate, becomes indistinguishable by local +operators from a polynomially large ensemble of weakly +entangled variational states in the thermodynamic limit. +The paper is organized as follows. In Sec. II, we in- +troduce (i) the statement of ETH and (ii) a class of +states which might be called microcanonical superposi- +tion states, which our converged variational states fall +under. We then review related works attempting to use +these states to estimate thermal averages and the rela- +tionship of this problem to ETH. In Sec. III we discuss +how averaging over an ensemble of these microcanoni- +cal superposition states could significantly improve how +well they can estimate microcanonical averages, and then +we detail the VME algorithm which can produce these +states. Finally in Sec. IV we present the numerical re- +sults starting with the behavior of the diagonal and off- +diagonal errors for various local operators, then the trace +distance, and finally the quantum resources like circuit +depth and entanglement. + +3 +FIG. 2. The energy-basis diagonal matrix elements ⟨E|A|E⟩ +of various local observables A acting in the middle of the +chain, plotted against energy density in the nonintegrable 1D +mixed-field Ising model, Eq. (4.1) with parameters J = 1, +hx = −1.05, and hz = −0.5. +Lighter blue colored points +are for system size N = 9 and darker blue points are for +N = 13. +Orange curves are coarse grained versions of the +N = 13 scatter plots which define the “smooth” function +A(E) in the thermodynamic limit. +II. +MOTIVATION +A. +Eigenstate Thermalization Hypothesis +Here we review relevant aspects of the ETH and some +recent works which attempt to exploit it to estimate ther- +mal averages. We assume a nonintegrable (i.e. chaotic) +Hamiltonian H which has a non-degenerate energy spec- +trum so that its eigenstates |E⟩ are uniquely labeled by +their energies E. Furthermore, we will assume that all +operators and states of interest are real in the energy ba- +sis for simplicity. The variant of ETH we consider was +formulated in Ref. [26] and proposes that in a quantum +chaotic system, the energy-basis matrix elements of ob- +servables have the form +⟨E| A |E′⟩ = δEE′A( ¯E) + D−1/2( ¯E)f( ¯E, ω)REE′ (2.1) +where ¯E = (E + E′)/2, ω = E − E′, D( ¯E) is the den- +sity of states at energy ¯E, A( ¯E) and f( ¯E, ω) approach +smooth functions in the thermodynamic limit, and REE′ +are order-one fluctuations. Examples of such functions +A( ¯E) are shown in Fig. 2 which demonstrates this for +local spin operators in the 1D mixed-field Ising model +(defined in Sec. IV). +The ansatz (2.1) captures several features of such ma- +trix elements that have been observed in numerical stud- +ies. Firstly, because the density of states is exponentially +large in system size, the off-diagonal matrix elements are +exponentially small. Secondly, the smooth function A(E) +is related to the statistical mechanical prediction for ⟨A⟩ +at average energy E; this function will play a central +role in our algorithm. Finally, the function f( ¯E, ω) con- +trols the approach to thermal equilibrium and is related +to other spectral properties of the observable [27]; this +function figures less prominently in our analysis. +To see how A(E) is related to a thermal average, con- +sider for example a broadened microcanonical ensemble +ρλ,δ centered on energy E = λ and of width O(δ) which +we will define more precisely at beginning of Sec. IV. +Under certain assumptions about the density of states +of the model and away from λ = 0 (which corresponds +to infinite temperature), and assuming the ETH ansatz +(2.1), we have in the thermodynamic limit that (see Ap- +pendix C for details) +A(λ) = ⟨A⟩mc + O(δ2/N) + O(D−1/2(λ)) +(2.2) +where ⟨A⟩mc = tr ρλ,δA. The ETH thus suggests that, +if one could prepare even a single eigenstate |λ⟩ of the +Hamiltonian with energy λ, then one could accurately +estimate thermal averages in sufficiently large systems. +However for a nonintegrable Hamiltonian, a generic ex- +cited eigenstate is volume-law entangled, and thus can- +not efficiently be prepared by classical algorithms nor by +VQE-type algorithms [28]. +Thus, this feature of ETH +does not appear practically useful, expect perhaps in the +case of an error corrected quantum computer. +B. +Microcanonical Superpositions +An alternative approach to using exact eigenstates for +computing thermal averages is using pure states of the +form +|ψ⟩ = +� +E +cE |E⟩ , +(2.3) +where either cE are exactly zero outside the energy win- +dow defined by |E − λ| ≤ δ, or the states satisfy the +weaker condition that ⟨ψ|(H − λ)2|ψ⟩ = O(δ2). We re- +fer to states of this type as “microcanonical superposi- +tion states” and they have been studied in the context of +thermal pure quantum (TPQ) states [29], the foundations +of quantum statistical mechanics [30, 31], algorithms for +analog quantum simulators [23], and tensor network al- +gorithms [24]. +The practical reason for considering these states is that +they appear to be significantly less entangled than exact +eigenstates. +In fact, there exist MPS-based numerical +constructions of them such that the maximum entan- +glement entropy across any cut scales as k/δ + log2 +√ +N +for some constant k [24] and N being the system size. +Thus, by choosing δ = O(1/log2N), such states can +have only O(log2N) entanglement, whereas a single ex- +cited eigenstate of a nonintegrable system is expected + + ϵ do +prepare |ψr(θ)⟩ = Up(θ) |ψ0 +r⟩; +measure C(θ) and {∂jC(θ)}j ; +update θ according to BFGS optimizer; +if Var(H) ≤ δ2 then +p∗ ← p; +θ∗ ← θ; +converged; +set |ψr⟩ = U(θ∗) |ψ0 +r⟩; +else +ϵ ← ϵ/2; +p ← p + 1; +We now describe the VME algorithm for preparing mi- +crocanonical superposition states |ψr⟩ discussed in the +previous section. At the beginning we fix a target energy + +7 +λ, and microcanonical window size +δ = ∆E +N N α, +(3.17) +where ∆E is the full energy bandwidth of the Hamilto- +nian H. In the mixed-field Ising model we later consider, +we find ∆E +N ≈ 3 independent of N. We focus our numer- +ical studies mainly on the case α = −1/2. We initialize +the QPU (see Fig. 1) in a random product state +|ψ0 +r⟩ = |ϕ1 +r⟩ |ϕ2 +r⟩ · · · |ϕN +r ⟩ , +(3.18) +with |ϕj +r⟩ = cos(ϕj +r) |0⟩ + sin(ϕj +r) |1⟩ and ϕj +r drawn from +the uniform distribution on [0, π), which we can expect +to have extensive energy variance [24, 32]. We then min- +imize the “folded-spectrum” cost function [9] +C(θ) = ⟨ψ(θ)|(H − λ)2|ψ(θ)⟩ , +(3.19) +until Var(H) = ⟨(H − ⟨H⟩)2⟩ ≤ δ2, obtaining the con- +verged variational state |ψr⟩. Note that C(θ) penalizes +both large energy variance and deviation of the average +energy from the target energy, since +C(θ) = Var(H) + ⟨H − λ⟩2, +(3.20) +but that the convergence criterion only concerns Var(H). +We find in practice that ⟨H − λ⟩2 is comparatively small +when N is large, so it is also possible to think of C(θ) ≲ δ2 +as the convergence criterion. +The optimization is repeated R times for different ini- +tial random product states to generate the variational +ensemble. Notice that this cost function C(θ) is zero if +and only if |ψ(θ)⟩ = |λ⟩, the eigenstate with energy λ +[37]. Unlike previous explorations [12, 28, 38] with this +cost function, we do not seek local or global minima, since +we minimize the cost function only until Var(H) ≤ δ2, +which is not a constraint on the gradient, but on the +value of the cost function itself. Furthermore, the unique +global minimum is a state completely different from the +one we target. +For simplicity we restrict our variational states to be +real in the computational basis (CB). Because we are +interested in the minimal circuit depth needed to pre- +pare the variational states, we employ a periodic struc- +ture ansatz (PSA) [14] circuit for which the number of +“layers” will be adaptively chosen by the algorithm. The +PSA with p layers is defined as +Up(θ) = +p +� +l=1 +V (θl), +(3.21) +where each layer is the unitary (see Fig. 3) +V (θl) = +N +� +j=1 +eiθl +jYj +N +� +j=1 +even +eiφl +jYjZj+1 +N +� +j=1 +odd +eiϕl +jYjZj+1 +(3.22) +and θ stands for the 2Np real parameters {θl +j, φl +j, ϕl +j}jl +and Yj, Zj are Pauli operators acting on qubit j. The +FIG. 3. Layer l of the ansatz circuit, Eq. (3.22), near qubit j. +“brickwall” form of this ansatz breaks down for odd N, +so in this case we add an additional gate to entangle the +ends of the chain. +That is, for odd N, we make the +replacement +N +� +j=1 +even +eiφl +jYjZj+1 → eiφl +1Y1ZN +N +� +j=1 +even +eiφl +jYjZj+1. +(3.23) +The operators appearing in the single layer unitary +V (θl) are chosen based on the findings of Ref. [39]. +There it is argued that the pool of 2N − 2 operators +P = {iYjZj+1}N−1 +j=1 ∪{iYj}N−1 +j=1 is “complete” in the sense +that for any state |ψ⟩, the set of states {Ak |ψ⟩}k form a +complete basis, where Ak are nested commutators of op- +erators in P, i.e. elements of the dynamical Lie algebra +[40] of P. We have added the extra gates YN and (for +odd N) Y1ZN to the pool, but clearly P ∪ {Y1YN, Y1} is +still complete in the above sense. +Since our convergence criterion is based on the value +of the cost function and the native convergence crite- +rion of a gradient based optimizer is based on the size of +the gradient, we “wrap” the optimizer in a simple loop +(see Algorithm 1) where we repeatedly interrupt the op- +timizer to check if the convergence criterion is satisfied, +which we accomplish by having it only minimize until the +gradient norm falls below a relatively large value ϵ which +starts at 101 and can only be decreased down to 10−3. +If the algorithm then cannot achieve convergence using +a p-layer ansatz by decreasing ϵ to 10−3, it adds another +layer p → p + 1 and repeats the procedure. +For the classical optimizer we employ the Broyden- +Fletcher-Goldfarb-Shanno (BFGS) optimizer which is +gradient-based. We therefore take advantage of the “pa- +rameter shift rule” [41] for computing analytic gradients +of the cost function. If the generators are Pauli strings +(hence squaring to I), then +∂ +∂θj +C(θ) = C(θ + π +4 ej) − C(θ − π +4 ej) +(3.24) + +Ry(j-3) +Ryz(Φj-3) +Ry(0§-2) +Ryz(βj-2) +Ry(0§-1) +Ryz(Φi-1) +Ry (0§) +Ryz(§) +Ry (0§+1) +Ryz(Φj+1) +Ry (0§+2) +Ryz(βi+2) +L8 +FIG. 4. Analysis of diagonal error in the VME, with operator A = Z⌊N/2⌋ taken as an example. In panel (a), the variational +ensemble diagonal matrix elements ρR(E) = ⟨E|ρR|E⟩ (blue) for N = 13, λ/N = −0.5, α = −1/2 [see Eq. (3.17)], and +R = 288 states in the ensemble. In black, the best Gaussian fit curve ρµ,σ(E) [see Eq. (4.2)], and in orange a coarse-grained +version of the blue scatter points for comparison. Panel (b) shows the diagonal error [see Eq. (3.2b)] with ρmc = ρλ,δ versus +R ≥ 12 for N = 11, 12, 13 where increasing N corresponds to darker blue data points. For comparison, we include in green the +ensemble-independent estimate δ|A′(λ)| = δ|a′(λ/N)|/N at N = 13, as well as the more accurate estimate χR [Eq. (3.8)] at +N = 13. Panel (c) includes the diagonal matrix elements AEE = ⟨E|A|E⟩ in blue, their coarse-graining in orange, and the best +fourth-order polynomial fit in black which defines a(E/N). This smooth function a is used to compute χR. Vertical dashed +lines show the scale of the microcanonical window as compared to the whole spectrum. +with ej a unit vector in the jth direction. Thus, during +the optimization both the cost function and its deriva- +tives can be measured using the QPU. +IV. +NUMERICAL RESULTS +We test the VME algorithm on the 1D Mixed Field +Ising Model (MFIM) with Hamiltonian +H = +N +� +j=1 +(JZjZj+1 + hx,jXj + hzZj) +(4.1) +and periodic boundary conditions so that site N + 1 +refers to site 1. To ensure there are no accidental degen- +eracies we consider weakly nonuniform transverse fields +hx,j = hx + rj, where rj ∈ [−0.01, 0.01] are drawn ran- +domly from the uniform distribution. We use a single +fixed configuration of the transverse fields for our numer- +ics. We fix parameters J = 1, hz = 0.5 and hx = −1.05 +such that the system is strongly nonintegrable [23]. In +this section we discuss the performance of the VME with +respect to this particular model. +A. +Diagonal variational ensemble +Here we characterize the nature of the converged varia- +tional states in the energy basis. To do so, we first define +a “broadened” microcanonical ensemble as was done in +Ref. [5]. This ensemble is of the form +ρλ,δ = D−1 +δ (λ)Gδ(H − λ) +(4.2) +where Gδ(x) = (2πδ2)−1/2e−x2/2δ2 is a normalized Gaus- +sian function, and Dδ(λ) = tr Gδ(H − λ) is the “broad- +ened” density of states evaluated at energy λ. Here, δ +corresponds to the convergence criterion for the VME al- +gorithm, i.e. Eq. (3.17) with α = −1/2. We will from now +on treat Dδ(λ) as a good approximation to the density of +states in the thermodynamic limit (see Appendix A for +further justification). +We claim that the variational algorithm 1 generates di- +agonal energy-basis matrix elements ρR(E) = ⟨E|ρR|E⟩ +approximating a broadened microcanonical ensemble. +Fig. 4(a) shows the variational ensemble diagonal energy- +basis matrix elements to which we fit the curve ρµ,σ(E) = +D−1(µ)Gσ(E−µ) with fitting parameters µ and σ (shown +in solid black). +Up to fluctuations from eigenstate to +eigenstate, we can see that the variational diagonal en- +semble is well described by the Gaussian best-fit. More +precisely, the coarse grained version, ρR(E), of the varia- +tional ensemble diagonal elements–where the fluctuations +are eliminated–agrees quite well with the Gaussian best- +fit. The coarse-grained curve is computed as follows: for +each E in the spectrum of H, ρR(E) is defined as the av- +erage of ⟨E′|ρR|E′⟩ over the K eigenenergies E′ nearest +to E. We set the “resolution” K = 64, except for E near +the edges of the spectrum where 1 ≤ K < 64. At the +largest system size of N = 13 and for an R = 288-state +ensemble, we list the best fit parameters µ and σ in Table +I. +Note that away from λ = 0, the variational ensembles +converge with µ slightly different than the target λ. This +is because the variational ensemble is generated by min- +imizing the first two moments of the operator (H − λ), +but the density of states determined by the underlying + +Pμ,(E) +diag +(a) +PR(E) +PR(E) +(b) +XR +(c) +AEE +AEE +a(E/N) +ER +N +0.003 +0.004 +0.75 +N +0.003 +0.50 +0.002 +0.25 +0.002 +0.001 +0.00 +0.001 +-0.25 +0.000 +0.000 +-0.6 +-0.4 +0 +100 +200 +300 +-1 +0 +1 +E/N +R +E/N9 +model is non-uniform. In fact if we assume the Gaus- +sian density of states discussed in Appendix A, we have +tr(ρµ,σH) ≈ µ−O(σ2 µ +N ), where we have assumed that σ +decreases with N so that higher order terms can be ne- +glected in the thermodynamic limit. +We can see that +when µ < 0, the ρµ,σ ensemble actually has average +energy larger than µ. In Appendix D we confirm this +statement quantitatively. As a consequence of this anal- +ysis, we conclude that the deviations in µ from λ are a +finite-size effect due to the non-constant density of states, +and not due to the fluctuations around the average curve +ρR(E). +In the last row of Table I, we see that all the σ are at +least about 15% smaller than δ. This is due to a simplifi- +cation we have made in the preceding analysis. Note the +form of ρR(E) in Fig. 3.2b(a) at the edges of the window; +the orange curve has more weight away from the window +than the Gaussian best-fit (in black). A more accurate +characterization of the ensemble might be, for example, +a sum of two Gaussian curves. We nonetheless opt to +consider the ensemble as roughly a single Gaussian peak +for simplicity. We discuss the quantitative consequences +of this simplification in Appendix C and explain why the +σ/δ shown in Table I are not closer to unity. +Up to fluctuations around the coarse-grained behavior +ρR(E) and the slight oversimplification of the single peak +Gaussian fit, we have thus established the form of the di- +agonal part of the variational ensemble. In the following +sections we will study the error in the VME estimate +of various observables. Clearly, we will need to choose +an appropriate microcanonical ensemble to compare to. +This ensemble is arguably ρµ,σ because it closely approx- +imates the (coarse-grained) diagonal ensemble ρR(E). +One could also imagine comparing to the diagonal en- +semble itself and focusing solely on the off-diagonal er- +ror as was done in [42]. However, imagining the VME +as a practical algorithm for computing broadened micro- +canonical ensemble averages, one could not know ahead +of time what µ and σ were. Furthermore we have shown +that the distinction between λ/N and µ/N is a finite-size +effect that is already small at N = 13. We will hence- +forth compare our numerical results to the ensemble ρλ,δ, +where λ and δ are precisely the parameters that were ini- +tially chosen before running the algorithm, i.e. we set +ρmc = ρλ,δ in Eq. (3.2b). +λ/N +µ/N +σ/δ +−0.750 −0.761 0.858 +−0.500 −0.511 0.831 +−0.250 −0.253 0.821 +0.000 +0.001 +0.832 +TABLE I. The N = 13 broadened microcanonical best fit +parameters (µ, σ) at various target energy densities λ/N. +B. +Diagonal error +We now briefly discuss the diagonal error with respect +to the broadened microcanonical estimates. +Fig. 4(b) +shows, for N = 11, 12, 13, the diagonal error versus +R ≥ 12 for the operator A = Z⌊N/2⌋ at λ/N = −0.5. +For comparison we plot a simple estimate of the error +(δ/N)|a′(λ/N)| at N = 13, as well as the more accu- +rate estimate χR defined via Eq. (3.7) which we calcu- +late numerically using the N = 13 variational ensemble. +We calculate a′(λ/N) using a fourth-order polynomial fit +to the (coarse-grained) graph of ⟨E| A |E⟩ versus E/N, +shown as the solid black line in panel (c) of Fig. 4. The +coarse-grained curve AEE is computed in the same way +as was ρR(E) in Sec. IV A, but here we use a resolution +of K = 32. +For this particular operator and energy density, it is +clear that the diagonal error decays with N and is an +order-one fraction of the rough estimate O(δ/N). Fur- +thermore we can see that for large R its behavior is well +captured by the expected estimate χR. +In Appendix F we present further numerical results +for the four local operators Z, ZZ, X, XX acting on the +central one or two sites of the chain at the energy den- +sities λ/N = −0.75, −0.5, −0.25, 0.0. At λ/N = −0.5, +all operators have the property that ϵdiag +R +decays with N +for large R, and the N = 13 values are consistent with +the estimate χR/N. At other energy densities the scal- +ing with N is not always so well established, but the +error for large R is always smaller than (δ/N)|a′(λ/N)|. +The value of χR also appears to generally be on the cor- +rect scale of ϵdiag +R +except for the operator XX, for which +χR/N undershoots the value of the diagonal error for +the higher energy densities. These various deviations are +likely due to ETH not yet having set in at such small sys- +tem sizes. In particular the N = 13 value of D−1/2(λ) is +never smaller than 0.04 for the energy densities we con- +sider, so that the ETH fluctuations, i.e. the third term +in equation (3.7), could be comparable to δ/N ≈ 0.06 at +N = 13 depending on the relative size of A(E) and the +ETH function f(E, 0). Furthermore, systematic correla- +tions in ⟨E|A|E′⟩ could also cause χR/N to undershoot +the diagonal error. +In any case, the diagonal error is always quite small +across all operators and energy densities, when compared +for example against the scale of the microcanonical fluc- +tuations themselves. For example, see Fig. 5(a) where in +purple we can see that even for a single variational state, +the diagonal ensemble estimate is highly accurate. We +observe this across every considered operator and energy +density, as shown in Appendix G. +C. +Off-diagonal error +We now turn to discuss the numerical details of the +off-diagonal error and how ensemble averaging reduces +it. First, Fig. 5(a) illustrates the main problem with us- + +10 +FIG. 5. Off-diagonal error in the VME algorithm, with A = X⌊N/2⌋ at λ/N = −0.5 taken as an example. In panel (a), the +broadened microcanonical ensemble expectation value ⟨A⟩λ,δ = tr(ρλ,δA) and error bars indicating the associated broadened +microcanonical standard deviations (obtained from ED) is plotted in blue as a function of N. We compare this to the estimate +in a single variational state, ⟨A⟩1 = tr(ρ1A), along with the corresponding diagonal ensemble estimation ⟨A⟩diag +1 +, i.e. Eq. (3.2d) +with R = 1. Note that ⟨A⟩diag +1 +can only be computed with ED, where off-diagonal contributions can be discarded by hand. In +panel (b) we show the off-diagonal error ϵoff +R [see Eq. (3.2c)] with increasing R for N = 11, 12, 13, where increasing N corresponds +to darker blue points as in Fig. 4. The orange curve, ϵR +off, is the N-averaged value of ϵoff +R as discussed in the text. The dashed +black line is a power-law best fit to ϵR +off, and for comparison we plot the “theoretical” error σRR−1/2 in green. Panel (c) shows +the probability density functions for N = 13 of the eigenvalues of ˆA and ˜A as defined in the text. +ing a single variational state with a large δ. We take as +an example the operator A = Z⌊N/2⌋ at the energy den- +sity λ/N = −0.5 and compare a single variational state +estimate ⟨A⟩1 = ⟨ψ1|A|ψ1⟩ to the smooth microcanon- +ical average ⟨A⟩λ,δ = tr(ρλ,δA). +The estimate is poor +even for the largest system size. In Fig. 5(b) we examine +how this error is reduced by averaging. +We plot on a +log-log scale the off-diagonal error ϵoff +R for N = 11, 12, 13 +versus R, anticipating a dephasing type of behavior as +discussed in Sec. III A. We can see that the error does +not clearly depend on N systematically, which gener- +ally holds across other operators and energy densities, +as shown in Appendix G). We attribute the lack of sys- +tematic N-independence for A = Z, ZZ, X to our choice +of only weakly scaling down δ with N. +Since the scale of the off-diagonal error does not sys- +tematically depend on N across a large range of R, we av- +erage it over N ∈ {9, 10, 11, 12, 13} to improve the statis- +tics. This curve, ϵRoff, is shown versus R in orange in +Fig. 5(b); we also show a corresponding power-law best- +fit of the form Rm in dashed black. For most A and λ we +observe a clear power law decay with best-fit parameter +m close to −1/2. This is also consistent with the scale σR +defined by Eq. (3.15), with the corresponding curve plot- +ted in green. Here, σR is computed using ˜A defined on +an energy window of half-width 3δ [see Eq. (3.10)]. We +justify numerically this approximation of computing σR +based only on the matrix elements of ρR and A in energy +eigenstates near λ in Appendix E. We can see that the +off-diagonal error decays with R up to some fluctuations, +and that the empirically computed prefactor σR sets the +correct scale of the error. +The R−1/2 dephasing result is not universal however; +in particular the N-averaged best fit curve for the opera- +tor XX appears to be less well described by a power law +Rm, in particular at energy density −0.25, see Fig. 18 +in Appendix G. Power law fits still appear to generally +agree with the decay; with −0.51 ≤ m ≤ −0.10 depend- +ing on energy density as shown in Appendix G. We do +not currently understand this behavior; it could be co- +herence between ⟨ψr| XX |ψr⟩ for different r, or it could +be that |E(xr)| > 0 in whatever distribution from which +xr is being sampled for the operator XX (recall the dis- +cussion of Sec. III). The distinction between XX and the +other operators is that it is not coupled to energy density. +This particular operator is also the one whose diagonal +error appears to be less well described by the ETH predic- +tion of χR/N, suggesting XX may have matrix elements +⟨E|XX|E′⟩ that are somehow more correlated than the +other three considered operators. The operators X and +ZZ also appear to decay only as R−0.36 and R−0.41, re- +spectively, at λ/N = −0.75, but this energy density is +approaching the low-energy tail of the spectrum, where +worse statistics are expected. +We do find conclusively +that for the operators Z, ZZ, X and away from the edges +of the spectrum, there is a clear R−1/2 power law decay +in ϵRoff. +Ref. [33] inferred correlations between energy-basis +matrix elements of local operators A by the form of the +eigenvalue statistics of certain sub-matrices of A. +To +help understand the nature of the off-diagonal error, in +Fig. 5(c) we also examine the eigenvalue distribution of + +0.49 +(a) +1 +,s +(b) +(c) +ff +OR·R-1/2 +2.5 +0.75 +2-3 +2.0 +0.50 +2-6 +1.5 +0.25 +2-9 +1.0 +0.00 +2-12 +0.5 +-0.25 +0.0 +5 +7 +9 +11 +13 +21 +23 +25 +27 +0 +1 +N +R +eigenvalue11 +the operator ˜A defined on an energy window of half-width +3δ and centered on energy λ, i.e. as in Eq. (3.10) but +with W slightly expanded to accommodate the tails of +the roughly Gaussian variational states. See Appendix E +for a graphical representation of how this energy win- +dow is defined. For comparison we also show in Fig. 5(c) +the eigenvalue distribution of the operator ˆA, which is +simply the operator A with its energy-basis diagonal ele- +ments deleted. There are a number of interesting qualita- +tive properties displayed by these eigenvalue distributions +that are relevant to the off-diagonal error in the VME. +Firstly, we observe that the eigenvalue distribution of +˜A does not appear to qualitatively change shape as N is +varied, except for a slight reduction in the total width for +increasing N, as demonstrated in Appendix E. Since it +is the operator ˜A which determines the off-diagonal error +for large R [43] the qualitative lack of N dependence +agrees with the fact that the off-diagonal error does not +depend on N in a systematic way at the system sizes we +examine. +Perhaps more interestingly, we observe a correla- +tion between the single-state variational estimates in +Fig. 5(a) as N is varied and the eigenvalue distribution of +˜A. When ⟨A⟩1 over/under-estimates the microcanonical +value across many system sizes, the eigenvalue distribu- +tion is biased to the right/left of zero. For further evi- +dence that this correlation is not an artifact of this energy +density or the choice of operator, see Appendix G. On +the one hand, this is not surprising since roughly speak- +ing, ˜A determines the off-diagonal error via Eq. (3.10), +and suggests that the variational states are in some sense +sampling from the eigenvalue distribution of ˜A. On the +other hand, due to |ψr⟩ being pseudo-random but not +fully Haar-random, we have not found a rigorous analyt- +ical argument for the connection between Fig. 5(a) and +Fig. 5(c). At all energy densities, the eigenvalue distri- +bution of � +XX is much flatter than the other three con- +sidered operators, which could be related to the fact that +the off-diagonal error for A = XX does not decay with +R in the same way the other operators do. +We conclude the discussion of the off-diagonal error +with some observations about the eigenvalue statistics of +the full-spectrum off-diagonal operator ˆA. Even though +the variational states in principle have support on the en- +tire energy spectrum, we can see that it is the statistics of +˜A and not of ˆA that are correlated with the off-diagonal +error, further justifying the truncation to a local energy +window. Interestingly, we can see that ˆA still has a sim- +ilar spectral form to that of a Pauli string with eigenval- +ues ±1, but the otherwise highly degenerate peaks have +been smeared out by removing the diagonal energy-basis +elements. In Appendix G we show the eigenvalue distri- +butions of ˆA for other A, and note that XX looks the +most similar to that of a Pauli string, i.e. its peaks have +been broadened the least. +When the window is reduced to the scale δ, the distri- +bution becomes less similar to that of a Pauli operator, +and we can expect that for a fixed N, as δ → 0, the +FIG. 6. In blue, smooth microcanonical averages ⟨A⟩λ,δ = +tr(ρλ,δA) and their thermal fluctuations, Eq. (4.3). In orange, +the corresponding variational estimates for α = −1/2 and +R = 288 with standard error. Averages and their error are +plotted as a function of system size N at a fixed energy density +of λ/N = −0.5. +spectrum approaches that of a random matrix, i.e. the +semi-circle law [25, 33, 44]. The fact that the eigenvalue +distribution is so far from a semi-circle law on the scale +δ = O(N −1/2) further confirms that ⟨E|A|E′⟩ are not +effectively independently distributed and thus we cannot +rely on randomness of the matrix elements alone to make +the off-diagonal error small. +D. +Explicit microcanonical estimates and trace +distance +Having examined in some detail the scaling of the ab- +solute diagonal and off-diagonal errors, we now take a +step back and consider what the overall statistics of the +variational estimates look like when compared to the mi- +crocanonical averages and their associated microcanon- +ical fluctuations. +For example, in Fig. 6 we show for +various system sizes the variational estimate tr(ρRA) for +R = 288 along with the standard error. This is compared +against the broadened microcanonical average calculated +from ED, with error bars indicating one microcanonical +standard deviation +(∆A)λ,δ = +� � +E +⟨E|ρλ,δ|E⟩ ⟨E|A|E⟩2 +�1/2 +(4.3) +which is another scale to which the error can be com- +pared. +Running the VME two different times yields two dif- +ferent variational ensembles ρR and ρ′ +R. The orange er- + +0.0 +0.0 +11 ++1111 +-0.2 +-0.2 +-0.4 +(z) +(ZZ) +0.4 +0.2 +0.3 +111111 +0.0 +0.2 +0.1 +-0.2 +0.0 +R +(b) +>.s +(c) +R +4 +3.0 +王 +0.5 +3 +2.5 +0.4 +2 +2.0 +0.3 +1 +>/N +1.5 +2 +4 +9 +2 +4 +9 +9 +11 +13 +IS] +ISI +N(SUN)R +5 +1.0 - +T +4 - +0.8 - +0.6 - +3 - +0.4 - +2 +0.2 +1 +0.6 +0.8 +1.0 +0.6 +0.8 +1.0 +[αl +[a]14 +V. +CONCLUSION AND OUTLOOK +In this work we propose a VQA for estimating Gaus- +sian microcanonical averages of local operators at inter- +mediate energy density. +Given the target average en- +ergy λ and a microcanonical width of O(N −1/2), the +variational algorithm evolves random product states into +weakly entangled states whose diagonal ensembles are +approximately Gaussian-microcanonical on average. We +have systematically examined what we call the diagonal +and off-diagonal contributions to the error in this estima- +tion, and found that the latter is the dominant source of +error. The off-diagonal error is on the one hand paramet- +rically reduced by ensemble averaging, but on the other +hand is not always small as compared to, for example, +characteristic microcanonical fluctuations. +For a fixed R, we find that more accurate estimates +are produced at intermediate target energy densities +λ/N = −0.75, −0.5 as compared to the higher energy +densities λ/N = −0.25, 0. We also find that the algo- +rithm performs better for operators coupled to energy +density (i.e., ones appearing in the Hamiltonian). +In +particular, for all energy densities but the lowest one of +−0.75, the parametric suppression of the off-diagonal er- +ror scales with the size R of the variational ensemble +roughly as Rm with −0.59 ≤ m ≤ −0.45 for the opera- +tors Z, X, and ZZ, but with −0.28 ≤ m ≤ −0.10 for the +operator XX, which is also generally less well described +by a power law. It appears that XX is also an outlier +from the perspective of the diagonal ETH, so it is possible +that this operator happens to be “slower” to thermalize +and therefore less well-described by the ETH—indeed, +since it is not coupled to energy density it is expected +that the thermalization behavior of this operator should +differ from the other three considered here [25]. +We have also examined the performance of the algo- +rithm in an observable-independent way by computing +the trace distance between the subsystem variational en- +semble and the subsystem microcanonical one, which +appears to continually decrease with N when we take +R = O(N 2) and λ/N = −0.75, −0.5, whereas for λ/N = +−0.25, 0 there is not a consistent decay. We find this re- +sult interesting because for other finite temperature VQA +methods, intermediate temperatures (as opposed to infi- +nite temperature) are more difficult to simulate [17, 19]. +Since the number of variational parameters appears to +scale roughly as O(N 2), this suggests that a classical op- +timizer could handle the optimization at larger system +sizes. The main bottleneck in the classical simulation of +our proposed VQA is the repeated evaluation of the cost +function; it would be useful to implement a more sophis- +ticated classical simulation of the QPU, for example by +tensor network methods so that larger systems could be +reached. This would help to decide if the trend in the +trace distance continues for larger N. +The quantum resources demanded by the algorithm +are mild; the variational ensemble generates area-law en- +tangled states when run at any target energy with target +energy variance δ2 = O(N −1), and these states require +shallow—in particular (roughly) linearly deep—quantum +circuits to prepare. A caveat to the efficiency, however, +is that the variational optimization requires many cost- +function evaluations to converge, on the order of 105 at +N = 13, implying that the algorithm run on a real de- +vice could have to make a prohibitive number of mea- +surements, which is a problem shared by other VQAs +[10]. +In Sec. IV A we saw in the thermodynamic limit the +diagonal error vanishes as O(δ/N) so that even prod- +uct states whose typical energy width is O( +√ +N) should +suffice for a vanishing diagonal error, provided the ETH +holds. The decay with R of the off-diagonal error follows +when the quantities xr are uncorrelated and are sampled +from a distribution with E[xr] = 0. It seems feasible that +even random product states can lead to a R−1/2 scaling +of off-diagonal error, albeit with a prefactor that could be +large or even increase with N. It would be enlightening +to see under what conditions these above assumptions +could be justified analytically, for example in a random +quantum circuit model [47] or with random MPS [48]. +As far as variational algorithms are concerned, the +VME could be considered as an example of a broader +class of VQAs where the convergence criterion is based +on the value of the cost function rather than its gradient. +Furthermore, the smallness of the cost function at con- +vergence is only O[1/poly(N)], so it would be interesting +to study in such cases if the well known barren-plateau +problem [13, 14] is either less significant or not present +at all. +ACKNOWLEDGMENTS +The authors acknowledge valuable discussions with +Ronak Tali, Milan Kornjaˇca, Yihua Qiang, Ana-Marija +Nedi´c, Niladri Gomes, and Yong-Xin Yao. This material +is based upon work supported by the National Science +Foundation under Grant No. DMR-2038010. +Appendix A: Density of states +Let Gy(x) = (2πy2)−1/2e−x2/2y2 be a normalized Gaussian window function with zero mean and standard deviation +y. We maintain the convention that f(Q) = � +q f(q) |q⟩ ⟨q| for some Hermitian operator Q with eigenvalues q. We find +numerically that the broadened density of states Dδ(λ) = tr Gδ(H−λ), with broadening parameter δ = (∆E/N)N −1/2 + +15 +FIG. 10. The broadened density of states of the MFIM plotted against N −1/2E so that the form of the curves is N-independent. +At N = 13, a Gaussian best-fit yields parameters γ = ∆2/N = 2.47 and ¯E/∆ = −0.03 at N = 13. The theoretical value +calculated directly from H with γth = ∆2 +th/N = 2.35 is also plotted and seen to agree well. The prefactor k depends on the +curve and is chosen so that the maximum value of each curve is 1 (the factor is (2π∆2)1/2 when the width is ∆). +is smooth (away from the tails of the spectrum) and can be approximated by a Gaussian of the form 2NG∆(E) = +2N(2π∆2)−1/2e−E2/2∆2 with ∆2 = γN and γ becoming N-independent in the thermodynamic limit. The density of +states and a Gaussian best fit curve are shown in Fig. 10. Following the method discussed in the Appendix of [49], +we can check if this approximation is reasonable by estimating the parameter γ from the Hamiltonian directly. Note +that in terms of the supposed form of the density of states and in the thermodynamic limit the following equality +should hold +tr(H2) = +� ∞ +−∞ +dE D(E)E2 = 2NγN. +(A1) +With periodic boundary conditions it can be checked for the Hamiltonian (4.1) that tr(H2) = 2N[(1+h2 +z)N +� +j h2 +xj], +but since the random fields hxj have been chosen to all be within 1% of the central value hx, we can safely approximate +tr(H2) ≈ 2N(1 + h2 +x + h2 +z)N, yielding the theoretical estimate γth = 1 + h2 +x + h2 +z = 2.35, which is within about 5% +of the best fit value of γ. We also checked that the density of states is roughly unaffected when it is computed using +using the broadening parameter σ (with which the variational states converge) instead of the broadening parameter +δ. +Appendix B: Off-diagonal contribution assuming independent identically distributed random variables +In Section. II, we claimed that should the order-one fluctuations REE′ be actual independent and identically +distributed random variables, then the off-diagonal contribution to equation Eq. (2.4) would be typically O(D−1/2(λ)). +To see this, assume for E > E′ that REE′ are samples from an underlying distribution satisfying E[REE′] = 0, +E[R2 +EE′] = 1, and E[REE′RE′′E′′′] = 0 for E ̸= E′′, E′ ̸= E′′′ and E′′ > E′′′. We restrict the energies to the upper +triangular part of the R matrix since REE′ = RE′E for energy-basis real observables. Let +x1 = 2 +� +E>E′∈W +cEcE′ ⟨E|A|E′⟩ = 2 +� +E>E′∈W +cEcE′D−1/2( ¯E)f( ¯E, ω)REE′, +(B1) +where in the second equality we have inserted the ETH matrix element ansatz. The above is notation consistent with +Section. III A, where x1 is the “off-diagonal error” for a single microcanonical superposition state supported only on +the microcanonical window W. Clearly we have E[x1] = 0 and +E[x2 +1] = 4 +� +E>E′∈W +E′′>E′′′∈W +cEcE′cE′′cE′′′D−1/2( ¯E)D−1/2( ¯ +E′′)f( ¯E, ω)f( ¯ +E′′, ω′′)E[REE′RE′′E′′′] +(B2) +where ¯E′ = (E′′ + E′′′)/2 and ω′′ = E′′ − E′′′. The independence assumption collapses the quadruple sum giving +E[x2 +1] = 4 +� +E>E′∈W +c2 +Ec2 +E′D−1( ¯E)f 2( ¯E, ω). +(B3) +Normalization of the state implies c2 +E = O(1/n) where n is the number of eigenenergies in W, and the double +sum runs over n(n − 1) = O(n2) energies. The function f is expected to be order one [50], so altogether we have +E[x2 +1] = O(D−1(λ)) since λ is a typical energy in the window. In the thermodynamic limit, the central limit theorem +implies that with high probability, x1 ≲ O(D−1/2(λ)). + +1.5 +kDs(E)/2N (N=11) +— kGA(E-E) (N= 13 +k Ds(E)/2N (N=13) +— kGAth(E)(N=13) +1.0 +0.5 +0.0 +1 +-4 +-2 +0 +2 +4 +6 +N-1/2E16 +Appendix C: Expectation value of (H − λ) in broadened microcanonical ensemble +Here we justify Eq. (2.2), which expresses the relation between the smooth function A(E) appearing in the ETH +matrix element ansatz and the microcanonical expectation value ⟨A⟩λ,δ, by considering the expectation value of +(H − λ) in the smooth microcanonical ensemble. We take the density of states to be the Gaussian function described +in Appendix A. Under such an assumption, the first few terms of the Taylor series for the density of states D(E) = +2N(2π∆2)−1/2e−E2/2∆2 near energy λ read +D(E)/D(λ) = 1 − λ +∆2 (E − λ) + +�� λ +∆2 +�2 +− 1 +∆2 +� +(E − λ)2 +2 ++ 1 +∆4 +� +3λ − λ3 +∆2 +� (E − λ)3 +6 ++ · · · . +(C1) +Here, ∆2 = γN with γ ≈ 1 + h2 +x + h2 +z, and we have not assumed that E − λ is small in any sense yet, only that the +density of states admits a Taylor expansion in the thermodynamic limit. If we ignore any error incurred in replacing +sums by integrals in the thermodynamic limit, where the energy bandwidth approaches infinity and the level spacing +zero, the first moment of (H − λ) in the broadened microcanonical ensemble ρλ,δ = � +E D−1(λ)Gδ(E − λ) |E⟩ ⟨E| +reads +tr[(H − λ)ρλ,δ] = −δ2λ +∆2 + 3δ4 +6∆4 +� +3λ − λ3 +∆2 +� ++ · · · +(C2) +We can then use these formulae to estimate the expectation value of an ETH-obeying operator in this broadened +microcanonical ensemble. In doing so, an important consequence of the ETH is that the smooth function A(E) should +be expressible as a function of energy density in the thermodynamic limit, see Fig. 2 in the main text. In this paper we +consider only Pauli-string-type observables. In this case, note that A(E) = a(E/N) is O(1) because A(E) is defined +via (a best fit curve to) the averaging procedure +1 +K +� +E′ +⟨E′|A|E′⟩ +(C3) +over K eigenstates near |E⟩, and | ⟨E′|A|E′⟩ | ≤ 1 for Pauli strings. Now since A(E) = a(E/N), it follows that +a(x) = O(1) and +dA +dE +���� +E += 1 +N +da +dx +���� +E/N += O +� 1 +N +� +, +(C4) +and similarly for higher derivatives. Now consider a broadened microcanonical ensemble at energy λ, i.e. ρλ,δ = +D−1(λ)Gδ(H − λ). Then, the expectation value of the smooth ETH function in this ensemble is obtained by going to +the continuum and combining Eqs. (C1) and (C2). The result reads +� +dE D(E) +D(λ) Gδ(E − λ)A(E) = A(λ) + O(δ2/N). +(C5) +From this and the ETH ansatz follows Eq. (2.2). +Appendix D: Additional discussion of diagonal ensemble +In this appendix we explain the deviations in µ and σ from λ and δ, respectively, when fitting the converged +variational ensembles ρR to the best fit curves ρµ,σ. As was described in Sec. IV A of the main text, µ will under- +shoot λ because the density of states is non-uniform. We can confirm this more precisely as follows. Using the best fit +parameters to the variational ensemble (i.e. the values in Table I), we treat the spectrum as continuous and compute +the numerical integral tr[ρµ,σ(H − λ)] ≈ −0.042 at λ/N = −0.5. Doing the same for the ensemble whose central +energy is λ, we find that the value of tr[ρλ,δ(H − λ)] ≈ 0.142. In the latter calculation, we emphasize that this value +is roughly the same whether using δ or σ for the width of the Gaussian. Thus, the ensemble with central energy µ +actually minimizes the operator (H − λ) much better than the ensemble with central energy λ. Thus the deviation in +µ from λ is a finite-size effect due to a non-uniform density of states. +We now address the deviations in σ from δ, which we claim to be due to the slight “non-Gaussianity” of ρR(E), +i.e. the excess weight outside the Gaussian window that we see in Fig. 4. Best-fit curves aside, we first check that the + +17 +FIG. 11. In shaded gray, the off-diagonal only sub-matrix ˜A of A relevant to off-diagonal error for microcanonical superpositions +whose weight is mostly within s standard deviations δ of the central energy λ. Matrix elements are shown on the energy scale +rather than the eigenvalue index scale. +FIG. 12. For various operators A, the effect of expanding the window to include s standard deviations around λ for N = 13, +R = 288, and when λ/N = −0.5. We see that at s = 3, we have captured basically all of ˆσR as shown by the percentages. +Other energy densities are unremarkable except that at λ/N = −0.75 only about 97% of XX is captured. +fluctuations of ρR(E) around ρR(E) contribute negligibly to the expectation value of (H − λ)2. We directly compute +tr[ρR(H)(H −λ)2] ≈ 0.92 δ2, and we can see that this value is consistent with the actual ensemble average value of the +cost function, tr[ρR(H − λ)2] ≈ 0.90 δ2. The fact that these values are slightly less than δ2 can be attributed to the +convergence criterion only requiring that the variance (which is approximately the cost) be at most δ2. Now comparing +this to the Gaussian model best-fit ensemble,which predicts a cost function value of only tr[ρµ,σ(H − λ)2] ≈ 0.69δ2, +we see that it undershoots δ2 since it is missing the contribution from the excess energy weight. + +A +E=E' +E +(+ sd,^+ sd) +A +(>- so,入- sd) +E'ZZ +0.5 +99.21% +0.4 +99.12% +0.3 +Z +- +X +XX +OR +0.5 +0.4 +98.51% +0.3 +98.04% +1.5 +2.5 +3.5 +1.5 +2.5 +3.518 +FIG. 13. At various energy densities λ/N, the maximum singular value of ˜A for various A, with ˜A the 3δ large sub-matrix of +A and δ = O(N −1/2), i.e. with s = 3 as in Fig. 11. +FIG. 14. For the operator A = X at energy density λ/N = −0.5, probability density functions of the eigenvalues of ˜A as in +the orange histogram of Fig. 5(c) in the main text, but here shown for N = 10 in blue and for N = 13 in transparent orange. +Appendix E: Justification for replacing ˆA by ˜A and further properties of ˜A +In this Appendix we justify deleting certain energy basis matrix elements of A based on the form of the variational +states, for the purposes of gaining some intuition about the nature of the off-diagonal error. Let the operator ˆA be A +with its energy basis diagonal elements set to zero, i.e., +⟨E| ˆA|E′⟩ = +� +⟨E|A|E′⟩ +if E ̸= E′ +0 +if E = E′. +(E1) + +-0.75 +-0.50 +-0.25 += 0.000 +N +N +N +N +1.5 +Z +ZZ +1.4 - +1.3 - +1.2 +1.1 +1.0 +1.5 +X +XX +1.4 +1.3 +1.2 +1.1 +1.0 +7 +9 +11 +13 +7 +9 +11 +13 +N +N2.5 +A @ N=10 +A @ N=13 +2.0 +1.5 +1.0 +0.5 +0.0 +-1.0 +-0.5 +0.0 +0.5 +eigenvalue19 +At this point we also define ˜A for general s, where s is the number of standard deviations δ around λ that are not +deleted: +⟨E| ˜A|E′⟩ = +� +� +� +� +� +⟨E|A|E′⟩ +if E ̸= E′, |E − λ| ≤ sδ, +and |E′ − λ| ≤ sδ +0 +otherwise. +(E2) +An equivalent definition is given graphically in Fig. 11. Regardless of the energy support of the |ψr⟩, the off-diagonal +error (3.2c) can be written exactly as +ϵoff +R = 1 +R +���� +� +r +⟨ψr| ˆA|ψr⟩ +����. +(E3) +Assume the error can be understood roughly as a random walk with ϵoff +R ≈ ˆσRR−1/2, where +ˆσ2 +R = 1 +R +� +r +⟨ψr| ˆA|ψr⟩ +2 . +(E4) +Since the ensemble of variational states |ψr⟩ approximates a Gaussian microcanonical ensemble near with average +energy near λ on average [see Fig. 4(a)], we can anticipate that the value of ⟨ψr| ˆA|ψr⟩ +2 will also be unaffected on +average by replacing ˆA with ˜A when ˜A has a sufficiently large energy support. Because of the established Gaussian +form, we might expect that two standard deviations around λ is always sufficient to capture 95% of the average +absolute error. However there are fluctuations around this behavior, and the coarse grained variational states have +some excess energy weight beyond the Gaussian best-fit curve. Furthermore, the variational ensembles are not exactly +centered on λ. Thus, we justify replacing ˆA with an appropriately chosen ˜A numerically as follows. In Fig. 12 we +compare σR when computed on a window of size 2s × 2s (see Fig. 11 for clarification) and ˆσR, which is computed +from the entire spectrum. We show on the plots the fraction of ˆσR that is captured by σR at s = 3, which we consider +to be sufficiently large to capture basically all of ˆσR. +For the 3δ truncated operators ˜A we have just discussed, in this appendix we also consider how the maximum +singular value of ˜A, i.e. the larger of |λmax( ˜A)|, |λmin( ˜A)| scales with N. The results are shown in Fig. 13. These +results are used to justify the upper bound Eq. (3.16) in cases when the random-walk behavior holds. We also consider +how the eigenvalue statistics qualitatively vary with N, with an example shown in Fig. 14. +Appendix F: Additional numerical data for diagonal error +Fig. 15 in this Appendix shows the diagonal error in the VME estimate for operators Z, ZZ, X, XX acting in the +middle of the chain. As stated in the main text, the results for energy density λ/N = −0.5 appear to agree best with +the prediction of ETH that the error should decay as 1/N and agree with χR/N [see Eq. (3.8)] for large R. While in +general such a clear scaling with N is missing, all cases show that the N = 13 diagonal error is never larger than some +order one fraction of the rough estimate a′(λ/N)δ/N, with XX at λ/N = 0 showing the case where the diagonal +error comes closest to the estimate. We also note that the ETH prediction χR/N for the diagonal error is always on +the correct scale of the error for N = 13 for operators coupled to energy density. The operator XX is also an outlier +in this sense–χR/N significantly underestimates the actual error except for λ/N = −0.5. + +20 +FIG. 15. Diagonal error in the VME. Curves are interpreted identically to those in Fig. 4(b) in the main text, except to bring +the estimate (δ/N)|a′(λ/N)| down to scale we plot instead in some cases (δ/4N)|a′(λ/N)|. The latter are shown in orange +instead of green to indicate the use of a constant scale factor. + +Z +ZZ +X +XX +0.030 +0.003 +^/N: +=-0.75 +0.030 +0.025 +0.025 +0.002 +0.020 +0.025 +0.020 +0.015 +0.020 +0.015 +0.001 +0.010 +0.015 +0.010 +0.005 +0.010 +0.000 +0.030 +X/N= -0.50 +0.004 +0.030 +0.015 +0.025 +0.025 +0.003 +0.020 +0.020 +0.010 +0.002 +0.015 +0.015 +0.010 +0.001 +0.010 +0.005 +0.005 +0.000 +0.005 +0.008 +^/N= -0.25 +0.006 +0.0020 +0.006 +0.006 +0.0015 +0.004 +0.004 - +0.0010 +0.004 +0.002 +0.0005 +0.002 +0.002 +0.0000 +0.000 +0.003 +0.003 +>/N= 0.000 +0.006 +0.006 +0.002 +0.002 +0.004 +0.004 +0.001 +0.001 +0.002 +0.002 +0.000 +0.000 +0.000 +0.000 +0 +200 +0 +200 +0 +200 +0 +200 +R +R +R +R21 +Appendix G: Additional numerical data for off-diagonal error +FIG. 16. Off-diagonal error in the VME for observables A = Z, ZZ, X, XX at λ/N = −0.75. See the caption of Fig. 5 in the +main text for further explanation of what is shown in the plots. +In Sec. IV of the main text, Fig. 5 demonstrates the behavior of the off-diagonal error for the operator Z acting on +the middle of the chain at the energy density λ/N = −0.5. Due to the small system sizes and statistical fluctuations +present in our analysis, we find it appropriate to include further numerical data that establishes the trends we observed +in that section. We show the equivalent of Fig. 5 for λ/N = −0.75, −0.5, −0.25, 0 and A = Z, ZZ, X, XX acting on + +Z(/N +-0.75) +0.5 - +2 - +0.0 +2-8- +-0.5 +2-11 +-0.52 +m +OR·R-1/2 +0 +0.0 +2.0 +2-1 +ZZ +-0.2 - +1.5 +-0.4 - +1.0 +-0.6 +0.5 +-0.8 +2-13 _ +0.41 +OR·R-1/2 +0.0 +A=X +0.6 +2.0 +0.5 +2-6 - +1.5 +0.4 +0.3 +2-9 _ +1.0 +0.2 +0.5 +2-12 +-0.36 +0.1 +OR·R-1/2 +0.0 +0.4 : +A=XX +5 +2-4 _ +0.2 +4 +2-7 +3 +F 0'0 +2 +-0.2 +2-13 ~ +-0.51 +-m= +1 +OR·R-1/2 +0 +5 +7 +9 +11 +13 +21 +23 +25 +27 +-1 +0 +1 +N +R +eigenvalue22 +FIG. 17. Off-diagonal error in the VME for observables A = Z, ZZ, X, XX at λ/N = −0.5. See the caption of Fig. 5 in the +main text for further explanation of what is shown in the plots. +the central one or two sites of the chain. The results are shown in Figs. 16,17,18,and 19, respectively. +The most important feature of these results in favor of the efficacy of the VME algorithm is that in every case, the +off-diagonal error ϵoff +R decays with R across various system sizes, albeit not always monotonically, for example see the +N = 13 (darkest blue) off-diagonal error for XX at λ/N = 0 in the bottom row of Fig. 19. To mitigate statistical +fluctuations we have considered the system-size-averaged off-diagonal error ϵoff +R , and we can see that it decays with R +as Rm with m ≈ −1/2, except at the lowest energy density; in particular see the cases of ZZ, X at λ/N = −0.75 in + +20 +Z(/N +:-0.50) +0.5 +2-3- +2-6 - +2 +0.0 +2-9 - +2-12 - +-0.5 +-0.57 +-m +2-15 - +OR·R-1/2 +0 +2-1 +2.0 +0.4 +A=ZZ +0.2 +2-4 - +1.5 +0.0 +2-7 - +1.0 - +-0.2 +2-10 _ +-0.4 +32 +0.5 +2-13 +-0.51 +-0.6 +OR·R-1/2 +0.0 +2.5 +0.8 - +A=X +0.6 +2.0 +0.4 +2-6 _ +1.5 +0.2 +2-9 +1.0 +0.0 +-0.2 +2-12- +0.5 +-m= +-0.49 +OR·R-1/2 +-0.4 +0.0 +0.6 +A=XX +5 +0.4 +2-4 - +4 +0.2 +3 +2-6 - +0.0 +2 +2-8 - +-0.10 +-0.2 +-m= +1 +OR·R-1/2 +0 +5 +7 +9 +11 +13 +21 +23 +25 +27 +-1 +0 +1 +N +R +eigenvalue23 +FIG. 18. Off-diagonal error in the VME for observables A = Z, ZZ, X, XX at λ/N = −0.25. See the caption of Fig. 5 in the +main text for further explanation of what is shown in the plots. +Fig. 16 which only have best fit parameters m = −0.36, −0.41. +A second observation that should be made from these plots is that the distribution of eigenvalues of ˜A (shown as +the orange probability density in the far right column of each plot) appears to determine whether the variational +estimate over- or underestimates the microcanonical value (shown in orange and blue scatter points respectively in +the far left column). While there is only a qualitative correlation, it does occur in every observed case. The effect +appears to be more extreme at lower energy densities where the histograms are less symmetric. + +20 +A=Z(>/N=-( +-0.25) +3 +0.5 +2-2 +2-4. +2 +0.0 +2-6 +-0.5 +ERff +1 +2-8 +-m +-0.55 +OR·R-1/2 +0 +2-1 +A=ZZ +2.0 +0.50 +0.25 +2-4 +1.5 +0.00 +2-7 +. +1.0 +-0.25 +2-10 +-0.50 +0.5 +-m: +0.59 +2-13 +-0.75 +OR·R-1/2 +0.0 +0.6 +A=X +2-2 +2.0 +0.4 +2-4 +0.2 +1.5 +2-6 +0.0 +2-8 +1.0 +-0.2 +2-10 +J02 +0.5 +m= +0.47 +-0.4 +2-12 +OR·R-1/2 +0.0 +A=XX +2-2 +5 +0.2 +0.0 +2-4 +3 +-0.2 +2-6 +2 +-0.4 +2-8 +-m= +0.12 +1 +OR·R-1/2 +0 +5 +7 +9 +11 +13 +21 +23 +25 +27 +-1 +0 +1 +N +R +eigenvalue24 +FIG. 19. Off-diagonal error in the VME for observables A = Z, ZZ, X, XX at λ/N = 0.0. See the caption of Fig. 5 in the main +text for further explanation of what is shown in the plots. +The final point of discussion for this section concerns XX as an outlier. The N-averaged error ϵoff +R does not decrease +with R as a power law in every case for A = XX, unlike the other operators. In particular, see XX at λ/N = −0.5 +in the bottom row of Fig. 17, where the orange curve is not so well fit by the dashed black power-law best-fit. We +can also see that XX is an outlier in terms of its eigenvalue statistics: both ˆA and ˜A differ from the other operators. +The operator ˆA has an eigenvalue spectrum closer to that of a Pauli string than the other operators; the peaks at ±1 +are narrower and taller. The eigenvalue distribution of ˜A is also flatter than the others, the distinction being most + +A= Z (>/N= 0.000) +3 +0.5 +2-4 +2-7 +2 +0.0 +2-10 +1 +-0.5 +m +-0.48 +2-13 +OR·R-1/2 +0 +2.0 +A=ZZ +0.2 +2-2 +1.5 +0.0 +2-5 +-0.2 +1.0 +-0.4 +0.5 +2-11 +:-0.45 +-0.6 +m 二 +OR·R- +-1/2 +0.0 +2-1 +A=X +0.50 +2.0 +2-3 +0.25 +2-5 - +1.5 +0.00 +2-7 - +1.0 +-0.25 +2-9 +-0.50 +0.5 +m= +-0.59 +2-11 +OR·R-1/2 +-0.75 +0.0 +2-2 +A=XX +0.2 +5 +2-4 +4 +0.0 +2-6 +3 - +-0.2 +2-8 +2 - +-0.4 - +2-10 +m= +-0.28 +1 +-0.6 +2-12 +R·R-1/2 +0 +5 +7 +9 +11 +13 +21 +23 +25 +27 +-1 +0 +1 +N +R +eigenvalue25 +pronounced at λ/N = −0.5. +Appendix H: Additional numerical data for expectation values +FIG. 20. Ensemble-averaged VME observable expectation values (orange) and broadened microcanonical averages (blue) plotted +versus system size for the full range of energy densities and operators considered in this work. See the caption of Fig. 6 in the +main text for further explanation of what is shown in the plots. +Here we show in Fig. 20 the VME estimates for the four local observables A = Z, ZZ, X, XX acting on the +central one or two sites of the chain. We show these estimates when targeting four different energy densities λ/N = + +/N= -0.50 +-0.3 +-0.3 - +0.3 - +0.2 +0.2 +0.1 +0.1 +0.1 +0.2 +F 00 +0.0 +0.0 +0.1 +-0.1 +-0.1 - +-0.1 +0.0 +-0.2 +0.2 - +-0.2 - +-0.3 +-0.3 - +入/N= -0.25 +-0.1 - +0.3 +0.2 - +0.1 +0.1 - +0.2 +0.1 - +0.0 +0.0 - +0.1 - +-0.1 - +0.0- +-0.1 - +0.0 - +-0.2 +-0.1 +-0.3 +-0.2 +-0.1 - +-0.2 - +/ N= 0.000 +-0.4 - +5791113 +5 +7 +91113 +¥７９1113 +９11 13 +N +N +N +N26 +−0.75, −0.5, −0.25, 0 and when R = 288. +As we observed in Sec. IV, the off-diagonal error generally does not +systematically depend on N. 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Rev. +Lett. 128, 060601 (2022). + diff --git a/m9E2T4oBgHgl3EQfzQh_/content/tmp_files/load_file.txt b/m9E2T4oBgHgl3EQfzQh_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4084937eba912a0bf31654ef4e6c2dec41753f55 --- /dev/null +++ b/m9E2T4oBgHgl3EQfzQh_/content/tmp_files/load_file.txt @@ -0,0 +1,1726 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf,len=1725 +page_content='Variational Microcanonical Estimator Kl´ee Pollock,1 Peter P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Orth,1, 2, 3 and Thomas Iadecola1, 2, ∗ 1Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 2Ames National Laboratory, Ames, Iowa 50011, USA 3Department of Physics, Saarland University, 66123 Saarbr¨ucken, Germany (Dated: January 11, 2023) We propose a variational quantum algorithm for estimating microcanonical expectation values in models obeying the eigenstate thermalization hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Using a relaxed criterion for convergence of the variational optimization loop, the algorithm generates weakly entangled superpositions of eigenstates at a given target energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' An ensemble of these variational states is then used to estimate microcanonical averages of local operators, with an error whose dominant contribution decreases as a power law in the size of the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' We apply the algorithm to the one-dimensional mixed-field Ising model, where it converges for ansatz circuits of depth roughly linear in system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The most accurate thermal estimates are produced for intermediate energy densities and for local operators that appear in the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In our error analysis, we find connections with recent works investigating the underpinnings of the eigenstate thermalization hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In particular, the failure of energy-basis matrix elements of local operators to behave as independent random variables is a potential source of error that the algorithm can overcome by averaging over an ensemble of variational states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' INTRODUCTION Calculating the ground state and thermal equilibrium properties of large and complex quantum systems re- mains a central task in contemporary quantum physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' While for integrable systems analytical techniques can often solve the problem, in generic nonintegrable sys- tems such methods do not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In the last two decades however, efficient numerical methods have been devel- oped to calculate ground-state and thermal properties in settings where the target state is only modestly entan- gled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Tensor network (TN) methods exploit the locality of physical Hamiltonians, in particular their area-law en- tangled ground states [1], to find efficient representations of the wavefunction via truncated matrix product states on classical hardware [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Additionally, these efficient representations can be extended to Gibbs states at finite temperature via matrix product operators [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Exam- ples of algorithms based on TNs include the minimally entangled typical thermal state (METTS) algorithm [4] for estimating canonical averages, and an algorithm for estimating microcanonical averages using time evolving block decimation (TEBD) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In higher than one spa- tial dimension however, the TN contraction step becomes hard [6], so that classical algorithms may not be sufficient for the simulation of even weakly entangled quantum sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' It has long been believed that quantum computers are the natural platform to simulate quantum systems [7], but to exploit their full power it is likely that deep quan- tum circuits and error correction will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Cur- rently, we have noisy intermediate scale quantum (NISQ) devices that cannot yet implement deep circuits with high ∗ iadecola@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='edu FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The VME algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In step (0), the QPU is ini- tialized in a random product state |ψ0 r⟩ (r = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' , R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The VQA repeats steps (1) and (2) that optimize the cost function C(θ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='1) to “squeeze” the state onto a microcanon- ical window of size δ as shown in step (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Steps (0-3) are repeated to produce a pseudo-random ensemble of states |ψr⟩ which for large N and R can be used to approximate micro- canonical averages of local operators A as in step (4), where ρR = 1 R � r |ψr⟩ ⟨ψr|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' fidelity, but which can still demonstrate the potential for quantum computing in cases where low-depth circuits are sufficient [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' There is thus a significant need to develop algorithms that can take advantage of these NISQ de- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Originating with the variational quantum eigensolver (VQE) [9], one class of algorithms that can potentially achieve this goal in some cases are the hybrid quantum- classical variational quantum algorithms (VQAs) [10– arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='04129v1 [quant-ph] 10 Jan 2023 1《E>12 QPU O(VN) (0) 01 3 02 I《E[r>|2 (3) 04 0(8) E (2) 0 (1) C(0) (4) tr(pRA) CPU CPU/QPU2 12], which employ a digital quantum computer aided by a classical optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Although generic VQAs suf- fer from the well known barren plateau problem [13– 15] which suggests unscalablility in full generality, there is evidence that VQAs can calculate the ground state of certain Hamiltonians using only polynomial quan- tum resources, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' by using the Hamiltonian variational ansatz for the transverse field Ising model [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Re- cent works have also considered using VQAs to pre- pare Gibbs states using cost functions such as the rel- ative entropy or relative free energy between the cur- rent state and target state [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' strategies to overcome the costly evaluation of the entropic term have also been proposed [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Other finite-temperature VQAs prepare thermofield-double (TFD) states, which require doubling the number of qubits in the physical system being simulated—for example the algorithm of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' [20] can prepare the TFD state of free fermions efficiently at any inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Alternative quantum algo- rithms for preparing thermal states include a quantum version of the minimally-entangled typical thermal states algorithm (QMETTS) that involves imaginary time evo- lution on quantum hardware [21], and an algorithm based on random quantum circuits [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In this work, we task a VQA with calculating mi- crocanonical averages of local observables in a one- dimensional (1D) nonintegrable spin model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Our work is partially inspired by analog quantum simulation [23] and classical tensor network [24] algorithms for estimating microcanonical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The algorithm takes advan- tage of the eigenstate thermalization hypothesis (ETH), in particular the “diagonal” ETH which states that in a nonintegrable model the energy-basis diagonal matrix elements ⟨E|A|E⟩ of an observable A approach a smooth function A(E) in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The algorithm, which we call the variational micro- canonical estimator (VME), works as follows (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' We initialize the QPU in a random product state [step (0)] |ψ0 r⟩, whose energy variance is typically extensive in N (the number of sites) [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Given a target energy λ and microcanonical window size δ, a classical optimizer is then tasked with minimizing the cost function C(θ) = ⟨ψ(θ)| (H − λ)2 |ψ(θ)⟩ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='1) [steps (1) and (2)] originally proposed in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' How- ever, instead of trying to reach a local or global min- imum, we stop the optimization as soon as Var(H) = ⟨(H − ⟨H⟩)2⟩ ≤ δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' This produces states whose energy support is roughly limited to the microcanonical window of interest [step (3)], and the resulting variational states |ψr⟩ are then used to compute the expectation of a local observable A by averaging ⟨ψr|A|ψr⟩ over R variational states [step (4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The ensemble average in step (4) en- ables a parametric reduction in the error and is essential to the algorithm’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' We benchmark the VME algorithm on a nonintegrable Ising chain by comparing its estimates for local ob- servables to corresponding Gaussian microcanonical en- semble predictions obtained from exact diagonalization (ED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' For (i) local operators appearing as terms the Hamiltonian, which we refer to as local operators cou- pled to energy density as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' [25], and (ii) for target energies λ in the bulk of the spectrum we conjecture that the absolute error in the VME estimate scales as ϵR ≲ O(Rm) + O(δ/N) + O(D−1/2(λ)) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='2) in the thermodynamic limit and for large R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Here, D(λ) is the density of states at the target energy λ and m ≈ −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The last two terms in this formula are predicted by ETH and the first term we give a heuristic argument for that we substantiate with numerical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' At the lowest energy density considered, which is near the edge of the spectrum, we find instead that m ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='36 for A coupled to energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' For A not coupled to energy density, the O(Rm) scaling appears to be less well established but power law fits yield −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='51 ≤ m ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' We then generalize the problem to the reduced state of small subsystems of the chain and find numerically that when choosing R = O(N 2) and for certain λ, the VME appears to approach the corresponding microcanonical state in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The states prepared by the VME are consistent with area law entanglement for a fixed N, and require roughly linearly deep quantum circuits to prepare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' We find that every random initial product state is able to converge, which we attribute to the fact that the algorithm does not seek global minima of the cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' An additional distinction from other current VQAs for preparing mixed states is that we pre- pare pure states one at a time, thus avoiding storage of a large ensemble of pure quantum states in a quantum memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Should the decrease in the trace distance with R = O(N 2) continue to hold for larger N than we access in this work, an interesting consequence of the VME algo- rithm would be that the microcanonical ensemble, which involves at least one (via ETH) highly entangled (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' vol- ume law) eigenstate, becomes indistinguishable by local operators from a polynomially large ensemble of weakly entangled variational states in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' II, we in- troduce (i) the statement of ETH and (ii) a class of states which might be called microcanonical superposi- tion states, which our converged variational states fall under.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' We then review related works attempting to use these states to estimate thermal averages and the rela- tionship of this problem to ETH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' III we discuss how averaging over an ensemble of these microcanoni- cal superposition states could significantly improve how well they can estimate microcanonical averages, and then we detail the VME algorithm which can produce these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Finally in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' IV we present the numerical re- sults starting with the behavior of the diagonal and off- diagonal errors for various local operators, then the trace distance, and finally the quantum resources like circuit depth and entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The energy-basis diagonal matrix elements ⟨E|A|E⟩ of various local observables A acting in the middle of the chain, plotted against energy density in the nonintegrable 1D mixed-field Ising model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='1) with parameters J = 1, hx = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='05, and hz = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Lighter blue colored points are for system size N = 9 and darker blue points are for N = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Orange curves are coarse grained versions of the N = 13 scatter plots which define the “smooth” function A(E) in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' MOTIVATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Eigenstate Thermalization Hypothesis Here we review relevant aspects of the ETH and some recent works which attempt to exploit it to estimate ther- mal averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' We assume a nonintegrable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' chaotic) Hamiltonian H which has a non-degenerate energy spec- trum so that its eigenstates |E⟩ are uniquely labeled by their energies E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Furthermore, we will assume that all operators and states of interest are real in the energy ba- sis for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The variant of ETH we consider was formulated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' [26] and proposes that in a quantum chaotic system, the energy-basis matrix elements of ob- servables have the form ⟨E| A |E′⟩ = δEE′A( ¯E) + D−1/2( ¯E)f( ¯E, ω)REE′ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='1) where ¯E = (E + E′)/2, ω = E − E′, D( ¯E) is the den- sity of states at energy ¯E, A( ¯E) and f( ¯E, ω) approach smooth functions in the thermodynamic limit, and REE′ are order-one fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Examples of such functions A( ¯E) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' 2 which demonstrates this for local spin operators in the 1D mixed-field Ising model (defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The ansatz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='1) captures several features of such ma- trix elements that have been observed in numerical stud- ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Firstly, because the density of states is exponentially large in system size, the off-diagonal matrix elements are exponentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Secondly, the smooth function A(E) is related to the statistical mechanical prediction for ⟨A⟩ at average energy E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' this function will play a central role in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Finally, the function f( ¯E, ω) con- trols the approach to thermal equilibrium and is related to other spectral properties of the observable [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' this function figures less prominently in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' To see how A(E) is related to a thermal average, con- sider for example a broadened microcanonical ensemble ρλ,δ centered on energy E = λ and of width O(δ) which we will define more precisely at beginning of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Under certain assumptions about the density of states of the model and away from λ = 0 (which corresponds to infinite temperature), and assuming the ETH ansatz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='1), we have in the thermodynamic limit that (see Ap- pendix C for details) A(λ) = ⟨A⟩mc + O(δ2/N) + O(D−1/2(λ)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='2) where ⟨A⟩mc = tr ρλ,δA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The ETH thus suggests that, if one could prepare even a single eigenstate |λ⟩ of the Hamiltonian with energy λ, then one could accurately estimate thermal averages in sufficiently large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' However for a nonintegrable Hamiltonian, a generic ex- cited eigenstate is volume-law entangled, and thus can- not efficiently be prepared by classical algorithms nor by VQE-type algorithms [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Thus, this feature of ETH does not appear practically useful, expect perhaps in the case of an error corrected quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Microcanonical Superpositions An alternative approach to using exact eigenstates for computing thermal averages is using pure states of the form |ψ⟩ = � E cE |E⟩ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content='3) where either cE are exactly zero outside the energy win- dow defined by |E − λ| ≤ δ, or the states satisfy the weaker condition that ⟨ψ|(H − λ)2|ψ⟩ = O(δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' We re- fer to states of this type as “microcanonical superposi- tion states” and they have been studied in the context of thermal pure quantum (TPQ) states [29], the foundations of quantum statistical mechanics [30, 31], algorithms for analog quantum simulators [23], and tensor network al- gorithms [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' The practical reason for considering these states is that they appear to be significantly less entangled than exact eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' In fact, there exist MPS-based numerical constructions of them such that the maximum entan- glement entropy across any cut scales as k/δ + log2 √ N for some constant k [24] and N being the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9E2T4oBgHgl3EQfzQh_/content/2301.04129v1.pdf'} +page_content=' Thus, by choosing δ = O(1/log2N), such states can have only O(log2N) entanglement, whereas a single ex- cited eigenstate of a nonintegrable system is expected 1. By Lemma 2.4, we see that the Toeplitz operator Tϕ is bounded +on Lp(T). Clearly, �ϕ ∈ Lq(T) implies that ϕ′ ∈ Lq(T). Then using Lemma 2.6, we have +clos{ϕ(w) : w ∈ T} = clos[σp(Tϕ)] ⊂ σ(Tϕ). +To prove the reverse inclusion, it is sufficient to show that Tϕ − λI is invertible on Lp(T) if λ is not in the +closure of the range of ϕ, where I is the identity operator on Lp(T). Observe that λ /∈ clos{ϕ(v) : v ∈ T} +implies that λ is not an eigenvalue of Tϕ. This means that Tϕ − λI is an injective from Lp(T) to Lp(T). +According to the Banach inverse operator theorem, we need only to show that Tϕ − λI is a surjective on +Lp(T) if λ /∈ clos{ϕ(v) : v ∈ T}. +Suppose that +inf +v∈T |ϕ(v) − λ| ⩾ δ +for some positive constant δ. Let δ0 = min{δ, |λ|}. Then we have that δ0 > 0 and +|ϕ(v) − λ| ⩾ δ0 +and +|ϕ−(v) − λ| ⩾ δ0 +for all v ∈ T. Letting g be a function in Lp(T), we define +f := 1 +λ +� +∆ +� +ϕ +ϕ − λ∇g +� +− g +� +. +(3.1) +In the following, we will show that f ∈ Lp(T) and f is the preimage of g under Tϕ − λI. +To show f ∈ Lp(T), we need only to prove that +∆ +� +ϕ +ϕ − λ∇g +� +∈ Lp(T). +Let us show � +� +ϕ +ϕ−λ +� +∈ Lq(T) first. Noting that +� +� +ϕ +ϕ − λ +� +(v) = (|v| + 1) +� +ϕ +ϕ − λ +�′ +(v) = (|v| + 1) +� +ϕ(v) +ϕ(v) − λ − +ϕ−(v) +ϕ−(v) − λ +� + +TOEPLITZ OPERATORS +5 +for v ∈ T, we have +���� +� +� +ϕ +ϕ − λ +����� +q += +� � +v∈T +���(|v| + 1) +� +ϕ(v) +ϕ(v) − λ − +ϕ−(v) +ϕ−(v) − λ +���� +q� 1 +q += +� � +v∈T +���(|v| + 1) +� +ϕ(v) +ϕ(v) − λ − +ϕ−(v) +ϕ(v) − λ + +ϕ−(v) +ϕ(v) − λ − +ϕ−(v) +ϕ−(v) − λ +���� +q� 1 +q +. +Then triangle inequality gives that +���� +� +� +ϕ +ϕ − λ +����� +q +⩽ +� � +v∈T +���(|v| + 1)ϕ(v) − ϕ−(v) +ϕ(v) − λ +��� +q� 1 +q ++ +� � +v∈T +���(|v| + 1) ϕ−(v)[ϕ(v) − ϕ−(v)] +[ϕ−(v) − λ][ϕ(v) − λ] +��� +q� 1 +q += +� � +v∈T +���(|v| + 1) +ϕ′(v) +ϕ(v) − λ +��� +q� 1 +q ++ +� � +v∈T +���(|v| + 1) +ϕ′(v)ϕ−(v) +[ϕ−(v) − λ][ϕ(v) − λ] +��� +q� 1 +q +⩽ 1 +δ0 +� � +v∈T +��(|v| + 1)ϕ′(v) +��q� 1 +q + 1 +δ2 +0 +� � +v∈T +��(|v| + 1)ϕ−(v)ϕ′(v) +��q� 1 +q +⩽ 1 +δ0 +� � +v∈T +��(|v| + 1)ϕ′(v) +��q� 1 +q + ∥ϕ∥∞ +δ2 +0 +� � +v∈T +��(|v| + 1)ϕ′(v) +��q� 1 +q += +� 1 +δ0 ++ ∥ϕ∥∞ +δ2 +0 +� +∥�ϕ∥q. +This shows that � +� +ϕ +ϕ−λ +� +∈ Lq(T), since �ϕ ∈ Lq(T). +Next, we are going to show that ∆ +� +ϕ +ϕ−λ∇g +� +∈ Lp(T). Noting that +ϕ +ϕ−λ and +� +ϕ +ϕ−λ +�′ are both in Lq(T), +we obtain by Lemma 2.3 that +∆ +� +ϕ +ϕ − λ∇g +� += T +ϕ +ϕ−λ g = +� +ϕ +ϕ − λ +� +−g + ∆ +�� +ϕ +ϕ − λ +�′ +g +� +. +Furthermore, we have +���∆ +�� +ϕ +ϕ − λ +�′ +g +���� +p ⩽ +���∆ +�� +ϕ +ϕ − λ +�′ +g +���� +1 = +� +v∈T +����∆ +�� +ϕ +ϕ − λ +�′ +g +� +(v) +���� +⩽ +� +v∈T +� +u∈Sv +���� +� +ϕ +ϕ − λ +�′ +(u)g(u) +���� += +� +v∈T +(|v| + 1) +���� +� +ϕ +ϕ − λ +�′ +(v)g(v) +���� += +� +v∈T +���� +� +� +ϕ +ϕ − λ +� +(v) +���� |g(v)| +⩽ ∥g∥p +���� +� +� +ϕ +ϕ − λ +����� +q +, +to get that ∆ +�� +ϕ +ϕ−λ +�′g +� +∈ Lp(T). Since +ϕ +ϕ−λ ∈ Lq(T) implies that +ϕ +ϕ−λ ∈ L∞(T), we obtain that +� +ϕ +ϕ−λ +� +− +is also bounded. It follows that +� +ϕ +ϕ−λ +� +−g is in Lp(T), which gives that ∆ +� +ϕ +ϕ−λ∇g +� +∈ Lp(T). Therefore, we +obtain that the function defined in (3.1) is in Lp(T). +To finish the proof, it remains to verify that +(Tϕ − λI)f = g. + +6 +MINGMEI HUANG, XIAOYAN ZHANG, AND XIANFENG ZHAO +Indeed, we have by Lemma 2.3 that +ϕ +ϕ − λ∇g ∈ L1(T), +since g ∈ Lp(T) and +ϕ +ϕ−λ, +� +ϕ +ϕ−λ +�′ ∈ Lq(T). It follows from (3.1) and Lemma 2.2 that +∇f = ∇ +� +1 +λ +� +∆ +� +ϕ +ϕ − λ∇g +� +− g +�� += 1 +λ +ϕ +ϕ − λ∇g − ∇g +λ = +∇g +ϕ − λ. +This yields that +(Tϕ − λI)f = Tϕf − λf = ∆(ϕ∇f) − λf += ∆ +� +ϕ +ϕ − λ∇g +� +− ∆ +� +ϕ +ϕ − λ∇g +� ++ g += g, +as desired. This completes the proof of Theorem 3.1. +□ +Now we give a description for the spectrum of Tϕ : L∞(T) → L∞(T) with ϕ, ϕ′ ∈ L1(T) in the following +theorem. +Theorem 3.2. If ϕ and ϕ′ are both in L1(T), then Tϕ is bounded on L∞(T) and in this case we have +σ(Tϕ) = clos{ϕ(v) : v ∈ T}. +Proof. Lemma 2.5 tells us that Tϕ : L∞(T) → L∞(T) is bounded if ϕ ∈ L1(T) and ϕ′ ∈ L1(T). According +to the proof of Theorem 3.1, we need only to show that +� +ϕ +ϕ − λ +� +−g +and +∆ +�� +ϕ +ϕ − λ +�′ +g +� +are both in L∞(T) if λ /∈ clos{ϕ(v) : v ∈ T} and g ∈ L1(T). In fact, +���� +� +ϕ +ϕ − λ +� +−g +���� +∞ += sup +v∈T +���� +� +ϕ +ϕ − λ +� +−(v)g(v) +���� ⩽ ∥g∥∞ +δ0 +∥ϕ∥∞, +where δ0 is the constant defined in the proof of Theorem 3.1. Moreover, since +����∆ +�� +ϕ +ϕ − λ +�′ +g +����� +∞ += sup +v∈T +���� +� +u∈Sv +g(u) +� +ϕ +ϕ − λ +�′ +(u) +���� +⩽ sup +v∈T +� +u∈Sv +|g(u)| +���� +ϕ(u) +ϕ(u) − λ − +ϕ−(u) +ϕ−(u) − λ +���� +⩽ |λ| ∥g∥∞ +δ2 +0 +sup +v∈T +� � +u∈Sv +|ϕ(u) − ϕ−(u)| +� += |λ| ∥g∥∞ +δ2 +0 +∥ϕ′∥1, +we obtain that +� +ϕ +ϕ−λ +� +−g ∈ L∞(T) and ∆ +�� +ϕ +ϕ−λ +�′g +� +∈ L∞(T). +The rest of the proof of this theorem is similar to the previous one, so we omit the details. This finishes +the proof of Theorem 3.2. +□ +Remark 3.3. The conclusions in Theorems 3.1 and 3.2 are comparable to the corresponding results for +analytic Toeplitz operators on the Hardy and Bergman spaces of the open unit disk, see the books [15] and +[24] for more information. + +TOEPLITZ OPERATORS +7 +4. The positivity of Toeplitz operators +We say that a bounded linear operator T is self-adjoint (positive) on a complex Hilbert space H if +⟨Tx, x⟩H is real (nonnegative) for every x in H . In this section, we study when a bounded Toeplitz +operator is self-adjoint (positive) on the Hilbert space L2(T). +Observing that the adjoint of the Topelitz operator Tϕ does not equal Tϕ in general, which leads to certain +difficulties in the study of the self-adjointness and positivity of Toeplitz operators on L2(T). However, we +are able to obtain a complete characterization for the self-adjoint Toeplitz operators on L2(T) by using +their point spectra. +Theorem 4.1. Suppose that ϕ is a function in L2(T) such that �ϕ ∈ L2(T). Then the following three +conditions are equivalent: +(1) Tϕ is positive on L2(T); +(2) Tϕ is self-adjoint on L2(T); +(3) ϕ(v) = 0 for all v ∈ T. +Proof. Clearly, (1) ⇒ (2) is trivial and (3) ⇒ (1) follows immediately from Lemma 2.3. To complete the +proof of this theorem, it remains to show that (2) ⇒ (3). +Now we suppose that Tϕ is self-adjoint on L2(T). Then we have by Lemma 2.6 that ϕ must be real- +valued, since +{ϕ(v) : v ∈ T} = σp(Tϕ) ⊂ σ(Tϕ) ⊂ R +or +{0} ∪ {ϕ(v) : v ∈ T} = σp(Tϕ) ⊂ σ(Tϕ) ⊂ R. +Let f be any function in L2(T). Since �ϕ ∈ L2(T), we see that the series � +v∈T +[Tϕf(v)]f(v) is absolutely +convergent. Now using Lemma 2.3 again gives +⟨Tϕf, f⟩ = +� +v∈T +[Tϕf(v)]f(v) = +� +v∈T +� +f(v)ϕ−(v)f(v) + ∆(fϕ′)(v)f(v) +� += +� +v∈T +|f(v)|2ϕ−(v) + +� +v∈T +f(v)∆(fϕ′)(v). +(4.1) +Furthermore, we have by the definitions of ϕ′ and the operator ∆ that +� +v∈T +f(v)∆(fϕ′)(v) += +� +v∈T +� +f(v) +� +u∈Sv +(fϕ′)(u) +� += +� +v∈T +� +f(v) +� +u∈Sv +f(u) +� +ϕ(u) − ϕ−(u) +�� += +� +v∈T +|f(v)|2ϕ(v) − +� +v∈T +|f(v)|2ϕ−(v) + +� +v∈T +� +f(v) +� +u∈Sv\{v} +� +f(u)ϕ(u) − f(u)ϕ−(u) +�� +. +(4.2) +We first show that ϕ is a constant function. Otherwise, we can choose two vertices ξ and η in T such +that ϕ(ξ) ̸= ϕ(η). Since the path joining ξ and η is a finite sequence of distinct vertices, we may assume +that ξ ∼ η. In other words, we obtain that ξ ∈ Chi(η) or η ∈ Chi(ξ). Without loss of generality, we may +assume that η ∈ Chi(ξ). Let us consider the function g : T → C defined by +g(v) = + + + + + + + + + +1, +if v = ξ, +i, +if v = η, +0, +otherwise. + +8 +MINGMEI HUANG, XIAOYAN ZHANG, AND XIANFENG ZHAO +It follows from (4.1) and (4.2) that +⟨Tϕg, g⟩ = |g(ξ)|2ϕ(ξ) + |g(η)|2ϕ(η) + g(ξ) +� +g(η)ϕ(η) − g(η)ϕ(η−) +� += |g(ξ)|2ϕ(ξ) + |g(η)|2ϕ(η) + g(ξ)g(η) +� +ϕ(η) − ϕ(η−) +� += +� +ϕ(ξ) + ϕ(η) +� ++ +� +ϕ(η) − ϕ(ξ) +� +i. +This means that ⟨Tϕg, g⟩ /∈ R, which contradicts that Tϕ is self-adjoint. Thus ϕ is a constant function. +However, the condition ϕ ∈ L2(T) yields that ϕ must be zero. This completes the proof. +□ +5. Finite rank Toeplitz operators +The final section is devoted to establishing a equivalent characterization for finite rank Toeplitz operators +on Lp(T), where 1 ⩽ p ⩽ ∞. More concretely, we will show in the following theorem that the Toeplitz +operator Tϕ having finite rank if and only if {v ∈ T : ϕ(v) ̸= 0} is a finite subset of T. +Theorem 5.1. Let 1 ⩽ p ⩽ ∞ and ϕ be a function in Lq(T) such that ϕ′ ∈ Lq(T) and the Toeplitz operator +Tϕ is bounded on Lp(T), where 1 +p + 1 +q = 1. Then Tϕ has finite rank on Lp(T) if and only if ϕ(v) = 0 except +for finitely many points v ∈ T. +Proof. We show the sufficiency first. Suppose that ϕ(v) = 0 for all v ∈ T except v1, v2, · · · , vn. For any +f ∈ Lp(T), we have +Tϕf(v) = ∆(ϕ∇f)(v) = +� +u∈Sv +ϕ(u)∇f(u), +v ∈ T. +It follows that Tϕf(v) = 0 for all v ∈ T with |v| > max{|vk| : k = 1, 2, · · · , n}. Let +W := +� +w ∈ T : |w| ⩽ max +1⩽k⩽n |vk| +� +. +Then W is a finite set and its elements can be enumerated by +W = {w1, w2, · · · , wN}. +Fix a vertex ξ ∈ T and define +eξ(v) = + + + +1, +if v = ξ, +0, +otherwise. +(5.1) +Since Tϕf is a vector in Lp(T), it can be expressed as +Tϕf = c1ew1 + c2ew2 + · · · + cNewN +with constants c1, c2, · · · , cN. This implies that +range(Tϕ) ⊂ span +� +ew1, ew2, · · · , ewN +� +, +to obtain that Tϕ is of finite rank on Lp(T). +To show the necessity, we suppose that Tϕ is a finite rank operator on Lp(T) and +range(Tϕ) ⊂ span +� +ev1, ev2, · · · , evn +� +, + +TOEPLITZ OPERATORS +9 +where vk is a vertex of T and evk is defined in (5.1). According to Lemma 2.3 and the computations in +(4.2), we obtain that +Tϕew(v) = ew(v)ϕ(v−) + +� +u∈Sv +ew(u)[ϕ(u) − ϕ(u−)] += ew(v)ϕ(v−) + ew(v)[ϕ(v) − ϕ(v−)] + +� +u∈Sv\{v} +ew(u)[ϕ(u) − ϕ(u−)] += ew(v)ϕ(v) + +� +u∈Sv\{v} +ew(u)[ϕ(u) − ϕ(u−)] +(5.2) +for every v ̸= o and w ∈ T. +If ϕ does not vanish at infinitely many vertices of T, then we can choose a vertex w ∈ T such that +|w| > max +� +|v1|, |v2|, · · · , |vn| +� +and ϕ(w) ̸= 0. It follows from (5.2) that +Tϕew(w) = ϕ(w). +We claim that Tϕew /∈ span +� +ev1, ev2, · · · , evn +� +. Indeed, if +Tϕew = c1ev1 + c2ev2 + · · · + cnevn +for some scalars c1, c2, · · · , cn, then we would have +0 = c1ev1(w) + c2ev2(w) + · · · + cnevn(w) = Tϕew(w) = ϕ(w) ̸= 0, +which is a contradiction. This yields that +dim +� +range(Tϕ) +� +> n. +But this contradicts that the dimension of range(Tϕ) is less than or equal to n. Therefore, the proof of +Theorem 5.1 is finished. +□ +Remark 5.2. It is worth noting that Theorem 5.1 is analogous to the corresponding results for finite rank +Toeplitz operators on the Bergman and Fock spaces, see [13] and [25, Theorem 6.42] respectively. +Acknowledgment. This work was partially supported by NSFC (grant number: 11701052) and Chongqing +Natural Science Foundation (cstc2019jcyj-msxmX0337). The third author was partially supported by the +Fundamental Research Funds for the Central Universities (grant numbers: 2020CDJQY-A039, 2020CDJ- +LHSS-003). +References +[1] Agrawal, A., Berge, A., Colbert-Pollack, S., Mart´ınez-Avenda˜no, R. A., Sliheet, E.: Norms, kernels and eigenvalues of +some infinite graphs. arXiv:1812.08276 (2018) +[2] Allen, R. F., Colonna, F., Easley, G. R.: Multiplication operators between Lipschitz-type spaces on a tree. Int. J. Math. +Sci. 36, Article ID: 472495 (2011) +[3] Allen, R. F., Colonna, F., Easley, G. R.: Multiplication operators on the iterated logarithmic Lipschitz spaces of a tree. +Mediterr. J. Math. 9, 575-600 (2012) +[4] Allen, R. F., Colonna, F., Easley, G. R.: Multiplication operators on the weighted Lipschitz space of a tree. J. Operator +Theory 69, 209-231 (2013) +[5] Allen, R. F., Colonna, F., Easley, G. R.: Composition operators on the Lipschitz space of a tree. Mediterr. J. Math. 11, +97-108 (2014) +[6] Allen, R. F., Colonna, F., Prudhom, A.: Multiplication operators between iterated logarithmic Lipschitz spaces of a tree. +Mediterr. J. Math. 14, Paper No. 212 (2017) +[7] Allen, R. F., Pons, M. A.: Composition operators on weighted Banach spaces of a tree. Bull. Malays. Math. Sci. Soc. 41, +1805-1818 (2018) +[8] Allen, R. F., Pons, M. A.: Weighted composition operators on discrete weighted Banach spaces. Acta Sci. Math. (Szeged) +(2022) https://doi.org/10.1007/s44146-022-00051-w + +10 +MINGMEI HUANG, XIAOYAN ZHANG, AND XIANFENG ZHAO +[9] Cartier, P.: Fonctions harmoniques sur un arbre, in: Symposia Mathematica, vol. IX, Convegno di Calcolo delle Proba- +bilit`a, INDAM, Rome, 1971. Academic Press, London (1972) +[10] Cartier, P.: G´eom´etrie et analyse sur les arbres, in: S´eminaire Bourbaki, 24`eme ann´ee (1971/1972), Exp. No. 407, in: +Lecture Notes in Math., vol. 317. Springer, Berlin, pp. 123-140 (1973) +[11] Colonna, F., Easley, G. R.: Multiplication operators on the Lipschitz space of a tree. Integral Equ. Oper. Theory 68, +391-411 (2010) +[12] Colonna, F., Easley, G. R.: Multiplication operators between the Lipschitz space and the space of bounded functions on +a tree. Mediterr. J. Math. 9, 423-438 (2012) +[13] Luecking, D.: Finite rank Toeplitz operators on the Bergman space. Proc. Amer. Math. Soc. 136, 1717-1723 (2008) +[14] Colonna, F., Mart´ınez-Avenda˜no, R. A.: Some class of operators with symbol on the Lipschitz space of a tree. Mediterr. +J. Math. 14, Paper No. 18 (2017) +[15] Douglas, R.: Banach Algebra Techniques in Operator Theory, second edition, Graduate Texts in Mathematics, vol. 179. +Springer, New York (1998) +[16] Jab�lo´nski, Z. J., Jung, I. B., Stochel, J.: Weighted shifts on directed trees. Mem. Amer. Math. Soc. 216, 1017 (2012) +[17] Mart´ınez-Avenda˜no, R. A.: Hypercyclicity of shifts on weighted Lp spaces of directed trees. J. Math. Anal. Appl. 446, +823-842 (2017) +[18] Mart´ınez-Avenda˜no, R. A., Rivera-Guasco, E.: The forward and backward shift on the Lipschitz space of a tree. Integral +Equ. Oper. Theory 92, Paper No. 3 (2020) +[19] Muthukumar, P., Ponnusamy, S.: Composition operators on Hardy spaces of the homogenous rooted trees. Monatsh. +Math. 192, 721-743 (2020) +[20] Pavone, M.: Chaotic composition operators on trees. Houston J. Math. 18, 47-56 (1992) +[21] Pavone, M.: Toeplitz operators on discrete groups. J. Operator Theory 27, 359-384 (1992) +[22] Pavone, M.: Partially ordered groups, almost invariant sets, and Toeplitz operators. J. Funct. Anal. 113, 1-18 (1993) +[23] Zhang, X., Zhao, X.: Toeplitz operator and the operator ∇ on the Lp space of a tree. J. Math. Anal. Appl. 516, Paper +No. 126540 (2022) +[24] Zhu, K.: Operator Theory in Function Spaces, 2nd edn, Math. Surveys Monogr., vol. 138. American Mathematical Society +(2007) +[25] Zhu, K.: Analysis on Fock Spaces, vol. 263. Springer-Verlag, New York (2012) +1 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P. R. China +Email address: huangmm0909@163.com +2 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P. R. China +Email address: xiaoyanzhang@cqu.edu.cn +3 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P. R. China +Email address: xianfengzhao@cqu.edu.cn + diff --git a/nNE_T4oBgHgl3EQf7hx2/content/tmp_files/load_file.txt b/nNE_T4oBgHgl3EQf7hx2/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c0e07f4efd345a414d2e642defe8bbd920f62675 --- /dev/null +++ b/nNE_T4oBgHgl3EQf7hx2/content/tmp_files/load_file.txt @@ -0,0 +1,532 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf,len=531 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='08370v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='FA] 20 Jan 2023 TOEPLITZ OPERATORS ON Lp-SPACES OF A TREE MINGMEI HUANG, XIAOYAN ZHANG, AND XIANFENG ZHAO Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let T be a rooted, countable infinite tree without terminal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In the present paper, we characterize the spectra, self-adjointness and positivity of Toeplitz operators on the spaces of p-summable functions on T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Moreover, we obtain a necessary and sufficient condition for Toeplitz operators to have finite rank on such function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Introduction Let us begin with the preliminary definitions, concepts and relevant notation on trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' A graph G is a pair (V, E) consisting of a countably-infinite set of vertices V and a set of edges E ⊂ � (u, v) : u, v ∈ V, u ̸= v � (which is a subset of V × V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' If (u, v) ∈ E, then we say that u and v are adjacent (or neighbors) and we denote this by u ∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' For each vertex v, the degree of v is the number of vertices adjacent to v, which is denoted by deg(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' We say that G is locally finite if the degree of every vertex is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' We call that a vertex v is a terminal vertex if v has a unique neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' A path of length n joining two vertices u and v is a finite sequence of (n + 1) distinct vertices u = u0 ∼ u1 ∼ u2 ∼ · · · ∼ un = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' By a tree T we mean a locally finite, connected and simply-connected graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=', for each pair of vertices there is one and only one path between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' We shall identify T with the collection of its vertices V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In the present paper, the tree we consider has a distinguished vertex, which we call the root of T and we denote it by o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The length of a path joining the root o and a vertex v is called the length of v and is denoted by |v|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Given a tree T rooted at o and a vertex v ∈ T, a vertex w is called a descendant of v if v lies in the unique path from o to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In this case, the vertex v is called an ancestor of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' We denote by v− the neighbor of a vertex v which is an ancestor of v, and the vertex v is called a child of v−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' For each w ∈ T, we denote the set of all children of w by Chi(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Given a vertex v, the sector determined by v is the set Sv consisting of v and all its descendants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In particular, So = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let 1 ⩽ p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The function space Lp(T) of T is defined as the set of functions f : T → C such that � v∈T |f(v)|p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' It is easy to show that every Lp(T) is a Banach space when endowed with the norm ∥f∥p := � � v∈T |f(v)|p � 1 p , since it is isomorphic to the p-summable sequence space ℓp(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Moreover, the dual space of Lp(T) with 1 < p < ∞ can be identified with Lq(T) under the sesquilinear dual pairing ⟨f, g⟩ := � v∈T f(v)g(v), f ∈ Lp(T), g ∈ Lq(T), Date: January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 47A30, 47B37, 05C05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Toeplitz operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' positivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' finite rank operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 1 2 MINGMEI HUANG, XIAOYAN ZHANG, AND XIANFENG ZHAO where q = p p−1 is the conjugate exponent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In particular, L2(T) is a Hilbert space with the obvious inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' When p = ∞, the space L∞(T) is the collection of bounded functions f on T equipped with the supremum norm ∥f∥∞ = sup � |f(v)| : v ∈ T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In addition, we say that q = ∞ is the conjugate exponent of p = 1 and that q = 1 is the conjugate exponent of p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The interest of the study of operators on infinite trees is motivated mainly by the research in harmonic analysis dealing with the spectrum of the Laplace operator on discrete structures, one can consult [9] and [10] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Linear operators on discrete structures other than Laplacian have been studied by many authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' For instance: Toeplitz operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Pavone studied the properties of Toeplitz operators on a discrete group and discussed the extent to which they paraller the properties of Topelitz operators on the Hardy space of the unit circle, see [21, 22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In [14], Colonna and Mart´ınez-Avenda˜no defined the Toeplitz operator on Lp-spaces of a tree, where 1 ⩽ p ⩽ ∞, and then characterized the boundedness and the eigenvalues of such Toeplitz operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Motivated by the paper [14], Zhang and Zhao [23] established several sufficient conditions for Toeplitz operators to be compact on Lp-spaces of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Composition operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Pavone [20] characterized the class of hypercyclic composition operators on the Lp-space of the Poisson boundary of a homogeneous tree, and showed that such operators are actually chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Allen, Colonna and Easley [5] investigated the boundedness, compactness and spectra of the composition operators on the Lipschitz space of a tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Later, Allen and Pons ([7, 8]) obtained the characterizations for the (weighted) composition operators with closed range, and to be bounded, compact, bounded below, invertible and Fredholm on weighted Banach spaces of a tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In addition, the bounded, compact, invertible, isometric composition operators on Hardy spaces of the homogenous rooted trees were investigated in the recent paper [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Shift operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The operator-norm estimate, spectral properties, hyponormality, subnormality and hypercyclicity of (weighted) shift operators on certain function spaces of a tree were well-studied in [16, 17, 18];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Moreover, the norm estimate, kernels and eigenvalues of the shift operator (usually called the adjacency operator) on the Lp-space of a graph were systematically investigated in the paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Multiplication operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The boundedness, compactness, invertibility, hypercyclicity and essential norm of multiplication operators on Lipschitz-type spaces of a tree were discussed in a series of papers of Allen, Colonna, Easley and Prudhom [2, 3, 4, 6, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' However, little is known about the spectral properties, self-adjointness and positivity of classical oper- ators on discrete structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The purpose of this paper is to study some fundamental properties about Toeplitz operators on Lp-spaces of a tree, such as the spectrum, self-adjointness, positivity and finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Before stating the definition of the Toeplitz operator on Lp-spaces of a tree, we need to introduce the fol- lowing two important linear operators on such function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' For any function f : T → C, the operator ∇ is defined by (∇f)(u) := f(u) − � v−=u f(v), u ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Recall that the operator ∆ is the transformation with domain L1(T) defined by (∆f)(u) := � v∈Su f(v), f ∈ L1(T), u ∈ T, where Su is the sector determined by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' For more details concerning the above two operators, one can consult [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let 1 ⩽ p ⩽ ∞ and ϕ be a complex-valued function on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The Toeplitz operator on Lp(T) with symbol ϕ is defined by Tϕf := ∆(ϕ∇f), f ∈ Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Some useful properties and important conclusions about ∇, ∆ and Toeplitz operators on Lp(T) will be included in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' TOEPLITZ OPERATORS 3 The main part of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In Section 3, we will study the spectra of some classes of bounded Topelitz operators on the Banach spaces Lp(T) with 1 ⩽ p ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' More precisely, we will show in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2 that the spectrum of the Topelitz operator is equal to the closure of the range of its symbol under certain mild assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This answers an open question posed by Colonna and Mart´ınez- Avenda˜no [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In Section 4, we investigate the self-adjointness and positivity of Topelitz operators on the Hilbert space L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Surprisingly, we find that there is no nontrivial self-adjoint Topelitz operator on L2(T), see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1 for the detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Section 5 is devoting to the characterization for finite rank Toeplitz operators on Lp(T) with 1 ⩽ p ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In fact, we will prove that the Toeplitz operator on Lp(T) is of finite rank if and only if the support of its symbol is a finite set, see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1 for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Preliminary In this section, we recall some important definitions and known results which will be required in the next three sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' We shall make use of the following derivative of a function on a tree repeatedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Given a complex-valued function f on a tree T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' We define the function f− on T by f−(v) = \uf8f1 \uf8f2 \uf8f3 f(v−), if v ̸= o, 0, if v = o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In addition, the function f ′ on T is defined by f ′(v) = f(v) − f−(v), v ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Recall that the operators ∇ and ∆ were introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The relationship between these two operators on L1(T) is given by the following lemma, see [14, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='7] if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' If f ∈ L1(T), then ∆(∇f) = f and ∇(∆f) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The following expression for the Toeplitz operator on Lp(T) is quite useful in the study of the bound- edness, compactness, positivity and spectra of Toeplitz operators, which was proved in [14, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let 1 < p ⩽ ∞ and let q be the conjugate exponent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' If ϕ and ϕ′ are both in Lq(T) and f ∈ Lp(T), then we have that ϕ∇f ∈ L1(T) and Tϕf = fϕ− + ∆(fϕ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The next two lemmas were established in [14, Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='6], which give nice sufficient conditions for the boundedness of Toeplitz operators on Lp(T) with 1 ⩽ p ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Assume that 1 ⩽ p < ∞ and q is the conjugate exponent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' If the functions ϕ and �ϕ are both in Lq(T), then the Toeplitz operator Tϕ is bounded on Lp(T), where �ϕ is defined by �ϕ(v) := (|v| + 1)ϕ′(v), v ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' If ϕ and ϕ′ are both in L1(T), then the Toeplitz operator Tϕ is bounded on L∞(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Recall that [14, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='10] obtained a characterization on the point spectrum (the set of eigenvalues) of the Toeplitz operator on Lp(T) in terms of the range of its symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' For the sake of convenience, we quote this theorem in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let 1 ⩽ p ⩽ ∞ and let q be the conjugate exponent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Suppose that ϕ, ϕ′ ∈ Lq(T) and the Toeplitz operator Tϕ is bounded on Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Then: (a) σp(Tϕ) = {ϕ(w) : w ∈ T} if there is no nonzero function g ∈ Lp(T) with ∇g = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' (b) σp(Tϕ) = {0} ∪ {ϕ(w) : w ∈ T} if there is a nonzero function g ∈ Lp(T) with ∇g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 4 MINGMEI HUANG, XIAOYAN ZHANG, AND XIANFENG ZHAO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The spectra of Toeplitz operators The study of the spectral properties of operators on infinite trees is relatively new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In this section, we investigate the spectra of Toeplitz operators on Lp(T) with 1 ⩽ p ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let q be the conjugate exponent of p and ϕ be a function in Lq(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Based on the characterization for eigenvalues of Toeplitz operators (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='6), we show that the spectrum of the Toeplitz operator Tϕ : Lp(T) → Lp(T) with 1 ⩽ p < ∞ equals the closure of ϕ(T) if �ϕ is also in Lq(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Recall that the function �ϕ was introduced in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' On the other hand, we also prove that the spectrum of the Toeplitz operator Tϕ : L∞(T) → L∞(T) is coincided with the closure of ϕ(T) when ϕ′ ∈ L1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let us consider the case that 1 ⩽ p < ∞ first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' As the spectrum of the Topelitz operator on L1(T) was obtained in [14] via the multiplication operator with the same symbol, it is sufficient for us to discuss the case of 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let 1 ⩽ p < ∞ and q be the conjugate exponent of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Suppose that ϕ and �ϕ are both in Lq(T), where �ϕ(v) = (|v| + 1)ϕ′(v), v ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Then Tϕ is bounded on Lp(T) and in this case we have σ(Tϕ) = clos{ϕ(v) : v ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' As we mentioned above, the conclusion for the case of p = 1 was obtained in Section 7 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' So we need only to consider the case for p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='4, we see that the Toeplitz operator Tϕ is bounded on Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Clearly, �ϕ ∈ Lq(T) implies that ϕ′ ∈ Lq(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Then using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='6, we have clos{ϕ(w) : w ∈ T} = clos[σp(Tϕ)] ⊂ σ(Tϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' To prove the reverse inclusion, it is sufficient to show that Tϕ − λI is invertible on Lp(T) if λ is not in the closure of the range of ϕ, where I is the identity operator on Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Observe that λ /∈ clos{ϕ(v) : v ∈ T} implies that λ is not an eigenvalue of Tϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This means that Tϕ − λI is an injective from Lp(T) to Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' According to the Banach inverse operator theorem, we need only to show that Tϕ − λI is a surjective on Lp(T) if λ /∈ clos{ϕ(v) : v ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Suppose that inf v∈T |ϕ(v) − λ| ⩾ δ for some positive constant δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let δ0 = min{δ, |λ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Then we have that δ0 > 0 and |ϕ(v) − λ| ⩾ δ0 and |ϕ−(v) − λ| ⩾ δ0 for all v ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Letting g be a function in Lp(T), we define f := 1 λ � ∆ � ϕ ϕ − λ∇g � − g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1) In the following, we will show that f ∈ Lp(T) and f is the preimage of g under Tϕ − λI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' To show f ∈ Lp(T), we need only to prove that ∆ � ϕ ϕ − λ∇g � ∈ Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let us show � � ϕ ϕ−λ � ∈ Lq(T) first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Noting that � � ϕ ϕ − λ � (v) = (|v| + 1) � ϕ ϕ − λ �′ (v) = (|v| + 1) � ϕ(v) ϕ(v) − λ − ϕ−(v) ϕ−(v) − λ � TOEPLITZ OPERATORS 5 for v ∈ T, we have ���� � � ϕ ϕ − λ ����� q = � � v∈T ���(|v| + 1) � ϕ(v) ϕ(v) − λ − ϕ−(v) ϕ−(v) − λ ���� q� 1 q = � � v∈T ���(|v| + 1) � ϕ(v) ϕ(v) − λ − ϕ−(v) ϕ(v) − λ + ϕ−(v) ϕ(v) − λ − ϕ−(v) ϕ−(v) − λ ���� q� 1 q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='Then triangle inequality gives that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='ϕ − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='⩽ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='v∈T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='���(|v| + 1)ϕ(v) − ϕ−(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='ϕ(v) − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='v∈T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='���(|v| + 1) ϕ−(v)[ϕ(v) − ϕ−(v)] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='[ϕ−(v) − λ][ϕ(v) − λ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='v∈T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='���(|v| + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='ϕ′(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='ϕ(v) − λ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='v∈T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='���(|v| + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='ϕ′(v)ϕ−(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='[ϕ−(v) − λ][ϕ(v) − λ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='⩽ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='δ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='v∈T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��(|v| + 1)ϕ′(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��q� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='v∈T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��(|v| + 1)ϕ−(v)ϕ′(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��q� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='⩽ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='δ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='v∈T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��(|v| + 1)ϕ′(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��q� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q + ∥ϕ∥∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='v∈T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��(|v| + 1)ϕ′(v) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='��q� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='δ0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='+ ∥ϕ∥∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='∥�ϕ∥q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This shows that � � ϕ ϕ−λ � ∈ Lq(T), since �ϕ ∈ Lq(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Next, we are going to show that ∆ � ϕ ϕ−λ∇g � ∈ Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Noting that ϕ ϕ−λ and � ϕ ϕ−λ �′ are both in Lq(T), we obtain by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='3 that ∆ � ϕ ϕ − λ∇g � = T ϕ ϕ−λ g = � ϕ ϕ − λ � −g + ∆ �� ϕ ϕ − λ �′ g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Furthermore, we have ���∆ �� ϕ ϕ − λ �′ g ���� p ⩽ ���∆ �� ϕ ϕ − λ �′ g ���� 1 = � v∈T ����∆ �� ϕ ϕ − λ �′ g � (v) ���� ⩽ � v∈T � u∈Sv ���� � ϕ ϕ − λ �′ (u)g(u) ���� = � v∈T (|v| + 1) ���� � ϕ ϕ − λ �′ (v)g(v) ���� = � v∈T ���� � � ϕ ϕ − λ � (v) ���� |g(v)| ⩽ ∥g∥p ���� � � ϕ ϕ − λ ����� q , to get that ∆ �� ϕ ϕ−λ �′g � ∈ Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Since ϕ ϕ−λ ∈ Lq(T) implies that ϕ ϕ−λ ∈ L∞(T), we obtain that � ϕ ϕ−λ � − is also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' It follows that � ϕ ϕ−λ � −g is in Lp(T), which gives that ∆ � ϕ ϕ−λ∇g � ∈ Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Therefore, we obtain that the function defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1) is in Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' To finish the proof, it remains to verify that (Tϕ − λI)f = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 6 MINGMEI HUANG, XIAOYAN ZHANG, AND XIANFENG ZHAO Indeed, we have by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='3 that ϕ ϕ − λ∇g ∈ L1(T), since g ∈ Lp(T) and ϕ ϕ−λ, � ϕ ϕ−λ �′ ∈ Lq(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2 that ∇f = ∇ � 1 λ � ∆ � ϕ ϕ − λ∇g � − g �� = 1 λ ϕ ϕ − λ∇g − ∇g λ = ∇g ϕ − λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This yields that (Tϕ − λI)f = Tϕf − λf = ∆(ϕ∇f) − λf = ∆ � ϕ ϕ − λ∇g � − ∆ � ϕ ϕ − λ∇g � + g = g, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This completes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' □ Now we give a description for the spectrum of Tϕ : L∞(T) → L∞(T) with ϕ, ϕ′ ∈ L1(T) in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' If ϕ and ϕ′ are both in L1(T), then Tϕ is bounded on L∞(T) and in this case we have σ(Tϕ) = clos{ϕ(v) : v ∈ T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='5 tells us that Tϕ : L∞(T) → L∞(T) is bounded if ϕ ∈ L1(T) and ϕ′ ∈ L1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' According to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1, we need only to show that � ϕ ϕ − λ � −g and ∆ �� ϕ ϕ − λ �′ g � are both in L∞(T) if λ /∈ clos{ϕ(v) : v ∈ T} and g ∈ L1(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In fact, ���� � ϕ ϕ − λ � −g ���� ∞ = sup v∈T ���� � ϕ ϕ − λ � −(v)g(v) ���� ⩽ ∥g∥∞ δ0 ∥ϕ∥∞, where δ0 is the constant defined in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Moreover, since ����∆ �� ϕ ϕ − λ �′ g ����� ∞ = sup v∈T ���� � u∈Sv g(u) � ϕ ϕ − λ �′ (u) ���� ⩽ sup v∈T � u∈Sv |g(u)| ���� ϕ(u) ϕ(u) − λ − ϕ−(u) ϕ−(u) − λ ���� ⩽ |λ| ∥g∥∞ δ2 0 sup v∈T � � u∈Sv |ϕ(u) − ϕ−(u)| � = |λ| ∥g∥∞ δ2 0 ∥ϕ′∥1, we obtain that � ϕ ϕ−λ � −g ∈ L∞(T) and ∆ �� ϕ ϕ−λ �′g � ∈ L∞(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The rest of the proof of this theorem is similar to the previous one, so we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This finishes the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The conclusions in Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2 are comparable to the corresponding results for analytic Toeplitz operators on the Hardy and Bergman spaces of the open unit disk, see the books [15] and [24] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' TOEPLITZ OPERATORS 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The positivity of Toeplitz operators We say that a bounded linear operator T is self-adjoint (positive) on a complex Hilbert space H if ⟨Tx, x⟩H is real (nonnegative) for every x in H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In this section, we study when a bounded Toeplitz operator is self-adjoint (positive) on the Hilbert space L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Observing that the adjoint of the Topelitz operator Tϕ does not equal Tϕ in general, which leads to certain difficulties in the study of the self-adjointness and positivity of Toeplitz operators on L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' However, we are able to obtain a complete characterization for the self-adjoint Toeplitz operators on L2(T) by using their point spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Suppose that ϕ is a function in L2(T) such that �ϕ ∈ L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Then the following three conditions are equivalent: (1) Tϕ is positive on L2(T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' (2) Tϕ is self-adjoint on L2(T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' (3) ϕ(v) = 0 for all v ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Clearly, (1) ⇒ (2) is trivial and (3) ⇒ (1) follows immediately from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' To complete the proof of this theorem, it remains to show that (2) ⇒ (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Now we suppose that Tϕ is self-adjoint on L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Then we have by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='6 that ϕ must be real- valued, since {ϕ(v) : v ∈ T} = σp(Tϕ) ⊂ σ(Tϕ) ⊂ R or {0} ∪ {ϕ(v) : v ∈ T} = σp(Tϕ) ⊂ σ(Tϕ) ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let f be any function in L2(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Since �ϕ ∈ L2(T), we see that the series � v∈T [Tϕf(v)]f(v) is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Now using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='3 again gives ⟨Tϕf, f⟩ = � v∈T [Tϕf(v)]f(v) = � v∈T � f(v)ϕ−(v)f(v) + ∆(fϕ′)(v)f(v) � = � v∈T |f(v)|2ϕ−(v) + � v∈T f(v)∆(fϕ′)(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1) Furthermore, we have by the definitions of ϕ′ and the operator ∆ that � v∈T f(v)∆(fϕ′)(v) = � v∈T � f(v) � u∈Sv (fϕ′)(u) � = � v∈T � f(v) � u∈Sv f(u) � ϕ(u) − ϕ−(u) �� = � v∈T |f(v)|2ϕ(v) − � v∈T |f(v)|2ϕ−(v) + � v∈T � f(v) � u∈Sv\\{v} � f(u)ϕ(u) − f(u)ϕ−(u) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2) We first show that ϕ is a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Otherwise, we can choose two vertices ξ and η in T such that ϕ(ξ) ̸= ϕ(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Since the path joining ξ and η is a finite sequence of distinct vertices, we may assume that ξ ∼ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' In other words, we obtain that ξ ∈ Chi(η) or η ∈ Chi(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Without loss of generality, we may assume that η ∈ Chi(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let us consider the function g : T → C defined by g(v) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1, if v = ξ, i, if v = η, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 8 MINGMEI HUANG, XIAOYAN ZHANG, AND XIANFENG ZHAO It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2) that ⟨Tϕg, g⟩ = |g(ξ)|2ϕ(ξ) + |g(η)|2ϕ(η) + g(ξ) � g(η)ϕ(η) − g(η)ϕ(η−) � = |g(ξ)|2ϕ(ξ) + |g(η)|2ϕ(η) + g(ξ)g(η) � ϕ(η) − ϕ(η−) � = � ϕ(ξ) + ϕ(η) � + � ϕ(η) − ϕ(ξ) � i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This means that ⟨Tϕg, g⟩ /∈ R, which contradicts that Tϕ is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Thus ϕ is a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' However, the condition ϕ ∈ L2(T) yields that ϕ must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Finite rank Toeplitz operators The final section is devoted to establishing a equivalent characterization for finite rank Toeplitz operators on Lp(T), where 1 ⩽ p ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' More concretely, we will show in the following theorem that the Toeplitz operator Tϕ having finite rank if and only if {v ∈ T : ϕ(v) ̸= 0} is a finite subset of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let 1 ⩽ p ⩽ ∞ and ϕ be a function in Lq(T) such that ϕ′ ∈ Lq(T) and the Toeplitz operator Tϕ is bounded on Lp(T), where 1 p + 1 q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Then Tϕ has finite rank on Lp(T) if and only if ϕ(v) = 0 except for finitely many points v ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' We show the sufficiency first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Suppose that ϕ(v) = 0 for all v ∈ T except v1, v2, · · · , vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' For any f ∈ Lp(T), we have Tϕf(v) = ∆(ϕ∇f)(v) = � u∈Sv ϕ(u)∇f(u), v ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' It follows that Tϕf(v) = 0 for all v ∈ T with |v| > max{|vk| : k = 1, 2, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Let W := � w ∈ T : |w| ⩽ max 1⩽k⩽n |vk| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Then W is a finite set and its elements can be enumerated by W = {w1, w2, · · · , wN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Fix a vertex ξ ∈ T and define eξ(v) = \uf8f1 \uf8f2 \uf8f3 1, if v = ξ, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1) Since Tϕf is a vector in Lp(T), it can be expressed as Tϕf = c1ew1 + c2ew2 + · · · + cNewN with constants c1, c2, · · · , cN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This implies that range(Tϕ) ⊂ span � ew1, ew2, · · · , ewN � , to obtain that Tϕ is of finite rank on Lp(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' To show the necessity, we suppose that Tϕ is a finite rank operator on Lp(T) and range(Tϕ) ⊂ span � ev1, ev2, · · · , evn � , TOEPLITZ OPERATORS 9 where vk is a vertex of T and evk is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' According to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='3 and the computations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2), we obtain that Tϕew(v) = ew(v)ϕ(v−) + � u∈Sv ew(u)[ϕ(u) − ϕ(u−)] = ew(v)ϕ(v−) + ew(v)[ϕ(v) − ϕ(v−)] + � u∈Sv\\{v} ew(u)[ϕ(u) − ϕ(u−)] = ew(v)ϕ(v) + � u∈Sv\\{v} ew(u)[ϕ(u) − ϕ(u−)] (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2) for every v ̸= o and w ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' If ϕ does not vanish at infinitely many vertices of T, then we can choose a vertex w ∈ T such that |w| > max � |v1|, |v2|, · · · , |vn| � and ϕ(w) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' It follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2) that Tϕew(w) = ϕ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' We claim that Tϕew /∈ span � ev1, ev2, · · · , evn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Indeed, if Tϕew = c1ev1 + c2ev2 + · · · + cnevn for some scalars c1, c2, · · · , cn, then we would have 0 = c1ev1(w) + c2ev2(w) + · · · + cnevn(w) = Tϕew(w) = ϕ(w) ̸= 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This yields that dim � range(Tϕ) � > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' But this contradicts that the dimension of range(Tϕ) is less than or equal to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Therefore, the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1 is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' It is worth noting that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='1 is analogous to the corresponding results for finite rank Toeplitz operators on the Bergman and Fock spaces, see [13] and [25, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='42] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' This work was partially supported by NSFC (grant number: 11701052) and Chongqing Natural Science Foundation (cstc2019jcyj-msxmX0337).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' The third author was 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=': Analysis on Fock Spaces, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' Springer-Verlag, New York (2012) 1 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' China Email address: huangmm0909@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='com 2 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' China Email address: xiaoyanzhang@cqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='cn 3 College of Mathematics and Statistics, Chongqing University, Chongqing, 401331, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content=' China Email address: xianfengzhao@cqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/nNE_T4oBgHgl3EQf7hx2/content/2301.08370v1.pdf'} diff --git a/o9E4T4oBgHgl3EQfvA3U/content/tmp_files/2301.05238v1.pdf.txt b/o9E4T4oBgHgl3EQfvA3U/content/tmp_files/2301.05238v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..02dd485671da059107c5fa7b69553bc0f6d2ce20 --- /dev/null +++ b/o9E4T4oBgHgl3EQfvA3U/content/tmp_files/2301.05238v1.pdf.txt @@ -0,0 +1,2749 @@ +Quantum criticality under decoherence or weak measurement +Jong Yeon Lee,1 Chao-Ming Jian,2 and Cenke Xu3 +1Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 +2Department of Physics, Cornell University, Ithaca, New York 14853 +3Department of Physics, University of California, Santa Barbara, CA 93106 +Decoherence inevitably happens when a quantum state is exposed to its environment, which can +affect quantum critical points (QCP) in a nontrivial way. As was pointed out in recent literature on +(1 + 1)d conformal field theory (CFT) [1], the effect of weak measurement can be mathematically +mapped to the problem of boundary CFT. In this work, we focus on the (2 + 1)d QCPs, whose +boundary and defect effects have attracted enormous theoretical and numerical interests very re- +cently. We focus on decoherence caused by weak measurements with and without post-selecting the +measurement outcomes. Our main results are: (1) for an O(N) Wilson-Fisher QCP under weak +measurement with post-selection, an observer would in general observe two different types of bound- +ary/defect criticality with very different behaviors from the well-known Wilson-Fisher fixed points; +in particular, it is possible to observe the recently proposed exotic “extraordinary-log” correlation. +(2) An extra quantum phase transition can be driven by decoherence, if we consider quantities non- +linear with the decohered density matrix, such as the Renyi entropy. We demonstrate the connection +between this transition to the information-theoretic transition driven by an error in the toric code +model. (3) When there is no post-selection, though correlation functions between local operators +remain the same as the undecohered pure state, nonlocal operators such as the “disorder operator” +would have qualitatively distinct behaviors; and we also show that the decoherence can lead to +confinement. +I. +INTRODUCTION +When a quantum state is exposed to an environment, +it is being constantly probed and “measured”, forming +entanglement with the degrees of freedom in the envi- +ronment. If one is ignorant of the environment and the +measurement outcome is lost, it amounts to tracing out +the environment’s degrees of freedom and the original +pure quantum state becomes a mixed state, which re- +sults in the loss of coherent quantum information. This +process is referred to as quantum decoherence, and it is +the bridge between the quantum mechanics that governs +the microscopic nature, and our classical macroscopic +world [2]. More generally, if a quantum state is weakly +measured and the measurement outcome is still accessi- +ble, one can consider the effect of post-selecting the mea- +surement outcome and study process that is more gen- +eral than decoherence. In recent years there has been a +surge of progress in simulating quantum states of matter +with nontrivial entanglement on platforms summarized +as the Noisy Intermediate-Scale Quantum (NISQ) tech- +nology [3], including simulating exotic quantum many- +body states such as topological order, spin liquids, and +symmetry protected topological states [4–8], which have +long been discussed in condensed matter physics. +In +these platforms, decoherence can happen due to various +reasons, which motivated recent inspection of the fate of +SPT states under decoherence [9–12] (related studies mo- +tivated from other contexts were also conducted [13, 14]). +Quantum criticality represents another class of quan- +tum many-body states with peculiar and universal en- +tanglement. Recently a class of (1 + 1)d conformal field +theory (CFT), i.e., the Luttinger liquid under weak mea- +surements has been studied, and it was pointed out that +the effect of weak measurements can be mathematically +mapped to the problem of the boundary of the CFT [1], +which is a subject that was studied extensively in the +past [15]. The same trick was used in the recent study +of SPT states under decoherence [12], which exploited +the observation that the wave function of the SPT states +can be mapped to the partition function of the boundary +states of the system after a space-time rotation [16, 17]. +In this work, we focus on quantum critical points in +(2 + 1)d. The connection between the decohered QCPs +(or QCPs under weak measurement) in the bulk and the +boundary criticality still holds, but the boundary criti- +cality of (2 + 1)d QCPs is a subject that has only been +carefully studied very recently, and it has attracted enor- +mous interests from both the theoretical and numerical +communities [18–30]. It has been understood since long +back that there exists an ordinary boundary condition of +a (2 + 1)d Wilson-Fisher QCP (or a 3d classical Wilson- +Fisher critical point), where the Landau order param- +eter φ has a scaling dimension ∆b +φ > 1, which is far +greater than the bulk scaling dimension of the order pa- +rameter (which is slightly greater than 1/2). Only re- +cently it has become clear that at the boundary of an +O(N) Wilson-Fisher critical point, in addition to the +well-known ordinary boundary criticality, there is a so- +called “extraordinary-log” boundary criticality, meaning +the correlation function of the order parameter at the +boundary reads +⟨φ(0)φ(x)⟩ ∼ +1 +(ln |x|)q , +(1) +where q depends on N. This peculiar scaling was pro- +posed theoretically [25, 26] and recently confirmed nu- +merically in Monte Carlo simulations [28, 29]. +arXiv:2301.05238v1 [cond-mat.stat-mech] 12 Jan 2023 + +2 +Our current work will bridge these two directions that +are under active studying, and we will demonstrate that +a (2+1)d QCP under decoherence or weak measurements +naturally exhibits the peculiar boundary criticality stud- +ied recently. This work is organized as follows: +In section II, we develop the general formalism of ana- +lyzing decohered quantum critical states, including the +general connection between the decohered or weakly- +measured bulk state and the boundary/defect criticality. +In section III we discuss the (2 + 1)d O(N) Wilson- +Fisher quantum critical points under decoherence or +weak measurements, and in section III A we focus on the +quantities linear with the density matrix, which can be +observed directly in experiments. We demonstrate that +weak measurements generally render the observed quan- +tities rather different from the bulk QCP (with certain +post-selection that preserves the O(N) symmetry in the +mixed state ensemble), and we may observe the exotic +extraordinary-log correlation mentioned above. +In section III B we discuss quantities nonlinear with +the density matrix, which reveals a lot more structures of +the mixed state density matrix of the critical state under +decoherence. In particular, we discuss a quantum infor- +mation phase transition that can be diagnosed through +the 2nd Renyi entropy of the decohered system, along +with other correlations defined in this section. +Here we would like to point out an important differ- +ence between the physical scenarios to be considered in +section III A and III B. In section III A we discuss physics +under weak measurements on quantities such as energy +density, and we allow a post-selection on the measure- +ment outcomes. Section III B considers quantities non- +linear with the density matrix where post-selection is not +needed in this scenario; hence it can genuinely correspond +to physics under decoherence due to coupling to the en- +vironment. +In section III D we demonstrate that there is an ex- +plicit lattice model with an information transition driven +by the strength of decoherence (or weak measurement) +analogous to the decoherence-driven transition in sec- +tion III B. We also show that the transition in this lattice +model is dual to an information transition in the toric +code model, which is related to the error threshold of the +toric code [31]. +If we forbid post-selection, local correlation functions +of the (locally) decohered density matrix would remain +largely unchanged from the undecohered pure state. In +recent years the nonlocal disorder operator has become +a very important diagnosis of quantum states of matter. +In section IV we demonstrate that the nonlocal disorder +operator can still have qualitatively different behavior +from the undecohered pure state density matrix, even if +there is no post-selection. In particular, we show that in +several examples, decoherence can lead to confinement. +II. +GENERAL FORMALISM +In order to discuss quantum states under decoherence, +one approach is to first explicitly derive the ground state +wave function |Ψ⟩ in either exactly soluble lattice mod- +els or effective field theories, then construct the density +matrix ˆρD under decoherence [12]. +This approach re- +lies on an explicit derivation of the ground state wave +function. +The ground state wave function can be de- +rived for gapped states such as SPT phases [9, 16, 17], +and also gapless phases with a Gaussian (free boson) La- +grangian, such as the (1 + 1)d Luttinger liquid [1], or +the Rokhsar-Kivelson (RK) point in (2+1)d models [32]. +But for general interacting theories, such as systems near +the Wilson-Fisher quantum critical points, deriving the +ground state wave function is cumbersome. In this sec- +tion, we follow a more general procedure to study inter- +acting systems under decoherence. +Let us prepare an interacting quantum state which is +the ground state of a Hamiltonian, whose pure state den- +sity matrix is given as ˆρ0. After the quantum state is +prepared, we turn off the Hamiltonian and expose the +system to local decoherence. For example, for a lattice +model of qubits, a decohered density matrix may be rep- +resented as +ˆρD = E[ˆρ0], +E = +� +x +Ex, +Ex[ˆρ0] = (1 − p)ˆρ0 + pZxˆρ0Zx. +(2) +where ˆρD describes a mixed state (ensemble) and E is +given as the composition of local decoherence channels +Ex. +One interpretation of this decoherence channel is +that, there is a certain probability for the environment +to measure qubits in the Z-basis (weak measurements) +at each location x, and the measurement outcomes are +“lost”, which amounts to the dephasing noise in the study +of quantum circuits. A generic decoherence channel maps +a density matrix ˆρ0 into ˆρD = � +m Kmˆρ0K† +m, where +{Km} is a set of Kraus operators satisfying the condition +� +m K† +mKm = 11. If we interpret a decoherence channel +to be induced by weak measurements, we can consider a +post-selection followed by the channel where the resulting +density matrix is given as +ˆρD +P ≡ +P[ˆρD] +tr{P[ˆρD]}, +P[ˆρD] ≡ +� +m∈P +Kmˆρ0K† +m +(3) +where P is a generalized projection onto a subset of mea- +surement outcomes P, i.e., post-selection. +In this work, we will mostly study systems at or close +to a QCP, hence we will use a coarse-grained continuous +space rather than a lattice model. In the coarse-grained +formalism, the density matrix of a pure state is given by +the following imaginary time path-integral +[ˆρ0]φ1(x),φ2(x) = ⟨φ1(x)|Ψ⟩⟨Ψ|φ2(x)⟩ +∼ lim +β→∞ +� +φ(x,0)=φ1(x) +φ(x,β)=φ2(x) +Dφ(x, τ) exp(−S), +(4) + +3 +where S = +� β +0 dτdx L(φ) is the bulk action of the system +and x = (x1, ..., xd) is the spatial coordinate. +Follow- +ing Ref. 1 and 12, a class of decoherence problem in the +coarse-grained continuous space can be converted into +the following imaginary-time path-integral: +[ˆρD]φ1(x),φ2(x) ∼ lim +β→∞ +� +φ(x,0)=φ1(x) +φ(x,β)=φ2(x) +Dφ(x, τ) exp +� +−S − Sint� +; +S = +� β +0 +dτdx L(φ), +Sint = +� +dx Lint(φ(x, 0), φ(x, β)). +(5) +The effect of decoherence is captured by an extra inter- +action term Lint in the Lagrangian, and Lint has no tem- +poral integral, i.e., it is a “long range” interaction in the +Euclidean temporal direction, between fields at τ = 0 and +τ = β. The coupling constant in Lint is controlled by the +strength of decoherence (or strength of weak measure- +ment) p. A natural choice of Lint favors configurations +with φ(x, 0) ∼ φ(x, β), meaning it enhances the weight +of the diagonal components of the density matrix, which +drives the system into a mixed state density matrix. +The most important factor that determines the form +of Lint is still its symmetry. Let us label the symmetry +of the original Lagrangian L as G, and assume that the +original pure state without decoherence is a symmetric +state under G. Then there are two types of symmetry +constraints on Lint. If the environment is weakly “mea- +suring” quantities that are invariant under G, then Lint +must be invariant under a “doubled” symmetry trans- +formation, i.e., it is invariant under symmetry transfor- +mation G on φ(x, 0) and φ(x, β) separately. In the lan- +guage of Ref. 12, this is the case that preserves the dou- +bled Gu × Gl symmetry, where Gu and Gl correspond +to the upper and lower symmetry in the formalism of +doubled Hilbert space using the Choi-Jamiolkowski iso- +morphism [33, 34], which maps any density matrix to a +pure ket-state in the doubled Hilbert space. However, if +the environment is measuring quantities that carry a non- +trivial representation of G, but eventually we sum over +the measurement outcomes within the same representa- +tion of G with equal weight, meaning the symmetry G is +broken in each quantum trajectory but still preserved in +an average sense, then Lint is only invariant under sym- +metry G, which is the diagonal subgroup of Gu × Gl, +and it corresponds to a simultaneous transformation of +φ(x, 0) and φ(x, β). +In Eq. (2), if the original pure state is invariant un- +der a Z2 symmetry action � +x Xx then the pure state +density matrix ˆρ0 is invariant under a doubled Zu +2 × Zl +2 +symmetry, where Zu +2 and Zl +2 correspond to the left and +right operations of the Z2 symmetry. 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+AhJm9GOD7VLNa3gyCc6Esq0xI161Y3x6bglMFBryOjt2096a7vstyvltev+HKvyJk9uiSyjbCd54dtmChsrY4Ta53YrU6sdXI34Sr179iJ8MAlNnKvrH7jfHsQPR3s/P7T5u6z5jr7KPg2+C74PoiCn4Pd4CA4Cc4CHOjgr+Dv4J/+bj/r8365kj580Ph8HXSe/vw/T +923g= ÿ +{mi}œP +AL3XicjZbPb9s2FMfV7ldn70 +e6HXcRFhTYITOsrOmGnZoGaxMUQbK4SYtarkFRlExYlBiKSu0SPO42DNhpwP6aXbf/YP/NHm06schkiJDA5Pt8Hyk9Pj4y4QWtZb/75273/wYcf3fu40/3k08+37j/xVldNQKTU1wVlXiVoJoUtCSnks +qCvOKCIJYU5GUy3TP85QURNa3KF3LOyYihvKQZxUiCabzxMK4bNlaxCuNJzREm6t+b4dLHbIxvbJFxhLDHy3DYz3e2Oz3+osn9BuRbWwG9jke3+/8HqcVbhgpJS5QXQ+jPpcjhYSkuC6Ezc1gZmKCdDaJ +aIkXqkFp+nwdgScOsEvBfynBhXfdQfDKvKa5bw6gCSTKr56xtRawGW+JIWZUSUepOp2XOBeITime64w7BkJx45qXc96h4bkgZNo2Js4rsaQVFRvdgyk7JhVBJHvAxJyzSBZRWCZG0rLHoCa+y41xD8CU +kdY9U7W8cNjL7YaRoyRtJSrxcgqwpQlmFJovClAqCZTGHBsKCwiqGeIEwhJyDYqyVtcMYbKVMUk0SpmSTVT8ZY2H7kOkwMDIZxJog60bkPMkJhqBQ3z1SopGgIpmNIyVzuR1vpBWz5rywV850q946tP7c 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+xo6xDnK90u56RM6exmMTkvV+Kfzo2MnF8vzGi+Ej6l+c0DSpSsdC9Q8j+6CkPRXOiGthAMRZ6MVL8XfbcFR9vOlmnphevIcRaQDE3hxIq3FI0QWVO7E5ZlSMoltp/hYtFGl5w8JAVJEGK8pxcs3ugLCZvbiHEWCt8G2GWaZXdRjiBESGCNi/2e7jnKspKsArCsfjeH3XMpnA+Rqz +ZjGyz94g1ANzo8xpyP5a0SAlEjquQKLmSmA6juDoYrELjvR49XpnTwae6B3KWFv1bvfpoSvbJyYi+5ca03UlYp6uS06g64b/GaCH1TAzceDh5fw0Id7l3DPg68yOCwrsVRUdRjLCZF+/Xy9Jqvh8nWDrG4HCp6udfVoOG8IOayhcQ8PISLkDAZ7fhQs1T2gmfud6tjiep1q5t +jE3DK4KBXkVbp6013dR7pfL61d8sVvmhTm6BDKNq+vl9uJ6a2OU7FyKq51KlZO7iZcpv4NOxEeuMRG7pXVb5xt96JHvZ2fH24+fmKvs/eCr4Kvg2+CKPg+eBzsB8fBaYCDP4O/gr+Df7rj7i/dX7u/LaV371ifL4PW0/3jP16uguo= ÿ +{mi}œP +FIG. 1. +Euclidean spacetime diagram. The expectation +value of operators amounts to evaluating the path-integral +in the imaginary space-time with a defect inserted at the d- +dimensional slab between τ = 0+ and τ = β−. Microscop- +ically, the defect (non-trivial Sint) is captured by the sum- +mation over Kraus operators for a local decoherence channel. +No post-selection corresponds to P being all possible labels of +{mi}. Since � +m K† +mKm = 1, if the inserted operator O(φ) is +the product of local operators, its expectation value (correla- +tion function) would only acquire a constant correction. With +post-selection, the set P is constrained, and the effect of the +summation of P corresponds to a non-trivial defect inserted +at τ = 0 (or equivalently τ = β), and connections to recently +studied boundary/defect criticality of (2 + 1)d QCP can be +made. +der the simultaneous action of both Zu +2 and Zl +2, hence +ˆρD is invariant under a diagonal Z2 symmetry. Here the +Zu +2 ×Zl +2 symmetry is explicitly broken down to Z2 by the +decoherence; later we will also discuss an example where +the decoherence preserves the doubled symmetry but the +doubled symmetry can be spontaneously broken down to +the diagonal Z2. +Now suppose we would like to compute the expectation +value of a certain quantity O(ˆφ) that is a composite of ˆφ. +The expectation value is linear with the density matrix, +and it can be evaluated in the path integral form: +tr{ˆρDO(ˆφ)} ∼ +� +φ(x,0)=φ(x,β) +Dφ(x, τ) +×O(φτ=0) exp +� +−S − Sint� +, +(6) +where we have identified the field configurations φ(x, 0) +and φ(x, β) due to the trace. +If there is no post- + +4 +y +τ +τ = 0, β +τ = β/2 +(a) +(b) +y +τ +y = 0, L +y = L/2 +FIG. 2. +Euclidean spacetime diagram and its Wick ro- +tated version. The evaluation of the second Renyi entropy +of ˆρD becomes a path-integral in the Euclidean space-time +with an interaction between fields at τ = 0 and τ = β/2. +In the Wick-rotated picture, such an interaction corresponds +to the long-range interaction between fields at y = 0 and +y = L/2, where L = β. +Note that the first coordinate of +x = (x, y) is not shown in the diagram. +selection at all, one can easily show that tr{ˆρDO(ˆφ)} +and tr{ˆρ0O(ˆφ)} have essentially the same behavior (ex- +cept for some local corrections) as long as O(ˆφ) is +a product of local operators [35], as pictorially illus- +trated in Fig. 1. For example, the correlation function +tr{ˆρD ˆφ(x1)ˆφ(x2)} should have the same scaling in space +as tr{ˆρ0 ˆφ(x1)ˆφ(x2)}. In the field theory language, this +corresponds to Sint being trivial when φ(x, 0) = φ(x, β). +But if there is some weak post-selection by P even on +quantities that are singlet under G, tr{ˆρD +P ˆφ(x1)ˆφ(x2)} +can be very different from tr{ˆρ0 ˆφ(x1)ˆφ(x2)}. With post- +selection, Sint remains non-trivial even at φ(x, 0) = +φ(x, β), which effectively corresponds to the insertion of +defects at the slab τ = 0 (or τ = β) in the path-integral +in the (d+1)−dimensional Euclidean space-time. This is +where we can make a connection to all the recent studies +on boundary criticality for systems with d = 2 [36], and +the desired decohered correlation functions become the +correlation functions restricted on the plane-like defect. +The expectation value of an operator is linear with the +density matrix, which is directly related to experimen- +tal observables. But a lot of information of the quantum +system is encoded in quantities that are nonlinear with +the density matrix. The most famous example of such is +the von Neumann entropy of the density matrix. In this +work, we will evaluate quantities such as the 2nd Renyi +entropy of ˆρD, which is an analogue of the von Neumann +entropy, and it also provides an approximate character- +ization [37] of the amount of quantum information lost +due to entangling with the environment: +S(2) = − log tr{(ˆρD)2} +∼ − log lim +β→∞ +� +Dφ(x, τ) × +exp +� +−S − +� +dx Lint(φ(x, 0), φ(x, β/2)) +� +. +(7) +This calculation is schematically shown in Fig. 2, which +amounts to evaluating the partition function and free en- +ergy of the system with an interaction between fields at +imaginary time τ = 0 and τ = β/2. Or we can also rotate +the space-time (assuming there is a Lorentz symmetry), +then the problem becomes evaluating the partition func- +tion of the system with nonlocal interaction in space, but +constant in time. +Other quantities of interest include tr{(ˆρD)2O(ˆφ)} or +tr{ˆρDO(ˆφ)ˆρDO(ˆφ)}. +We will show that these quanti- +ties nonlinear with ˆρD would reveal some novel quan- +tum phase transitions. In the example we will discuss +in the next section, the decoherence respects the doubled +Zu +2 ×Zl +2 symmetry; but when we increase the decoherence +strength, the 2nd Renyi entropy of the system (which +captures the information loss to the environment) may +encounter singularity at a critical decoherence strength, +which corresponds to the spontaneous symmetry break- +ing from Zu +2 × Zl +2 to the diagonal Z2. +III. +DECOHERED WILSON-FISHER CRITICAL +POINT +In this section, we study two different scenarios for the +system under weak measurements: In Sec. III A, we keep +measurement outcomes for post-selections and show that +the resulting correlation functions linear in the density +matrix ˆρD +P exhibit novel behaviors. In Sec. III B, we ig- +nore measurement outcomes, which corresponds to gen- +uine decoherence; although correlation functions linear +in ˆρD exhibit an ordinary Wilson-Fisher behavior in this +case, we show that quantities non-linear in ˆρD still exhibit +novel behaviors including the extraordinary-log critical- +ity and information-theoretic phase transition. +A. +Quantities linear with ˆρD +P +We consider physics near the O(N) Wilson-Fisher fixed +point in (2 + 1)d, where x = (x, y) is a two-dimensional +spatial coordinate. First, we would like to consider quan- +tities linear with ˆρD +P , for example the correlation func- +tion tr{ˆρD +P ˆφ(0) · ˆφ(x)}. To evaluate quantities such as +tr{ˆρD +P O( ˆφ(x))}, it boils down to evaluating path-integral +Eq. 6. Since we would like to keep at least the diagonal +O(N) symmetry, Sint must be a function of the magni- +tude of the order parameter |φ| at τ = 0, which is always +equal to |φ| at τ = β due to the trace. The simplest term +(and most relevant term) of Sint is +Sint = +� +dxdy ε|φ(x, 0)|2. +(8) +The term Sint can be interpreted as weakly measuring the +energy density or the order parameter φ of the system, +followed by post-selection. +Although there is only one simple term in Sint, the +physical consequence is already rather nontrivial. Sint is + +5 +ϕu +a(0) +ϕl +a(0) +ϕu +b(x) +ϕl +b(x) +ϕu +a(0) +ϕl +a(x) +ϕu +a(0) +ϕu +a(x) +C(2)(x) ∼ +C(2) +X (x) ∼ +C(2) +XI (x) ∼ +(a) +(b) +(c) +⟨⟨ +⟨⟨ +⟨⟨ +⟨⟨ +⟨⟨ +⟨⟨ +A +ALvnicjZbdbts2FMfV7quz95Fu2NVuhAUFdpEZVrZ0w6SZmsTFEGyuEmLWm5AUZREWJQYikrtEnyIPcJutyfa2+xIplKLTIYIMEye3/+ +Q0uHhISOe0qOx/eu/Bhx9/MmDTwfDz7/4suNh1+dV2UtMDnDZV6KVxGqSE4LciapzMkrLghiU5eRvP9hr+8IqKiZfFCLjmZMZQWN +KEYSTBdbHwTnhMh/VBk5Zvf4G+vSHPiX2xsjkfj9vHdRmAam5Ti4eDv4M4xLXjBQS56iqpsGYy5lCQlKcEz0I64pwhOcoJVNoFoiRaqb +a9f+I7DEflIK+BXSb63rHopny4riqjeMypEki2rJ+lbEKrBFlpSVMRGFHgx65lQgnlG80AN7CIZk5ozbvJz1DjVPBSHzvjGyXonVuaSif +Kv9npkUNaOSWOJVSHqmDNZNCJL0rbCqESyi5V5B8DMSW8ayzvfOK1l8stM0YLXkhR4tQRJnfuy9Js08WMqCJb5EhoICwqr6OMCYQlJBM +MVZC3uGQMFbEKSaRVyKJyocIt3XzkOowOGwjhjCJ1qHUfYobEXCtoNF+torwm2g9jWqRqJ9BaP+rLF325gO/s1Du+sxMLJg6syerMFJS +wVZ58fv8bHjuwYnLuVrlNu0SgvAJawnkqVodoFqTM78p8SWgcVRHTJbBRZrxitYRj0NZhC1iKnNJqrWAgm9YsJFU0ZMjhcdRzNqeGnLp +oYNHRc4Oeu4gbxF0UGxS7KDUodGlQZcuyg3KXfTMoGcuWtAuJtBy6NzA+Q3xNKhykTRIuogxw5jD9l83COqghHSDjm5KnLU9K4Kb4tHk +wITgUbgFWZS0mdDXIc473R7npIjp4nYxuSw68e+XjYxc3ixMaNoJn9L09gElijrdCxT9j67EUDRb3dQUgqlIo5kaj4Ift8ajnZ2tpqVb15 +nlLCBQt4cSi3S1GipSYndKVIyiW2n2FqzYNrzh4yBKSIEZpSm7YPVAWozd3EGKsFb6LMEm0Su4izGBEiKDJi4MRHtmKohSshHC03/ur +DtkczseA1ZuBaY4esxqAHX1eQe6HkuYxgUhCxZIVL8XNB1LcHzV7oJjfdG93vmTiSN6hxLWV73be3pkyw5IE5GDa03TtSViGa9LTqFrh/ +8EZoI/lPsn7tocXcMjF+5fw30HvkrgsCzFSlFWfigzIt36+XpNVtHiNlVczhU9Gqvq0nNeU6ayxYS/8ILkKiyWjLhzZLlVw5BMVcKm7 +Y4nqdaudYxk4JXDQq0Cr7ae9Ne3Lfer5XUr/ur6CAMI1N4ju3l+2IaJ/M5qOeWdU36jU9452Ztwlfq37ER4BIb2FdWt3G+PQoej3b+G +lz94m5zj7wvW+873Au9nb9c78E68Mw97yvL+9v7Z7g7TIZsWK6k9+8Zn6+93jNc/Ae5gHZn +ÎflDÍÍ +A +ALvnicjZbdbts2FMfV7quz95Fu2NVuhAUFdpEZVrZ0w6SZmsTFEGyuEmLWm5AUZREWJQYikrtEnyIPcJutyfa2+xIplKLTIYIMEye3/+ +Q0uHhISOe0qOx/eu/Bhx9/MmDTwfDz7/4suNh1+dV2UtMDnDZV6KVxGqSE4LciapzMkrLghiU5eRvP9hr+8IqKiZfFCLjmZMZQWN +KEYSTBdbHwTnhMh/VBk5Zvf4G+vSHPiX2xsjkfj9vHdRmAam5Ti4eDv4M4xLXjBQS56iqpsGYy5lCQlKcEz0I64pwhOcoJVNoFoiRaqb +a9f+I7DEflIK+BXSb63rHopny4riqjeMypEki2rJ+lbEKrBFlpSVMRGFHgx65lQgnlG80AN7CIZk5ozbvJz1DjVPBSHzvjGyXonVuaSif +Kv9npkUNaOSWOJVSHqmDNZNCJL0rbCqESyi5V5B8DMSW8ayzvfOK1l8stM0YLXkhR4tQRJnfuy9Js08WMqCJb5EhoICwqr6OMCYQlJBM +MVZC3uGQMFbEKSaRVyKJyocIt3XzkOowOGwjhjCJ1qHUfYobEXCtoNF+torwm2g9jWqRqJ9BaP+rLF325gO/s1Du+sxMLJg6syerMFJS +wVZ58fv8bHjuwYnLuVrlNu0SgvAJawnkqVodoFqTM78p8SWgcVRHTJbBRZrxitYRj0NZhC1iKnNJqrWAgm9YsJFU0ZMjhcdRzNqeGnLp +oYNHRc4Oeu4gbxF0UGxS7KDUodGlQZcuyg3KXfTMoGcuWtAuJtBy6NzA+Q3xNKhykTRIuogxw5jD9l83COqghHSDjm5KnLU9K4Kb4tHk +wITgUbgFWZS0mdDXIc473R7npIjp4nYxuSw68e+XjYxc3ixMaNoJn9L09gElijrdCxT9j67EUDRb3dQUgqlIo5kaj4Ift8ajnZ2tpqVb15 +nlLCBQt4cSi3S1GipSYndKVIyiW2n2FqzYNrzh4yBKSIEZpSm7YPVAWozd3EGKsFb6LMEm0Su4izGBEiKDJi4MRHtmKohSshHC03/ur +DtkczseA1ZuBaY4esxqAHX1eQe6HkuYxgUhCxZIVL8XNB1LcHzV7oJjfdG93vmTiSN6hxLWV73be3pkyw5IE5GDa03TtSViGa9LTqFrh/ +8EZoI/lPsn7tocXcMjF+5fw30HvkrgsCzFSlFWfigzIt36+XpNVtHiNlVczhU9Gqvq0nNeU6ayxYS/8ILkKiyWjLhzZLlVw5BMVcKm7 +Y4nqdaudYxk4JXDQq0Cr7ae9Ne3Lfer5XUr/ur6CAMI1N4ju3l+2IaJ/M5qOeWdU36jU9452Ztwlfq37ER4BIb2FdWt3G+PQoej3b+G +lz94m5zj7wvW+873Au9nb9c78E68Mw97yvL+9v7Z7g7TIZsWK6k9+8Zn6+93jNc/Ae5gHZn +ÎflDÍÍ +A +ALvnicjZbdbts2FMfV7quz95Fu2NVuhAUFdpEZVrZ0w6SZmsTFEGyuEmLWm5AUZREWJQYikrtEnyIPcJutyfa2+xIplKLTIYIMEye3/+ +Q0uHhISOe0qOx/eu/Bhx9/MmDTwfDz7/4suNh1+dV2UtMDnDZV6KVxGqSE4LciapzMkrLghiU5eRvP9hr+8IqKiZfFCLjmZMZQWN +KEYSTBdbHwTnhMh/VBk5Zvf4G+vSHPiX2xsjkfj9vHdRmAam5Ti4eDv4M4xLXjBQS56iqpsGYy5lCQlKcEz0I64pwhOcoJVNoFoiRaqb +a9f+I7DEflIK+BXSb63rHopny4riqjeMypEki2rJ+lbEKrBFlpSVMRGFHgx65lQgnlG80AN7CIZk5ozbvJz1DjVPBSHzvjGyXonVuaSif +Kv9npkUNaOSWOJVSHqmDNZNCJL0rbCqESyi5V5B8DMSW8ayzvfOK1l8stM0YLXkhR4tQRJnfuy9Js08WMqCJb5EhoICwqr6OMCYQlJBM +MVZC3uGQMFbEKSaRVyKJyocIt3XzkOowOGwjhjCJ1qHUfYobEXCtoNF+torwm2g9jWqRqJ9BaP+rLF325gO/s1Du+sxMLJg6syerMFJS +wVZ58fv8bHjuwYnLuVrlNu0SgvAJawnkqVodoFqTM78p8SWgcVRHTJbBRZrxitYRj0NZhC1iKnNJqrWAgm9YsJFU0ZMjhcdRzNqeGnLp +oYNHRc4Oeu4gbxF0UGxS7KDUodGlQZcuyg3KXfTMoGcuWtAuJtBy6NzA+Q3xNKhykTRIuogxw5jD9l83COqghHSDjm5KnLU9K4Kb4tHk +wITgUbgFWZS0mdDXIc473R7npIjp4nYxuSw68e+XjYxc3ixMaNoJn9L09gElijrdCxT9j67EUDRb3dQUgqlIo5kaj4Ift8ajnZ2tpqVb15 +nlLCBQt4cSi3S1GipSYndKVIyiW2n2FqzYNrzh4yBKSIEZpSm7YPVAWozd3EGKsFb6LMEm0Su4izGBEiKDJi4MRHtmKohSshHC03/ur +DtkczseA1ZuBaY4esxqAHX1eQe6HkuYxgUhCxZIVL8XNB1LcHzV7oJjfdG93vmTiSN6hxLWV73be3pkyw5IE5GDa03TtSViGa9LTqFrh/ +8EZoI/lPsn7tocXcMjF+5fw30HvkrgsCzFSlFWfigzIt36+XpNVtHiNlVczhU9Gqvq0nNeU6ayxYS/8ILkKiyWjLhzZLlVw5BMVcKm7 +Y4nqdaudYxk4JXDQq0Cr7ae9Ne3Lfer5XUr/ur6CAMI1N4ju3l+2IaJ/M5qOeWdU36jU9452Ztwlfq37ER4BIb2FdWt3G+PQoej3b+G +lz94m5zj7wvW+873Au9nb9c78E68Mw97yvL+9v7Z7g7TIZsWK6k9+8Zn6+93jNc/Ae5gHZn +ÎflDÍÍ +FIG. 3. +Correlation functions in the doubled Hilbert +space. The red and blue lines represent the upper and lower +Hilbert spaces respectively in the doubled Hilbert space (See +Sec. III D). +a 2d interface in a (2 + 1)d cylindrical space-time with +extra mass ε for the order parameter, and obviously ε is +always a relevant perturbation, as the scaling dimension +of |φ|2 is always smaller than 2 at the O(N) Wilson- +Fisher fixed point. When ε > 0, Sint suppresses the fields +at τ = 0, and “cuts” the connection between τ = 0+ +and τ = β−, i.e. +τ = 0+ and τ = β− become two +boundaries with ordinary boundary condition. +In this +case, the correlation function tr{ˆρD +P +ˆφ(0) · ˆφ(x)} scales +as [38] +tr{ˆρD +P ˆφ(0) · ˆφ(x)} ∼ +1 +|x|2∆b +φ , +(9) +where +∆b +φ = 1 + 2 +3N + O +� 1 +N 2 +� +(10) +is the boundary scaling dimension of the order parameter +φ to the first order expansion of 1/N. Note that the value +of ∆b +φ is far larger than the bulk scaling dimension of φ. +When ε < 0, the latest progress of the boundary crit- +icality indicates that Sint will drive the interface τ = 0 +into an extraordinary-log criticality, which implies that +the correlation function becomes +tr{ˆρD +P ˆφ(0) · ˆφ(x)} ∼ +1 +(ln |x|)q . +(11) +Please note that, for a single exposed boundary against +the vacuum, the extraordinary-log criticality exists only +for N < Nc, with the critical Nc estimated around Nc ∼ +5 [26]; but for an interface defect inserted in space-time, +the theoretical prediction is that [36] Nc → ∞. +B. +Quantities nonlinear with ˆρD +Now we evaluate quantities nonlinear with ˆρD. +Al- +though our formalism can be straightforwardly general- +ized for any higher orders of ˆρD, here we focus on the +quantities that are quadratic in ˆρD. The quantities of +interest include the 2nd Renyi entropy S(2), the corre- +lation function C(2)(x), the “crossed” correlation func- +tion C(2) +X (x), and the “crossed-Ising” correlation function +C(2) +XI(x) defined as follows: +S(2) = − log tr{(ˆρD)2}, +C(2)(x) ∼ tr{(ˆρD)2 ˆφ(0) · ˆφ(x)}, +C(2) +X (x) ∼ +� +a +tr{ˆρD ˆφa(0)ˆρD ˆφa(x)} +C(2) +XI(x) ∼ +� +a̸=b +tr{ˆρD ˆφa(0)ˆφb(x)ˆρD ˆφa(0)ˆφb(x)}. (12) +The expression of the correlation functions above need to +be divided by the purity of the decohered density matrix +tr{(ρD)2} to be properly normalized. We note that these +correlation functions are distinguished by their represen- +tation under O(N)u × O(N)l symmetry, which becomes +clear in the doubled Hilbert space as in Fig. 3. +As an example, we consider the following interaction +term Sint in Eq. 7: +Sint = +� +dxdy +� +W (|φ(x, 0)|, |φ(x, β/2)|) +− w (φ(x, 0) · φ(x, β/2))2 � +. +(13) +Here W is still a function of the magnitude of the order +parameter at τ = 0 and τ = β/2. The two terms in Eq. 13 +correspond to two different types of weak measurements +(decoherence) channels. The first term still corresponds +to weakly measuring the energy density of the system, +which preserves the doubled symmetry of the system, +resulting in the coupling +� +dxdy W (|φ(x, 0)|, |φ(x, β/2)|) += +� +dxdy ε +2(|φ(x, 0)|2 + |φ(x, β/2)|2) + ... +(14) +where the “...” part includes quartic and higher-order +terms in |φ(x, 0)| and |φ(x, β/2)|. The second w term +can be rewritten as +− w (φ(x, 0) · φ(x, β/2))2 ∼ +− w +N +� +a,b=1 +(Qab(x, 0)Qab(x, β/2)) + · · · , +(15) +where Qab = φaφb− 1 +N δab|φ|2 is the traceless rank-2 sym- +metric tensor of the O(N) vector φ, and the ellipsis are +terms that only depend on |φ| and can be absorbed into +W. The w term can be interpreted as a weak measure- +ment on the tensor Qab, and eventually all the measure- +ment outcomes are summed with equal weight. The w +term breaks the SO(N)u × SO(N)l down to the diagonal +SO(N), but still preserves the Zu +2 × Zl +2 symmetry, where +Z2 corresponds to changing the sign of φ. + +6 +One can perform the Wick rotation in the (y, τ) plane, +so the two interfaces at τ = 0, β/2 become spatial inter- +faces at y = 0 and L/2, with L = β. The interaction +term then becomes +Sint = +� +dτdx W +� +|φ(x, τ)|y=0, |φ(x, τ)|y=L/2 +� +− w +� +φ(x, τ)y=0 · φ(x, τ)y=L/2 +�2 . +(16) +The first term W will either explicitly include an extra +mass term ε|φ|2 at the interface y = 0 and y = L/2, or +generate the mass term through renormalization group +flow. +Then depending on the sign of ε, there can be +three possible scenarios: +(1) If ε > 0, the extra mass term ε|φ|2 at the interface +y = 0 and y = L/2 is a relevant perturbation. +The +role of this extra mass term is to “cut” the system into +two halves: the region from y ∈ (0, L/2) and region y ∈ +(L/2, L ∼ 0). +The relevant mass term will make the +two interfaces at y = 0 and y = L/2 both at ordinary +boundary criticality. +In this scenario, w is obviously an irrelevant perturba- +tion since both interfaces y = 0 and y = L/2 have ordi- +nary boundary criticality, and the scaling dimension of φ +is greater than 1 at the interfaces. Hence the directions +of φ at the two interfaces are pretty much uncorrelated +to each other. Then we expect the correlation function, +the crossed-Ising correlation, and the crossed-correlation +functions to behave as +C(2) ∼ tr{(ˆρD)2 ˆφ(0) · ˆφ(x)} ∼ +1 +|x|2∆b +φ , +C(2) +XI ∼ +� +a̸=b +tr{ˆρD ˆφa(0)ˆφb(x)ˆρD ˆφa(0)ˆφb(x)} ∼ 0, +C(2) +X ∼ +� +a +tr{ˆρD ˆφa(0)ˆρD ˆφa(x)} ∼ 0. +(17) +For example, the crossed-correlation C(2) +X +corresponds +to the correlation function between order parameters at +the two different interfaces; since w is irrelevant when +ε > 0, the directions of order parameters at the two in- +terfaces are uncorrelated, hence C(2) +X +should vanish in +the limit β, L → ∞. +Another way to perceive these +results is that, since w is irrelevant, the system should +have a full SO(N)u × SO(N)l symmetry in the infrared, +but the correlation function C(2) +X +and C(2) +XI break the +SO(N)u × SO(N)l symmetry, hence they must vanish. +(2) In the case with ε < 0, the extra local mass term +ε is still a relevant perturbation, and it flows to the +extraordinary-log criticality. Then the w term (we as- +sume w > 0) is a very relevant perturbation, and it will +flow to w → ∞. +The fate of the system with strong w can be perceived +through a mean field decoupling of Lint. We first recog- +nize that the w term is analogous to the coupling between +two sets of N´eel order parameters in the J1 − J2 Heisen- +berg model on the square lattice, which has applications +in the context of frustrated magnets and iron-pnictides +superconductors [39–43]. Guided by the previous studies +in these contexts, the most natural mean field decoupling +of the w term is +−w +� +φ(x, τ)y=0 · φ(x, τ)y=L/2 +�2 ∼ +−2wΦ(x, τ) +� +φ(x, τ)y=0 · φ(x, τ)y=L/2 +� ++wΦ(x, τ)2. +(18) +Here we have introduced an Ising order parameter +Φ(x, τ) ∼ +� +φ(x, τ)y=0 · φ(x, τ)y=L/2 +� +. The order param- +eter Φ is analogous to the nematic order parameter in the +J1 − J2 Heisenberg model on the square lattice, and the +phase with large w is likely a phase with condensation +of Φ, which spontaneously breaks Zu +2 × Zl +2 down to the +diagonal Z2. +The condensation of Φ will “pin” ˆφy=0 and ˆφy=L/2 +along the parallel direction. With a nonzero condensate +of Φ, the three correlation functions mentioned above +behave like +C(2) ∼ tr{(ˆρD)2 ˆφ(0) · ˆφ(x)} ∼ +1 +(ln |x|)q , +C(2) +XI ∼ +� +a̸=b +tr{ˆρD ˆφa(0)ˆφb(x)ˆρD ˆφa(0)ˆφb(x)} +∼ Const, +C(2) +X ∼ +� +a +tr{ˆρD ˆφa(0)ˆρD ˆφa(x)} ∼ +1 +(ln |x|)q . +(19) +The crossed-Ising correlation saturates to a nonzero con- +stant in the limit of large |x| due to the condensate of +Φ, which also leads to an extraordinary-log correlation +of crossed-correlation function. When w flows to infinity +and Φ condenses, there is only one O(N) symmetry in +the infrared; hence all these correlations can be nonzero. +(3) Now if ε = 0, with large but finite N, the scaling +dimension of the traceless rank-2 symmetric tensor Qab +is ∆ = 1+32/(3π2N) to the leading order of 1/N expan- +sion. This means that w is weakly irrelevant with scaling +dimension [w] = −64/(3π2N). Then the beta function +of w should be +β(w) = dw +d ln l = − A +N w + Cw2, +(20) +where A = 64/(3π2). The constant C can be extracted +through some OPE calculation, or simply a one-loop cal- +culation in the large−N limit. Since C is positive, there +is a fixed point at finite w∗, beyond which w will flow +strongly, and the order parameter Φ defined above will +condense. +C. +The phase diagram +In phase diagram Fig. 4 we summarize the results +related to (ˆρD)2 discussed in this section. +The order- +disorder transition of Φ should extend to the region with + +7 +FIG. 4. +Phase Diagram. The global phase diagram of +decohered Wilson-Fisher critical point in terms of quantities +nonlinear in ˆρD. +ε > 0, where there is a competition between ε which +drives the interfaces to the ordinary boundary critical- +ity and w which drives the crossed-Ising transition. In +this phase diagram, the critical wc is a function of ε: +wc(ε) − wc(0) ∼ ε∆w/∆ε, and ∆w,ε are the scaling di- +mensions of parameter w and ε. +Since Φ(x) ∼ φ(x, 0) · φ(x, β/2) is the crossed-Ising +order parameter, the crossed-Ising correlation function +should show a transition from short-range to correla- +tion while increasing w. If we consider tr{(ˆρD)2} as the +partition function, and the 2nd Renyi entropy S(2) = +− log tr{(ˆρD)2} as the free energy, the nature of the tran- +sition at w = wc and ε > 0 should belong to a 2d +classical Ising universality class. +Without coupling to +other modes with nonlocal correlations in the infrared, +the order-disorder transition of an Ising order parameter +in the 2d space should belong to the 2d Ising universality +class, and here we should consider the perturbation of +the coupling w on top of the 2d Ising transition. In fact, +if we start with a 2d classical Ising transition of order +parameter Φ, the coupling −wΦ(x) (φ(x, 0) · φ(x, β/2)) +is irrelevant knowing the fact that the scaling dimension +∆b +φ > 1 for the ordinary boundary criticality at ε > 0. +Hence at w = wc and ε > 0, the crossed-Ising correlation +function should scale as +C(2) +XI(r) ∼ +1 +|r|1/4 . +(21) +Also, when we increase w across wc, the 2nd Renyi en- +tropy S(2) = − log tr{(ˆρD)2} should have the same sin- +gularity as the free energy of the classical 2d Ising model. +D. +Lattice Model and Doubled Hilbert Space +We have shown that a decoherence channel whose +Kraus operators are symmetric under O(N) can drive +an Ising-type phase transition that spontaneously breaks +the Zu +2 × Zl +2 ⊂ O(N)u × O(N)l symmetry down to the +diagonal Z2 symmetry for quantities nonlinear in the den- +sity matrix. However, most physical quantities are linear + paramagnet +ℤu +2 + coupling +∼ ZuZuZlZl + paramagnet +ℤl +2 +Toric code +Toric code +Kramers +Wannier +Duality + Anyon tunneling +∼ +doubled +toric code +single +toric code +symmetric + symmetry +broken (SSB) +ℤ2 +p(2) +c +p(2) +c +AL8nicjZbPb9s2FMfV7ldn70e6HXcRFhTtgMywsqUbdmoarE1QBMniJg1ieQFUTJhUmIoyo +0r8I/Ydbdh1wH7a3bYZftX9iRTqUmQwQYJt/n+0jp8fGRkWC0UMPh3fuvPue+9/cO/DXv+jz/5dO3+ZydFXkpMjnHOcnkaoYIwmpFjRUjp0ISxCNGXkWznZq/mhNZ0Dx7qRaCTDhKM5pQjBSYztf2QoaylBH/0dl5Nd8otd/8M/3V0vKwNT1sbKFcqsOCcj9U5FJXuE8K5Q+X1sfDobN47uNwDTWPfMcnt/v/RLGOS45yRmqCjGwV +CoSYWkopgR3QvLgiEZyglY2hmiJNiUjXfrP0HYIn9Jfwy5TfWFc9KjFdFBQXnWEqhuCdiwXvWhEvwBZUp7HRGa61+uYU4nElOJL3bOH4EhNnXHrl7PeoRSpJGTWNUbWK/GSKSrz19rvmElWcqIJV6GpGOawlpLSZKuFTIhgoW3AsI/pTEljEvWfcbx6VKvp9UNBOlIhleLkFSMl/lfp1afkwlwYotoIGwpLCKPp4ibCBIShMvIa5 +5yjLK5CEukq5F+WYUbuv7IVRjt1RDCGUXVntZdiDmSM1Bo/7qKmIl0X4Y0ytgKt9YOu/LIrl/CdrXrLVR+biSGrj+2JRytw1FBJVvnBW3zg+K7AkUvFChU2LdIMcA7riVQu61Q1SZn/iNiy8DiqPa4rQKLNeMclGPgwlELeLVeh1Va4GkXjLpoKmHBkcLjuO5sjwIxeNDBq56IVBL1wkDBIuig2KXZQalLrowqALFzGDmIueG/Tc +RZe0jQm0HDozcHZNPA0qXKQMUi7i3DusJ2zGpnaDR1dlzhrexYE18WjzoERwYNwA7IoaTKhq0NCtLptIUgW08ubxeQia8U/XtQycnG9MKFpK3xG05sHVChqdS9R9D+6HEPRbHRjUwjGMo0m1XAQfLMxHGxtbdQt3bhOLGcJCRSK+lBCzC5FUzgNidkpbTmCYqndV5g3aTgX4KFySIYpSm5ZvdAWYx+voUQY13h2wiTRFfJbYRTGBEiaPJi +d4AHtiLJc8hHM3/qBDPoPzMeDlemCag8e8BGBHXxSQ+6GiLCYQSejYAoXKt4K6YwkO5s0uONDn7eudPB05ojco4V3Vm+1n+7Zsl9QR2b3S1F1bIhfxquQIunb4D2Em+EPMP3TXZv8K7rtw5wruOPA0gcMyl0tFXsANa0qUWz/PVmQFzW6SFaWAQ0Uv93o1KoVgpL5sIbnw9+EiJOuMtnxovVTAo58UgVC6fZYonrVaufYFJwSOirQFe +bdT3prm9T7pfL61Z8uV3fKGA9mp5vl6Eya6unBaTqx1Ytc6sdbJ3oTL1L9hJ8IDl9jAvrK6jZPNQfB4sPXTt+tPnpr7D3vC+9L75EXeN95T7xd79A79rD3p/eX94/3b1/1f+3/1v9Kb17x/h87nWe/h/AdOi1w=È(Zv,uZv,l)(ZvÕ,uZvÕ,l)Í ≥ const +AMFnicjZbPb9s2FMeVdD86ez/S7biLsKBAB2SGlS3dsFPTYG2CIkgWN2mQyA0oipIJkxJDUaldgX/E +bvtrdht2HbDT/pTd9mjTqUmWwQYJt/n+0jp8fGRiWC0Uv3+3yur957/4MP73/U6X78yaefrT34/KQqa4nJMS5ZKU8TVBFGC3KsqGLkVEiCeMLIq2S8Y/irKyIrWhYv1VSQIUd5QTOKkQLTxRqPGSpyRsJHsZBletGQMKZFGOeIc/Q6FkQKHZ6BeaPWX/+/iIEolvMR4ryMFZkoiRvcFlUSl+srfd7/dkT+o3INtYD+xePOj8EqclrjkpFGaoqs6jvlD +DBklFMSO6E9cVEQiPU7OoVkgTqphM4uLDh+CJQ2zUsKvUOHMuzRiNG0orhqDdMwBO9cTXnbingFtsSR8jIlstCdTsucSyRGFE90x2CIzXyxjUv57xDLXJyLhtTJxX4jVTVJZvdNgyk6LmVBFHPA9JyzSCfJCSZG0rZEsCyeG4VxD8EUkdY1mz9je1yr7YdjQtSKFHi+BFnNQlWGJv3ClEqCFZtCA2FJYRVDPEISYQVJCkMV5A0uIauKtIlJopuYJ+ +WkiTe0+chlmOwZCOFMkmZP6zbEHMmxbqBhvrpJWE10GKe0yJutSGv9sC2ftOUSvnOh3vLVx3ZiyOpjd+LBEhzMqCTL/OAdPvB8l+DAp2KJCpdWeQG4hPVEqpRmFzTG5M1/RFwZWDzVHndVYHFmvIJl1OfREKW8GbdRNVZIKnTPqojlHFsfzjqc5svzIRwOLBj56YdELHwmLhI9Si1If5RblPrq06NJHzCLmo+cWPfRhC5iAi2Pji0c3xBPiyofKYuUj +zi3jHts58wgW7uho02Jc7ZnRbApHiYHBgT34g3IomyWCW0dEmKh2xaCFCmd3C4ml8VC/NOlkZHLm4UZzRfCZzS/fUCFkoXuJUr+Q1diKJoz3bktBOcyT4ZNvxd9u9HvbW1tmJaeuQ4dZwkJFAtzKCHmlqIRnIbE7pRFOYJiqf1XuJql4ZUAD1VCEqQoz8kNuwfKYvL6DkKMdYPvIswy3WR3EY5gRIigzYvdHu65iqKUvIRwzL73Rx3zMZyPEa/XI9vsPeY1 +ADf6oLcjxVlKYFIQscVKFS/E5iOIzi4mu2CA32xeL2TpwNP9BZlvK16u/1s35XtEhOR3WuN6boSOU2XJUfQdcN/CDPBH2Lhob82+9dw34c713DHg6cZHJalnCvKCm5YI6L8+nm2JKvMZe1mWVULOFT0fK83g1oIRsxlC8lpuA8XIWky2vGhZqlGFRz5pImE0otjieplq5tjI3DK4KBvIt1smnrSXt9ZuZ8vr1/x5ba5UcIAi6ulnebTZjo+sLpOLGFE7v +RiS2c3E04T/1bdiI8cImN3Cur3zjZ7EWPe1s/f7f+5Km9zt4Pvgy+Ch4FUfB98CTYDQ6D4wAHfwX/rKyu3Ov+2v2t+3v3j7l0dcX6fBG0nu6f/wKMiJjv +È( +Ÿ +eœ“‹ +Ze,u)( +Ÿ +eœ“‹ +Ze,l)Í ≥ const +FIG. 5. +Choi Isomorphism and Duality. Schematic di- +agrams illustrate how decohered mixed states map into the +coupled bilayer system under the Choi-Jamiolkowski isomor- +phism. +Through the Kramers-Wannier duality, Z2 param- +agnet under symmetric decoherence (left) maps into the Z2 +toric code under dephasing noise (right). Accordingly, their +phase diagrams in the doubled Hilbert space is also dual to +each other. +At p > p(2) +c , both sides are characterized by +non-vanishing correlation functions of operators, which cor- +responds to (left) the formation of mean-field for Zv,uZv,l +operator or (right) condensation of a pair of e-anyons euel. +in the density matrix, and this calls for a proper phys- +ical interpretation of these quantities. +As we will see, +our crossed-Ising transition is closely related to a better +known information transition in the context of topologi- +cal surface codes. +In the following, we use another model to illustrate the +essential physics of such a spontaneous symmetry break- +ing (SSB) from Zu +2 × Zl +2 to the diagonal Z2. Instead of +starting with a state at quantum criticality, we consider +a concrete lattice model with a trivially disordered state +|Ω0⟩ = |+⟩⊗N on a L × L square lattice with qubits de- +fined on vertices. +We consider the following quantum +channel as a symmetric local decoherence model: +Ee=(v,v′) : ρ → (1 − p)ρ + pZvZv′ρZvZv′, +(22) +where “e” labels the edge between the nearest-neighbor +pair of sites v and v′. The decoherence channel is given +as the composition of local channels, E = � +e Ee. To pro- +ceed further, we apply the Choi-Jamiolkowski isomor- +phism to map the density matrix into the pure state +and decoherence channel into the operator in the doubled +Hilbert space. Under this mapping, the Choi state of the +pure state density matrix ρ0 = |Ω0⟩⟨Ω0| is denoted as +∥ρ0⟩⟩ ≡ |Ω0⟩|Ω0⟩, which is nothing but a vectorized den- +sity matrix, and the Choi operator of the decoherence +channel is given as +ˆE = +� +e=(v,v′) +(1 − 2p)1/2eτZv,uZv′,uZv,lZv′,l +(23) +where tanh τ = p/(1 − p). As illustrated in Fig. 5, the +isomorphism effectively maps the density matrix into the + +crossed-Ising +order +extraordinary-log + +crossed-Ising order +ordinary +<0 +> +0 +88 +pure state in the bilayer system. Now, when the system +is subject to the decoherence, we can show that ∥ρD⟩⟩ = +ˆE∥ρ0⟩⟩ is the ground state wavefunction of the following +local Hamiltonian in the doubled-Hilbert space [9, 12]: +Htot = Hu + Hl + Hint +Hu(l) = − cosh 2τ +� +v +Xv,u(l) +Hint = +� +v +cosh4 4τ +� +v′∈v +(1 − Zv,uZv′,uZv,lZv′,l tanh 4τ), +(24) +where v′ ∈ v means that v′ is a nearest-neighbor site of v. +At p = τ = 0, it reduces into two decoupled disordered +states. At p > 0, the decoherence gives rise to the (lo- +cal) coupling between two layers that may induce a phase +transition into an SSB phase that breaks the Zu +2 ×Zl +2 into +the diagonal Z2 symmetry. The (unnormalized) ground- +state can be written as +ˆE∥ρ0⟩⟩ = (1 − p)2Nv � +l +(tanh τ)|l||∂l⟩ ⊗ |∂l⟩ +(25) +where the summation is taken over all string config- +urations l on the edges of the square lattice, |∂l⟩ ≡ +� +v∈∂l Zv|Ω0⟩, and Nv = L2 is the number of vertices +in the square lattice. We stress that ˆE∥ρ0⟩⟩ is normalized +in the sense that its corresponding density matrix ρD +(under the Choi-Jamiolkowski isomorphism) is normal- +ized with tr{ρD} = 1 in the single Hilbert space while it +is unnormalized as a state in the doubled Hilbert space. +Then, it is straightforward to show that the norm of the +wavefunction ˆE∥ρ0⟩⟩, i.e., the purity of the density ma- +trix tr{(ρD)2} is equivalent to the partition function of +the 2d Ising model at the temperature β = 2τ, as explic- +itly derived in Appendix. A 6. Accordingly, at β = 0.441, +which corresponds to p(2) +c += 0.178, the wavefunction in +the doubled Hilbert space undergoes the transition of 2d +Ising universality. Here the superscript in p implies that +the transition happens in the quantity that involves the +product of two density matrices. +The SSB transition between the Zu +2 × Zl +2-symmetric +phase and the phase with only the diagonal Z2 symmetry +in the doubled Hilbert space is captured by the suscepti- +bility χ to an external symmetry-breaking field coupled +to O = � +v Zv,uZv,l (in the doubled Hilbert space): +χ ≡ 1 +L2 +� +⟨⟨O2⟩⟩ − ⟨⟨O⟩⟩2� += 1 +L2 +�⟨⟨ρD∥O2∥ρD⟩⟩ +⟨⟨ρD∥ρD⟩⟩ +− ⟨⟨ρD∥O∥ρD⟩⟩2 +⟨⟨ρD∥ρD⟩⟩2 +� += +� +v,v′ +tr{ρDZvZv′ρDZvZv′} +L2tr{(ρD)2} +− +�� +v tr{ρDZvρDZv} +L tr{(ρD)2} +�2 +(26) +which is given by the summation over connected corre- +lation functions of the order parameter Zv,uZv,l of this +SSB transition in the doubled Hilbert space, related to +the crossed-Ising correlation function C(2) +XI in Eq. (19). χ +is closely related to the crossed-Ising correlation functions +in Eq. (19). χ diverges as |p − p(2) +c |−7/4 at the SSB tran- +sition. The exponent 7/4 is the signature of the 2d Ising +universality class of this SSB transition. +Note that ⟨⟨O⟩⟩ +in the definition of χ is the order parameter that takes a +non-zero value in the symmetry-broken phase and should +be evaluated with δ = 0+ after the thermodynamical +limit is taken. We elaborate more on this in the next +section. +In the doubled Hilbert space, the definition of χ and +the associated notion of the external symmetry-breaking +field are standard for generic condensed matter systems +with global symmetries. In Sec. III E, we explain their +physical meaning at the level of the density matrix ρD). +E. +Correlation Functions and Physical Meanings +In the previous sections, we have defined several cor- +relation functions or physical quantities which are non- +linear in decohered density matrices. Accordingly, these +quantities are not directly accessible from experiments, +raising questions about what they really mean physically. +In order to connect these quantities to experiments, we +consider the signatures of the transition in the proba- +bility distribution of measurement outcomes. More pre- +cisely, we will show that χ in Eq. (26) corresponds to the +sensitivity of the decohered mixed state against small +perturbations. This, in turn, is related to the amount +of information that can be obtained from measuring the +mixed state. +To proceed, we define the notion of distance between +two different density matrices as the following: +DJ(ρ, σ) ≡ tr(ρ log ρ − ρ log σ + σ log σ − σ log ρ). (27) +This quantity is the quantum generalization of the Jef- +freys divergence [44] (or symmetrized quantum relative +entropy), which quantifies the distance between two den- +sity matrices. It satisfies good properties to be a valid +metric between mixed states: (i) non-negative, (ii) van- +ishing iff ρ = σ, and (iii) monotonically decreasing under +the application of quantum channels. +In order to discuss the “sensitivity” of the decohered +density matrix, we define an infinitesimal variation of the +original density matrix under symmetry breaking “per- +turbation”, defined as the following quantum channel: +Mδ,v : ρ → (1 − δ)ρ + δZvρZv +ρδ ≡ Mδ[ρ], +Mδ ≡ +� +v +Mδ,v. +(28) +which amounts to the application of weak measurement +in the Z-basis. +Hence, δ also serves as a symmetry- +breaking field in the doubled Hilbert space. +Then, +D(δ) ≡ DJ(ρD, ρD +δ ) quantifies the difference between + +9 +the unperturbed and perturbed decohered density ma- +trices. Accordingly, how fast the distance changes with δ +tells how sensitive the decohered system is against weak +measurement in the Z-basis (or how informative the Z- +measurement is). This is captured by the second deriva- +tive of the distance, which is related to the (classical) +Fisher Information as F ≡ ∂2 +δD(δ) +�� +δ=0 [45]. +However, the expression in Eq. (27) is very challeng- +ing to evaluate. To proceed, we generalize the Jeffreys +distance in a way similar to Ref. 46: +D(n)(ρ, σ) ≡ +1 +n − 1 +� +log tr(ρn) + log tr(σn) +− log tr(ρσn−1) − log tr(σρn−1) +� +(29) +This n-th Jeffreys distance is symmetric and non- +negative [47]. Furthermore, in the limit n → 1+, D(n) → +DJ. At n = 2, we find that the expression behaves like +an overlap between Choi states of two density matrices +in the doubled Hilbert space description: +D(2)(ρ, σ) ≡ − log +� +⟨⟨ρ∥σ⟩⟩ +� +⟨⟨ρ∥ρ⟩⟩ · ⟨⟨σ∥σ⟩⟩ +� +, +(30) +With this definition, we evaluate the derivatives of the +“distance” between ρD and its perturbed version ρD +δ with +respect to δ (See Appendix. B). This quantity exhibits in- +teresting behaviors when δ → 0 and L → ∞. The two +different orders of limit lead to different results which +have natural interpretations from a standard condensed +matter perspective and from a quantum information per- +spective respectively. +In the limit where the thermodynamical limit L → ∞ +is taken first and δ → 0+ afterward, the conventional +choice of limit for a condensed matter system experienc- +ing SSB, the second-order derivative ∂2 +δD(2) produces a +quantity proportional to the susceptibility: +lim +δ→0+ lim +L→∞ +1 +L2 ∂2 +δD(2) = χ, +(31) +which diverges at the SSB phase transition. Based on +the nature of this SSB transition and the interpretation +of δ as the symmetry-breaking field, we conclude that the +decohered density matrix undergoes a qualitative change +(with respect to this distance) as a function of p. Beyond +critical p(2) +c , the density matrix changes significantly un- +der weak measurements in the Z-basis. +However, when we consider taking the limit δ → 0 be- +fore the thermodynamical limit, the second-order deriva- +tive produces a quantity that aligns with the information- +theoretic intuition: +lim +L→∞ lim +δ→0 +1 +L4 ∂2 +δD(2) = +� +v,v′ +tr{ρDZvZv′ρDZvZv′} +L4tr{(ρD)2} += M, +(32) +where M can be interpreted as the expectation value of +the squared order parameter, which vanishes in the para- +magnetic phase but acquires a finite value in the SSB +phase. Since this SSB transition belongs to the 2d Ising +universality class, the standard phenomenology of the +Ising model can be directly translated into the language +of our study. In an Ising ferromagnet, the magnetic order +of the ground state of the system is very sensitive to an +infinitesimal external Zeeman field. This well-known fea- +ture corresponds to the sensitivity of the density matrix +to the decoherence in the Zv basis (as in Eq. (28)), in +the phase where the Zu +2 × Zl +2 symmetry is spontaneously +broken down to the diagonal Z2. This is quantified by the +value of ∂2 +δD(2)�� +δ=0 (where δ is set to 0 before L grows +large) scaling as L4 in the SSB phase, which aligns with +the Fisher information of the GHZ state in the context +of quantum metrology [48]. +It is straightforward to argue that the second deriva- +tives of the distance D(n>2) in both limits would also +exhibit similar behavior across a certain critical value +(that depends on n), particularly because tr((ρD)n) is +mathematically equivalent to the partition function of +the coupled Ising model which is expected to undergo a +transition from a paramagnetic to a ferromagnetic phase +potentially at a different critical strength p(n) +c +≥ p(2) +c . Ac- +cordingly, we expect the behavior of ∂2 +δD(n) to extrapo- +late in the limit n → 1. Therefore, there should be two +different phases separated by a critical point at strength +p = p(1) +c . This is indeed the case, as we will see by using +a dual description of the model. +F. +Duality, Intrinsic transition, and Decodability +Under the Kramers-Wannier (KW) duality, the Z2 +paramagnet under symmetric decoherence maps to the +toric code under dephasing noise. +The toric code un- +der dephasing noise, in turn, is well known to exhibit an +information transition at critical noise strength, beyond +which the quantum information (logical qubits) stored in +the toric code is not decodable [31]. This hints at an inti- +mate connection between the criticality discussed above +and a well-known information transition. +First, we illustrate how the doubled Hilbert space for- +malism provides us some insights into the transition in +the decohered toric code. Under the KW duality (See +Appendix. A 5), the Hamiltonian in Eq. (24) maps to the +two copies of toric code coupled by local anyon tunnel- +ing terms, as schematically shown in Fig. 5. At critical +strength of the tunneling, the anyon condenses and the +topological order reduces into a single toric code. The +critical behavior is associated with a Higgs transition +signified by the development of the anyon condensation +amplitude ∼ ⟨⟨ρD∥ˆγuˆγl∥ρD⟩⟩, where ˆγu/l = � +e∈γ⊥ Ze,u/l +creates a pair of e-anyon at the end of the string γ⊥. In +turn, this implies that ˆρD and ˆγˆρDˆγ have an apprecia- +ble overlap and they become less and less distinguishable +for p > p(2) +c +in the doubled Hilbert space. Without us- +ing the KW duality, the existence of the criticality can +be directly seen by calculating the purity of the deco- + +10 +hered toric code, which is given by the partition func- +tion of the 2d Ising model at β = tanh−1(1 − 2p)2 as +shown in Appendix. A 3. +Note that the corresponding +Ising model goes from an ordered phase to a disordered +phase as we increase p, which is the behavior opposite to +that of Eq. (25). However, both undergo transitions at +the same critical point p(2) +c += 0.178 as expected from the +KW duality. +As a next step, we calculate the von Neumann entropy +of the decohered toric code state, a quantity highly non- +linear in the density matrix that enters the expression +in Eq. (27). Following the calculations in Appendix. A 2, +we can show that the von Neumann entropy of the de- +cohered density matrix is proportional to the free en- +ergy of the random bond Ising model along the Nishi- +mori line [49, 50] at β = tanh−1(1 − 2p), which is known +to be critical at p = 0.1094 [51]. Therefore, the transi- +tion behavior indeed extends down to the n → 1 limit +for a toric code under dephasing noise, and by the du- +ality, for a paramagnetic under symmetric decoherence. +As remarked, the transition behavior in the limit n → 1 +corresponds to the singularity of the Fisher information, +a well-known information-theoretic quantity. We remark +that this series of transitions for different n is intrin- +sic to the decohered density matrix (and the distance +in use), as these transitions are associated with sponta- +neous symmetry breaking. Interestingly, the n → 1 limit +of this intrinsic transition coincides with the decodability +transition point demonstrated in Ref. 31, which is based +on a certain decoding procedure. Such a connection may +imply that the intrinsic transition point provides an up- +per bound for some information retrieval protocols to be +successful in this setting. +IV. +NON-LOCAL OPERATORS +As +we +mentioned +previously, +without +any +post- +selection, the correlation function for local operators lin- +ear with the decohered density matrix should have the +same scaling as the undecohered correlation function. +But in this section, we will show that nonlocal quantities +can still have qualitatively different behaviors even if we +only consider expectation values linear with the density +matrix. +A. +1d Quantum Rotor +To illustrate the behavior of nonlocal operators un- +der decoherence, let us start with the 1d systems with +a description in terms of the quantum rotor, such as the +spin-1/2 chain. In terms of the Abelian bosonization, the +N´eel and valence bond solid (VBS) order parameters of +a spin-1/2 chain are represented as +(N x, N y, N z, V ) ∼ (sin θ, cos θ, sin φ, cos φ) .(33) +Under decoherence or weak measurement of (for exam- +ple) local operator ei ˆφ, the density matrix becomes +ˆρD = E[ˆρ0], +E = +� +x +Ex, +Ex[ˆρ0] ∼ (1 − p)ˆρ0 + p +2 ei ˆφ(x)ˆρ0e−i ˆφ(x) ++ p +2 e−i ˆφ(x)ˆρ0ei ˆφ(x). +(34) +ˆρ0 is the undecohered density matrix of the spin-1/2 +chain, and ˆρD = E[ˆρ0] still keeps tr[ˆρD] = 1. +We can first evaluate the correlation function of local +operator O(r): +CD(r) = tr{ˆρD ˆO(r) ˆO(0)} = tr{ˆρ0E[ ˆO(r) ˆO(0)]}, +Ex[ ˆO(r) ˆO(0)] ∼ (1 − p) ˆO(r) ˆO(0) + p +2 ei ˆφ(x) ˆO(r) ˆO(0)e−i ˆφ(x) + p +2 e−i ˆφ(x) ˆO(r) ˆO(0)ei ˆφ(x) +(35) +For local bosonic order parameters, such as the N´eel and +VBS, although they may have nontrivial commutation +with ei ˆφ(x) at the vicinity of x, at long distance apart, +ˆO(r) and ei ˆφ(x) should commute. Therefore, the corre- +lation functions of local operators acquire only a con- +stant amount of corrections under decoherence, and any +local order parameter should still have the same power- +law with scaling dimensions of the undecohered spin-1/2 +chain. +However, for nonlocal operators, the situation can be +very different. +A particular family of nonlocal opera- +tors is the disorder operators [52–59] [60], which have +attracted great interests recently. These operators have +been used as an auxiliary diagnosis for the states of mat- +ter, especially for critical states of matter. For a 1d quan- +tum rotor, if we view ˆφ as the phase angle of a local boson +creation operator, then (∇xˆθ)/2π is the boson density ˆnφ. +The following operator is called a disorder operator +˜Or = exp +� +iα +� r +0 +dx ˆnφ(x) +� += exp +� +i α +2π (ˆθ(r) − ˆθ(0)) +� +. +(36) +If ˆφ carries a full U(1) symmetry, then α ∈ R; if ˆφ only +carries a ZN symmetry, then α = 2πk +N with k ∈ {1, ..., N}, +as ˆnφ is defined modulo N. +For example, for the spin-1/2 chain, if we view ˆφ as + +11 +a local operator, then eiˆθ/2 is a disorder operator of ˆφ, +and eiˆθ/2 plays two roles simultaneously: it first creates a +fractionalized spin-1/2 excitation (i.e. a spinon), it also +creates a domain wall of the VBS order parameter, i.e. +eiˆθ(r)/2 shifts ˆφ(x) → ˆφ(x) + π for x < r. Indeed, it is +well-known that a spin-1/2 is localized at the domain wall +between two VBSorders. We can evaluate the correlation +function of eiˆθ/2 for the decohered density matrix: +CD(r) = tr{ˆρDeiˆθ(r)/2 e−iˆθ(0)/2} +∼ e−r/ξtr{ˆρ0 eiˆθ(r)/2 e−iˆθ(0)/2}, +(37) +where tr{ˆρ0eiˆθ(r)/2 e−iˆθ(0)/2} is the correlation function of +eiˆθ/2 for undecohered spin-1/2 chain, and the “correlation +length” ξ is ξ ∼ −1/ ln(1 − 2p) for small p. +Hence our calculation for the 1d quantum rotor sys- +tem implies that, although the disorder operator has a +power-law correlation with the absence of decoherence, it +can be rendered short-ranged under decoherence. As we +will show in the next subsection, similar behavior of the +disorder operator happens in higher dimensions as well. +The spin-1/2 chain is also an example of spin liquid with +fractionalized spinon excitations since the spinon corre- +lation function decays as a power-law in the undecohered +spin-1/2 chain. But under decoherence or weak measure- +ment on the VBS order parameter the spin chain loses its +fractionalization, as the spinon operator decays exponen- +tially. This can be intuitively understood as the fact that, +if the VBS operator is “measured”, in each measurement +outcome the system is pinned to a certain particular VBS +pattern, which leads to confinement in this measurement +outcome. The confinement persists even if we average +over all measurement outcomes. +B. +(2 + 1)d quantum critical points with a U(1) or +ZN symmetry +Now let us consider a (2 + 1)d QCP or CFT with a +global U(1) symmetry, or ZN symmetry that can be em- +bedded into a U(1) that emerges in the infrared. This +U(1) symmetry is dual to a noncompact U(1) gauge +field [61–63]. +We always turn on decoherence on the +scalar boson creation operator which carries the U(1) +charge, or equivalently the monopole operator of the dual +U(1) gauge field. The decohered density matrix takes the +same form as Eq. 34: +ˆρD = E[ˆρ0], +E = +� +x +Ex, +Ex[ˆρ0] ∼ (1 − p)ˆρ0 + p +2 ei ˆφ(x)ˆρ0e−i ˆφ(x) ++ p +2 e−i ˆφ(x)ˆρ0ei ˆφ(x). +(38) +Here ei ˆφ(x) is the monopole operator, which creates a +scalar boson, or a 2π gauge flux at location x. +We evaluate the expectation value of the following +quantity defined for a closed loop C = ∂A: +˜OC = exp +� +� +x∈A,∂A=C +i2π +N ˆnφ(x) +� +. +(39) +In the undecohered density matrix, and in the dual for- +malism, this quantity reduces to the evaluation of the +Wilson loop: +⟨ ˜OC⟩ ∼ ⟨exp +� i +N +� +C +dx · ˆa(x) +� +⟩, +(40) +and as was shown previously, it should obey a perimeter +law, with a universal logarithmic contribution from the +sharp corners of the loop C [56, 57, 64]. The coefficient +of the universal logarithmic contribution arising from the +corner is proportional to the universal conductivity of the +scalar boson current at the (2+1)d CFT in the AC limit +ω/T → ∞. +However, under decoherence, the operator ˜OC will shift +ˆφ(x) by angle 2π/N for x ∈ A. Hence we expect the +decoherence to change the behavior of ⟨ ˜OC⟩ significantly: +⟨ ˜OC⟩ ∼ +� +(1 − p) + p cos +�2π +N +��A +, +(41) +namely ⟨ ˜OC⟩ should now decay with an area law. Just +like the 1d example discussed in the previous subsection, +an area law decay of the Wilson loop is a sign of confine- +ment. It means that the vortex of the U(1) boson, which +is also the gauge charge of the dual gauge field ˆa, should +be confined under decoherence of the scalar boson cre- +ation operator. Here we would like to remark that, one of +the tools for diagnosing fractionalization and deconfine- +ment is the dynamic structure factor, where the fraction- +alization would lead to a continuum [65–67]. Computing +real-time dynamics is beyond the current set-up of our +current manuscript as it requires the formalism that in- +volves the Lindbladian. Here we use the behavior of the +Wilson loop as the sign of confinement/deconfinement. +V. +SUMMARY AND FUTURE DIRECTIONS +In this study, we examined the effects of decoherence +and weak measurement on quantum critical points in +(2+1)d space-time. We found that this problem is math- +ematically equivalent to the boundary or defect critical- +ity of (2 + 1)d conformal field theories, which have been +extensively researched in recent years. Our results indi- +cate that when a QCP is exposed to decoherence or weak +measurement, observers may observe peculiar behaviors, +including the extraordinary-log correlation recently dis- +covered in the context of boundary criticality. Addition- +ally, as the strength of decoherence or weak measurement +increases, the system can experience an information- +theoretic transition that is captured by quantities non- +linear in the decohered density matrix. This transition is + +12 +linked to spontaneous symmetry breaking when we con- +sider quantities to the n-th power of the density matrix; +in particular using the “doubled formalism”, we show +that for n = 2, this transition belongs to the 2d Ising +universality class. +There are many related directions that are very much +worth exploring in the future. We list two such directions +as follows: +(1) Unconventional quantum criticality under decoher- +ence: It is known that in the world of quantum many- +body systems, there are two types of quantum critical +points: conventional and unconventional. Conventional +QCPs correspond to quantum phase transitions between +a disordered state with a direct product structure and +a state that breaks a certain symmetry. +This type of +QCPs has classical analogs such as the Wilson-Fisher +fixed points discussed in this work. But it is known that +there is another large class of unconventional QCPs with- +out a simple classical analogue [68]. On the other hand, +unconventional QCPs are those that do not have a simple +classical analog and may involve transitions between two +ordered phases with different symmetries, or between an +ordered phase and a topological order. The most well- +known example of unconventional QCPs is the deconfined +QCP [69, 70], which has many desirable phenomena such +as deconfinement and a duality web as was summarized +in Ref. 71. It is reasonable to expect that the unconven- +tional QCPs under decoherence can also be mapped to +certain boundary criticality problems, and it is going to +be an unusual boundary criticality with unconventional +QCP in the bulk. As we have already seen in the current +work, decoherence may be at odds with deconfinement, +as deconfinement is often signified by nonlocal operators +such as the disordered operators or the Wilson loops, +whose behavior can be strongly affected by decoherence. +It would be interesting to study the fate of unconven- +tional QCPs under decoherence in general in the future. +(2) The Strange Correlator: The notion of a strange +correlator was originally proposed as a tool to diagnose +SPT states using their bulk wave functions [17], rather +than edge states. The strange correlator is defined as the +following quantity +CS(r) = ⟨Ω| ˆO(0) ˆO(r)|Ψ⟩ +⟨Ω|Ψ⟩ +, +(42) +where |Ψ⟩ is the wave function that awaits diagnosis, and +|Ω⟩ is the trivial direct product disordered state with the +same symmetry G and Hilbert space as |Ψ⟩. +ˆO is an +order parameter that carries a nontrivial representation +of G. The arguments given in Ref. 17 suggest that, al- +though the ordinary correlation functions in both |Ψ⟩ and +|Ω⟩ must be short-ranged, this strange correlator Eq. 42 +must have either long-ranged or power-law correlation, +for 1d and 2d states. +In the past decade, the strange +correlator has been used as a tool for both conceptual +understanding and numerical diagnosis for SPT states +and also topological states [72–93]. +One of the future directions worth pursuing is the +strange correlator between a quantum critical state |Ω⟩, +and an SPT wave function |Ψ⟩. Let us still focus on two- +dimensional systems. Using the formalism developed in +this work, this problem may be mapped to the 2d inter- +face between a quantum critical point on the temporal +domain τ < 0, and an SPT state on the other domain +τ > 0 in the Euclidean space-time path-integral. +Un- +der Wick-rotation, the strange correlator is mapped to +the spatial interface between a quantum criticality and +an SPT state. This kind of interface has two types of +boundary effects: the boundary states arising from the +bulk topology, and also the boundary criticality originat- +ing from the bulk critical modes. This is a subject under +very active research lately, both theoretically and numer- +ically [18–22, 24, 27]. In particular, some novel interface +criticality especially a (1 + 1)d deconfined quantum crit- +ical point was identified in the literature [24]. One po- +tentially highly interesting direction in the future is to +analyze the strange correlator (and its generalized form +defined in Ref. 12) between quantum criticality and SPT +state, and explore the possible novel phenomena, espe- +cially when either the bulk quantum critical state |Ω⟩, or +the SPT state |Ψ⟩, or both are under decoherence. +ACKNOWLEDGMENTS +We thank Ehud Altman, Soonwon Choi, Matthew P. +A. Fisher, Sam Garrett, Yi-Zhuang You for inspiring dis- +cussions and previous collaborations. J.Y.L. is supported +by the Gordon and Betty Moore Foundation under the +grant GBMF8690 and by the National Science Founda- +tion under the grant PHY-1748958. C. X. acknowledges +the support from the Simons Foundation through the Si- +mons Investigator program. C.-M. J. is supported by a +faculty startup grant at Cornell University. +Note Added: While finishing up this work, we became +aware of an independent related work [94, 95], which +should appear on arXiv on the same day as our work. +Appendix A: Toric code under decoherence +The goal of this section is to show that the entan- +glement entropy of the toric code state under dephasing +noise is given by the free energy of the random bond Ising +model along the Nishimori line. We will show that such +a decohered state is dual to the disordered product state +under Z2 symmetric decoherence channel in Eq. (23). +First, the toric code Hamiltonian is defined as +H = − +� +v +� +e∋v +Ze − +� +p +� +e∈p +Xe += − +� +v +Av − +� +p +Bp +(A1) + +13 +The ground state is characterized by Av = Bp = 1. Fur- +thermore, on the torus, the ground state is 4-fold degen- +erate with two logical qubits. Logical qubits reside on the +space where the following effective Pauli operators act on +CX +i +≡ � +e∈Ci Xe and CZ +i ≡ � +e∈C⊥ +i Ze where Ci is a cycle +along the i-th axis; while Ci is along the bond, C⊥ +i crosses +the bond. Note that {CX +1 , CZ +2 } = {CZ +1 , CX +2 } = 0, while +[CZ +i , CX +i ] = 0. As an example we choose to study one +of the four ground states denoted as |ψtc⟩, whose pure +state density matrix is ρtc = |ψtc⟩⟨ψtc|. The ground state +is the eigenstate of the qubits CX +i |ψtc⟩ = ai|ψtc⟩, with +a1 = a2 = 1. +1. +Decomposition +We consider the toric code ground state decohered un- +der the following channel: +Ee : ρ → (1 − p)ρ + pZeρZe, +E = +� +e +Ee +(A2) +In order to understand the structure of E[ρtc], first we +evaluate the matrix elements of the decohered toric code +density matrix. To do the job, consider ρs,s′ ≡ |Ωs′⟩⟨Ωs| +where |Ωs⟩ is a generic product state characterized by +s = {se}, se = ±1: +|Ωs⟩ ≡ +� +e +Z(1−se)/2|+⟩⊗2Nv, +(A3) +where Nv = L2 is the number of vertices. Then +⟨Ωs|E[ρtc]|Ωs′⟩ = tr(ρs,s′E[ρtc]) = tr(E[ρs,s′]ρtc) +(A4) +where E[ρs,s′] is given as +ρs,s′ = +1 +22Nv +� +e +(1 + seXe)Z(1−ses′ +e)/2 +e +E[ρs,s′] = +1 +22Nv +� +e +(1 + se(1 − 2p)Xe)Z(1−ses′ +e)/2 +e +(A5) +For tr(E[ρs,s′]ρtc) not to vanish, ∂(s · s′) = 0 so that +product of Ze does not create anyons. In such a case, +the product of Z-strings always commutes with a loop of +X-strings along the bond. In fact, we can show that +⟨ψtc| +� +e∈l +Xe +� +e +Z(1−ses′ +e)/2 +e +|ψtc⟩ = F(l, s · s′)δ∂l,0δ∂(s·s′),0 +(A6) +where s · s′ defines a non-trivial link configuration along +the dual link whenever it takes a negative value. +For +⟨CX,Y,Z +1,2 +⟩ = cx,y,z +1,2 +we have +F(l, s · s′) = ⟨ψtc|CX +h(l)CZ +h(s·s′)|ψtc⟩ +(A7) +where h(l) ∈ π1(T2) is the element of the homotopy group +of the torus. For the simplest case where CX +1,2 = 1, we +remark that F(l, s · s′) is non-zero iff h(s, s′) is trivial. +Then, the overlap in Eq. (A4) is given as +tr(E[ρs,s′]ρtc) = +1 +22Nv +� +l +(1 − 2p)|l| � +e∈l +se +× ⟨ψtc| +� +e∈l +Xe +� +e +Z(1−ses′ +e)/2 +e +|ψtc⟩ += +δh(s·s′),1 +2Nv(2 cosh β)2Nv ZRBIM[s, β] +(A8) +where the summation is over all possible edge configura- +tion l, tanh β = (1 − 2p), and +ZRBIM[s, β] ≡ +� +σ +� +e=(v,v′) +eβseσvσv′ , +(A9) +which turns out to be the partition function of an Ising +model with random signs, specified by {se}, in the near- +neighbor spin-spin interaction. At p = 0, β−1 = 0, i.e., +zero temperature limit, It has an interesting consequence: +the matrix element ⟨Ωs|E[ρtc]|Ωs′⟩ vanishes unless s and +s′ belong to the same equivalence class, i.e., ∂(s·s′) = 0. +Furthermore, if s ∼ s′, then ZRBIM[s, β] = ZRBIM[s′, β]. +Accordingly, E[ρs] is block-diagonal. +Therefore, E[ρtc] +decomposes as the following: +E[ρtc] = +� +s,s′ +ρs,s′tr(ρs,s′E[ρtc]) += +1 +2Nv(2 cosh β)2Nv +� +m +ZRBIM[sm, β] ρm. (A10) +where the summation is taken over the equivalence class +of s, denoted by m; the equivalence class is defined as +∂(s · s′) = 0. sm is the representative of the equivalence +class m, and ρm is defined as +ρm ≡ +� +s,s′∼sm +|Ωs⟩⟨Ωs′|. +(A11) +Therefore, the decohered density matrix has the following +block-diagonal structure (in X-basis) +E[ρtc] = +� +����� +B1 +0 +0 +· · · +0 +B2 +0 +· · · +0 +0 +B3 · · · +... +... +... +... +� +����� +(A12) +where each block is 2Nv−1 by 2Nv−1 dimensional matrix +labeled by the equivalence class m and its entries are all +equal, i.e., +Bi ∝ +� +��� +1 1 1 · · · +1 1 1 · · · +1 1 1 · · · +... +... +... +... +� +��� +� +�� +� +2Nv−1 +� +� +� +� +� +� +� +2Nv−1 = 2Nv−1|φm⟩⟨φm| +(A13) + +14 +where |φm⟩ = +1 +√ +2Nv−1 +� +s∼sm |Ωs⟩. +There are total +2Nv+1 equivalence classes originated from (Nv − 1) in- +dependent stabilizers Bp and two logical operators CX +i . +2. +Entanglement Entropy +After reorganizing terms, we get +ρD +tc = +� +m +pm|φm⟩⟨φm|, +pm = +ZRBIM[sm, β] +2 · (2 cosh β)2Nv . +(A14) +Note that the entanglement entropy has a very interest- +ing structure: +S = −tr(ρD +tc ln ρD +tc) += − +� +m +pm log pm +∝ − +� +m +Z[sm, β] log Z[sm, β] +(A15) +which is nothing but a disorder averaged random bond +Ising model’s free energy along the Nishimori line, whose +transition point is located at pc = 0.1094 [51]. Therefore, +there is an intrinsic phase transition of the entanglement +entropy of the decohered toric code state at pc = 0.1094, +which coincides with the decodability transition point ob- +tained in Ref. 31. +3. +Purity +Interestingly, the purity of the decohered density ma- +trix maps to the partition function of the Ising model: +(Here C ≡ (22 · (2 cosh β)4Nv)−1): +tr((ρD +tc)2) = C +� +m +Z2 +RBIM[sm, β] = C +� +s +Z2 +RBIM[s, β] +2Nv−1 += C(cosh β)4Nv +2Nv−1 +� +s +� +σ +� +σ′ +� +L1,L2 +(1 − 2p)|L1|+|L2| +× +� +e∈L1 +se +� +e′∈L2 +se′ +� +i∈∂L1 +σi +� +j∈∂L2 +σ′ +j +(A16) +where Li is the link configuration defined on the square +lattice. This expression can be further simplified by the +following: +tr((ρD +tc)2) = +1 +24Nv+Nv+1 +� +σ +� +σ′ +� +L1,L2 +(1 − 2p)|L1|+|L2| +× 22NvδL1,L2 +� +i∈∂L1 +σi +� +j∈∂L2 +σ′ +j += +1 +24Nv+Nv+1 +� +L1,L2 +(1 − 2p)|L1|+|L2| · 22NvδL1,L2 +× 2Nvδ∂L1,0 · 2Nvδ∂L2,0 += +1 +2Nv+1 +� +L +(1 − 2p)2|L|δ∂L,0 += +ZFIM[β′] +2(2 cosh β′)2Nv , +tanh β′ = (1 − 2p)2 +(A17) +where ZFIM[β′] = ZRBIM[1, β′] is the partition function +of the ferromagnetic Ising model. In the last equality, we +used that +� +∂γ=0 +(tanh β)|γ| � +e∈γ +se = +ZRBIM[s, β] +2Nv(cosh β)2Nv . +(A18) +Since the ferromagnetic 2d Ising model has a transition +at β = ln +� +1 + +√ +2 +� +/2 = 0.441, correlation functions in the +doubled Hilbert space (in the next section) would exhibit +a critical behavior at p(2) +c += 0.178. +4. +Choi Isomorphism +One may study the decohered toric code state in the +doubled Hilbert space under Choi isomorphism. +The +decohered density matrix maps into the following Choi +state: +∥E[ρtc]⟩⟩ = +� +i +|i⟩ ⊗ (E[ρtc]|i⟩) = +� +m +pm|φm⟩|φm⟩ +(A19) +where we used Eq. (A14). The dephasing channel maps +to the following Choi operator in the doubled Hilbert +space: +ˆE = +� +e +(1 − 2p)1/2eτZe,uZe,l, +tanh τ = +p +1 − p +(A20) +which can be considered as an imaginary time evolu- +tion by an Ising Hamiltonian. +Note that cosh τ += +(1 − p)/√1 − 2p and sinh τ = p/√1 − 2p. Then, we see +that +ˆEXe,u ˆE−1 = Xe,ue−2τZe,uZe,l +⇒ +ˆEBp ˆE−1 = Bp +� +e∈p +e−2τZe,uZe,l +(A21) + +15 +Now, consider the following parent Hamiltonian: +Hparent = 1 +2 +� +v +(1 − Av)†(1 − Av) ++ 1 +2 +� +p +(1 − Bp)†(1 − Bp) +(A22) +for some α, β > 0. Following the procedure in Ref. 12, +we can show that the Choi state ∥E[ρtc]⟩⟩ = ˆE∥ρtc⟩⟩ is the +ground state of the following Hamiltonian: +ˆHD = ˆHD +u + ˆHD +l + ˆHD +int +ˆHD +u = − +� +v +Av,u − +� +p +cosh(2τ)Bp,u +ˆHD +int = +� +p +� +e∈p +(cosh 4τ − Ze,uZe,l sinh 4τ) +(A23) +In this doubled system, the coupling ˆHD +int breaks two mi- +croscopic magnetic one-form symmetries into their diag- +onal subgroup, and we expect the phase transition from a +doubled toric code order to a single toric code order. The +transition should be captured by the condensation of e- +anyons, which is diagnosed by non-vanishing expectation +values of +⟨( +� +e∈γ⊥ +Ze,u)( +� +e∈γ⊥ +Ze,l)⟩ ∼ const, +(A24) +where γ⊥ is the open string defined along the dual lattice. +5. +Kramers-Wannier Duality +Under Kramers-Wannier duality, the toric code maps +to the trivially disordered state, |Ω0⟩ = |+⟩⊗N on the +vertices of the dual lattice. +The mapping is explicitly +given as the following: +� +e∋v +Xe ↔ Xv +Ze⊥(v,v′) ↔ ZvZv′ +(A25) +Note that in this dual model, the system has two 0-form +Z2 symmetries even under decoherence. Also, v that la- +bels the vertices in the dual lattice labels the plaquette in +the original lattice. Since � +v Xv is dual to � +p Bp = 1 in +the toric code, the mapped states must be symmetric un- +der the 0-form Z2 symmetry. The above relation makes +it clear that the dephasing channel in Eq. (A2) maps to +the Z2 symmetric decoherence channel in Eq. (23). +Now, applying the Kramers-Wannier duality on the +doubled Hamiltonian in Eq. (A23), we can obtain the +Hamiltonian for the dual model in the doubled Hilbert +space as the following: +ˆHD = ˆHD +u + ˆHD +l + ˆHD +int +ˆHD +u = −2 cosh 2τ +� +v +Xv +ˆHD +int = 2 +� +v +� +v′∈v +(cosh 4τ − Zv,uZv′,uZv,lZv′,l sinh 4τ) +(A26) +In this model, The transition should be captured by +the development of an order parameter that breaks off- +diagonal Zu +2 × Zl +2 symmetry, which is diagnosed by non- +vanishing expectation values of +⟨(Zv,uZv,l)(Zv′,uZv′,l)⟩ ∼ const +(A27) +for any well separated v and v′. +6. +Purity calculation +As stated in the main text, the (unnormalized) ground- +state of the above Hamiltonian in the doubled Hilbert +space is given as +ˆE∥ρ0⟩⟩ = (1 − p)2Nv � +l +(tanh τ)|l||∂l⟩ ⊗ |∂l⟩. +(A28) +The norm of this wavefunction is given as +tr{(ρD)2} = ⟨⟨ρD∥ρD⟩⟩ ∝ +� +l1,l2 +δ∂l1,∂l2(tanh τ)|l1|+|l2| +∝ +� +{sv} +� � +e +� +1 + tanh τ +� +v∈∂e +sv +��2 +∝ +� +{sv} +� +e +� +1 + tanh 2τ +� +v∈∂e +sv +� +∝ ZFIM[2τ] +(A29) +where we used the following identity: +δ∂l1,∂l2 = +1 +2Nv +� +sv∈{±1} +� +v∈∂l1 +sv +� +v′∈∂l2 +sv′. +(A30) +Note that at p = p(2) +c , 2τ here agrees with β′ from +Eq. (A17), which establishes the self duality in the 2d +Ising model. +Appendix B: Derivatives of the Distance +In this section, we calculate the second derivative of +the distance defined in Eq. (30) for the decohered density +matrix ρD = E[|Ω0⟩⟨Ω0|] from Eq. (23). The perturba- +tion is defined by the following channel +Mδ,v : ρ → (1 − δ)ρ + δZvρZv +ρδ ≡ Mδ[ρ], +Mδ ≡ +� +v +Mδ,v. +(B1) + +16 +By defining h such that tanh h = δ/(1 − δ) (eh = +1/ +√ +1 − 2δ), the channel can be mapped to the follow- +ing operator under Choi isomorphism: +ˆ +Mδ ≡ +� +v +(1 − 2δ)1/2ehZv,uZv,l +(B2) +Note that ∂h/∂δ = 1/(1 − 2δ). Then, we can show that +∂δD(2) = ⟨⟨ρD +δ ∥∂δρD +δ ⟩⟩ +⟨⟨ρD +δ ∥ρD +δ ⟩⟩ +− ⟨⟨ρD∥∂δρD +δ ⟩⟩ +⟨⟨ρD∥ρD +δ ⟩⟩ +∂2 +δD(2) = ⟨⟨∂δρD +δ ∥∂δρD +δ ⟩⟩ +⟨⟨ρD +δ ∥ρD +δ ⟩⟩ +− +� +⟨⟨ρD∥∂δρD +δ ⟩⟩ +⟨⟨ρD∥ρD +δ ⟩⟩ +�2 +(B3) +Let O ≡ � +v Zv,uZv,l. Then, at δ → 0, we evaluate that +∂δMδ +�� +δ→0 = (O − L2) +(B4) +Furthermore, exactly at δ = 0, ⟨⟨O⟩⟩ +�� +δ=0 = 0 due to the +symmetric nature of the initial decohered density matrix +under symmetric decoherence channel. Then, plugging +Eq. (B4) into the Eq. (B3), it is straightforward to show +that the first derivative of the distance vanishes as ex- +pected, and the second derivative, depending on the or- +der of limit, would be given as Eq. (31) or Eq. (32). +[1] S. J. Garratt, Z. Weinstein, and E. Altman, Measure- +ments conspire nonlocally to restructure critical quantum +states (2022). +[2] W. H. Zurek, Decoherence, einselection, and the quan- +tum origins of the classical, Rev. Mod. Phys. 75, 715 +(2003). +[3] J. Preskill, Quantum Computing in the NISQ era and +beyond, Quantum 2, 79 (2018). +[4] C. Song, D. Xu, P. Zhang, J. Wang, Q. Guo, W. Liu, +K. Xu, H. Deng, K. Huang, D. Zheng, S.-B. Zheng, +H. Wang, X. Zhu, C.-Y. Lu, and J.-W. 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Vishwanath, Diag- +nostics of mixed-state topological order and breakdown +of quantum memory, To appear (2023). + diff --git a/o9E4T4oBgHgl3EQfvA3U/content/tmp_files/load_file.txt b/o9E4T4oBgHgl3EQfvA3U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..437aa0bb648ecb94de9fa7ae50f770e325f6f784 --- /dev/null +++ b/o9E4T4oBgHgl3EQfvA3U/content/tmp_files/load_file.txt @@ -0,0 +1,1381 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf,len=1380 +page_content='Quantum criticality under decoherence or weak measurement Jong Yeon Lee,1 Chao-Ming Jian,2 and Cenke Xu3 1Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 2Department of Physics, Cornell University, Ithaca, New York 14853 3Department of Physics, University of California, Santa Barbara, CA 93106 Decoherence inevitably happens when a quantum state is exposed to its environment, which can affect quantum critical points (QCP) in a nontrivial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As was pointed out in recent literature on (1 + 1)d conformal field theory (CFT) [1], the effect of weak measurement can be mathematically mapped to the problem of boundary CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In this work, we focus on the (2 + 1)d QCPs, whose boundary and defect effects have attracted enormous theoretical and numerical interests very re- cently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We focus on decoherence caused by weak measurements with and without post-selecting the measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Our main results are: (1) for an O(N) Wilson-Fisher QCP under weak measurement with post-selection, an observer would in general observe two different types of bound- ary/defect criticality with very different behaviors from the well-known Wilson-Fisher fixed points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' in particular, it is possible to observe the recently proposed exotic “extraordinary-log” correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (2) An extra quantum phase transition can be driven by decoherence, if we consider quantities non- linear with the decohered density matrix, such as the Renyi entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We demonstrate the connection between this transition to the information-theoretic transition driven by an error in the toric code model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (3) When there is no post-selection, though correlation functions between local operators remain the same as the undecohered pure state, nonlocal operators such as the “disorder operator” would have qualitatively distinct behaviors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' and we also show that the decoherence can lead to confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' INTRODUCTION When a quantum state is exposed to an environment, it is being constantly probed and “measured”, forming entanglement with the degrees of freedom in the envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' If one is ignorant of the environment and the measurement outcome is lost, it amounts to tracing out the environment’s degrees of freedom and the original pure quantum state becomes a mixed state, which re- sults in the loss of coherent quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This process is referred to as quantum decoherence, and it is the bridge between the quantum mechanics that governs the microscopic nature, and our classical macroscopic world [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' More generally, if a quantum state is weakly measured and the measurement outcome is still accessi- ble, one can consider the effect of post-selecting the mea- surement outcome and study process that is more gen- eral than decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In recent years there has been a surge of progress in simulating quantum states of matter with nontrivial entanglement on platforms summarized as the Noisy Intermediate-Scale Quantum (NISQ) tech- nology [3], including simulating exotic quantum many- body states such as topological order, spin liquids, and symmetry protected topological states [4–8], which have long been discussed in condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In these platforms, decoherence can happen due to various reasons, which motivated recent inspection of the fate of SPT states under decoherence [9–12] (related studies mo- tivated from other contexts were also conducted [13, 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Quantum criticality represents another class of quan- tum many-body states with peculiar and universal en- tanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Recently a class of (1 + 1)d conformal field theory (CFT), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', the Luttinger liquid under weak mea- surements has been studied, and it was pointed out that the effect of weak measurements can be mathematically mapped to the problem of the boundary of the CFT [1], which is a subject that was studied extensively in the past [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The same trick was used in the recent study of SPT states under decoherence [12], which exploited the observation that the wave function of the SPT states can be mapped to the partition function of the boundary states of the system after a space-time rotation [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In this work, we focus on quantum critical points in (2 + 1)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The connection between the decohered QCPs (or QCPs under weak measurement) in the bulk and the boundary criticality still holds, but the boundary criti- cality of (2 + 1)d QCPs is a subject that has only been carefully studied very recently, and it has attracted enor- mous interests from both the theoretical and numerical communities [18–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' It has been understood since long back that there exists an ordinary boundary condition of a (2 + 1)d Wilson-Fisher QCP (or a 3d classical Wilson- Fisher critical point), where the Landau order param- eter φ has a scaling dimension ∆b φ > 1, which is far greater than the bulk scaling dimension of the order pa- rameter (which is slightly greater than 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Only re- cently it has become clear that at the boundary of an O(N) Wilson-Fisher critical point, in addition to the well-known ordinary boundary criticality, there is a so- called “extraordinary-log” boundary criticality, meaning the correlation function of the order parameter at the boundary reads ⟨φ(0)φ(x)⟩ ∼ 1 (ln |x|)q , (1) where q depends on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This peculiar scaling was pro- posed theoretically [25, 26] and recently confirmed nu- merically in Monte Carlo simulations [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='05238v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='stat-mech] 12 Jan 2023 2 Our current work will bridge these two directions that are under active studying, and we will demonstrate that a (2+1)d QCP under decoherence or weak measurements naturally exhibits the peculiar boundary criticality stud- ied recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This work is organized as follows: In section II, we develop the general formalism of ana- lyzing decohered quantum critical states, including the general connection between the decohered or weakly- measured bulk state and the boundary/defect criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In section III we discuss the (2 + 1)d O(N) Wilson- Fisher quantum critical points under decoherence or weak measurements, and in section III A we focus on the quantities linear with the density matrix, which can be observed directly in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We demonstrate that weak measurements generally render the observed quan- tities rather different from the bulk QCP (with certain post-selection that preserves the O(N) symmetry in the mixed state ensemble), and we may observe the exotic extraordinary-log correlation mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In section III B we discuss quantities nonlinear with the density matrix, which reveals a lot more structures of the mixed state density matrix of the critical state under decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In particular, we discuss a quantum infor- mation phase transition that can be diagnosed through the 2nd Renyi entropy of the decohered system, along with other correlations defined in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Here we would like to point out an important differ- ence between the physical scenarios to be considered in section III A and III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In section III A we discuss physics under weak measurements on quantities such as energy density, and we allow a post-selection on the measure- ment outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Section III B considers quantities non- linear with the density matrix where post-selection is not needed in this scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' hence it can genuinely correspond to physics under decoherence due to coupling to the en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In section III D we demonstrate that there is an ex- plicit lattice model with an information transition driven by the strength of decoherence (or weak measurement) analogous to the decoherence-driven transition in sec- tion III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We also show that the transition in this lattice model is dual to an information transition in the toric code model, which is related to the error threshold of the toric code [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' If we forbid post-selection, local correlation functions of the (locally) decohered density matrix would remain largely unchanged from the undecohered pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In recent years the nonlocal disorder operator has become a very important diagnosis of quantum states of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In section IV we demonstrate that the nonlocal disorder operator can still have qualitatively different behavior from the undecohered pure state density matrix, even if there is no post-selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In particular, we show that in several examples, decoherence can lead to confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' GENERAL FORMALISM In order to discuss quantum states under decoherence, one approach is to first explicitly derive the ground state wave function |Ψ⟩ in either exactly soluble lattice mod- els or effective field theories, then construct the density matrix ˆρD under decoherence [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This approach re- lies on an explicit derivation of the ground state wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The ground state wave function can be de- rived for gapped states such as SPT phases [9, 16, 17], and also gapless phases with a Gaussian (free boson) La- grangian, such as the (1 + 1)d Luttinger liquid [1], or the Rokhsar-Kivelson (RK) point in (2+1)d models [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' But for general interacting theories, such as systems near the Wilson-Fisher quantum critical points, deriving the ground state wave function is cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In this sec- tion, we follow a more general procedure to study inter- acting systems under decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Let us prepare an interacting quantum state which is the ground state of a Hamiltonian, whose pure state den- sity matrix is given as ˆρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' After the quantum state is prepared, we turn off the Hamiltonian and expose the system to local decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' For example, for a lattice model of qubits, a decohered density matrix may be rep- resented as ˆρD = E[ˆρ0], E = � x Ex, Ex[ˆρ0] = (1 − p)ˆρ0 + pZxˆρ0Zx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (2) where ˆρD describes a mixed state (ensemble) and E is given as the composition of local decoherence channels Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' One interpretation of this decoherence channel is that, there is a certain probability for the environment to measure qubits in the Z-basis (weak measurements) at each location x, and the measurement outcomes are “lost”, which amounts to the dephasing noise in the study of quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A generic decoherence channel maps a density matrix ˆρ0 into ˆρD = � m Kmˆρ0K† m, where {Km} is a set of Kraus operators satisfying the condition � m K† mKm = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' If we interpret a decoherence channel to be induced by weak measurements, we can consider a post-selection followed by the channel where the resulting density matrix is given as ˆρD P ≡ P[ˆρD] tr{P[ˆρD]}, P[ˆρD] ≡ � m∈P Kmˆρ0K† m (3) where P is a generalized projection onto a subset of mea- surement outcomes P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', post-selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In this work, we will mostly study systems at or close to a QCP, hence we will use a coarse-grained continuous space rather than a lattice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the coarse-grained formalism, the density matrix of a pure state is given by the following imaginary time path-integral [ˆρ0]φ1(x),φ2(x) = ⟨φ1(x)|Ψ⟩⟨Ψ|φ2(x)⟩ ∼ lim β→∞ � φ(x,0)=φ1(x) φ(x,β)=φ2(x) Dφ(x, τ) exp(−S), (4) 3 where S = � β 0 dτdx L(φ) is the bulk action of the system and x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', xd) is the spatial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Follow- ing Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 1 and 12, a class of decoherence problem in the coarse-grained continuous space can be converted into the following imaginary-time path-integral: [ˆρD]φ1(x),φ2(x) ∼ lim β→∞ � φ(x,0)=φ1(x) φ(x,β)=φ2(x) Dφ(x, τ) exp � −S − Sint� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' S = � β 0 dτdx L(φ), Sint = � dx Lint(φ(x, 0), φ(x, β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (5) The effect of decoherence is captured by an extra inter- action term Lint in the Lagrangian, and Lint has no tem- poral integral, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', it is a “long range” interaction in the Euclidean temporal direction, between fields at τ = 0 and τ = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The coupling constant in Lint is controlled by the strength of decoherence (or strength of weak measure- ment) p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A natural choice of Lint favors configurations with φ(x, 0) ∼ φ(x, β), meaning it enhances the weight of the diagonal components of the density matrix, which drives the system into a mixed state density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The most important factor that determines the form of Lint is still its symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Let us label the symmetry of the original Lagrangian L as G, and assume that the original pure state without decoherence is a symmetric state under G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then there are two types of symmetry constraints on Lint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' If the environment is weakly “mea- suring” quantities that are invariant under G, then Lint must be invariant under a “doubled” symmetry trans- formation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', it is invariant under symmetry transfor- mation G on φ(x, 0) and φ(x, β) separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the lan- guage of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 12, this is the case that preserves the dou- bled Gu × Gl symmetry, where Gu and Gl correspond to the upper and lower symmetry in the formalism of doubled Hilbert space using the Choi-Jamiolkowski iso- morphism [33, 34], which maps any density matrix to a pure ket-state in the doubled Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' However, if the environment is measuring quantities that carry a non- trivial representation of G, but eventually we sum over the measurement outcomes within the same representa- tion of G with equal weight, meaning the symmetry G is broken in each quantum trajectory but still preserved in an average sense, then Lint is only invariant under sym- metry G, which is the diagonal subgroup of Gu × Gl, and it corresponds to a simultaneous transformation of φ(x, 0) and φ(x, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (2), if the original pure state is invariant un- der a Z2 symmetry action � x Xx then the pure state density matrix ˆρ0 is invariant under a doubled Zu 2 × Zl 2 symmetry, where Zu 2 and Zl 2 correspond to the left and right operations of the Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' But since the deco- herence is caused by weakly measuring the Ising variable which carries a nontrivial representation of Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' the de- cohered density matrix ˆρD is no longer invariant under separate Zu 2 or Zl 2 action,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' it is instead only invariant un- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='AL2HicjZbdbts2FMfV7quz95Ful7sRFhTYReZ2dINu2oarE0QBMniJg1quQElUTJhUaIpKrVLENjdsM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='sN2NPsdnuIvc0OZSq1yGSoAMPk+f1JiucHipiOanEcPjvnbvPve+x/c+7DX/+jTz7duP/ZeVXWPMZncZmX/CJCFc5Jgc8ETm+YBwjGuX4eTb0/z5FeYVKYtnYsnwhKsICmJkQDT5cY3h5eSqpcyvGJTVIiSyjBWYa5Un4n9co8VcKY7c2BwOhs3ju43ANDY985xc3u/9HiZlXFNciDhHVTUOhkxMJOKCxDlWvbCuMEPxDGV4DM0CUVxNZLMz5T8AS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} 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+page_content='{mi}œP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Euclidean spacetime diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The expectation value of operators amounts to evaluating the path-integral in the imaginary space-time with a defect inserted at the d- dimensional slab between τ = 0+ and τ = β−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Microscop- ically, the defect (non-trivial Sint) is captured by the sum- mation over Kraus operators for a local decoherence channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' No post-selection corresponds to P being all possible labels of {mi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Since � m K† mKm = 1, if the inserted operator O(φ) is the product of local operators, its expectation value (correla- tion function) would only acquire a constant correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' With post-selection, the set P is constrained, and the effect of the summation of P corresponds to a non-trivial defect inserted at τ = 0 (or equivalently τ = β), and connections to recently studied boundary/defect criticality of (2 + 1)d QCP can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' der the simultaneous action of both Zu 2 and Zl 2, hence ˆρD is invariant under a diagonal Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Here the Zu 2 ×Zl 2 symmetry is explicitly broken down to Z2 by the decoherence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' later we will also discuss an example where the decoherence preserves the doubled symmetry but the doubled symmetry can be spontaneously broken down to the diagonal Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Now suppose we would like to compute the expectation value of a certain quantity O(ˆφ) that is a composite of ˆφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The expectation value is linear with the density matrix, and it can be evaluated in the path integral form: tr{ˆρDO(ˆφ)} ∼ � φ(x,0)=φ(x,β) Dφ(x, τ) ×O(φτ=0) exp � −S − Sint� , (6) where we have identified the field configurations φ(x, 0) and φ(x, β) due to the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' If there is no post- 4 y τ τ = 0, β τ = β/2 (a) (b) y τ y = 0, L y = L/2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Euclidean spacetime diagram and its Wick ro- tated version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The evaluation of the second Renyi entropy of ˆρD becomes a path-integral in the Euclidean space-time with an interaction between fields at τ = 0 and τ = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the Wick-rotated picture, such an interaction corresponds to the long-range interaction between fields at y = 0 and y = L/2, where L = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Note that the first coordinate of x = (x, y) is not shown in the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' selection at all, one can easily show that tr{ˆρDO(ˆφ)} and tr{ˆρ0O(ˆφ)} have essentially the same behavior (ex- cept for some local corrections) as long as O(ˆφ) is a product of local operators [35], as pictorially illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' For example, the correlation function tr{ˆρD ˆφ(x1)ˆφ(x2)} should have the same scaling in space as tr{ˆρ0 ˆφ(x1)ˆφ(x2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the field theory language, this corresponds to Sint being trivial when φ(x, 0) = φ(x, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' But if there is some weak post-selection by P even on quantities that are singlet under G, tr{ˆρD P ˆφ(x1)ˆφ(x2)} can be very different from tr{ˆρ0 ˆφ(x1)ˆφ(x2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' With post- selection, Sint remains non-trivial even at φ(x, 0) = φ(x, β), which effectively corresponds to the insertion of defects at the slab τ = 0 (or τ = β) in the path-integral in the (d+1)−dimensional Euclidean space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This is where we can make a connection to all the recent studies on boundary criticality for systems with d = 2 [36], and the desired decohered correlation functions become the correlation functions restricted on the plane-like defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The expectation value of an operator is linear with the density matrix, which is directly related to experimen- tal observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' But a lot of information of the quantum system is encoded in quantities that are nonlinear with the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The most famous example of such is the von Neumann entropy of the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In this work, we will evaluate quantities such as the 2nd Renyi entropy of ˆρD, which is an analogue of the von Neumann entropy, and it also provides an approximate character- ization [37] of the amount of quantum information lost due to entangling with the environment: S(2) = − log tr{(ˆρD)2} ∼ − log lim β→∞ � Dφ(x, τ) × exp � −S − � dx Lint(φ(x, 0), φ(x, β/2)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (7) This calculation is schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 2, which amounts to evaluating the partition function and free en- ergy of the system with an interaction between fields at imaginary time τ = 0 and τ = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Or we can also rotate the space-time (assuming there is a Lorentz symmetry), then the problem becomes evaluating the partition func- tion of the system with nonlocal interaction in space, but constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Other quantities of interest include tr{(ˆρD)2O(ˆφ)} or tr{ˆρDO(ˆφ)ˆρDO(ˆφ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We will show that these quanti- ties nonlinear with ˆρD would reveal some novel quan- tum phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the example we will discuss in the next section, the decoherence respects the doubled Zu 2 ×Zl 2 symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' but when we increase the decoherence strength, the 2nd Renyi entropy of the system (which captures the information loss to the environment) may encounter singularity at a critical decoherence strength, which corresponds to the spontaneous symmetry break- ing from Zu 2 × Zl 2 to the diagonal Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' DECOHERED WILSON-FISHER CRITICAL POINT In this section, we study two different scenarios for the system under weak measurements: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' III A, we keep measurement outcomes for post-selections and show that the resulting correlation functions linear in the density matrix ˆρD P exhibit novel behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' III B, we ig- nore measurement outcomes, which corresponds to gen- uine decoherence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' although correlation functions linear in ˆρD exhibit an ordinary Wilson-Fisher behavior in this case, we show that quantities non-linear in ˆρD still exhibit novel behaviors including the extraordinary-log critical- ity and information-theoretic phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Quantities linear with ˆρD P We consider physics near the O(N) Wilson-Fisher fixed point in (2 + 1)d, where x = (x, y) is a two-dimensional spatial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' First, we would like to consider quan- tities linear with ˆρD P , for example the correlation func- tion tr{ˆρD P ˆφ(0) · ˆφ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' To evaluate quantities such as tr{ˆρD P O( ˆφ(x))}, it boils down to evaluating path-integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Since we would like to keep at least the diagonal O(N) symmetry, Sint must be a function of the magni- tude of the order parameter |φ| at τ = 0, which is always equal to |φ| at τ = β due to the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The simplest term (and most relevant term) of Sint is Sint = � dxdy ε|φ(x, 0)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (8) The term Sint can be interpreted as weakly measuring the energy density or the order parameter φ of the system, followed by post-selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Although there is only one simple term in Sint, the physical consequence is already rather nontrivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Sint is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='ϕu ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='lz94m5zj7wvW+873Au9nb9c78E68Mw97yvL+9v7Z7g7TIZsWK6k9+8Zn6+93jNc/Ae5gHZn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='ÎflDÍÍ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Correlation functions in the doubled Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The red and blue lines represent the upper and lower Hilbert spaces respectively in the doubled Hilbert space (See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' III D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' a 2d interface in a (2 + 1)d cylindrical space-time with extra mass ε for the order parameter, and obviously ε is always a relevant perturbation, as the scaling dimension of |φ|2 is always smaller than 2 at the O(N) Wilson- Fisher fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' When ε > 0, Sint suppresses the fields at τ = 0, and “cuts” the connection between τ = 0+ and τ = β−, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' τ = 0+ and τ = β− become two boundaries with ordinary boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In this case, the correlation function tr{ˆρD P ˆφ(0) · ˆφ(x)} scales as [38] tr{ˆρD P ˆφ(0) · ˆφ(x)} ∼ 1 |x|2∆b φ , (9) where ∆b φ = 1 + 2 3N + O � 1 N 2 � (10) is the boundary scaling dimension of the order parameter φ to the first order expansion of 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Note that the value of ∆b φ is far larger than the bulk scaling dimension of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' When ε < 0, the latest progress of the boundary crit- icality indicates that Sint will drive the interface τ = 0 into an extraordinary-log criticality, which implies that the correlation function becomes tr{ˆρD P ˆφ(0) · ˆφ(x)} ∼ 1 (ln |x|)q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (11) Please note that, for a single exposed boundary against the vacuum, the extraordinary-log criticality exists only for N < Nc, with the critical Nc estimated around Nc ∼ 5 [26];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' but for an interface defect inserted in space-time, the theoretical prediction is that [36] Nc → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Quantities nonlinear with ˆρD Now we evaluate quantities nonlinear with ˆρD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Al- though our formalism can be straightforwardly general- ized for any higher orders of ˆρD, here we focus on the quantities that are quadratic in ˆρD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The quantities of interest include the 2nd Renyi entropy S(2), the corre- lation function C(2)(x), the “crossed” correlation func- tion C(2) X (x), and the “crossed-Ising” correlation function C(2) XI(x) defined as follows: S(2) = − log tr{(ˆρD)2}, C(2)(x) ∼ tr{(ˆρD)2 ˆφ(0) · ˆφ(x)}, C(2) X (x) ∼ � a tr{ˆρD ˆφa(0)ˆρD ˆφa(x)} C(2) XI(x) ∼ � a̸=b tr{ˆρD ˆφa(0)ˆφb(x)ˆρD ˆφa(0)ˆφb(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (12) The expression of the correlation functions above need to be divided by the purity of the decohered density matrix tr{(ρD)2} to be properly normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We note that these correlation functions are distinguished by their represen- tation under O(N)u × O(N)l symmetry, which becomes clear in the doubled Hilbert space as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As an example, we consider the following interaction term Sint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 7: Sint = � dxdy � W (|φ(x, 0)|, |φ(x, β/2)|) − w (φ(x, 0) · φ(x, β/2))2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (13) Here W is still a function of the magnitude of the order parameter at τ = 0 and τ = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The two terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 13 correspond to two different types of weak measurements (decoherence) channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The first term still corresponds to weakly measuring the energy density of the system, which preserves the doubled symmetry of the system, resulting in the coupling � dxdy W (|φ(x, 0)|, |φ(x, β/2)|) = � dxdy ε 2(|φ(x, 0)|2 + |φ(x, β/2)|2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (14) where the “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..” part includes quartic and higher-order terms in |φ(x, 0)| and |φ(x, β/2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The second w term can be rewritten as − w (φ(x, 0) · φ(x, β/2))2 ∼ − w N � a,b=1 (Qab(x, 0)Qab(x, β/2)) + · · · , (15) where Qab = φaφb− 1 N δab|φ|2 is the traceless rank-2 sym- metric tensor of the O(N) vector φ, and the ellipsis are terms that only depend on |φ| and can be absorbed into W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The w term can be interpreted as a weak measure- ment on the tensor Qab, and eventually all the measure- ment outcomes are summed with equal weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The w term breaks the SO(N)u × SO(N)l down to the diagonal SO(N), but still preserves the Zu 2 × Zl 2 symmetry, where Z2 corresponds to changing the sign of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 6 One can perform the Wick rotation in the (y, τ) plane, so the two interfaces at τ = 0, β/2 become spatial inter- faces at y = 0 and L/2, with L = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The interaction term then becomes Sint = � dτdx W � |φ(x, τ)|y=0, |φ(x, τ)|y=L/2 � − w � φ(x, τ)y=0 · φ(x, τ)y=L/2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (16) The first term W will either explicitly include an extra mass term ε|φ|2 at the interface y = 0 and y = L/2, or generate the mass term through renormalization group flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then depending on the sign of ε, there can be three possible scenarios: (1) If ε > 0, the extra mass term ε|φ|2 at the interface y = 0 and y = L/2 is a relevant perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The role of this extra mass term is to “cut” the system into two halves: the region from y ∈ (0, L/2) and region y ∈ (L/2, L ∼ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The relevant mass term will make the two interfaces at y = 0 and y = L/2 both at ordinary boundary criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In this scenario, w is obviously an irrelevant perturba- tion since both interfaces y = 0 and y = L/2 have ordi- nary boundary criticality, and the scaling dimension of φ is greater than 1 at the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Hence the directions of φ at the two interfaces are pretty much uncorrelated to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then we expect the correlation function, the crossed-Ising correlation, and the crossed-correlation functions to behave as C(2) ∼ tr{(ˆρD)2 ˆφ(0) · ˆφ(x)} ∼ 1 |x|2∆b φ , C(2) XI ∼ � a̸=b tr{ˆρD ˆφa(0)ˆφb(x)ˆρD ˆφa(0)ˆφb(x)} ∼ 0, C(2) X ∼ � a tr{ˆρD ˆφa(0)ˆρD ˆφa(x)} ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (17) For example, the crossed-correlation C(2) X corresponds to the correlation function between order parameters at the two different interfaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' since w is irrelevant when ε > 0, the directions of order parameters at the two in- terfaces are uncorrelated, hence C(2) X should vanish in the limit β, L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Another way to perceive these results is that, since w is irrelevant, the system should have a full SO(N)u × SO(N)l symmetry in the infrared, but the correlation function C(2) X and C(2) XI break the SO(N)u × SO(N)l symmetry, hence they must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (2) In the case with ε < 0, the extra local mass term ε is still a relevant perturbation, and it flows to the extraordinary-log criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then the w term (we as- sume w > 0) is a very relevant perturbation, and it will flow to w → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The fate of the system with strong w can be perceived through a mean field decoupling of Lint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We first recog- nize that the w term is analogous to the coupling between two sets of N´eel order parameters in the J1 − J2 Heisen- berg model on the square lattice, which has applications in the context of frustrated magnets and iron-pnictides superconductors [39–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Guided by the previous studies in these contexts, the most natural mean field decoupling of the w term is −w � φ(x, τ)y=0 · φ(x, τ)y=L/2 �2 ∼ −2wΦ(x, τ) � φ(x, τ)y=0 · φ(x, τ)y=L/2 � +wΦ(x, τ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (18) Here we have introduced an Ising order parameter Φ(x, τ) ∼ � φ(x, τ)y=0 · φ(x, τ)y=L/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The order param- eter Φ is analogous to the nematic order parameter in the J1 − J2 Heisenberg model on the square lattice, and the phase with large w is likely a phase with condensation of Φ, which spontaneously breaks Zu 2 × Zl 2 down to the diagonal Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The condensation of Φ will “pin” ˆφy=0 and ˆφy=L/2 along the parallel direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' With a nonzero condensate of Φ, the three correlation functions mentioned above behave like C(2) ∼ tr{(ˆρD)2 ˆφ(0) · ˆφ(x)} ∼ 1 (ln |x|)q , C(2) XI ∼ � a̸=b tr{ˆρD ˆφa(0)ˆφb(x)ˆρD ˆφa(0)ˆφb(x)} ∼ Const, C(2) X ∼ � a tr{ˆρD ˆφa(0)ˆρD ˆφa(x)} ∼ 1 (ln |x|)q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (19) The crossed-Ising correlation saturates to a nonzero con- stant in the limit of large |x| due to the condensate of Φ, which also leads to an extraordinary-log correlation of crossed-correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' When w flows to infinity and Φ condenses, there is only one O(N) symmetry in the infrared;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' hence all these correlations can be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (3) Now if ε = 0, with large but finite N, the scaling dimension of the traceless rank-2 symmetric tensor Qab is ∆ = 1+32/(3π2N) to the leading order of 1/N expan- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This means that w is weakly irrelevant with scaling dimension [w] = −64/(3π2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then the beta function of w should be β(w) = dw d ln l = − A N w + Cw2, (20) where A = 64/(3π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The constant C can be extracted through some OPE calculation, or simply a one-loop cal- culation in the large−N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Since C is positive, there is a fixed point at finite w∗, beyond which w will flow strongly, and the order parameter Φ defined above will condense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The phase diagram In phase diagram Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 4 we summarize the results related to (ˆρD)2 discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The order- disorder transition of Φ should extend to the region with 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Phase Diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The global phase diagram of decohered Wilson-Fisher critical point in terms of quantities nonlinear in ˆρD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' ε > 0, where there is a competition between ε which drives the interfaces to the ordinary boundary critical- ity and w which drives the crossed-Ising transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In this phase diagram, the critical wc is a function of ε: wc(ε) − wc(0) ∼ ε∆w/∆ε, and ∆w,ε are the scaling di- mensions of parameter w and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Since Φ(x) ∼ φ(x, 0) · φ(x, β/2) is the crossed-Ising order parameter, the crossed-Ising correlation function should show a transition from short-range to correla- tion while increasing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' If we consider tr{(ˆρD)2} as the partition function, and the 2nd Renyi entropy S(2) = − log tr{(ˆρD)2} as the free energy, the nature of the tran- sition at w = wc and ε > 0 should belong to a 2d classical Ising universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Without coupling to other modes with nonlocal correlations in the infrared, the order-disorder transition of an Ising order parameter in the 2d space should belong to the 2d Ising universality class, and here we should consider the perturbation of the coupling w on top of the 2d Ising transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In fact, if we start with a 2d classical Ising transition of order parameter Φ, the coupling −wΦ(x) (φ(x, 0) · φ(x, β/2)) is irrelevant knowing the fact that the scaling dimension ∆b φ > 1 for the ordinary boundary criticality at ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Hence at w = wc and ε > 0, the crossed-Ising correlation function should scale as C(2) XI(r) ∼ 1 |r|1/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (21) Also, when we increase w across wc, the 2nd Renyi en- tropy S(2) = − log tr{(ˆρD)2} should have the same sin- gularity as the free energy of the classical 2d Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Lattice Model and Doubled Hilbert Space We have shown that a decoherence channel whose Kraus operators are symmetric under O(N) can drive an Ising-type phase transition that spontaneously breaks the Zu 2 × Zl 2 ⊂ O(N)u × O(N)l symmetry down to the diagonal Z2 symmetry for quantities nonlinear in the den- sity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' most physical quantities are linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='paramagnet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='ℤu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='coupling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='∼ ZuZuZlZl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='paramagnet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='ℤl ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Toric code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Toric code ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Kramers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Wannier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Duality ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Anyon tunneling ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='È( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Ÿ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='eœ“‹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='u)( Ÿ eœ“‹ Ze,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='l)Í ≥ const FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Choi Isomorphism and Duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Schematic di- agrams illustrate how decohered mixed states map into the coupled bilayer system under the Choi-Jamiolkowski isomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Through the Kramers-Wannier duality, Z2 param- agnet under symmetric decoherence (left) maps into the Z2 toric code under dephasing noise (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Accordingly, their phase diagrams in the doubled Hilbert space is also dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' At p > p(2) c , both sides are characterized by non-vanishing correlation functions of operators, which cor- responds to (left) the formation of mean-field for Zv,uZv,l operator or (right) condensation of a pair of e-anyons euel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' in the density matrix, and this calls for a proper phys- ical interpretation of these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As we will see, our crossed-Ising transition is closely related to a better known information transition in the context of topologi- cal surface codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the following, we use another model to illustrate the essential physics of such a spontaneous symmetry break- ing (SSB) from Zu 2 × Zl 2 to the diagonal Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Instead of starting with a state at quantum criticality, we consider a concrete lattice model with a trivially disordered state |Ω0⟩ = |+⟩⊗N on a L × L square lattice with qubits de- fined on vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We consider the following quantum channel as a symmetric local decoherence model: Ee=(v,v′) : ρ → (1 − p)ρ + pZvZv′ρZvZv′, (22) where “e” labels the edge between the nearest-neighbor pair of sites v and v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The decoherence channel is given as the composition of local channels, E = � e Ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' To pro- ceed further, we apply the Choi-Jamiolkowski isomor- phism to map the density matrix into the pure state and decoherence channel into the operator in the doubled Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Under this mapping, the Choi state of the pure state density matrix ρ0 = |Ω0⟩⟨Ω0| is denoted as ∥ρ0⟩⟩ ≡ |Ω0⟩|Ω0⟩, which is nothing but a vectorized den- sity matrix, and the Choi operator of the decoherence channel is given as ˆE = � e=(v,v′) (1 − 2p)1/2eτZv,uZv′,uZv,lZv′,l (23) where tanh τ = p/(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 5, the isomorphism effectively maps the density matrix into the crossed-Ising order extraordinary-log + crossed-Ising order ordinary <0 > 0 88 pure state in the bilayer system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Now, when the system is subject to the decoherence, we can show that ∥ρD⟩⟩ = ˆE∥ρ0⟩⟩ is the ground state wavefunction of the following local Hamiltonian in the doubled-Hilbert space [9, 12]: Htot = Hu + Hl + Hint Hu(l) = − cosh 2τ � v Xv,u(l) Hint = � v cosh4 4τ � v′∈v (1 − Zv,uZv′,uZv,lZv′,l tanh 4τ), (24) where v′ ∈ v means that v′ is a nearest-neighbor site of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' At p = τ = 0, it reduces into two decoupled disordered states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' At p > 0, the decoherence gives rise to the (lo- cal) coupling between two layers that may induce a phase transition into an SSB phase that breaks the Zu 2 ×Zl 2 into the diagonal Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The (unnormalized) ground- state can be written as ˆE∥ρ0⟩⟩ = (1 − p)2Nv � l (tanh τ)|l||∂l⟩ ⊗ |∂l⟩ (25) where the summation is taken over all string config- urations l on the edges of the square lattice, |∂l⟩ ≡ � v∈∂l Zv|Ω0⟩, and Nv = L2 is the number of vertices in the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We stress that ˆE∥ρ0⟩⟩ is normalized in the sense that its corresponding density matrix ρD (under the Choi-Jamiolkowski isomorphism) is normal- ized with tr{ρD} = 1 in the single Hilbert space while it is unnormalized as a state in the doubled Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then, it is straightforward to show that the norm of the wavefunction ˆE∥ρ0⟩⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', the purity of the density ma- trix tr{(ρD)2} is equivalent to the partition function of the 2d Ising model at the temperature β = 2τ, as explic- itly derived in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Accordingly, at β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='441, which corresponds to p(2) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='178, the wavefunction in the doubled Hilbert space undergoes the transition of 2d Ising universality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Here the superscript in p implies that the transition happens in the quantity that involves the product of two density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The SSB transition between the Zu 2 × Zl 2-symmetric phase and the phase with only the diagonal Z2 symmetry in the doubled Hilbert space is captured by the suscepti- bility χ to an external symmetry-breaking field coupled to O = � v Zv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='uZv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='l (in the doubled Hilbert space): χ ≡ 1 L2 � ⟨⟨O2⟩⟩ − ⟨⟨O⟩⟩2� = 1 L2 �⟨⟨ρD∥O2∥ρD⟩⟩ ⟨⟨ρD∥ρD⟩⟩ − ⟨⟨ρD∥O∥ρD⟩⟩2 ⟨⟨ρD∥ρD⟩⟩2 � = � v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='v′ tr{ρDZvZv′ρDZvZv′} L2tr{(ρD)2} − �� v tr{ρDZvρDZv} L tr{(ρD)2} �2 (26) which is given by the summation over connected corre- lation functions of the order parameter Zv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='uZv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='l of this SSB transition in the doubled Hilbert space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' related to the crossed-Ising correlation function C(2) XI in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' χ is closely related to the crossed-Ising correlation functions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' χ diverges as |p − p(2) c |−7/4 at the SSB tran- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The exponent 7/4 is the signature of the 2d Ising universality class of this SSB transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Note that ⟨⟨O⟩⟩ in the definition of χ is the order parameter that takes a non-zero value in the symmetry-broken phase and should be evaluated with δ = 0+ after the thermodynamical limit is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We elaborate more on this in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the doubled Hilbert space, the definition of χ and the associated notion of the external symmetry-breaking field are standard for generic condensed matter systems with global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' III E, we explain their physical meaning at the level of the density matrix ρD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Correlation Functions and Physical Meanings In the previous sections, we have defined several cor- relation functions or physical quantities which are non- linear in decohered density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Accordingly, these quantities are not directly accessible from experiments, raising questions about what they really mean physically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In order to connect these quantities to experiments, we consider the signatures of the transition in the proba- bility distribution of measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' More pre- cisely, we will show that χ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (26) corresponds to the sensitivity of the decohered mixed state against small perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This, in turn, is related to the amount of information that can be obtained from measuring the mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' To proceed, we define the notion of distance between two different density matrices as the following: DJ(ρ, σ) ≡ tr(ρ log ρ − ρ log σ + σ log σ − σ log ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (27) This quantity is the quantum generalization of the Jef- freys divergence [44] (or symmetrized quantum relative entropy), which quantifies the distance between two den- sity matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' It satisfies good properties to be a valid metric between mixed states: (i) non-negative, (ii) van- ishing iff ρ = σ, and (iii) monotonically decreasing under the application of quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In order to discuss the “sensitivity” of the decohered density matrix, we define an infinitesimal variation of the original density matrix under symmetry breaking “per- turbation”, defined as the following quantum channel: Mδ,v : ρ → (1 − δ)ρ + δZvρZv ρδ ≡ Mδ[ρ], Mδ ≡ � v Mδ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (28) which amounts to the application of weak measurement in the Z-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Hence, δ also serves as a symmetry- breaking field in the doubled Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then, D(δ) ≡ DJ(ρD, ρD δ ) quantifies the difference between 9 the unperturbed and perturbed decohered density ma- trices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Accordingly, how fast the distance changes with δ tells how sensitive the decohered system is against weak measurement in the Z-basis (or how informative the Z- measurement is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This is captured by the second deriva- tive of the distance, which is related to the (classical) Fisher Information as F ≡ ∂2 δD(δ) �� δ=0 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' However, the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (27) is very challeng- ing to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' To proceed, we generalize the Jeffreys distance in a way similar to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 46: D(n)(ρ, σ) ≡ 1 n − 1 � log tr(ρn) + log tr(σn) − log tr(ρσn−1) − log tr(σρn−1) � (29) This n-th Jeffreys distance is symmetric and non- negative [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Furthermore, in the limit n → 1+, D(n) → DJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' At n = 2, we find that the expression behaves like an overlap between Choi states of two density matrices in the doubled Hilbert space description: D(2)(ρ, σ) ≡ − log � ⟨⟨ρ∥σ⟩⟩ � ⟨⟨ρ∥ρ⟩⟩ · ⟨⟨σ∥σ⟩⟩ � , (30) With this definition, we evaluate the derivatives of the “distance” between ρD and its perturbed version ρD δ with respect to δ (See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This quantity exhibits in- teresting behaviors when δ → 0 and L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The two different orders of limit lead to different results which have natural interpretations from a standard condensed matter perspective and from a quantum information per- spective respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the limit where the thermodynamical limit L → ∞ is taken first and δ → 0+ afterward, the conventional choice of limit for a condensed matter system experienc- ing SSB, the second-order derivative ∂2 δD(2) produces a quantity proportional to the susceptibility: lim δ→0+ lim L→∞ 1 L2 ∂2 δD(2) = χ, (31) which diverges at the SSB phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Based on the nature of this SSB transition and the interpretation of δ as the symmetry-breaking field, we conclude that the decohered density matrix undergoes a qualitative change (with respect to this distance) as a function of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Beyond critical p(2) c , the density matrix changes significantly un- der weak measurements in the Z-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' However, when we consider taking the limit δ → 0 be- fore the thermodynamical limit, the second-order deriva- tive produces a quantity that aligns with the information- theoretic intuition: lim L→∞ lim δ→0 1 L4 ∂2 δD(2) = � v,v′ tr{ρDZvZv′ρDZvZv′} L4tr{(ρD)2} = M, (32) where M can be interpreted as the expectation value of the squared order parameter, which vanishes in the para- magnetic phase but acquires a finite value in the SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Since this SSB transition belongs to the 2d Ising universality class, the standard phenomenology of the Ising model can be directly translated into the language of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In an Ising ferromagnet, the magnetic order of the ground state of the system is very sensitive to an infinitesimal external Zeeman field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This well-known fea- ture corresponds to the sensitivity of the density matrix to the decoherence in the Zv basis (as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (28)), in the phase where the Zu 2 × Zl 2 symmetry is spontaneously broken down to the diagonal Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This is quantified by the value of ∂2 δD(2)�� δ=0 (where δ is set to 0 before L grows large) scaling as L4 in the SSB phase, which aligns with the Fisher information of the GHZ state in the context of quantum metrology [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' It is straightforward to argue that the second deriva- tives of the distance D(n>2) in both limits would also exhibit similar behavior across a certain critical value (that depends on n), particularly because tr((ρD)n) is mathematically equivalent to the partition function of the coupled Ising model which is expected to undergo a transition from a paramagnetic to a ferromagnetic phase potentially at a different critical strength p(n) c ≥ p(2) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Ac- cordingly, we expect the behavior of ∂2 δD(n) to extrapo- late in the limit n → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Therefore, there should be two different phases separated by a critical point at strength p = p(1) c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This is indeed the case, as we will see by using a dual description of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Duality, Intrinsic transition, and Decodability Under the Kramers-Wannier (KW) duality, the Z2 paramagnet under symmetric decoherence maps to the toric code under dephasing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The toric code un- der dephasing noise, in turn, is well known to exhibit an information transition at critical noise strength, beyond which the quantum information (logical qubits) stored in the toric code is not decodable [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This hints at an inti- mate connection between the criticality discussed above and a well-known information transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' First, we illustrate how the doubled Hilbert space for- malism provides us some insights into the transition in the decohered toric code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Under the KW duality (See Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A 5), the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (24) maps to the two copies of toric code coupled by local anyon tunnel- ing terms, as schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' At critical strength of the tunneling, the anyon condenses and the topological order reduces into a single toric code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The critical behavior is associated with a Higgs transition signified by the development of the anyon condensation amplitude ∼ ⟨⟨ρD∥ˆγuˆγl∥ρD⟩⟩, where ˆγu/l = � e∈γ⊥ Ze,u/l creates a pair of e-anyon at the end of the string γ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In turn, this implies that ˆρD and ˆγˆρDˆγ have an apprecia- ble overlap and they become less and less distinguishable for p > p(2) c in the doubled Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Without us- ing the KW duality, the existence of the criticality can be directly seen by calculating the purity of the deco- 10 hered toric code, which is given by the partition func- tion of the 2d Ising model at β = tanh−1(1 − 2p)2 as shown in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Note that the corresponding Ising model goes from an ordered phase to a disordered phase as we increase p, which is the behavior opposite to that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' However, both undergo transitions at the same critical point p(2) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='178 as expected from the KW duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As a next step, we calculate the von Neumann entropy of the decohered toric code state, a quantity highly non- linear in the density matrix that enters the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Following the calculations in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A 2, we can show that the von Neumann entropy of the de- cohered density matrix is proportional to the free en- ergy of the random bond Ising model along the Nishi- mori line [49, 50] at β = tanh−1(1 − 2p), which is known to be critical at p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='1094 [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Therefore, the transi- tion behavior indeed extends down to the n → 1 limit for a toric code under dephasing noise, and by the du- ality, for a paramagnetic under symmetric decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As remarked, the transition behavior in the limit n → 1 corresponds to the singularity of the Fisher information, a well-known information-theoretic quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We remark that this series of transitions for different n is intrin- sic to the decohered density matrix (and the distance in use), as these transitions are associated with sponta- neous symmetry breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Interestingly, the n → 1 limit of this intrinsic transition coincides with the decodability transition point demonstrated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 31, which is based on a certain decoding procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Such a connection may imply that the intrinsic transition point provides an up- per bound for some information retrieval protocols to be successful in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' NON-LOCAL OPERATORS As we mentioned previously, without any post- selection, the correlation function for local operators lin- ear with the decohered density matrix should have the same scaling as the undecohered correlation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' But in this section, we will show that nonlocal quantities can still have qualitatively different behaviors even if we only consider expectation values linear with the density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 1d Quantum Rotor To illustrate the behavior of nonlocal operators un- der decoherence, let us start with the 1d systems with a description in terms of the quantum rotor, such as the spin-1/2 chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In terms of the Abelian bosonization, the N´eel and valence bond solid (VBS) order parameters of a spin-1/2 chain are represented as (N x, N y, N z, V ) ∼ (sin θ, cos θ, sin φ, cos φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (33) Under decoherence or weak measurement of (for exam- ple) local operator ei ˆφ, the density matrix becomes ˆρD = E[ˆρ0], E = � x Ex, Ex[ˆρ0] ∼ (1 − p)ˆρ0 + p 2 ei ˆφ(x)ˆρ0e−i ˆφ(x) + p 2 e−i ˆφ(x)ˆρ0ei ˆφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (34) ˆρ0 is the undecohered density matrix of the spin-1/2 chain, and ˆρD = E[ˆρ0] still keeps tr[ˆρD] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We can first evaluate the correlation function of local operator O(r): CD(r) = tr{ˆρD ˆO(r) ˆO(0)} = tr{ˆρ0E[ ˆO(r) ˆO(0)]}, Ex[ ˆO(r) ˆO(0)] ∼ (1 − p) ˆO(r) ˆO(0) + p 2 ei ˆφ(x) ˆO(r) ˆO(0)e−i ˆφ(x) + p 2 e−i ˆφ(x) ˆO(r) ˆO(0)ei ˆφ(x) (35) For local bosonic order parameters, such as the N´eel and VBS, although they may have nontrivial commutation with ei ˆφ(x) at the vicinity of x, at long distance apart, ˆO(r) and ei ˆφ(x) should commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Therefore, the corre- lation functions of local operators acquire only a con- stant amount of corrections under decoherence, and any local order parameter should still have the same power- law with scaling dimensions of the undecohered spin-1/2 chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' However, for nonlocal operators, the situation can be very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A particular family of nonlocal opera- tors is the disorder operators [52–59] [60], which have attracted great interests recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' These operators have been used as an auxiliary diagnosis for the states of mat- ter, especially for critical states of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' For a 1d quan- tum rotor, if we view ˆφ as the phase angle of a local boson creation operator, then (∇xˆθ)/2π is the boson density ˆnφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The following operator is called a disorder operator ˜Or = exp � iα � r 0 dx ˆnφ(x) � = exp � i α 2π (ˆθ(r) − ˆθ(0)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (36) If ˆφ carries a full U(1) symmetry, then α ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' if ˆφ only carries a ZN symmetry, then α = 2πk N with k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', N}, as ˆnφ is defined modulo N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' For example, for the spin-1/2 chain, if we view ˆφ as 11 a local operator, then eiˆθ/2 is a disorder operator of ˆφ, and eiˆθ/2 plays two roles simultaneously: it first creates a fractionalized spin-1/2 excitation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' a spinon), it also creates a domain wall of the VBS order parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' eiˆθ(r)/2 shifts ˆφ(x) → ˆφ(x) + π for x < r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Indeed, it is well-known that a spin-1/2 is localized at the domain wall between two VBSorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We can evaluate the correlation function of eiˆθ/2 for the decohered density matrix: CD(r) = tr{ˆρDeiˆθ(r)/2 e−iˆθ(0)/2} ∼ e−r/ξtr{ˆρ0 eiˆθ(r)/2 e−iˆθ(0)/2}, (37) where tr{ˆρ0eiˆθ(r)/2 e−iˆθ(0)/2} is the correlation function of eiˆθ/2 for undecohered spin-1/2 chain, and the “correlation length” ξ is ξ ∼ −1/ ln(1 − 2p) for small p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Hence our calculation for the 1d quantum rotor sys- tem implies that, although the disorder operator has a power-law correlation with the absence of decoherence, it can be rendered short-ranged under decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As we will show in the next subsection, similar behavior of the disorder operator happens in higher dimensions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The spin-1/2 chain is also an example of spin liquid with fractionalized spinon excitations since the spinon corre- lation function decays as a power-law in the undecohered spin-1/2 chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' But under decoherence or weak measure- ment on the VBS order parameter the spin chain loses its fractionalization, as the spinon operator decays exponen- tially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This can be intuitively understood as the fact that, if the VBS operator is “measured”, in each measurement outcome the system is pinned to a certain particular VBS pattern, which leads to confinement in this measurement outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The confinement persists even if we average over all measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (2 + 1)d quantum critical points with a U(1) or ZN symmetry Now let us consider a (2 + 1)d QCP or CFT with a global U(1) symmetry, or ZN symmetry that can be em- bedded into a U(1) that emerges in the infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This U(1) symmetry is dual to a noncompact U(1) gauge field [61–63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We always turn on decoherence on the scalar boson creation operator which carries the U(1) charge, or equivalently the monopole operator of the dual U(1) gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The decohered density matrix takes the same form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 34: ˆρD = E[ˆρ0], E = � x Ex, Ex[ˆρ0] ∼ (1 − p)ˆρ0 + p 2 ei ˆφ(x)ˆρ0e−i ˆφ(x) + p 2 e−i ˆφ(x)ˆρ0ei ˆφ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (38) Here ei ˆφ(x) is the monopole operator, which creates a scalar boson, or a 2π gauge flux at location x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We evaluate the expectation value of the following quantity defined for a closed loop C = ∂A: ˜OC = exp � � x∈A,∂A=C i2π N ˆnφ(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (39) In the undecohered density matrix, and in the dual for- malism, this quantity reduces to the evaluation of the Wilson loop: ⟨ ˜OC⟩ ∼ ⟨exp � i N � C dx · ˆa(x) � ⟩, (40) and as was shown previously, it should obey a perimeter law, with a universal logarithmic contribution from the sharp corners of the loop C [56, 57, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The coefficient of the universal logarithmic contribution arising from the corner is proportional to the universal conductivity of the scalar boson current at the (2+1)d CFT in the AC limit ω/T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' However, under decoherence, the operator ˜OC will shift ˆφ(x) by angle 2π/N for x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Hence we expect the decoherence to change the behavior of ⟨ ˜OC⟩ significantly: ⟨ ˜OC⟩ ∼ � (1 − p) + p cos �2π N ��A , (41) namely ⟨ ˜OC⟩ should now decay with an area law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Just like the 1d example discussed in the previous subsection, an area law decay of the Wilson loop is a sign of confine- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' It means that the vortex of the U(1) boson, which is also the gauge charge of the dual gauge field ˆa, should be confined under decoherence of the scalar boson cre- ation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Here we would like to remark that, one of the tools for diagnosing fractionalization and deconfine- ment is the dynamic structure factor, where the fraction- alization would lead to a continuum [65–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Computing real-time dynamics is beyond the current set-up of our current manuscript as it requires the formalism that in- volves the Lindbladian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Here we use the behavior of the Wilson loop as the sign of confinement/deconfinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' SUMMARY AND FUTURE DIRECTIONS In this study, we examined the effects of decoherence and weak measurement on quantum critical points in (2+1)d space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We found that this problem is math- ematically equivalent to the boundary or defect critical- ity of (2 + 1)d conformal field theories, which have been extensively researched in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Our results indi- cate that when a QCP is exposed to decoherence or weak measurement, observers may observe peculiar behaviors, including the extraordinary-log correlation recently dis- covered in the context of boundary criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Addition- ally, as the strength of decoherence or weak measurement increases, the system can experience an information- theoretic transition that is captured by quantities non- linear in the decohered density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This transition is 12 linked to spontaneous symmetry breaking when we con- sider quantities to the n-th power of the density matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' in particular using the “doubled formalism”, we show that for n = 2, this transition belongs to the 2d Ising universality class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' There are many related directions that are very much worth exploring in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We list two such directions as follows: (1) Unconventional quantum criticality under decoher- ence: It is known that in the world of quantum many- body systems, there are two types of quantum critical points: conventional and unconventional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Conventional QCPs correspond to quantum phase transitions between a disordered state with a direct product structure and a state that breaks a certain symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This type of QCPs has classical analogs such as the Wilson-Fisher fixed points discussed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' But it is known that there is another large class of unconventional QCPs with- out a simple classical analogue [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' On the other hand, unconventional QCPs are those that do not have a simple classical analog and may involve transitions between two ordered phases with different symmetries, or between an ordered phase and a topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The most well- known example of unconventional QCPs is the deconfined QCP [69, 70], which has many desirable phenomena such as deconfinement and a duality web as was summarized in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' It is reasonable to expect that the unconven- tional QCPs under decoherence can also be mapped to certain boundary criticality problems, and it is going to be an unusual boundary criticality with unconventional QCP in the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As we have already seen in the current work, decoherence may be at odds with deconfinement, as deconfinement is often signified by nonlocal operators such as the disordered operators or the Wilson loops, whose behavior can be strongly affected by decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' It would be interesting to study the fate of unconven- tional QCPs under decoherence in general in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (2) The Strange Correlator: The notion of a strange correlator was originally proposed as a tool to diagnose SPT states using their bulk wave functions [17], rather than edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The strange correlator is defined as the following quantity CS(r) = ⟨Ω| ˆO(0) ˆO(r)|Ψ⟩ ⟨Ω|Ψ⟩ , (42) where |Ψ⟩ is the wave function that awaits diagnosis, and |Ω⟩ is the trivial direct product disordered state with the same symmetry G and Hilbert space as |Ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' ˆO is an order parameter that carries a nontrivial representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The arguments given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 17 suggest that, al- though the ordinary correlation functions in both |Ψ⟩ and |Ω⟩ must be short-ranged, this strange correlator Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 42 must have either long-ranged or power-law correlation, for 1d and 2d states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the past decade, the strange correlator has been used as a tool for both conceptual understanding and numerical diagnosis for SPT states and also topological states [72–93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' One of the future directions worth pursuing is the strange correlator between a quantum critical state |Ω⟩, and an SPT wave function |Ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Let us still focus on two- dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Using the formalism developed in this work, this problem may be mapped to the 2d inter- face between a quantum critical point on the temporal domain τ < 0, and an SPT state on the other domain τ > 0 in the Euclidean space-time path-integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Un- der Wick-rotation, the strange correlator is mapped to the spatial interface between a quantum criticality and an SPT state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This kind of interface has two types of boundary effects: the boundary states arising from the bulk topology, and also the boundary criticality originat- ing from the bulk critical modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This is a subject under very active research lately, both theoretically and numer- ically [18–22, 24, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In particular, some novel interface criticality especially a (1 + 1)d deconfined quantum crit- ical point was identified in the literature [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' One po- tentially highly interesting direction in the future is to analyze the strange correlator (and its generalized form defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 12) between quantum criticality and SPT state, and explore the possible novel phenomena, espe- cially when either the bulk quantum critical state |Ω⟩, or the SPT state |Ψ⟩, or both are under decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' ACKNOWLEDGMENTS We thank Ehud Altman, Soonwon Choi, Matthew P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Fisher, Sam Garrett, Yi-Zhuang You for inspiring dis- cussions and previous collaborations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' is supported by the Gordon and Betty Moore Foundation under the grant GBMF8690 and by the National Science Founda- tion under the grant PHY-1748958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' acknowledges the support from the Simons Foundation through the Si- mons Investigator program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' is supported by a faculty startup grant at Cornell University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Note Added: While finishing up this work, we became aware of an independent related work [94, 95], which should appear on arXiv on the same day as our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Appendix A: Toric code under decoherence The goal of this section is to show that the entan- glement entropy of the toric code state under dephasing noise is given by the free energy of the random bond Ising model along the Nishimori line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' We will show that such a decohered state is dual to the disordered product state under Z2 symmetric decoherence channel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' First, the toric code Hamiltonian is defined as H = − � v � e∋v Ze − � p � e∈p Xe = − � v Av − � p Bp (A1) 13 The ground state is characterized by Av = Bp = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Fur- thermore, on the torus, the ground state is 4-fold degen- erate with two logical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Logical qubits reside on the space where the following effective Pauli operators act on CX i ≡ � e∈Ci Xe and CZ i ≡ � e∈C⊥ i Ze where Ci is a cycle along the i-th axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' while Ci is along the bond, C⊥ i crosses the bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Note that {CX 1 , CZ 2 } = {CZ 1 , CX 2 } = 0, while [CZ i , CX i ] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' As an example we choose to study one of the four ground states denoted as |ψtc⟩, whose pure state density matrix is ρtc = |ψtc⟩⟨ψtc|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The ground state is the eigenstate of the qubits CX i |ψtc⟩ = ai|ψtc⟩, with a1 = a2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Decomposition We consider the toric code ground state decohered un- der the following channel: Ee : ρ → (1 − p)ρ + pZeρZe, E = � e Ee (A2) In order to understand the structure of E[ρtc], first we evaluate the matrix elements of the decohered toric code density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' To do the job, consider ρs,s′ ≡ |Ωs′⟩⟨Ωs| where |Ωs⟩ is a generic product state characterized by s = {se}, se = ±1: |Ωs⟩ ≡ � e Z(1−se)/2|+⟩⊗2Nv, (A3) where Nv = L2 is the number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then ⟨Ωs|E[ρtc]|Ωs′⟩ = tr(ρs,s′E[ρtc]) = tr(E[ρs,s′]ρtc) (A4) where E[ρs,s′] is given as ρs,s′ = 1 22Nv � e (1 + seXe)Z(1−ses′ e)/2 e E[ρs,s′] = 1 22Nv � e (1 + se(1 − 2p)Xe)Z(1−ses′ e)/2 e (A5) For tr(E[ρs,s′]ρtc) not to vanish, ∂(s · s′) = 0 so that product of Ze does not create anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In such a case, the product of Z-strings always commutes with a loop of X-strings along the bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In fact, we can show that ⟨ψtc| � e∈l Xe � e Z(1−ses′ e)/2 e |ψtc⟩ = F(l, s · s′)δ∂l,0δ∂(s·s′),0 (A6) where s · s′ defines a non-trivial link configuration along the dual link whenever it takes a negative value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' For ⟨CX,Y,Z 1,2 ⟩ = cx,y,z 1,2 we have F(l, s · s′) = ⟨ψtc|CX h(l)CZ h(s·s′)|ψtc⟩ (A7) where h(l) ∈ π1(T2) is the element of the homotopy group of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' For the simplest case where CX 1,2 = 1, we remark that F(l, s · s′) is non-zero iff h(s, s′) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then, the overlap in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A4) is given as tr(E[ρs,s′]ρtc) = 1 22Nv � l (1 − 2p)|l| � e∈l se × ⟨ψtc| � e∈l Xe � e Z(1−ses′ e)/2 e |ψtc⟩ = δh(s·s′),1 2Nv(2 cosh β)2Nv ZRBIM[s, β] (A8) where the summation is over all possible edge configura- tion l, tanh β = (1 − 2p), and ZRBIM[s, β] ≡ � σ � e=(v,v′) eβseσvσv′ , (A9) which turns out to be the partition function of an Ising model with random signs, specified by {se}, in the near- neighbor spin-spin interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' At p = 0, β−1 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', zero temperature limit, It has an interesting consequence: the matrix element ⟨Ωs|E[ρtc]|Ωs′⟩ vanishes unless s and s′ belong to the same equivalence class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', ∂(s·s′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Furthermore, if s ∼ s′, then ZRBIM[s, β] = ZRBIM[s′, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Accordingly, E[ρs] is block-diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Therefore, E[ρtc] decomposes as the following: E[ρtc] = � s,s′ ρs,s′tr(ρs,s′E[ρtc]) = 1 2Nv(2 cosh β)2Nv � m ZRBIM[sm, β] ρm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A10) where the summation is taken over the equivalence class of s, denoted by m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' the equivalence class is defined as ∂(s · s′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' sm is the representative of the equivalence class m, and ρm is defined as ρm ≡ � s,s′∼sm |Ωs⟩⟨Ωs′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A11) Therefore, the decohered density matrix has the following block-diagonal structure (in X-basis) E[ρtc] = � ����� B1 0 0 · · 0 B2 0 · · 0 0 B3 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' � ����� (A12) where each block is 2Nv−1 by 2Nv−1 dimensional matrix labeled by the equivalence class m and its entries are all equal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=', Bi ∝ � ��� 1 1 1 · · · 1 1 1 · · · 1 1 1 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' � ��� � �� � 2Nv−1 � � � � � � � 2Nv−1 = 2Nv−1|φm⟩⟨φm| (A13) 14 where |φm⟩ = 1 √ 2Nv−1 � s∼sm |Ωs⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' There are total 2Nv+1 equivalence classes originated from (Nv − 1) in- dependent stabilizers Bp and two logical operators CX i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Entanglement Entropy After reorganizing terms, we get ρD tc = � m pm|φm⟩⟨φm|, pm = ZRBIM[sm, β] 2 · (2 cosh β)2Nv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A14) Note that the entanglement entropy has a very interest- ing structure: S = −tr(ρD tc ln ρD tc) = − � m pm log pm ∝ − � m Z[sm, β] log Z[sm, β] (A15) which is nothing but a disorder averaged random bond Ising model’s free energy along the Nishimori line, whose transition point is located at pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='1094 [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Therefore, there is an intrinsic phase transition of the entanglement entropy of the decohered toric code state at pc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='1094, which coincides with the decodability transition point ob- tained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Purity Interestingly, the purity of the decohered density ma- trix maps to the partition function of the Ising model: (Here C ≡ (22 · (2 cosh β)4Nv)−1): tr((ρD tc)2) = C � m Z2 RBIM[sm, β] = C � s Z2 RBIM[s, β] 2Nv−1 = C(cosh β)4Nv 2Nv−1 � s � σ � σ′ � L1,L2 (1 − 2p)|L1|+|L2| × � e∈L1 se � e′∈L2 se′ � i∈∂L1 σi � j∈∂L2 σ′ j (A16) where Li is the link configuration defined on the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' This expression can be further simplified by the following: tr((ρD tc)2) = 1 24Nv+Nv+1 � σ � σ′ � L1,L2 (1 − 2p)|L1|+|L2| × 22NvδL1,L2 � i∈∂L1 σi � j∈∂L2 σ′ j = 1 24Nv+Nv+1 � L1,L2 (1 − 2p)|L1|+|L2| · 22NvδL1,L2 × 2Nvδ∂L1,0 · 2Nvδ∂L2,0 = 1 2Nv+1 � L (1 − 2p)2|L|δ∂L,0 = ZFIM[β′] 2(2 cosh β′)2Nv , tanh β′ = (1 − 2p)2 (A17) where ZFIM[β′] = ZRBIM[1, β′] is the partition function of the ferromagnetic Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' In the last equality, we used that � ∂γ=0 (tanh β)|γ| � e∈γ se = ZRBIM[s, β] 2Nv(cosh β)2Nv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A18) Since the ferromagnetic 2d Ising model has a transition at β = ln � 1 + √ 2 � /2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='441, correlation functions in the doubled Hilbert space (in the next section) would exhibit a critical behavior at p(2) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Choi Isomorphism One may study the decohered toric code state in the doubled Hilbert space under Choi isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The decohered density matrix maps into the following Choi state: ∥E[ρtc]⟩⟩ = � i |i⟩ ⊗ (E[ρtc]|i⟩) = � m pm|φm⟩|φm⟩ (A19) where we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The dephasing channel maps to the following Choi operator in the doubled Hilbert space: ˆE = � e (1 − 2p)1/2eτZe,uZe,l, tanh τ = p 1 − p (A20) which can be considered as an imaginary time evolu- tion by an Ising Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Note that cosh τ = (1 − p)/√1 − 2p and sinh τ = p/√1 − 2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then, we see that ˆEXe,u ˆE−1 = Xe,ue−2τZe,uZe,l ⇒ ˆEBp ˆE−1 = Bp � e∈p e−2τZe,uZe,l (A21) 15 Now, consider the following parent Hamiltonian: Hparent = 1 2 � v (1 − Av)†(1 − Av) + 1 2 � p (1 − Bp)†(1 − Bp) (A22) for some α, β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Following the procedure in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 12, we can show that the Choi state ∥E[ρtc]⟩⟩ = ˆE∥ρtc⟩⟩ is the ground state of the following Hamiltonian: ˆHD = ˆHD u + ˆHD l + ˆHD int ˆHD u = − � v Av,u − � p cosh(2τ)Bp,u ˆHD int = � p � e∈p (cosh 4τ − Ze,uZe,l sinh 4τ) (A23) In this doubled system, the coupling ˆHD int breaks two mi- croscopic magnetic one-form symmetries into their diag- onal subgroup, and we expect the phase transition from a doubled toric code order to a single toric code order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The transition should be captured by the condensation of e- anyons, which is diagnosed by non-vanishing expectation values of ⟨( � e∈γ⊥ Ze,u)( � e∈γ⊥ Ze,l)⟩ ∼ const, (A24) where γ⊥ is the open string defined along the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Kramers-Wannier Duality Under Kramers-Wannier duality, the toric code maps to the trivially disordered state, |Ω0⟩ = |+⟩⊗N on the vertices of the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The mapping is explicitly given as the following: � e∋v Xe ↔ Xv Ze⊥(v,v′) ↔ ZvZv′ (A25) Note that in this dual model, the system has two 0-form Z2 symmetries even under decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Also, v that la- bels the vertices in the dual lattice labels the plaquette in the original lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Since � v Xv is dual to � p Bp = 1 in the toric code, the mapped states must be symmetric un- der the 0-form Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The above relation makes it clear that the dephasing channel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A2) maps to the Z2 symmetric decoherence channel in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Now, applying the Kramers-Wannier duality on the doubled Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' we can obtain the Hamiltonian for the dual model in the doubled Hilbert space as the following: ˆHD = ˆHD u + ˆHD l + ˆHD int ˆHD u = −2 cosh 2τ � v Xv ˆHD int = 2 � v � v′∈v (cosh 4τ − Zv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='uZv′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='uZv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='lZv′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='l sinh 4τ) (A26) In this model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The transition should be captured by the development of an order parameter that breaks off- diagonal Zu 2 × Zl 2 symmetry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' which is diagnosed by non- vanishing expectation values of ⟨(Zv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='uZv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='l)(Zv′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='uZv′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content='l)⟩ ∼ const (A27) for any well separated v and v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Purity calculation As stated in the main text, the (unnormalized) ground- state of the above Hamiltonian in the doubled Hilbert space is given as ˆE∥ρ0⟩⟩ = (1 − p)2Nv � l (tanh τ)|l||∂l⟩ ⊗ |∂l⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A28) The norm of this wavefunction is given as tr{(ρD)2} = ⟨⟨ρD∥ρD⟩⟩ ∝ � l1,l2 δ∂l1,∂l2(tanh τ)|l1|+|l2| ∝ � {sv} � � e � 1 + tanh τ � v∈∂e sv ��2 ∝ � {sv} � e � 1 + tanh 2τ � v∈∂e sv � ∝ ZFIM[2τ] (A29) where we used the following identity: δ∂l1,∂l2 = 1 2Nv � sv∈{±1} � v∈∂l1 sv � v′∈∂l2 sv′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A30) Note that at p = p(2) c , 2τ here agrees with β′ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (A17), which establishes the self duality in the 2d Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Appendix B: Derivatives of the Distance In this section, we calculate the second derivative of the distance defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (30) for the decohered density matrix ρD = E[|Ω0⟩⟨Ω0|] from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' The perturba- tion is defined by the following channel Mδ,v : ρ → (1 − δ)ρ + δZvρZv ρδ ≡ Mδ[ρ], Mδ ≡ � v Mδ,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (B1) 16 By defining h such that tanh h = δ/(1 − δ) (eh = 1/ √ 1 − 2δ), the channel can be mapped to the follow- ing operator under Choi isomorphism: ˆ Mδ ≡ � v (1 − 2δ)1/2ehZv,uZv,l (B2) Note that ∂h/∂δ = 1/(1 − 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then, we can show that ∂δD(2) = ⟨⟨ρD δ ∥∂δρD δ ⟩⟩ ⟨⟨ρD δ ∥ρD δ ⟩⟩ − ⟨⟨ρD∥∂δρD δ ⟩⟩ ⟨⟨ρD∥ρD δ ⟩⟩ ∂2 δD(2) = ⟨⟨∂δρD δ ∥∂δρD δ ⟩⟩ ⟨⟨ρD δ ∥ρD δ ⟩⟩ − � ⟨⟨ρD∥∂δρD δ ⟩⟩ ⟨⟨ρD∥ρD δ ⟩⟩ �2 (B3) Let O ≡ � v Zv,uZv,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then, at δ → 0, we evaluate that ∂δMδ �� δ→0 = (O − L2) (B4) Furthermore, exactly at δ = 0, ⟨⟨O⟩⟩ �� δ=0 = 0 due to the symmetric nature of the initial decohered density matrix under symmetric decoherence channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' Then, plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (B4) into the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (B3), it is straightforward to show that the first derivative of the distance vanishes as ex- pected, and the second derivative, depending on the or- der of limit, would be given as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (31) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/o9E4T4oBgHgl3EQfvA3U/content/2301.05238v1.pdf'} +page_content=' [1] S.' metadata={'source': 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Pittman +400 Dowman Drive, Atlanta +Abstract +We greatly expand upon the results of Kochengin, Oliker and Tempeski [1] to +include results for uniqueness in the general case. We also include results for +existence in the rotationally symmetric case and the case where the target set +is sufficiently small. We also point out an error that was found in [1]. +Keywords: +partial differential equations, geometric optics, geometry +2020 MSC: 78A05, 35, 51, 53 +1. Introduction +Let O be the origin of R3, and let S2 be the unit sphere centered at O. We +treat points on S2 as unit vectors with initial points at O. Let an aperture be +a subset of S2; in our work, the aperture will be an open set. Physically, it +makes sense to consider O as the location of an anisotropic point source of light +such that rays of light are emitted in a set of directions defined by an aperture +D ⊆ S2. +Definition 1.1. Assume that we are given an aperture that is a connected open +set D ⊆ S2, and a function ρ : D → (0, ∞) that is continuous and almost +everywhere differentiable. Then a reflector is the set R = {mρ(m)|m ∈ D} ⊂ +R3. +Email address: dpittm2@emory.edu (Dylanger S. Pittman) +Preprint submitted to Inverse Problems +January 6, 2023 +arXiv:2301.02106v1 [math.AP] 5 Jan 2023 + +We first recall the classical law of reflection. Assume that we have a contin- +uous, almost everywhere differentiable, positive function ρ : D → (0, ∞) and a +corresponding reflector R = {mρ(m)|m ∈ D}. Suppose that a ray originating +from O in the direction m ∈ D is incident on the reflector R at the point mρ(m). +If ρ is differentiable at m, there is a unit vector, n(m), normal to the reflector +R at mρ(m). Therefore, by the reflection law of geometric optics, a ray from O +of direction m reflects off the point mρ(m) in the direction +y(m) = m − 2⟨m, n(m)⟩n(m) +(1) +where ⟨m, n(m)⟩ is the standard Euclidean inner product in R3 and n(m) is +oriented such that ⟨m, n(m)⟩ > 0 [2]. +Definition 1.2. Assume that we are given an aperture that is a connected open +set D ⊆ S2. Let U be an open subset of S2 such that D ⊆ U. Consider a function +ρ : U → (0, ∞) that is continuous and almost everywhere differentiable. Then a +refractor is the set R = {mρ(m)|m ∈ U} ⊂ R3. +Note that R = {mρ(m)|m ∈ D} can be considered as either a reflector or +a refractor. If R = {mρ(m)|m ∈ D} is considered as a refractor, the refracted +direction ˆy is determined by Snell’s law and is given as +ˆy(m) = cfm − +�� +1 − c2 +f(1 − ⟨m, n(m)⟩2) − cf⟨m, n(m)⟩ +� +n(m) +(2) +where cf denotes the refraction index. +We borrow the following motivation from [1]. Consider a two-sheeted hy- +perboloid of revolution with sheets B and H. Let O be the focus inside the +convex body bounded by the first sheet B and x the focus inside the convex +body bounded by the sheet H. Suppose that a point source of light is positioned +at O and the sheet H is a reflector. H has very special and important reflecting +properties. Specifically, if a ray of direction m from O is incident on a point +z ∈ H and is reflected in the direction y(m) as defined by (1), then the ray from +z of direction y(m) coincides with a ray from x of direction y(m). This means +that the focus x can be viewed, from a physical perspective, as a virtual source +2 + +of rays reflected off H. A two-dimensional analog of this situation is illustrated +in Figure 1. +This same situation can also be interpreted from a different point of view +allowing us to treat it geometrically as a refraction problem, rather than a +reflection problem. Now suppose a light ray of direction m from O strikes H +and ‘refracts’ such that the refracted direction is given by +ˆy = −y = −m + 2⟨m, n(m)⟩n(m). +(3) +Then since the refracted direction is the opposite of the reflected direction, every +ray of direction m that strikes the refractor H will cross the focus x. Equation +(3) can also be considered as the version equation of (2) where cf = −1. Under +the law of total energy conservation, the total energy ‘delivered’ by the refractor +H to the point x will be equal to the total energy produced by the source O. +We will only discuss this type of refraction, where cf = −1, for the rest of the +dissertation. +This interpretation of the reflection with a virtual source as a particular +case of refraction is convenient from a geometric point of view and we use this +terminology throughout this paper. Physically, however, it is more natural to +treat the point x as a virtual source. This would also be consistent with the +case of a distributed virtual source; which we focus on. +To quote [1]: with this terminology, the problem studied in this paper can +now be described as a problem of finding a convex refractor R which will refract a +given anisotropic bundle of rays from a source O in such a way that the refracted +rays are incident on a specified set in space and produce there, a given-in- +advance intensity distribution. More specifically, suppose that we have a system +consisting of an anisotropic point source at O, an aperture D, a nonnegative +g(m) ∈ L1(S2), a target set T ⊂ R3 \ {O}, and nonnegative integrable function +f defined on T. The problem consists of finding a refractor R which produces +the specified in advance f on T. Henceforth, we call this problem the refractor +problem. +The only previous work available with respect to this problem can be found +3 + +S2 centered at the origin O +light rays +Target set consisting of a single point +The hyperboloid H +Figure 1: Here is an illustration of a virtual source reflector system where the target set is +a single point. Note that all the rays of light reflect off of the hyperboloid H such that it +appears that the light is originating from the target point. +4 + +in [1]. In the paper, they develop a definition of a weak solution to a PDE +of Monge Amp`ere type; specifically, the PDE described by equation (4) in [1]. +They detail the construction of simply connected convex refractors and provide +an existence theorem for the case where the target set is discrete (Theorem +5.1). Due to the weak convergence of Dirac measures to Lebesgue measures, +one can create refractors that produce discrete irradiance distributions that are +arbitrarily close to a continuous distribution, like pixels in a photo. However, +this does not imply the existence of a refractor that produces a continuous +intensity distribution at the limit. This is expected for problems that can be +described by a fully nonlinear PDE of Monge-Amp´ere type [3]. However, if the +refractors are convex, due to the unique properties that convexity provides, one +can use the weak convergence of Dirac measures to Lebesgue measures to obtain +a refractor that produces a continuous irradiance distribution; see [4],[5]. +In this paper, we work on the weak formulation developed by [1], where we +develop existence and uniqueness results. Due to a mistake in [1] (see Section +8), Theorem 9 in [1] (that I later present as Theorem 5.1) is the only existence +theorem for the refractor problem. That theorem is hard to use, and, in its +current form, it cannot be extended to the continuous case. However, Theorem +9 in [1] can be used to prove another existence theorem for the discrete case +(Theorem 5.2) that, in turn, can be extended to the continuous case (Theorem +6.1). We use this result to then prove the existence of solutions for the rota- +tionally symmetric case (Theorem 7.1). Additionally, we prove a uniqueness +theorem for the case where the target set is finite (Theorem 4.1) and for the +general case (Theorem 4.2). +2. Hyperboloids of Revolution +We do all our work in R3. We denote S2 to be the unit sphere with the center +at O and kx = x/|x| for all x ∈ R3\{O}. We borrow much of this geometric setup +from [1]. Hyperboloids of revolution are of paramount importance when solving +the virtual-source reflector problem due to their unique optical properties. +5 + +Consider the rotationally symmetric hyperboloid of two sheets in R3 such +that one focus is O and the other is x; let H(x) be the branch of the hyperboloid +that has x as a focus. From now on, when we use the term hyperboloid, we are +only referring to this branch. +With each hyperboloid H(x) we associate its radial projection by rays from +the origin onto an open spherical disk D(x) ⊂ S2 and its polar radius +h ϵ(m) = +|x|(1 − ϵ2) +2ϵ(1 − ϵ⟨m, kx⟩), m ∈ D(x) +(4) +where ϵ is the eccentricity of the hyperboloid. Be aware that ϵ > 1 since we are +describing a hyperboloid. +Define Hϵ(x) to be the hyperboloid with eccentricity ϵ and focus x. We now +introduce a similar function h x,ϵ(m) which introduces x ∈ R3\{O} as a variable. +In this paper, we define hx,ϵ(m) = mh x,ϵ(m) for m ∈ D(x) and x ∈ R3 \ {O}. +Let Dϵ(x) ⊂ S2 be the preimage of Hϵ(x) under hx,ϵ, then Dϵ(x) = {m ∈ S2| 1 +ϵ < +⟨m, kx⟩}. Thus we can easily verify that Hϵ(x) = {hx,ϵ(m)|m ∈ Dϵ(x)}. +From a physical perspective, x being the focus means that all light from +the origin reflected off of the reflector H(x) appears to be originating from x, +making x a virtual source. +From the above work, we see by taking the eccentricity ϵ to infinity that +the shape of the hyperboloid becomes a plane, which is the directrix of the +hyperboloid, and Dϵ(x) and goes to the hemisphere oriented towards x. The +following two propositions summarize what I say precisely. +Proposition 2.1. As the eccentricity ϵ of Hϵ(x) goes to infinity, Dϵ(x) goes to +{m ∈ S2|⟨m, kx⟩ ≥ 0}. +Proposition 2.2. As the eccentricity ϵ of Hϵ(x) goes to infinity, the resultant +set is a plane represented by the equation ⟨x, y − x +2⟩ = 0 where y ∈ R3, or +equivalently by the polar radius equation r(m) = +|x| +2⟨m,kx⟩ where m ∈ {m ∈ +S2|⟨m, kx⟩ > 0}. +Observe that as the eccentricity ϵ goes to 1, we obtain a ray originating at +x going in the direction described by the vector kx. We call this a degenerate +6 + +hyperboloid. +An important property of hyperboloids can be described by the following +proposition. +Proposition 2.3. Let c > 0 and ϵ > 1 such that cϵ > 1. Then the hyperboloids +Hcϵ(x) and Hϵ(x) have the same foci: O and x. +The aforementioned property is important because a reflector Hϵ(x) will +reflect the light emitted from O so that the light appears to be emitted from +x; thus, making x a virtual source. Alternatively, a refractor Hϵ(x) will refract +the light emitted from O so that the light is delivered to x. These properties +are true no matter how large or small the eccentricity is; all that matters is the +location of the foci. +Let Ax = {m ∈ S2|⟨m, kx⟩ ≥ 0} and Aδ +x = {m ∈ S2|⟨m, kx⟩ ≥ δ} for δ ∈ R. +Observe that Ax = A0 +x, A1 +x = {kx}, Aδ +x = ∅ for δ > 1, and Aδ +x = A−1 +x += S2 for +δ ≤ −1. It is also clear that if δ1 ≤ δ2, then Aδ1 +x ⊆ Aδ2 +x with a strict inclusion +if δ1 < δ2 and δ1, δ2 ∈ [−1, 1]. So while δ only has practical significance while +taking values in [−1, 1], allowing it to take all values in R makes some of the +upcoming proofs easier. +By Propositions 2.1 and 2.2, we have the following statement. +Proposition 2.4. Let 0 < δ < 1 and B ⊆ Aδ +x. Then if ϵ > 1 +δ , hx,ϵ[B] ⊂ Hϵ(x). +In particular, 1 +ϵ < δ implies that Aδ +x ⊂ Dϵ(x), 1 +ϵ > δ implies that Dϵ(x) ⊂ Aδ +x, +and 1 +ϵ = δ implies that Int(Aδ +x) = Dϵ(x). +Also note the following definition. +Definition 2.1. For an element x ∈ R3 and a set A ⊂ R3, let the set Cx,A = +{at + x(1 − t)|t ∈ [0, 1], a ∈ A} be the union of all line segments from x to A +and Cx,A,∞ = {at + x(1 − t)|t ∈ [0, ∞), a ∈ A} be the union of all rays from x +that intersect A. +7 + +3. Convex Weak Solutions +We now have the background to construct and proceed with our discussion of +the weak solution. Keep in mind that this weak solution definition, apart from +some minor differences in notation, is identical to the weak solution defined in +[1]. +Let c = minx,y∈T ⟨kx, ky⟩, ℓ = minx∈T |x|, and L = maxx∈T |x|. Assume we +are given a set T ⊆ R3. We say that T satisfies Hypothesis H1 if the following +condition is met. +Hypothesis H1. T is a compact subset of R3 contained in a half space of R3, +ℓ > 0, and 2ℓc > L. +Note that this is Hypothesis H1 from [1]. +We also define a constant, +ϵ0 = ℓ + +√ +ℓ2 − 2Lℓc + L2 +2ℓc − L +, +(5) +that depends only on T. +We first assume we are given a target set T that satisfies Hypothesis H1. +Let ˜Hϵ(x) be the convex body bounded by Hϵ(x). Consider the aperture Dδγ +T = +Int +�� +x∈T Aδγ +x +� +where δγ = +1 +ϵ0+γ for some γ > 0. We then define a simply +connected refractor over Dδγ +T as the boundary of the intersection of the convex +bodies bounded by hyperboloids. Specifically, +R = ∂h where h = +� +x∈T +˜Hϵx(x) +(6) +where each ϵx ≥ ϵ′ ≥ ϵ0 + γ = +1 +δγ . Observe that +R = +� +m sup +x∈T h x,ϵx(m) +�����m ∈ Int +� � +x∈T +A +1 +ϵx +x +�� +. +(7) +Note that Dδγ +T ⊆ Int +�� +x∈T A +1 +ϵx +x +� +, and supx∈T h x,ϵx is twice differentiable al- +most everywhere by Alexandrov’s theorem [6]; thus R may be considered a +refactor per Definition 1.2. Let +Rϵ′ +convex(T) +(8) +8 + +be the set of all such refractors. Please note that by Lemma 1 in [1], the set +Rϵ′ +convex(T) is nonempty. +Note the following definition. +Definition 3.1. A hyperboloid H(x) is said to be supporting to a set Q ⊂ R3 +at a point z ∈ ∂Q if the convex body ˜H(x) bounded by H(x) contains Q and +z ∈ H(x) ∩ ∂Q. +For a subset ω ⊆ T and a refractor R ∈ Rϵ′ +convex(T) put +M(ω) = {z ∈ R| there exists x ∈ ω such that H(x) is supporting to R at z}. +(9) +The intersection of Dδγ +T with the image of the set M(ω) under radial projection +on S2 we call the visibility set of ω and denote it by Vconvex(ω). By Lemma 4 +of [1], this set Vconvex(ω) is measurable for all Borel sets ω ⊆ T. +For m ∈ Dδγ +T let r(m) be the set of points of intersection between the refrac- +tor R and the ray of direction m originating at O. The possibly multivalued +map αconvex : Dδγ +T → T, +αconvex(m) = {x ∈ T| there exists H(x) supporting to R at r(m)} +(10) +is called the refractor map. +Assume we are given a nonnegative g ∈ L1(S2). Let us define for measurable +X ⊆ S2 +µg(X) = +� +X +g(m)dσ(m) +(11) +where σ denotes the standard measure on S2. Assume that g ≡ 0 outside of +Dδγ +T . +In order to formulate and solve the refractor problem (in the framework of +weak solutions to be defined below), we need to define a measure representing +the energy generated by g and redistributed by a refractor R ∈ Rϵ′ +convex(T). +Define for any refractor R ∈ Rϵ′ +convex(T), +Gconvex(ω) = µg(Vconvex(ω)) +(12) +9 + +which we will deem the energy function. It can be shown that G is a finite +measure on the Borel σ-algebra of T. +Let F be a nonnegative, finite, Borel measure on Borel subsets of T. We +say that a refractor R ∈ Rϵ′ +convex(T) is a convex weak solution to the refractor +problem if the refractor map αconvex determined by R is such that αconvex(m) ⊆ +T for all m ∈ Dδγ +T , and +F(ω) = Gconvex(ω) for any Borel set ω ⊆ T. +(13) +4. Uniqueness Theorems +We start with some uniqueness results. Note that Theorem 4.1 can be con- +sidered as a direct corollary to Theorem 4.2. +We include both as separate +statements and proofs; as discrete versions of the uniqueness theorems proved +to be of special interest in related problems; see [4] and [7]. We proceed with +the following lemma; which is shown in the proof of Lemma 2 in [1]. +Lemma 4.1. Let T be a target set that satisfies Hypothesis H1. Suppose we are +given positive real numbers γ and ϵ′ such that ϵ′ ≥ ϵ0 + γ where ϵ0 is defined by +(5). Let R ∈ Rϵ′ +convex(T). Then Vconvex(ω) is closed for all closed ω ⊆ T. +Before we proceed, note that if we write Vconvex(R; ω) for some Borel set +ω ⊆ T and some refractor R ∈ Rϵ′ +convex(T), this is specifically the visibility set +for the refractor R evaluated on the set ω. Similarly, if we write Gconvex(R; ω) +for some Borel set ω ⊆ T and some refractor R ∈ Rϵ′ +convex(T), this is specifically +the energy function for the refractor R evaluated on the set ω. We will be using +this when we are talking about multiple refractors and we need to specify the +energy function for each refractor. +Here we consider the case of the refractor problem (13) where the set T is +finite. We prescribe the measure F in (13) as a Dirac measure concentrated at +points in T. We now introduce notation for refractors in the discrete case. Let +T = {x1, x2, . . . , xk}. We set Hi = H(xi) and the eccentricity of Hi we denote +10 + +by ϵi. For the hyperboloids H1, . . . , Hk define the refractor +R = ∂ +� k� +i=1 +˜Hi +� +∈ Rϵ′ +convex(T). +(14) +Since each hyperboloid Hi is uniquely defined by its eccentricity ϵi, the +refractor R can be identified with the point with coordinates (ϵ1, ϵ2, . . . , ϵk) in +the region +ϵ1 ≥ ϵ′, ϵ2 ≥ ϵ′, . . . , ϵk ≥ ϵ′ +(15) +in k−dimensional euclidean space. Thus we can write a refractor R ∈ Rϵ′ +convex(T) +as (ϵ1, ϵ2, . . . , ϵk). We start with a uniqueness theorem for the discrete case. +Theorem 4.1. Let T = {x1, . . . , xk} be a collection of k distinct points that +satisfy Hypothesis H1. Suppose we are given positive real numbers γ and ϵ′ such +that ϵ′ ≥ ϵ0 + γ where ϵ0 is defined by (5). Assume we are given a nonnegative +g ∈ L1(S2) such that g > 0 inside Dδγ +T and g ≡ 0 outside Dδγ +T where δγ = +1 +ϵ0+γ . +Let f1, . . . , fk be a collection of positive real numbers such that +k +� +i=1 +fi = µg(Dδγ +T ). +(16) +Let R = (ϵ1, . . . , ϵk) and ˜R = ( ˜ϵ1, . . . , ˜ϵk) be refractors in Rϵ′ +convex(T) such that +Gconvex( ˜R; xi) = Gconvex(R; xi) = fi for all i ∈ [k]. +Then the inequality ˜ϵj ≥ ϵj for some j implies that ˜ϵi ≥ ϵi for all i ∈ [k]. +Furthermore, the equality ˜ϵj = ϵj for some j implies that ˜ϵi = ϵi for all i ∈ [k]. +Proof. Let J be a nonempty subset of [k] such that for any i ∈ J, ˜ϵi > ϵi, +and for any i ∈ [k] \ J, ˜ϵi ≤ ϵi. +Note that m ∈ Vconvex( ˜R; {xi|i ∈ J}) if +and only if there exists some i ∈ J such that h xi, ˜ϵi(m) ≥ h xℓ, ˜ϵℓ(m) for all +ℓ ∈ [k] \ J. For this m: since h xi,ϵi(m) > h xi, ˜ϵi(m) for all i ∈ J and h xℓ,ϵℓ(m) ≤ +h xℓ, ˜ϵℓ(m) for all ℓ ∈ [k] \ J, then there exists some i ∈ J such that h xi,ϵi(m) > +h xℓ,ϵℓ(m) for all ℓ ∈ [k] \ J. +Thus, any m ∈ Vconvex( ˜R; {xi|i ∈ J}) is an +interior point of Vconvex(R; {xi|i ∈ J}); in other words, Vconvex( ˜R; {xi|i ∈ J}) ⊆ +Int(Vconvex(R; {xi|i ∈ J})). Recall that, by Lemma 4.1, Vconvex( ˜R; {xi|i ∈ J}) +is closed and, since f1, . . . , fk are positive, Vconvex(R; {xi|i ∈ J}) is nonempty. +11 + +Then Int(Vconvex(R; {xi|i ∈ J}))\Vconvex( ˜R; {xi|i ∈ J}) is open and nonempty. +So µg(Vconvex(R; {xi|i ∈ J}) \ Vconvex( ˜R; {xi|i ∈ J})) > 0. Therefore we must +have +� +i∈J +fi = Gconvex( ˜R; {xi|i ∈ J}) < Gconvex(R; {xi|i ∈ J}) = +� +i∈J +fi +(17) +which is a contradiction because Gconvex( ˜R; xi) = Gconvex(R; xi) = fi for all +i ∈ [k]. The theorem is proved. +Observe that for all refractors R ∈ Rϵ′ +convex(T), there exists a function K : +T → [ϵ′, ∞) such that R = ∂(� +x∈T ˜HK(x)(x)). Since each hyperboloid ˜HK(x)(x) +is uniquely determined by K, the refactor R can be identified with the function +K : T → [ϵ′, ∞). Thus we can write a refractor R ∈ Rϵ′ +convex(T) as [K] where K : +T → [ϵ′, ∞); note that [K] = +� +m supx∈T h x,K(x)(m) +����m ∈ Int +�� +x∈T A +1 +K(x) +x +�� +. +Given a refractor R ∈ Rϵ′ +convex(T), we call K : T → [ϵ′, ∞) the maximal function +of R if R = +� +m maxx∈T h x,K(x)(m) +����m ∈ Int +�� +x∈T A +1 +K(x) +x +�� +. +We proceed +with the following lemma. +Lemma 4.2. Let R ∈ Rϵ′ +convex(T) be a refractor such that for all x ∈ T, +Vconvex({x}) is nonempty. Then there exists a K : T → [ϵ′, ∞) that is a the +maximal function of R. +Proof. By Lemma 2 in [1], T ⊂ � +x∈T ˜Hϵx(x). Therefore, by Definition 3.1, +that m ∈ Vconvex({x}) if and only if there exists a corresponding ϵ′ +x ≥ ϵ′ such +that h x,ϵ′x(m) = supx∈T h x,ϵx(m). Define K(x) = ϵ′ +x for all x ∈ T. Then R = +� +m maxx∈T h x,K(x)(m) +����m ∈ Int +�� +x∈T A +1 +K(x) +x +�� +. +We now conclude with a uniqueness theorem for more general measures and +target sets. +Theorem 4.2. Let T be a target set that satisfies Hypothesis H1. Let F be a +nonnegative, finite, Borel measure on Borel subsets of T. Suppose we are given +positive real numbers γ and ϵ′ such that ϵ′ ≥ ϵ0 + γ where ϵ0 is defined by (5). +12 + +Assume we are given a nonnegative g ∈ L1(S2) such that g > 0 inside Dδγ +T and +g ≡ 0 outside Dδγ +T where δγ = +1 +ϵ0+γ such that +F(T) = µg(Dδγ +T ). +(18) +Let R and ˜R be refractors in Rϵ′ +convex(T) such that for all nonempty Borel +ω ⊆ T: +F(ω) = Gconvex( ˜R; ω) = Gconvex(R; ω), Vconvex( ˜R; ω) ̸= ∅, and +Vconvex(R; ω) ̸= ∅. +Then there exists functions K : T → [ϵ′, ∞) and ˜K : T → [ϵ′, ∞) that are, +respectively, maximal functions of R and ˜R such that the inequality ˜K(x) ≥ +K(x) for some x ∈ T implies that ˜K(y) ≥ K(y) for all y ∈ T. Furthermore, the +equality ˜K(x) = K(x) for some x ∈ T implies that ˜K(y) = K(y) for all y ∈ T. +Proof. By Lemma 4.2, there exists functions K : T → [ϵ′, ∞) and ˜K : T → +[ϵ′, ∞) that are maximal functions of R and ˜R respectively. Note that R = [K] +and ˜R = [ ˜K]. +Let J be a nonempty closed subset of T such that for any x ∈ J, ˜K(x) > +K(x), and for any x ∈ T \ J, ˜K(x) ≤ K(x). Note that m ∈ Vconvex( ˜R; J) if +and only if there exists some z ∈ J such that h z, ˜ +K(z)(m) ≥ h z′, ˜ +K(z′)(m) for +all z′ ∈ T \ J. For this m: since h z,K(z)(m) > h z, ˜ +K(z)(m) for all z ∈ J and +h z′,K(z′)(m) ≤ h z′, ˜ +K(z′)(m) for all z′ ∈ T \ J, then there exists some z ∈ J such +that h z,K(z)(m) > h z′,K(z′)(m) for all z′ ∈ T \J. Thus, any m ∈ Vconvex( ˜R; J) is +an interior point of Vconvex(R; J); in other words, Vconvex( ˜R; J) ⊆ Int(Vconvex(R; J)). +Recall that, by Lemma 4.1, Vconvex( ˜R; J) is closed. Then Int(Vconvex(R; J)) \ +Vconvex( ˜R; J) is open and nonempty. So µg(Vconvex(R; J) \ Vconvex( ˜R; J)) > 0. +Therefore we must have +F(J) = Gconvex( ˜R; J) < Gconvex(R; J) = F(J) +(19) +which is a contradiction because F(ω) = Gconvex( ˜R; ω) = Gconvex(R; ω) for all +Borel ω ⊆ T. The theorem is proved. +13 + +5. Weak Solutions in the Discrete Case +Here we consider the case of the refractor problem (13) where the set T is +finite. We prescribe the measure F in (13) as a Dirac measure concentrated +at points in T. We recall notation for refractors in the discrete case. Let T = +{x1, x2, . . . , xk}. We set Hi = H(xi) and the eccentricity of Hi we denote by ϵi. +For the hyperboloids H1, . . . , Hk define the refractor +R = ∂ +� k� +i=1 +˜Hi +� +∈ Rϵ′ +convex(T). +(20) +Since each hyperboloid Hi is uniquely defined by its eccentricity ϵi, the +refractor R can be identified with the point with coordinates (ϵ1, ϵ2, . . . , ϵk) in +the region +ϵ1 ≥ ϵ′, ϵ2 ≥ ϵ′, . . . , ϵk ≥ ϵ′ +(21) +in k−dimensional euclidean space. Thus we can write a refractor R ∈ Rϵ′ +convex(T) +as (ϵ1, ϵ2, . . . , ϵk). We now recall Theorem 9 from [1]. +Theorem 5.1 (Theorem 9 in [1]). Let T = {x1, . . . , xk} be a collection of k +distinct points in R3 \ {O}, k > 2. Assume that T satisfies Hypothesis H1. Let +γ, ϵM, ϵmin, and ϵmax be positive real numbers such that ϵ0 + γ < ϵM ≤ ϵmin ≤ +ϵmax < ∞, where ϵ0 is defined by (5). +Assume we are given a nonnegative +g ∈ L1(S2) such that g ≡ 0 outside Dδγ +T +where δγ = +1 +ϵ0+γ . Let f1, . . . , fk be +nonnegative real numbers such that +k +� +i=1 +fi = µg(Dδγ +T ). +(22) +Suppose that there also exists some ℓ ∈ [k] such that for all i ∈ [k], i ̸= ℓ, +Gconvex(Rℓ; xi) ≤ fi +(23) +where Rℓ = (ϵ1 = ϵmax, ..., ϵℓ−1 = ϵmax, ϵℓ = ϵmin, ϵℓ+1 = ϵmax, ..., ϵk = ϵmax), +and +Gconvex(Rℓi; xℓ) < fℓ +(24) +14 + +where Rℓi = (ϵ1 = ϵmax, . . . , ϵi−1 = ϵmax, ϵi = ϵM, ϵi+1 = ϵmax, . . . , ϵℓ−1 = +ϵmax, ϵℓ = ϵmin, ϵℓ+1 = ϵmax, ..., ϵk = ϵmax). Then there exists a refractor R = +(ϵ1, . . . , ϵk) ∈ RϵM +convex(T) such that +Gconvex(R; xi) = fi for all i ∈ [k]. +(25) +This theorem inspires an obvious corollary. +Corollary 5.1. Let T = {x1, . . . , xk} be a collection of k distinct points in +R3 \ {O}, k > 2. Assume that T satisfies Hypothesis H1. Let γ, ϵM, ϵmin, and +ϵmax be positive real numbers such that ϵ0 + γ < ϵM ≤ ϵmin ≤ ϵmax < ∞, where +ϵ0 is defined by (5). Assume we are given a nonnegative g ∈ L1(S2) such that +g ≡ 0 outside Dδγ +T where δγ = +1 +ϵ0+γ . Let f1, . . . , fk be nonnegative real numbers +such that +k +� +i=1 +fi = µg(Dδγ +T ). +(26) +Suppose that there also exists some ℓ ∈ [k] where fℓ > 0 such that for all +i ∈ [k], i ̸= ℓ, +Gconvex(Rℓ; xi) = 0 +(27) +where Rℓ = (ϵ1 = ϵmax, ..., ϵℓ−1 = ϵmax, ϵℓ = ϵmin, ϵℓ+1 = ϵmax, ..., ϵk = ϵmax), +and +Gconvex(Rℓi; xℓ) = 0 +(28) +where Rℓi = (ϵ1 = ϵmax, . . . , ϵi−1 = ϵmax, ϵi = ϵM, ϵi+1 = ϵmax, . . . , ϵℓ−1 = +ϵmax, ϵℓ = ϵmin, ϵℓ+1 = ϵmax, ..., ϵk = ϵmax). Then there exists a refractor R = +(ϵ1, . . . , ϵk) ∈ RϵM +convex(T) such that +Gconvex(R; xi) = fi for all i ∈ [k]. +(29) +We will now use the above corollary to prove the following proposition. +15 + +Proposition 5.1. Let T = {x1, . . . , xk} be a collection of k distinct points in +R3 \{O} such that T satisfies Hypothesis H1 and minx,y∈T ⟨kx, ky⟩ = 1. Suppose +that we are given γ > 0 such that ϵ0 + γ < limt→K+ +1 +t−1 for K = maxx∈T |x| +minx∈T |x| and +ϵ0 is defined by (5). Assume we are given a nonnegative g ∈ L1(S2) such that +g ≡ 0 outside Dδγ +T where δγ = +1 +ϵ0+γ . Let f1, . . . , fk be nonnegative real numbers +such that +k +� +i=1 +fi = µg(Dδγ +T ) +(30) +and for the ℓ ∈ [k] where |xℓ| = maxy∈T |y|, fℓ > 0. +Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ +1 +t−1) such that we can construct +a convex, rotationally symmetric refractor R = (ϵ1, . . . , ϵk) ∈ RϵM +convex(T) where +Gconvex(R; xi) = fi for all i ∈ [k]. +(31) +Proof. Note that maxx,y⟨kx, ky⟩ = 1 implies that kx = ky for all x, y ∈ T, and +ϵ0 = 1 as defined by (5). The case where k = 1 is trivial; let k ≥ 2. Assume +that |xi| ≥ |xi+1| for all i ∈ [k − 1] and thus f1 > 0. Recall that for x ∈ T by +Proposition 2.2, h ϵ,x(m) → +|x| +2⟨m,kx⟩ as ϵ → ∞ for m ∈ Dδγ +T . +Observe that +|x1| +2⟨m, kx⟩ < +|xk|(1 − ϵ2 +M) +2ϵM(1 − ϵM⟨m, kx⟩) for m ∈ Dδγ +T , +(32) +if and only if +|x1| +2 +< |xk|(1 − ϵ2 +M) +2ϵM(1 − ϵM). +(33) +Thus we have that ϵM < +1 +K−1 where K = |x1| +|xk|. Note that by Hypothesis H1 +and the fact that k ≥ 2, we have 1 < K < 2 and 1 < +1 +K−1 < ∞. Thus we can +have that 1 = ϵ0 < ϵ0 + γ < ϵM < +1 +K−1. If k = 2, by the continuity implied +by Lemma 8 of [1], there exists a refractor R = (ϵ1, ϵ2) ∈ RϵM +convex(T) such that +Gconvex(R; xi) = fi for all i ∈ [k]. +If k > 2, we borrow language and notation from Corollary 5.1. By conti- +nuity, if ϵmin = ϵmax be sufficiently large such that +1 +K−1 < ϵmin = ϵmax, then, +assuming that ϵM < +1 +K−1, Gconvex(R1; xi) = 0 and Gconvex(R1i; x1) = 0 for +16 + +all i ∈ [k] such that i ̸= 1. Therefore by Corollary 5.1, there exists a refractor +R = (ϵ1, . . . , ϵk) ∈ RϵM +convex(T) such that Gconvex(R; xi) = fi for all i ∈ [k]. +The above proposition motivates our main result. +Theorem 5.2. Assume that we are given some w, W ∈ (0, ∞) where w > W +2 . +Given some m∗ ∈ S2, let +S(m∗, ξ) = {x ∈ R3|w ≤ |x| ≤ W, ⟨kx, m∗⟩ ≥ 1 − ξ} +(34) +where 1 − cos +� 1 +2 arccos +� W +2w +�� +> ξ > 0; note that S(m∗, ξ) satisfies Hypothesis +H1. Let T = {x1, . . . , xk} ⊂ S(m∗, ξ) be a collection of k distinct points. Recall +that δγ = +1 +ϵ0+γ where γ > 0 and ϵ0 is defined by (5). +Then there exists positive ξ and γ such that, for any nonnegative g ∈ L1(S2) +such that g ≡ 0 outside Dδγ +T +and any collection f1, . . . , fk of nonnegative real +numbers where +k +� +i=1 +fi = µg(Dδγ +T ) +(35) +and fℓ > 0 for the ℓ ∈ [k] where |xℓ| = maxy∈T |y|, there exists an ϵM > ϵ0 + γ +such that we can construct a refractor R = (ϵ1, . . . , ϵk) ∈ RϵM +convex(T) where +Gconvex(R; xi) = fi for all i ∈ [k]. +(36) +Proof. Observe that ξ → 0 implies that minx,y∈T ⟨kx, ky⟩ → 1. Assume that +|xℓ| = maxy∈T |y|. Let h T max,ϵ(m) = maxx∈T h x,ϵ(m) and PT max(m) = maxx∈T +|x| +2⟨m,kx⟩ +where m ∈ Dδγ +T . Note that h T max,ϵ(m) → h xℓ,ϵ(m) and PT max(m) → +|xℓ| +2⟨m,kxℓ⟩ +as miny∈T ⟨kxℓ, ky⟩ → 1. +We borrow language and notation from Corollary 5.1. Choose an ϵmin; then, +by continuity, there exists a ξ > 0 such that if minx,y∈T ⟨kx, ky⟩ ≥ 1 − ξ, then +we have PT max(m) < h xℓ,ϵmin(m) for all m ∈ Dδγ +T . Therefore by continuity +there exists an ϵmax such that PT max(m) < h T max,ϵmax(m) < h xℓ,ϵmin(m) for +all m ∈ Dδγ +T . +Observe that ϵ0 → 1 as minx,y∈T ⟨kx, ky⟩ → 1 and recall that a degenerate +hyperboloid has eccentricity 1. Then, for sufficiently small γ > 0, by continuity +17 + +we can choose ϵ0+γ < ϵM < ϵmin < ϵmax, such that PT max(m) < h xℓ,ϵmax(m) < +h T max,ϵmin(m) < h xℓ,ϵM (m) for all m ∈ Dδγ +T . +Then Gconvex(Rℓ; xi) = 0 and Gconvex(Rℓi; xℓ) = 0 for all i ∈ [k] such +that i ̸= ℓ. Then by Corollary 5.1, there exists a refractor R = (ϵ1, . . . , ϵk) ∈ +RϵM +convex(T) such that Gconvex(R; xi) = fi for all i ∈ [k]. +6. Weak Solutions in the General Case +In this section, we extend the results of Theorems 5.2 and 4.1 to the case +of more general sets T and energy distributions F. We consider the case where +prescribe the measure F as a Lebesgue measure over T, specifically +F(ω) = +� +ω +f(x)dλ(x) for any Borel set ω ⊆ T +(37) +for some given nonnegative function f ∈ L1(T); here λ is the Lebesgue measure +on T. +Theorem 6.1. Assume that we are given some w, W ∈ (0, ∞) where w > W +2 . +Given some m∗ ∈ S2, let +S(m∗, ξ) = {x ∈ R3|w ≤ |x| ≤ W, ⟨kx, m∗⟩ ≥ 1 − ξ} +(38) +where 1 − cos +� 1 +2 arccos +� W +2w +�� +> ξ > 0; note that S(m∗, ξ) satisfies Hypothesis +H1. Let T ⊆ S(m∗, ξ) be a closed set. Recall that δγ = +1 +ϵ0+γ where γ > 0 and +ϵ0 is defined by (5). +Then there exists positive ξ and γ such that, for any nonnegative f ∈ L1(T) +with a measure F defined by (37), and any nonnegative g ∈ L1(S2) where g ≡ 0 +outside Dδγ +T where +F(T) = µg(Dδγ +T ), +(39) +there exists an ϵM > ϵ0 + γ such that we can construct a convex refractor +R ∈ RϵM +convex(T) where R that is a convex weak solution to the refractor problem +(13). +18 + +The following argument is based on similar arguments made in [1], [5], and +[4]; specifically, the argument made for Theorem 13 in [1]. Even though Theorem +13 in [1] is incorrect1, the type of argument presented in its proof is broadly +applicable; thus it would be good to put the argument in a correct context. +Proof. The following argument follows the proof of Theorem 13 in [1] very closely +with some adjustments to fit into this new context. For the sake of the following +proof, recall that our definition of the energy function can also be considered as +a measure of T. +If µg(Dδγ +T ) = 0, then any refractor R ∈ RϵM +convex(T) will do. Assume that +µg(Dδγ +T ) > 0. +Since T is bounded, for any δ > 0 there exists an N ∈ N such that for each +k ≥ N there exists a partition of T into k Borel sets ωk +1, . . . , ωk +k such that +diam(ωk +i ) ≤ δ for any k ≥ N, i ∈ [k]. +(40) +For each k ∈ N, we choose an xk +i ∈ ωk +i for i ∈ [k], and put +F k +i = F(ωk +i ). +(41) +Define a measure F k on T by +F k(ω) = +� +xk +i ∈ω +F k +i for any Borel set ω ⊆ T. +(42) +Note that F k converges weakly to F as k → ∞. For each k, there exists +a nonempty S ⊆ [k] such that F k +i +> 0 for all i ∈ S. +Since {xk +i }i∈S and +{F k +i }i∈S satisfies the assumptions of Theorem 5.2, there exists a convex refrac- +tor Rk ∈ RϵM +convex({xk +i |i ∈ S}) ⊆ RϵM +convex({xk +i |i ∈ [k]}) ⊆ RϵM +convex(T) defined by +hyperboloids with an eccentricity greater than or equal to some ϵM > ϵ0 + γ +such that +Gconvex(Rk; xk +i ) = F k +i for i ∈ [k]. +(43) +1It should be noted that in [1], due to erroneous assumptions, there were issues with +attempts to construct refractors in the special case explored by Theorems 12 and 13; see +Section 8. +19 + +Let Gk be the measure on T defined by +Gk(ω) = +� +xk +i ∈ω +Gconvex(Rk; xk +i ) +(44) +then obviously F k ≡ Gk for all k ∈ N and consequently, Gk → F. To finish the +proof we need to construct a refractor R whose energy function, G, would be +the limit of measures Gk. This refractor is constructed in the following manner +as a limit of refractors Rk. +First, we note that since g ≡ 0 outside Dδγ +T , we only need to consider the +part of the refractor Rk ∩ CO,D +δγ +T ,∞. See Definition 2.1 as a refresher on the +meaning of CO,D +δγ +T ,∞. Also for some ϵM > ϵ0 + γ one can show that for any +R ∈ RϵM (T) +� +R ∩ CO,D +δγ +T ,∞ +� +⊆ B(O, b) for some b > 0 +(45) +where B(O, b) is the open ball centered at the origin O of radius b. Let us prove +this statement. We define +b = +max +x∈T,m∈DT +δγ h x,ϵM (m). +(46) +Since h x,ϵM is a continuous function and DT +δγ is compact, this definition is +correct and b < ∞. Thus for any ϵ > ϵM, h x,ϵ(m) ≤ b and (45) is proved. +For each of the refractors Rk we consider a bounded convex body +hk +b = hk ∩ CO,D +δγ +T ,∞ ∩ B(O, b) +(47) +where for each k ∈ N the set hk is defined by (6). +By Blaschke’s selection +theorem [8], there exists a subsequence of {hk +b} which we again denote by {hk +b}, +which converges to some convex body hb. +We show now that for each point r ∈ [∂(hb) ∩ CO,D +δγ +T ,∞] \ ∂(B(O, b)) there +exists a hyperboloid Hr(x) which is supporting to hb at point r. +Let r ∈ [∂(hb) ∩ CO,D +δγ +T ,∞] \ ∂(B(O, b)). Then there exists a sequence {rk} +that converges to r where each rk ∈ Rk. Let H(xk) be a supporting hyperboloid +to Rk at rk. Since T is compact, {xk} contains a subsequence, which we will +denote by {x∗ +k}, converging to some x ∈ T. The convex body ˜H(x∗ +k) bounded by +20 + +H(x∗ +k) contains the body hk +b. The corresponding sequence { ˜H(x∗ +k)} converges to +the body ˜Hr(x) containing hb and hk +b converges to hb. Therefore ˜Hr(x) contains +∂(hb). It follows that Hr(x) is supporting to hb at r. +We now define the refractor R = ∂(� +x∈T ˜Hr(x)) and show that the sequence +of measures Gk, that are equivalent to the energy functions corresponding to the +refractors Rk, converges weakly to the measure G, which is the energy function +of the refractor R. +Let αk +convex and αconvex be the refractor maps corresponding to Rk and R +respectively. By a theorem of Reidemeister on singularities on convex surfaces +(see [8]) the refractor maps αk +convex for k ∈ N and αconvex are single-valued +functions almost everywhere. Furthermore, for almost all m ∈ Dδγ +T the hyper- +boloids Hk supporting to Rk at points rk(m) converge to the hyperboloid H +supporting to R at the point r(m). Thus, αk +convex(m) converges to αconvex(m) +almost everywhere. +If given a set of cardinality one, {z}, let Ele({z}) = z. Let Y k(m) = {x ∈ +αk +convex(m)|x ∈ {xk +i }i∈[k]} and let Jk(m) ⊆ [k] be the set of indices such that +{xk +i |i ∈ Jk(m)} = Y k(m). Let z ∈ T, +Kk(m) = xk +min Jk(m), +(48) +and +K(m) = +� +� +� +� +� +Ele(αconvex(m)) +if |αconvex(m)| = 1 +z +if |αconvex(m)| > 1 +. +(49) +Then for any continuous function u on T we have +� +T +udGk = +� +D +δγ +T +u(Kk(m))dµg(m) −→ +� +D +δγ +T +u(K(m))dµg(m) = +� +T +udG (50) +as k → ∞, that is, the measures {Gk} converge weakly to G. +7. Rotationally Symmetric Convex Refractors on the Surface of a +Right Circular Cone +Both Theorems 6.1 and 5.1 inspire this next result. +21 + +Corollary 7.1. Let T be a closed set that satisfies Hypothesis H1 and minx,y∈T ⟨kx, ky⟩ = +1. +Assume that we are given γ > 0 such that ϵ0 + γ < limt→K+ +1 +t−1 for +K = +maxx∈T |x| +minx∈T |x| and ϵ0 is defined by (5). Also, assume we are given a non- +negative g ∈ L1(S2) such that g ≡ 0 outside Dδγ +T where δγ = +1 +ϵ0+γ . Suppose we +are given a nonnegative f ∈ L1(T) and a measure F defined by (37) such that +F(T) = µg(Dδγ +T ). +(51) +Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ +1 +t−1) such that we can construct +a convex rotationally symmetric refractor R ∈ RϵM +convex(T) where R that is a +convex weak solution to the refractor problem (13). +We will use the above corollary to create rotationally symmetric refractors +with the target set on a right circular cone. +Definition 7.1. Let k ≥ 2, d > 0, 1 > ξ > 0, and m∗, m′ ∈ S2 such that +⟨m∗, m′⟩ = 0. We create a Cartesian coordinate system centered at O where m∗ +is the direction of our z-axis, m′ is the direction of our x-axis, and m∗ × m′ is +the direction of our y-axis. Let (x, y, z)′ represent a point in this system. +Recall that, given a point (x, y, z)′ ∈ R3, there exists r ∈ [0, ∞), φ ∈ [0, π], +θ ∈ [0, 2π), such that +x = r cos θ sin φ +(52) +y = r sin θ sin φ +(53) +z = r cos φ. +(54) +Let Q be a closed subset of the interval (0, ∞). +Define the set of points +T ξ +k,Q(m∗, m′) as +�� +d cos +�2πj +k +� +sin (arccos(ξ)) , d sin +�2πj +k +� +sin (arccos(ξ)) , dξ +�′ +|j ∈ I, d ∈ Q +� +(55) +where I = {0, 1, . . . , k − 1}. +Define the set T ξ +∞,Q(m∗, m′) as +� +(d cos (θ) sin (arccos(ξ)) , d sin (θ) sin (arccos(ξ)) , dξ)′ |θ ∈ [0, 2π), d ∈ Q +� +. +(56) +22 + +Proposition 7.1. Let m∗, m′ ∈ S2 such that ⟨m∗, m′⟩ = 0. Let Q be a closed +subset of the interval (0, ∞), k ≥ 2, and 1 > ξ > 0 such that T = T ξ +k,Q(m∗, m′) +satisfies Hypothesis H1 such that ϵ0 < limt→K+ +1 +t−1 for K = maxx∈T |x| +minx∈T |x| where +ϵ0 is defined by (5). +Assume that we are given γ > 0 such that ϵ0 + γ < +limt→K+ +1 +t−1. Also, assume we are given a nonnegative g∗ ∈ L1(S2) that is +rotationally symmetric about the axis defined by the ray of direction m∗ origi- +nating at O. Let the function g ∈ L1(S2) be defined as g ≡ g∗ inside Dδγ +T +and +g ≡ 0 outside Dδγ +T where δγ = +1 +ϵ0+γ . +Suppose we are given a nonnegative f ∈ L1(T) such that for every d ∈ Q: +f +�� +d cos +�2πj +k +� +sin (arccos(ξ)) , d sin +�2πj +k +� +sin (arccos(ξ)) , dξ +�′� +(57) +is constant for all j ∈ {0, 1, . . . , k − 1}. Let F be the measure defined by (37) +and +F(T) = µg(Dδγ +T ). +(58) +Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ +1 +t−1) such that we can construct +a convex refractor R ∈ RϵM +convex(T) where R that is a convex weak solution to +the refractor problem (13). +Proof. We create a Cartesian coordinate system centered at O where m∗ is the +direction of our z-axis, m′ is the direction of our x-axis, and m∗ × m′ is the +direction of our y-axis. Let (x, y, z)′ represent a point in this system. +Recall that, given a point (x, y, z)′ ∈ R3, there exists r ∈ [0, ∞), φ ∈ [0, π], +θ ∈ [0, 2π), such that +x = r cos θ sin φ +(59) +y = r sin θ sin φ +(60) +z = r cos φ. +(61) +Let Tj = +�� +d cos +� 2πj +k +� +sin (arccos(ξ)) , d sin +� 2πj +k +� +sin (arccos(ξ)) , dξ +�′ |d ∈ Q +� +. +Note that T = � +i∈{0,1,...,k−1} Ti. +Let m1, m2 ∈ S2 such that ⟨m1, m2⟩ > −1 and L (m1, m2) be the shortest +arc on S2 between the points m1 and m2. +For all j ∈ {0, 1, . . . , k − 1}, let +23 + +Bj ⊂ Dδγ +T +be the set {t ∈ L (m∗, y)|y ∈ ∂(Dδγ +T ) ∩ ∂(Aδγ +x )} where x ∈ Tj. For +d ∈ Q, since T ξ +k,{d}(m∗, m′) defines the points of a regular k-gon centered at the +axis defined by the ray of direction m∗ originating at O, then µg(Bj) = µg(D +δγ +T ) +k +for all j. For all j ∈ {0, 1, . . . , k−1}, let gj be a function over S2 such that gj ≡ g +inside Int(Bj) and gj ≡ 0 outside Int(Bj). Note that µg(Int(Bj)) = µgj(Dδγ +Tj) +for all j. +Let mj ∈ S2 be the unit vector such that a ray originating from O of direction +mj intersects every point in Tj. Also, let fj be the restriction of the function +f to the set Tj and Fj be the corresponding measure as defined by (37). By +Proposition 7.1, for all j ∈ {0, 1, . . . , k − 1}, there exists a refractor Rj = +∂ +�� +d∈Q ˜Hϵd(dmj) +� +∈ RϵM +convex(Tj), that is rotationally symmetric about the +axis defined by the ray starting at O with direction mj, such that Rj that +is a convex weak solution to the refractor problem (13); i.e. where Fj(ω) = +µgj(V (Rj; ω)) for all Borel ω ⊆ T. +Since for x ∈ Rj, we have that |x| increases as ⟨kx, mj⟩ decreases for all +j ∈ {0, 1, . . . , k − 1}, then for j ∈ {0, 1, . . . , k − 1}, defining rj(m) as the point +of intersection between Rj and the ray originating from O in direction m, we +have |rj(m)| = maxi∈{0,1,...,k−1} |ri(m)| for all m ∈ Bj. +Thus +∂ +� +� +� +j∈{0,...,k−1} +� +� � +d∈Q +˜Hϵd(dmj) +� +� +� +� ∈ RϵM +convex(T) +(62) +is our refractor. +With an argument similar to that we use in the proof of Theorem 6.1, we +obtain the following result from Proposition 7.1. +Theorem 7.1. Let m∗, m′ ∈ S2 such that ⟨m∗, m′⟩ = 0. Let 1 > ξ > 0 and Q +be a closed subset of the interval (0, ∞) such that T = T ξ +∞,Q(m∗, m′) satisfies +Hypothesis H1 where ϵ0 < limt→K+ +1 +t−1 for K = maxx∈T |x| +minx∈T |x| and ϵ0 is defined by +(5). Assume that we are given γ > 0 such that ϵ0 + γ < limt→K+ +1 +t−1. Also, +assume we are given a nonnegative g ∈ L1(S2) that is rotationally symmetric +24 + +about the axis defined by the ray of direction m∗ originating at O such that g ≡ 0 +outside Dδγ +T where δγ = +1 +ϵ0+γ . +Assume we have a nonnegative f ∈ L1(T) such that for every d ∈ Q: +f +� +(d cos (θ) sin (arccos(ξ)) , d sin (θ) sin (arccos(ξ)) , dξ)′� +(63) +is constant for all θ ∈ [0, 2π). Let F be the measure defined by (37) and +F(T) = µg(Dδγ +T ). +(64) +Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ +1 +t−1) such that we can construct +a convex, rotationally symmetric refractor R ∈ RϵM +convex(T) where R that is a +convex weak solution to the refractor problem (13). +8. No Set of Points Satisfies Both Hypotheses H1 and H2 +In the paper of Kochengin et al. +[1], they specify two assumptions with +regards to the target set T that they call Hypothesis H1 and Hypothesis H2. +In Lemma 11, Theorem 12, and Theorem 13, they prove key results with the +assumption that both hypothesis hold for T. In this section we will prove that +Hypotheses H1 and H2 are inherently contradictory. We start by restating some +key definitions from [1] and proceed with the proof. +Let kx = +x +|x|. +We are given a set of points T in R3 \ {O}, where c = +minx,y∈T ⟨kx, ky⟩, ℓ = minx∈T |x|, and L = maxx∈T |x|. +We also use the definition of ϵ0 provided in Lemma 2 of [1]: +ϵ0 = ℓ + +√ +ℓ2 − 2Lℓc + L2 +2ℓc − L +. +(65) +In [1] we have Hypothesis H1 which is as follows. +Hypothesis H1. T is a compact subset of R3 contained in a half space of R3, +ℓ > 0, and 2ℓc > L. +Assume that T satisfies Hypothesis H1. By definition L ≥ ℓ > 0, therefore +we can write L = ℓ(1 + δ) where δ ≥ 0. We can also observe that L > 0 implies +25 + +that 2ℓc > 0. Thus c > 0. Observe that c is the cosine of the largest angle +between two points in T, then 1 ≥ c since cosine is bounded above by 1. +Copying directly from [1] we present Hypothesis H2 as follows. +Hypothesis H2. We say that T satisfies hypothesis H2 if +1. inequalities (22)-(24) in [1] hold for T, +2. for some number γ′ > 0 condition (28) in [1] is satisfied +As the title reveals, we prove that no set of points T satisfies both Hypotheses +H1 and H2. The inequalities I will focus on are (22) and (23), namely +2ℓ − Lϵ0 > 0 +(22) +and +ϵ0 > ℓϵ0 + +� +ℓ2ϵ2 +0 − 2ℓLϵ0 + L2ϵ2 +0 +2ℓ − Lϵ0 +. +(23) +The central idea of our main proof is showing that these two inequalities +contradict each other when given Hypothesis H1. +However, we first need to prove the following claim. +Claim 8.1. Let a set of points T satisfy Hypothesis H1, then ϵ0 ≥ 1. +Proof. Assume to the contrary that there exists a ℓ, L and c such that ϵ0 < 1 +and Hyopthesis H1 is also satisfied. Recall that we can write L = ℓ(1+δ) where +δ ≥ 0. Thus we can rewrite (65) as +ϵ0 = ℓ + +√ +ℓ2 − 2Lℓc + L2 +2ℓc − L += ℓ + +� +ℓ2 − 2ℓ2(1 + δ)c + ℓ2(1 + δ)2 +2ℓc − ℓ(1 + δ) += 1 + +� +1 − 2(1 + δ)c + (1 + δ)2 +2c − (1 + δ) +. +Thus we now have +26 + +1 > 1 + +� +1 − 2(1 + δ)c + (1 + δ)2 +2c − (1 + δ) +Hypotheses H1 tells us that 2ℓc − L > 0. Thus 2c − (1 + δ) is positive. We +now have that +2c − (1 + δ) > 1 + +� +1 − 2(1 + δ)c + (1 + δ)2 +⇒ 2(c − 1) − δ > +� +2(1 − c) + 2δ(1 − c) + δ2. +Since 1 ≥ c > 0, we have that 0 ≥ c − 1. Since δ ≥ 0, we have that the LHS of +the last inequality is non-positive and the RHS is non-negative. A contradiction +since ϵ0 ≥ 1. +Now for the main result. +Theorem 8.1. Let a set of points T satisfy Hypothesis H1, then T does not +satisfy Hypothesis H2. +Proof. Let us rewrite L = ℓ(1 + δ) when δ ≥ 0. +Thus we can rewrite (22) as +2 − (1 + δ)ϵ0 > 0. +and we can rewrite (23) as +ϵ0 > ϵ0 + +� +ϵ2 +0 − 2(1 + δ)ϵ0 + (1 + δ)2ϵ2 +0 +2 − (1 + δ)ϵ0 +. +Assume to the contrary that there exists a set T such that H2 can be satisfied. +Then there exists an ϵ0 and a δ that satisfies both inequalities (22) and (23). If +T satisfies inequality (22) thus we can obtain an equivalent inequality to (23): +ϵ0(2 − (1 + δ)ϵ0) − ϵ0 > +� +ϵ2 +0 − 2(1 + δ)ϵ0 + (1 + δ)2ϵ2 +0. +With a little bit of algebra on each side, we obtain +−δϵ2 +0 − (ϵ2 +0 − ϵ0) > +� +(1 + 2δ)(ϵ2 +0 − ϵ0) + ϵ2 +0δ2. +27 + +By the above Claim A.1, ϵ0 ≥ 1, thus ϵ2 +0 ≥ ϵ0. That combined with the fact +that δ ≥ 0, we have that the LHS is non-positive and the RHS is non-negative. +This would make the above inequality incorrect, thus we have a contradiction. +Thus when given H1, the inequality (23) is not valid when given inequal- +ity (22). Therefore the inequalities in H2 are contradictory, so H2 cannot be +satisfied. +9. Discussion +In this section, we proved existence theorems for the rotationally symmetric +case, Theorem 7.1. Rotationally symmetric cases are not only practically useful +because this case provides a model situation [9], but also because rotationally +symmetric solutions can be used to recover nonrotationally symmetric solutions +from irradiance distributions without special symmetry assumptions [10]. We +also proved an existence theorem for the case where the points in the target set +are sufficiently close to each other, Theorem 6.1. Theorem 6.1 has the potential +to lead to solutions for all kinds of exotic, Lebesgue measurable, target sets. +We also proved a uniqueness theorem for the case when the target set is finite, +Theorem 4.1, and the general case, Theorem 4.2. Thus, we make significant +progress on the original formulation of the refractor problem. +For Theorem 6.1, a possible avenue for further research would be to find +precise values for ξ and γ such that the theorem holds. +Another potential +avenue for research is to find an explicit algorithm to find proper hyperboloids +for the discrete case. In addition, the author believes that Theorems 6.1 and +7.1 provide indirect evidence for the following conjecture. +Conjecture 9.1. Let T be a target set that satisfies Hypothesis H1 and ϵ0 < +limt→K+ +1 +t−1 for K = maxx∈T |x| +minx∈T |x| when ϵ0 is defined by (5). Assume that we are +given a γ > 0 such that ϵ0 + γ < limt→K+ +1 +t−1. Also, assume we are given a +nonnegative g ∈ L1(S2) where g ≡ 0 outside Dδγ +T where δγ = +1 +ϵ0+γ . Assume we +are given a nonnegative f ∈ L1(T) and F is the measure defined by (37) such +28 + +that +F(T) = µg(Dδγ +T ). +(66) +Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ +1 +t−1) such that there exists a +convex refractor R ∈ RϵM +convex(T) where R is a convex weak solution to the +refractor problem (13). +Acknowledgements +The author would like to especially thank his advisor, Prof. Vladimir Oliker, +for introducing him to this problem. Also, the author would like to thank Prof. +David Borthwick for helping him with the presentation of this paper. +References +[1] S. Kochengin, V. Oliker, O. von Tempeski, On the design of reflectors with +prescribed distribution of virtual sources and intensities, Inverse Problems +14 (1998) 661–678. +[2] M. Born, E. Wolf, Principles of Optics: 60th Anniversary Edition, 7th +Edition, Cambridge University Press, 2019. +[3] V. Oliker, L. Prussner, On the numerical solution of the equation ∂2z +∂x2 ∂2z +∂y2 − +( ∂2z +∂x∂y)2 = f and its discretizations, i., Numerische Mathematik 54 (3) +(1989) 271–294. +[4] S. Kochengin, V. Oliker, Determination of reflector surfaces from near-field +scattering data, Inverse Problems 13 (2) (1997) 363–373. +[5] L. Caffarelli, V. Oliker, Weak solutions of one inverse problem in geometric +optics, Journal of Mathematical Sciences 154 (2008) 39–49. +[6] R. Howard, Alexandrov’s theorem on the second derivatives of convex func- +tions via rademacher’s theorem on the first derivatives of lipschitz functions, +http://https://people.math.sc.edu/howard/Notes/alex.pdf (1998). +29 + +[7] S. Kochengin, V. Oliker, Determination of reflector surfaces from near-field +scattering data ii. numerical solution ., Numer. Math. 79 (1998) 553–568. +[8] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, 2nd Edition, +Encyclopedia of Mathematics and its Applications, Cambridge University +Press, 2013. +[9] V. Oliker, On reconstructing a reflecting surface from the scattering data in +the geometric optics approximation, Inverse Problems 5 (1) (1989) 51–65. +[10] V. Oliker, Near radially symmetric solutions of an inverse problem in geo- +metric optics, Inverse Problems 3 (1987) 743–756. +30 + diff --git a/qNA0T4oBgHgl3EQfKf_Y/content/tmp_files/load_file.txt b/qNA0T4oBgHgl3EQfKf_Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4f8aff58fd050f06fe5c4a6b3eb88433b7e9f191 --- /dev/null +++ b/qNA0T4oBgHgl3EQfKf_Y/content/tmp_files/load_file.txt @@ -0,0 +1,780 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf,len=779 +page_content='Convex Solutions to the Virtual Source Reflector Problem Dylanger S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Pittman 400 Dowman Drive, Atlanta Abstract We greatly expand upon the results of Kochengin, Oliker and Tempeski [1] to include results for uniqueness in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We also include results for existence in the rotationally symmetric case and the case where the target set is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We also point out an error that was found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Keywords: partial differential equations, geometric optics, geometry 2020 MSC: 78A05, 35, 51, 53 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Introduction Let O be the origin of R3, and let S2 be the unit sphere centered at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We treat points on S2 as unit vectors with initial points at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let an aperture be a subset of S2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' in our work, the aperture will be an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Physically, it makes sense to consider O as the location of an anisotropic point source of light such that rays of light are emitted in a set of directions defined by an aperture D ⊆ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we are given an aperture that is a connected open set D ⊆ S2, and a function ρ : D → (0, ∞) that is continuous and almost everywhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then a reflector is the set R = {mρ(m)|m ∈ D} ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Email address: dpittm2@emory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='edu (Dylanger S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Pittman) Preprint submitted to Inverse Problems January 6, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='02106v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='AP] 5 Jan 2023 We first recall the classical law of reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we have a contin- uous, almost everywhere differentiable, positive function ρ : D → (0, ∞) and a corresponding reflector R = {mρ(m)|m ∈ D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Suppose that a ray originating from O in the direction m ∈ D is incident on the reflector R at the point mρ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' If ρ is differentiable at m, there is a unit vector, n(m), normal to the reflector R at mρ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Therefore, by the reflection law of geometric optics, a ray from O of direction m reflects off the point mρ(m) in the direction y(m) = m − 2⟨m, n(m)⟩n(m) (1) where ⟨m, n(m)⟩ is the standard Euclidean inner product in R3 and n(m) is oriented such that ⟨m, n(m)⟩ > 0 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we are given an aperture that is a connected open set D ⊆ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let U be an open subset of S2 such that D ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Consider a function ρ : U → (0, ∞) that is continuous and almost everywhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then a refractor is the set R = {mρ(m)|m ∈ U} ⊂ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that R = {mρ(m)|m ∈ D} can be considered as either a reflector or a refractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' If R = {mρ(m)|m ∈ D} is considered as a refractor, the refracted direction ˆy is determined by Snell’s law and is given as ˆy(m) = cfm − �� 1 − c2 f(1 − ⟨m, n(m)⟩2) − cf⟨m, n(m)⟩ � n(m) (2) where cf denotes the refraction index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We borrow the following motivation from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Consider a two-sheeted hy- perboloid of revolution with sheets B and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let O be the focus inside the convex body bounded by the first sheet B and x the focus inside the convex body bounded by the sheet H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Suppose that a point source of light is positioned at O and the sheet H is a reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' H has very special and important reflecting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Specifically, if a ray of direction m from O is incident on a point z ∈ H and is reflected in the direction y(m) as defined by (1), then the ray from z of direction y(m) coincides with a ray from x of direction y(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' This means that the focus x can be viewed, from a physical perspective, as a virtual source 2 of rays reflected off H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' A two-dimensional analog of this situation is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' This same situation can also be interpreted from a different point of view allowing us to treat it geometrically as a refraction problem, rather than a reflection problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Now suppose a light ray of direction m from O strikes H and ‘refracts’ such that the refracted direction is given by ˆy = −y = −m + 2⟨m, n(m)⟩n(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (3) Then since the refracted direction is the opposite of the reflected direction, every ray of direction m that strikes the refractor H will cross the focus x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Equation (3) can also be considered as the version equation of (2) where cf = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Under the law of total energy conservation, the total energy ‘delivered’ by the refractor H to the point x will be equal to the total energy produced by the source O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We will only discuss this type of refraction, where cf = −1, for the rest of the dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' This interpretation of the reflection with a virtual source as a particular case of refraction is convenient from a geometric point of view and we use this terminology throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Physically, however, it is more natural to treat the point x as a virtual source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' This would also be consistent with the case of a distributed virtual source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' which we focus on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' To quote [1]: with this terminology, the problem studied in this paper can now be described as a problem of finding a convex refractor R which will refract a given anisotropic bundle of rays from a source O in such a way that the refracted rays are incident on a specified set in space and produce there, a given-in- advance intensity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' More specifically, suppose that we have a system consisting of an anisotropic point source at O, an aperture D, a nonnegative g(m) ∈ L1(S2), a target set T ⊂ R3 \\ {O}, and nonnegative integrable function f defined on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The problem consists of finding a refractor R which produces the specified in advance f on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Henceforth, we call this problem the refractor problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The only previous work available with respect to this problem can be found 3 S2 centered at the origin O light rays Target set consisting of a single point The hyperboloid H Figure 1: Here is an illustration of a virtual source reflector system where the target set is a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that all the rays of light reflect off of the hyperboloid H such that it appears that the light is originating from the target point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 4 in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' In the paper, they develop a definition of a weak solution to a PDE of Monge Amp`ere type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' specifically, the PDE described by equation (4) in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' They detail the construction of simply connected convex refractors and provide an existence theorem for the case where the target set is discrete (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Due to the weak convergence of Dirac measures to Lebesgue measures, one can create refractors that produce discrete irradiance distributions that are arbitrarily close to a continuous distribution, like pixels in a photo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' However, this does not imply the existence of a refractor that produces a continuous intensity distribution at the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' This is expected for problems that can be described by a fully nonlinear PDE of Monge-Amp´ere type [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' However, if the refractors are convex, due to the unique properties that convexity provides, one can use the weak convergence of Dirac measures to Lebesgue measures to obtain a refractor that produces a continuous irradiance distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' see [4],[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' In this paper, we work on the weak formulation developed by [1], where we develop existence and uniqueness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Due to a mistake in [1] (see Section 8), Theorem 9 in [1] (that I later present as Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1) is the only existence theorem for the refractor problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' That theorem is hard to use, and, in its current form, it cannot be extended to the continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' However, Theorem 9 in [1] can be used to prove another existence theorem for the discrete case (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2) that, in turn, can be extended to the continuous case (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We use this result to then prove the existence of solutions for the rota- tionally symmetric case (Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Additionally, we prove a uniqueness theorem for the case where the target set is finite (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1) and for the general case (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Hyperboloids of Revolution We do all our work in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We denote S2 to be the unit sphere with the center at O and kx = x/|x| for all x ∈ R3\\{O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We borrow much of this geometric setup from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Hyperboloids of revolution are of paramount importance when solving the virtual-source reflector problem due to their unique optical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 5 Consider the rotationally symmetric hyperboloid of two sheets in R3 such that one focus is O and the other is x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' let H(x) be the branch of the hyperboloid that has x as a focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' From now on, when we use the term hyperboloid, we are only referring to this branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' With each hyperboloid H(x) we associate its radial projection by rays from the origin onto an open spherical disk D(x) ⊂ S2 and its polar radius h ϵ(m) = |x|(1 − ϵ2) 2ϵ(1 − ϵ⟨m, kx⟩), m ∈ D(x) (4) where ϵ is the eccentricity of the hyperboloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Be aware that ϵ > 1 since we are describing a hyperboloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Define Hϵ(x) to be the hyperboloid with eccentricity ϵ and focus x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We now introduce a similar function h x,ϵ(m) which introduces x ∈ R3\\{O} as a variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' In this paper, we define hx,ϵ(m) = mh x,ϵ(m) for m ∈ D(x) and x ∈ R3 \\ {O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let Dϵ(x) ⊂ S2 be the preimage of Hϵ(x) under hx,ϵ, then Dϵ(x) = {m ∈ S2| 1 ϵ < ⟨m, kx⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus we can easily verify that Hϵ(x) = {hx,ϵ(m)|m ∈ Dϵ(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' From a physical perspective, x being the focus means that all light from the origin reflected off of the reflector H(x) appears to be originating from x, making x a virtual source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' From the above work, we see by taking the eccentricity ϵ to infinity that the shape of the hyperboloid becomes a plane, which is the directrix of the hyperboloid, and Dϵ(x) and goes to the hemisphere oriented towards x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The following two propositions summarize what I say precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' As the eccentricity ϵ of Hϵ(x) goes to infinity, Dϵ(x) goes to {m ∈ S2|⟨m, kx⟩ ≥ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' As the eccentricity ϵ of Hϵ(x) goes to infinity, the resultant set is a plane represented by the equation ⟨x, y − x 2⟩ = 0 where y ∈ R3, or equivalently by the polar radius equation r(m) = |x| 2⟨m,kx⟩ where m ∈ {m ∈ S2|⟨m, kx⟩ > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Observe that as the eccentricity ϵ goes to 1, we obtain a ray originating at x going in the direction described by the vector kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We call this a degenerate 6 hyperboloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' An important property of hyperboloids can be described by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let c > 0 and ϵ > 1 such that cϵ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then the hyperboloids Hcϵ(x) and Hϵ(x) have the same foci: O and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The aforementioned property is important because a reflector Hϵ(x) will reflect the light emitted from O so that the light appears to be emitted from x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' thus, making x a virtual source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Alternatively, a refractor Hϵ(x) will refract the light emitted from O so that the light is delivered to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' These properties are true no matter how large or small the eccentricity is;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' all that matters is the location of the foci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let Ax = {m ∈ S2|⟨m, kx⟩ ≥ 0} and Aδ x = {m ∈ S2|⟨m, kx⟩ ≥ δ} for δ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Observe that Ax = A0 x, A1 x = {kx}, Aδ x = ∅ for δ > 1, and Aδ x = A−1 x = S2 for δ ≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' It is also clear that if δ1 ≤ δ2, then Aδ1 x ⊆ Aδ2 x with a strict inclusion if δ1 < δ2 and δ1, δ2 ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' So while δ only has practical significance while taking values in [−1, 1], allowing it to take all values in R makes some of the upcoming proofs easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2, we have the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let 0 < δ < 1 and B ⊆ Aδ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then if ϵ > 1 δ , hx,ϵ[B] ⊂ Hϵ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' In particular, 1 ϵ < δ implies that Aδ x ⊂ Dϵ(x), 1 ϵ > δ implies that Dϵ(x) ⊂ Aδ x, and 1 ϵ = δ implies that Int(Aδ x) = Dϵ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Also note the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For an element x ∈ R3 and a set A ⊂ R3, let the set Cx,A = {at + x(1 − t)|t ∈ [0, 1], a ∈ A} be the union of all line segments from x to A and Cx,A,∞ = {at + x(1 − t)|t ∈ [0, ∞), a ∈ A} be the union of all rays from x that intersect A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Convex Weak Solutions We now have the background to construct and proceed with our discussion of the weak solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Keep in mind that this weak solution definition, apart from some minor differences in notation, is identical to the weak solution defined in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let c = minx,y∈T ⟨kx, ky⟩, ℓ = minx∈T |x|, and L = maxx∈T |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume we are given a set T ⊆ R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We say that T satisfies Hypothesis H1 if the following condition is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' T is a compact subset of R3 contained in a half space of R3, ℓ > 0, and 2ℓc > L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that this is Hypothesis H1 from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We also define a constant, ϵ0 = ℓ + √ ℓ2 − 2Lℓc + L2 2ℓc − L , (5) that depends only on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We first assume we are given a target set T that satisfies Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let ˜Hϵ(x) be the convex body bounded by Hϵ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Consider the aperture Dδγ T = Int �� x∈T Aδγ x � where δγ = 1 ϵ0+γ for some γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We then define a simply connected refractor over Dδγ T as the boundary of the intersection of the convex bodies bounded by hyperboloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Specifically, R = ∂h where h = � x∈T ˜Hϵx(x) (6) where each ϵx ≥ ϵ′ ≥ ϵ0 + γ = 1 δγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Observe that R = � m sup x∈T h x,ϵx(m) �����m ∈ Int � � x∈T A 1 ϵx x �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (7) Note that Dδγ T ⊆ Int �� x∈T A 1 ϵx x � , and supx∈T h x,ϵx is twice differentiable al- most everywhere by Alexandrov’s theorem [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' thus R may be considered a refactor per Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let Rϵ′ convex(T) (8) 8 be the set of all such refractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Please note that by Lemma 1 in [1], the set Rϵ′ convex(T) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' A hyperboloid H(x) is said to be supporting to a set Q ⊂ R3 at a point z ∈ ∂Q if the convex body ˜H(x) bounded by H(x) contains Q and z ∈ H(x) ∩ ∂Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For a subset ω ⊆ T and a refractor R ∈ Rϵ′ convex(T) put M(ω) = {z ∈ R| there exists x ∈ ω such that H(x) is supporting to R at z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (9) The intersection of Dδγ T with the image of the set M(ω) under radial projection on S2 we call the visibility set of ω and denote it by Vconvex(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By Lemma 4 of [1], this set Vconvex(ω) is measurable for all Borel sets ω ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For m ∈ Dδγ T let r(m) be the set of points of intersection between the refrac- tor R and the ray of direction m originating at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The possibly multivalued map αconvex : Dδγ T → T, αconvex(m) = {x ∈ T| there exists H(x) supporting to R at r(m)} (10) is called the refractor map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let us define for measurable X ⊆ S2 µg(X) = � X g(m)dσ(m) (11) where σ denotes the standard measure on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that g ≡ 0 outside of Dδγ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' In order to formulate and solve the refractor problem (in the framework of weak solutions to be defined below), we need to define a measure representing the energy generated by g and redistributed by a refractor R ∈ Rϵ′ convex(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Define for any refractor R ∈ Rϵ′ convex(T), Gconvex(ω) = µg(Vconvex(ω)) (12) 9 which we will deem the energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' It can be shown that G is a finite measure on the Borel σ-algebra of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let F be a nonnegative, finite, Borel measure on Borel subsets of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We say that a refractor R ∈ Rϵ′ convex(T) is a convex weak solution to the refractor problem if the refractor map αconvex determined by R is such that αconvex(m) ⊆ T for all m ∈ Dδγ T , and F(ω) = Gconvex(ω) for any Borel set ω ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (13) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Uniqueness Theorems We start with some uniqueness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 can be con- sidered as a direct corollary to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We include both as separate statements and proofs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' as discrete versions of the uniqueness theorems proved to be of special interest in related problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' see [4] and [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We proceed with the following lemma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' which is shown in the proof of Lemma 2 in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T be a target set that satisfies Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Suppose we are given positive real numbers γ and ϵ′ such that ϵ′ ≥ ϵ0 + γ where ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let R ∈ Rϵ′ convex(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then Vconvex(ω) is closed for all closed ω ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Before we proceed, note that if we write Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω) for some Borel set ω ⊆ T and some refractor R ∈ Rϵ′ convex(T), this is specifically the visibility set for the refractor R evaluated on the set ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Similarly, if we write Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω) for some Borel set ω ⊆ T and some refractor R ∈ Rϵ′ convex(T), this is specifically the energy function for the refractor R evaluated on the set ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We will be using this when we are talking about multiple refractors and we need to specify the energy function for each refractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Here we consider the case of the refractor problem (13) where the set T is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We prescribe the measure F in (13) as a Dirac measure concentrated at points in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We now introduce notation for refractors in the discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , xk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We set Hi = H(xi) and the eccentricity of Hi we denote 10 by ϵi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For the hyperboloids H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , Hk define the refractor R = ∂ � k� i=1 ˜Hi � ∈ Rϵ′ convex(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (14) Since each hyperboloid Hi is uniquely defined by its eccentricity ϵi, the refractor R can be identified with the point with coordinates (ϵ1, ϵ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) in the region ϵ1 ≥ ϵ′, ϵ2 ≥ ϵ′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk ≥ ϵ′ (15) in k−dimensional euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus we can write a refractor R ∈ Rϵ′ convex(T) as (ϵ1, ϵ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We start with a uniqueness theorem for the discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , xk} be a collection of k distinct points that satisfy Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Suppose we are given positive real numbers γ and ϵ′ such that ϵ′ ≥ ϵ0 + γ where ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2) such that g > 0 inside Dδγ T and g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , fk be a collection of positive real numbers such that k � i=1 fi = µg(Dδγ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (16) Let R = (ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) and ˜R = ( ˜ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ˜ϵk) be refractors in Rϵ′ convex(T) such that Gconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then the inequality ˜ϵj ≥ ϵj for some j implies that ˜ϵi ≥ ϵi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Furthermore, the equality ˜ϵj = ϵj for some j implies that ˜ϵi = ϵi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let J be a nonempty subset of [k] such that for any i ∈ J, ˜ϵi > ϵi, and for any i ∈ [k] \\ J, ˜ϵi ≤ ϵi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that m ∈ Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) if and only if there exists some i ∈ J such that h xi, ˜ϵi(m) ≥ h xℓ, ˜ϵℓ(m) for all ℓ ∈ [k] \\ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For this m: since h xi,ϵi(m) > h xi, ˜ϵi(m) for all i ∈ J and h xℓ,ϵℓ(m) ≤ h xℓ, ˜ϵℓ(m) for all ℓ ∈ [k] \\ J, then there exists some i ∈ J such that h xi,ϵi(m) > h xℓ,ϵℓ(m) for all ℓ ∈ [k] \\ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus, any m ∈ Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) is an interior point of Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' in other words, Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) ⊆ Int(Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Recall that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) is closed and, since f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , fk are positive, Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 11 Then Int(Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}))\\Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) is open and nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' So µg(Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) \\ Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J})) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Therefore we must have � i∈J fi = Gconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) < Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' {xi|i ∈ J}) = � i∈J fi (17) which is a contradiction because Gconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Observe that for all refractors R ∈ Rϵ′ convex(T), there exists a function K : T → [ϵ′, ∞) such that R = ∂(� x∈T ˜HK(x)(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Since each hyperboloid ˜HK(x)(x) is uniquely determined by K, the refactor R can be identified with the function K : T → [ϵ′, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus we can write a refractor R ∈ Rϵ′ convex(T) as [K] where K : T → [ϵ′, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' note that [K] = � m supx∈T h x,K(x)(m) ����m ∈ Int �� x∈T A 1 K(x) x �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Given a refractor R ∈ Rϵ′ convex(T), we call K : T → [ϵ′, ∞) the maximal function of R if R = � m maxx∈T h x,K(x)(m) ����m ∈ Int �� x∈T A 1 K(x) x �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We proceed with the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let R ∈ Rϵ′ convex(T) be a refractor such that for all x ∈ T, Vconvex({x}) is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists a K : T → [ϵ′, ∞) that is a the maximal function of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By Lemma 2 in [1], T ⊂ � x∈T ˜Hϵx(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Therefore, by Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, that m ∈ Vconvex({x}) if and only if there exists a corresponding ϵ′ x ≥ ϵ′ such that h x,ϵ′x(m) = supx∈T h x,ϵx(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Define K(x) = ϵ′ x for all x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then R = � m maxx∈T h x,K(x)(m) ����m ∈ Int �� x∈T A 1 K(x) x �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We now conclude with a uniqueness theorem for more general measures and target sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T be a target set that satisfies Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let F be a nonnegative, finite, Borel measure on Borel subsets of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Suppose we are given positive real numbers γ and ϵ′ such that ϵ′ ≥ ϵ0 + γ where ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 12 Assume we are given a nonnegative g ∈ L1(S2) such that g > 0 inside Dδγ T and g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ such that F(T) = µg(Dδγ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (18) Let R and ˜R be refractors in Rϵ′ convex(T) such that for all nonempty Borel ω ⊆ T: F(ω) = Gconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω) = Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω), Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω) ̸= ∅, and Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists functions K : T → [ϵ′, ∞) and ˜K : T → [ϵ′, ∞) that are, respectively, maximal functions of R and ˜R such that the inequality ˜K(x) ≥ K(x) for some x ∈ T implies that ˜K(y) ≥ K(y) for all y ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Furthermore, the equality ˜K(x) = K(x) for some x ∈ T implies that ˜K(y) = K(y) for all y ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2, there exists functions K : T → [ϵ′, ∞) and ˜K : T → [ϵ′, ∞) that are maximal functions of R and ˜R respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that R = [K] and ˜R = [ ˜K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let J be a nonempty closed subset of T such that for any x ∈ J, ˜K(x) > K(x), and for any x ∈ T \\ J, ˜K(x) ≤ K(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that m ∈ Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J) if and only if there exists some z ∈ J such that h z, ˜ K(z)(m) ≥ h z′, ˜ K(z′)(m) for all z′ ∈ T \\ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For this m: since h z,K(z)(m) > h z, ˜ K(z)(m) for all z ∈ J and h z′,K(z′)(m) ≤ h z′, ˜ K(z′)(m) for all z′ ∈ T \\ J, then there exists some z ∈ J such that h z,K(z)(m) > h z′,K(z′)(m) for all z′ ∈ T \\J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus, any m ∈ Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J) is an interior point of Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' in other words, Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J) ⊆ Int(Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Recall that, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J) is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then Int(Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J)) \\ Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J) is open and nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' So µg(Vconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J) \\ Vconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J)) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Therefore we must have F(J) = Gconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J) < Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' J) = F(J) (19) which is a contradiction because F(ω) = Gconvex( ˜R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω) = Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω) for all Borel ω ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The theorem is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Weak Solutions in the Discrete Case Here we consider the case of the refractor problem (13) where the set T is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We prescribe the measure F in (13) as a Dirac measure concentrated at points in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We recall notation for refractors in the discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T = {x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , xk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We set Hi = H(xi) and the eccentricity of Hi we denote by ϵi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For the hyperboloids H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , Hk define the refractor R = ∂ � k� i=1 ˜Hi � ∈ Rϵ′ convex(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (20) Since each hyperboloid Hi is uniquely defined by its eccentricity ϵi, the refractor R can be identified with the point with coordinates (ϵ1, ϵ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) in the region ϵ1 ≥ ϵ′, ϵ2 ≥ ϵ′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk ≥ ϵ′ (21) in k−dimensional euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus we can write a refractor R ∈ Rϵ′ convex(T) as (ϵ1, ϵ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We now recall Theorem 9 from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 (Theorem 9 in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , xk} be a collection of k distinct points in R3 \\ {O}, k > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that T satisfies Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let γ, ϵM, ϵmin, and ϵmax be positive real numbers such that ϵ0 + γ < ϵM ≤ ϵmin ≤ ϵmax < ∞, where ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2) such that g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , fk be nonnegative real numbers such that k � i=1 fi = µg(Dδγ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (22) Suppose that there also exists some ℓ ∈ [k] such that for all i ∈ [k], i ̸= ℓ, Gconvex(Rℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) ≤ fi (23) where Rℓ = (ϵ1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=', ϵℓ−1 = ϵmax, ϵℓ = ϵmin, ϵℓ+1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=', ϵk = ϵmax), and Gconvex(Rℓi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xℓ) < fℓ (24) 14 where Rℓi = (ϵ1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵi−1 = ϵmax, ϵi = ϵM, ϵi+1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵℓ−1 = ϵmax, ϵℓ = ϵmin, ϵℓ+1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=', ϵk = ϵmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists a refractor R = (ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) ∈ RϵM convex(T) such that Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (25) This theorem inspires an obvious corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , xk} be a collection of k distinct points in R3 \\ {O}, k > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that T satisfies Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let γ, ϵM, ϵmin, and ϵmax be positive real numbers such that ϵ0 + γ < ϵM ≤ ϵmin ≤ ϵmax < ∞, where ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2) such that g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , fk be nonnegative real numbers such that k � i=1 fi = µg(Dδγ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (26) Suppose that there also exists some ℓ ∈ [k] where fℓ > 0 such that for all i ∈ [k], i ̸= ℓ, Gconvex(Rℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = 0 (27) where Rℓ = (ϵ1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=', ϵℓ−1 = ϵmax, ϵℓ = ϵmin, ϵℓ+1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=', ϵk = ϵmax), and Gconvex(Rℓi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xℓ) = 0 (28) where Rℓi = (ϵ1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵi−1 = ϵmax, ϵi = ϵM, ϵi+1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵℓ−1 = ϵmax, ϵℓ = ϵmin, ϵℓ+1 = ϵmax, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=', ϵk = ϵmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists a refractor R = (ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) ∈ RϵM convex(T) such that Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (29) We will now use the above corollary to prove the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 15 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , xk} be a collection of k distinct points in R3 \\{O} such that T satisfies Hypothesis H1 and minx,y∈T ⟨kx, ky⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Suppose that we are given γ > 0 such that ϵ0 + γ < limt→K+ 1 t−1 for K = maxx∈T |x| minx∈T |x| and ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume we are given a nonnegative g ∈ L1(S2) such that g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , fk be nonnegative real numbers such that k � i=1 fi = µg(Dδγ T ) (30) and for the ℓ ∈ [k] where |xℓ| = maxy∈T |y|, fℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ 1 t−1) such that we can construct a convex, rotationally symmetric refractor R = (ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) ∈ RϵM convex(T) where Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (31) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that maxx,y⟨kx, ky⟩ = 1 implies that kx = ky for all x, y ∈ T, and ϵ0 = 1 as defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The case where k = 1 is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that |xi| ≥ |xi+1| for all i ∈ [k − 1] and thus f1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Recall that for x ∈ T by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2, h ϵ,x(m) → |x| 2⟨m,kx⟩ as ϵ → ∞ for m ∈ Dδγ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Observe that |x1| 2⟨m, kx⟩ < |xk|(1 − ϵ2 M) 2ϵM(1 − ϵM⟨m, kx⟩) for m ∈ Dδγ T , (32) if and only if |x1| 2 < |xk|(1 − ϵ2 M) 2ϵM(1 − ϵM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (33) Thus we have that ϵM < 1 K−1 where K = |x1| |xk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that by Hypothesis H1 and the fact that k ≥ 2, we have 1 < K < 2 and 1 < 1 K−1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus we can have that 1 = ϵ0 < ϵ0 + γ < ϵM < 1 K−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' If k = 2, by the continuity implied by Lemma 8 of [1], there exists a refractor R = (ϵ1, ϵ2) ∈ RϵM convex(T) such that Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' If k > 2, we borrow language and notation from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By conti- nuity, if ϵmin = ϵmax be sufficiently large such that 1 K−1 < ϵmin = ϵmax, then, assuming that ϵM < 1 K−1, Gconvex(R1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = 0 and Gconvex(R1i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' x1) = 0 for 16 all i ∈ [k] such that i ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Therefore by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, there exists a refractor R = (ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) ∈ RϵM convex(T) such that Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The above proposition motivates our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we are given some w, W ∈ (0, ∞) where w > W 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Given some m∗ ∈ S2, let S(m∗, ξ) = {x ∈ R3|w ≤ |x| ≤ W, ⟨kx, m∗⟩ ≥ 1 − ξ} (34) where 1 − cos � 1 2 arccos � W 2w �� > ξ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' note that S(m∗, ξ) satisfies Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T = {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , xk} ⊂ S(m∗, ξ) be a collection of k distinct points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Recall that δγ = 1 ϵ0+γ where γ > 0 and ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists positive ξ and γ such that, for any nonnegative g ∈ L1(S2) such that g ≡ 0 outside Dδγ T and any collection f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , fk of nonnegative real numbers where k � i=1 fi = µg(Dδγ T ) (35) and fℓ > 0 for the ℓ ∈ [k] where |xℓ| = maxy∈T |y|, there exists an ϵM > ϵ0 + γ such that we can construct a refractor R = (ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) ∈ RϵM convex(T) where Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (36) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Observe that ξ → 0 implies that minx,y∈T ⟨kx, ky⟩ → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that |xℓ| = maxy∈T |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let h T max,ϵ(m) = maxx∈T h x,ϵ(m) and PT max(m) = maxx∈T |x| 2⟨m,kx⟩ where m ∈ Dδγ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that h T max,ϵ(m) → h xℓ,ϵ(m) and PT max(m) → |xℓ| 2⟨m,kxℓ⟩ as miny∈T ⟨kxℓ, ky⟩ → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We borrow language and notation from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Choose an ϵmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' then, by continuity, there exists a ξ > 0 such that if minx,y∈T ⟨kx, ky⟩ ≥ 1 − ξ, then we have PT max(m) < h xℓ,ϵmin(m) for all m ∈ Dδγ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Therefore by continuity there exists an ϵmax such that PT max(m) < h T max,ϵmax(m) < h xℓ,ϵmin(m) for all m ∈ Dδγ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Observe that ϵ0 → 1 as minx,y∈T ⟨kx, ky⟩ → 1 and recall that a degenerate hyperboloid has eccentricity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then, for sufficiently small γ > 0, by continuity 17 we can choose ϵ0+γ < ϵM < ϵmin < ϵmax, such that PT max(m) < h xℓ,ϵmax(m) < h T max,ϵmin(m) < h xℓ,ϵM (m) for all m ∈ Dδγ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then Gconvex(Rℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = 0 and Gconvex(Rℓi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xℓ) = 0 for all i ∈ [k] such that i ̸= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then by Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, there exists a refractor R = (ϵ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ϵk) ∈ RϵM convex(T) such that Gconvex(R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xi) = fi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Weak Solutions in the General Case In this section, we extend the results of Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 to the case of more general sets T and energy distributions F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We consider the case where prescribe the measure F as a Lebesgue measure over T, specifically F(ω) = � ω f(x)dλ(x) for any Borel set ω ⊆ T (37) for some given nonnegative function f ∈ L1(T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' here λ is the Lebesgue measure on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we are given some w, W ∈ (0, ∞) where w > W 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Given some m∗ ∈ S2, let S(m∗, ξ) = {x ∈ R3|w ≤ |x| ≤ W, ⟨kx, m∗⟩ ≥ 1 − ξ} (38) where 1 − cos � 1 2 arccos � W 2w �� > ξ > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' note that S(m∗, ξ) satisfies Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T ⊆ S(m∗, ξ) be a closed set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Recall that δγ = 1 ϵ0+γ where γ > 0 and ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists positive ξ and γ such that, for any nonnegative f ∈ L1(T) with a measure F defined by (37), and any nonnegative g ∈ L1(S2) where g ≡ 0 outside Dδγ T where F(T) = µg(Dδγ T ), (39) there exists an ϵM > ϵ0 + γ such that we can construct a convex refractor R ∈ RϵM convex(T) where R that is a convex weak solution to the refractor problem (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 18 The following argument is based on similar arguments made in [1], [5], and [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' specifically, the argument made for Theorem 13 in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Even though Theorem 13 in [1] is incorrect1, the type of argument presented in its proof is broadly applicable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' thus it would be good to put the argument in a correct context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The following argument follows the proof of Theorem 13 in [1] very closely with some adjustments to fit into this new context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For the sake of the following proof, recall that our definition of the energy function can also be considered as a measure of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' If µg(Dδγ T ) = 0, then any refractor R ∈ RϵM convex(T) will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that µg(Dδγ T ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Since T is bounded, for any δ > 0 there exists an N ∈ N such that for each k ≥ N there exists a partition of T into k Borel sets ωk 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , ωk k such that diam(ωk i ) ≤ δ for any k ≥ N, i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (40) For each k ∈ N, we choose an xk i ∈ ωk i for i ∈ [k], and put F k i = F(ωk i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (41) Define a measure F k on T by F k(ω) = � xk i ∈ω F k i for any Borel set ω ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (42) Note that F k converges weakly to F as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For each k, there exists a nonempty S ⊆ [k] such that F k i > 0 for all i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Since {xk i }i∈S and {F k i }i∈S satisfies the assumptions of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2, there exists a convex refrac- tor Rk ∈ RϵM convex({xk i |i ∈ S}) ⊆ RϵM convex({xk i |i ∈ [k]}) ⊆ RϵM convex(T) defined by hyperboloids with an eccentricity greater than or equal to some ϵM > ϵ0 + γ such that Gconvex(Rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xk i ) = F k i for i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (43) 1It should be noted that in [1], due to erroneous assumptions, there were issues with attempts to construct refractors in the special case explored by Theorems 12 and 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' see Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 19 Let Gk be the measure on T defined by Gk(ω) = � xk i ∈ω Gconvex(Rk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' xk i ) (44) then obviously F k ≡ Gk for all k ∈ N and consequently, Gk → F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' To finish the proof we need to construct a refractor R whose energy function, G, would be the limit of measures Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' This refractor is constructed in the following manner as a limit of refractors Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' First, we note that since g ≡ 0 outside Dδγ T , we only need to consider the part of the refractor Rk ∩ CO,D δγ T ,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' See Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 as a refresher on the meaning of CO,D δγ T ,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Also for some ϵM > ϵ0 + γ one can show that for any R ∈ RϵM (T) � R ∩ CO,D δγ T ,∞ � ⊆ B(O, b) for some b > 0 (45) where B(O, b) is the open ball centered at the origin O of radius b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let us prove this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We define b = max x∈T,m∈DT δγ h x,ϵM (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (46) Since h x,ϵM is a continuous function and DT δγ is compact, this definition is correct and b < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus for any ϵ > ϵM, h x,ϵ(m) ≤ b and (45) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For each of the refractors Rk we consider a bounded convex body hk b = hk ∩ CO,D δγ T ,∞ ∩ B(O, b) (47) where for each k ∈ N the set hk is defined by (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By Blaschke’s selection theorem [8], there exists a subsequence of {hk b} which we again denote by {hk b}, which converges to some convex body hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We show now that for each point r ∈ [∂(hb) ∩ CO,D δγ T ,∞] \\ ∂(B(O, b)) there exists a hyperboloid Hr(x) which is supporting to hb at point r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let r ∈ [∂(hb) ∩ CO,D δγ T ,∞] \\ ∂(B(O, b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists a sequence {rk} that converges to r where each rk ∈ Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let H(xk) be a supporting hyperboloid to Rk at rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Since T is compact, {xk} contains a subsequence, which we will denote by {x∗ k}, converging to some x ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The convex body ˜H(x∗ k) bounded by 20 H(x∗ k) contains the body hk b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The corresponding sequence { ˜H(x∗ k)} converges to the body ˜Hr(x) containing hb and hk b converges to hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Therefore ˜Hr(x) contains ∂(hb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' It follows that Hr(x) is supporting to hb at r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We now define the refractor R = ∂(� x∈T ˜Hr(x)) and show that the sequence of measures Gk, that are equivalent to the energy functions corresponding to the refractors Rk, converges weakly to the measure G, which is the energy function of the refractor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let αk convex and αconvex be the refractor maps corresponding to Rk and R respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By a theorem of Reidemeister on singularities on convex surfaces (see [8]) the refractor maps αk convex for k ∈ N and αconvex are single-valued functions almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Furthermore, for almost all m ∈ Dδγ T the hyper- boloids Hk supporting to Rk at points rk(m) converge to the hyperboloid H supporting to R at the point r(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus, αk convex(m) converges to αconvex(m) almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' If given a set of cardinality one, {z}, let Ele({z}) = z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let Y k(m) = {x ∈ αk convex(m)|x ∈ {xk i }i∈[k]} and let Jk(m) ⊆ [k] be the set of indices such that {xk i |i ∈ Jk(m)} = Y k(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let z ∈ T, Kk(m) = xk min Jk(m), (48) and K(m) = � � � � � Ele(αconvex(m)) if |αconvex(m)| = 1 z if |αconvex(m)| > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (49) Then for any continuous function u on T we have � T udGk = � D δγ T u(Kk(m))dµg(m) −→ � D δγ T u(K(m))dµg(m) = � T udG (50) as k → ∞, that is, the measures {Gk} converge weakly to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Rotationally Symmetric Convex Refractors on the Surface of a Right Circular Cone Both Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 inspire this next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 21 Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T be a closed set that satisfies Hypothesis H1 and minx,y∈T ⟨kx, ky⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we are given γ > 0 such that ϵ0 + γ < limt→K+ 1 t−1 for K = maxx∈T |x| minx∈T |x| and ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Also, assume we are given a non- negative g ∈ L1(S2) such that g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Suppose we are given a nonnegative f ∈ L1(T) and a measure F defined by (37) such that F(T) = µg(Dδγ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (51) Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ 1 t−1) such that we can construct a convex rotationally symmetric refractor R ∈ RϵM convex(T) where R that is a convex weak solution to the refractor problem (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We will use the above corollary to create rotationally symmetric refractors with the target set on a right circular cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let k ≥ 2, d > 0, 1 > ξ > 0, and m∗, m′ ∈ S2 such that ⟨m∗, m′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We create a Cartesian coordinate system centered at O where m∗ is the direction of our z-axis, m′ is the direction of our x-axis, and m∗ × m′ is the direction of our y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let (x, y, z)′ represent a point in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Recall that, given a point (x, y, z)′ ∈ R3, there exists r ∈ [0, ∞), φ ∈ [0, π], θ ∈ [0, 2π), such that x = r cos θ sin φ (52) y = r sin θ sin φ (53) z = r cos φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (54) Let Q be a closed subset of the interval (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Define the set of points T ξ k,Q(m∗, m′) as �� d cos �2πj k � sin (arccos(ξ)) , d sin �2πj k � sin (arccos(ξ)) , dξ �′ |j ∈ I, d ∈ Q � (55) where I = {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Define the set T ξ ∞,Q(m∗, m′) as � (d cos (θ) sin (arccos(ξ)) , d sin (θ) sin (arccos(ξ)) , dξ)′ |θ ∈ [0, 2π), d ∈ Q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (56) 22 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let m∗, m′ ∈ S2 such that ⟨m∗, m′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let Q be a closed subset of the interval (0, ∞), k ≥ 2, and 1 > ξ > 0 such that T = T ξ k,Q(m∗, m′) satisfies Hypothesis H1 such that ϵ0 < limt→K+ 1 t−1 for K = maxx∈T |x| minx∈T |x| where ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we are given γ > 0 such that ϵ0 + γ < limt→K+ 1 t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Also, assume we are given a nonnegative g∗ ∈ L1(S2) that is rotationally symmetric about the axis defined by the ray of direction m∗ origi- nating at O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let the function g ∈ L1(S2) be defined as g ≡ g∗ inside Dδγ T and g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Suppose we are given a nonnegative f ∈ L1(T) such that for every d ∈ Q: f �� d cos �2πj k � sin (arccos(ξ)) , d sin �2πj k � sin (arccos(ξ)) , dξ �′� (57) is constant for all j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let F be the measure defined by (37) and F(T) = µg(Dδγ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (58) Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ 1 t−1) such that we can construct a convex refractor R ∈ RϵM convex(T) where R that is a convex weak solution to the refractor problem (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We create a Cartesian coordinate system centered at O where m∗ is the direction of our z-axis, m′ is the direction of our x-axis, and m∗ × m′ is the direction of our y-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let (x, y, z)′ represent a point in this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Recall that, given a point (x, y, z)′ ∈ R3, there exists r ∈ [0, ∞), φ ∈ [0, π], θ ∈ [0, 2π), such that x = r cos θ sin φ (59) y = r sin θ sin φ (60) z = r cos φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (61) Let Tj = �� d cos � 2πj k � sin (arccos(ξ)) , d sin � 2πj k � sin (arccos(ξ)) , dξ �′ |d ∈ Q � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that T = � i∈{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=',k−1} Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let m1, m2 ∈ S2 such that ⟨m1, m2⟩ > −1 and L (m1, m2) be the shortest arc on S2 between the points m1 and m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For all j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , k − 1}, let 23 Bj ⊂ Dδγ T be the set {t ∈ L (m∗, y)|y ∈ ∂(Dδγ T ) ∩ ∂(Aδγ x )} where x ∈ Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For d ∈ Q, since T ξ k,{d}(m∗, m′) defines the points of a regular k-gon centered at the axis defined by the ray of direction m∗ originating at O, then µg(Bj) = µg(D δγ T ) k for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For all j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , k−1}, let gj be a function over S2 such that gj ≡ g inside Int(Bj) and gj ≡ 0 outside Int(Bj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Note that µg(Int(Bj)) = µgj(Dδγ Tj) for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let mj ∈ S2 be the unit vector such that a ray originating from O of direction mj intersects every point in Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Also, let fj be the restriction of the function f to the set Tj and Fj be the corresponding measure as defined by (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, for all j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , k − 1}, there exists a refractor Rj = ∂ �� d∈Q ˜Hϵd(dmj) � ∈ RϵM convex(Tj), that is rotationally symmetric about the axis defined by the ray starting at O with direction mj, such that Rj that is a convex weak solution to the refractor problem (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' where Fj(ω) = µgj(V (Rj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' ω)) for all Borel ω ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Since for x ∈ Rj, we have that |x| increases as ⟨kx, mj⟩ decreases for all j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , k − 1}, then for j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' , k − 1}, defining rj(m) as the point of intersection between Rj and the ray originating from O in direction m, we have |rj(m)| = maxi∈{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=',k−1} |ri(m)| for all m ∈ Bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus ∂ � � � j∈{0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=',k−1} � � � d∈Q ˜Hϵd(dmj) � � � � ∈ RϵM convex(T) (62) is our refractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' With an argument similar to that we use in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, we obtain the following result from Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let m∗, m′ ∈ S2 such that ⟨m∗, m′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let 1 > ξ > 0 and Q be a closed subset of the interval (0, ∞) such that T = T ξ ∞,Q(m∗, m′) satisfies Hypothesis H1 where ϵ0 < limt→K+ 1 t−1 for K = maxx∈T |x| minx∈T |x| and ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we are given γ > 0 such that ϵ0 + γ < limt→K+ 1 t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Also, assume we are given a nonnegative g ∈ L1(S2) that is rotationally symmetric 24 about the axis defined by the ray of direction m∗ originating at O such that g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume we have a nonnegative f ∈ L1(T) such that for every d ∈ Q: f � (d cos (θ) sin (arccos(ξ)) , d sin (θ) sin (arccos(ξ)) , dξ)′� (63) is constant for all θ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let F be the measure defined by (37) and F(T) = µg(Dδγ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (64) Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ 1 t−1) such that we can construct a convex, rotationally symmetric refractor R ∈ RϵM convex(T) where R that is a convex weak solution to the refractor problem (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' No Set of Points Satisfies Both Hypotheses H1 and H2 In the paper of Kochengin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' [1], they specify two assumptions with regards to the target set T that they call Hypothesis H1 and Hypothesis H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' In Lemma 11, Theorem 12, and Theorem 13, they prove key results with the assumption that both hypothesis hold for T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' In this section we will prove that Hypotheses H1 and H2 are inherently contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We start by restating some key definitions from [1] and proceed with the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let kx = x |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We are given a set of points T in R3 \\ {O}, where c = minx,y∈T ⟨kx, ky⟩, ℓ = minx∈T |x|, and L = maxx∈T |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We also use the definition of ϵ0 provided in Lemma 2 of [1]: ϵ0 = ℓ + √ ℓ2 − 2Lℓc + L2 2ℓc − L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (65) In [1] we have Hypothesis H1 which is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' T is a compact subset of R3 contained in a half space of R3, ℓ > 0, and 2ℓc > L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that T satisfies Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' By definition L ≥ ℓ > 0, therefore we can write L = ℓ(1 + δ) where δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We can also observe that L > 0 implies 25 that 2ℓc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Observe that c is the cosine of the largest angle between two points in T, then 1 ≥ c since cosine is bounded above by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Copying directly from [1] we present Hypothesis H2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Hypothesis H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We say that T satisfies hypothesis H2 if 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' inequalities (22)-(24) in [1] hold for T, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' for some number γ′ > 0 condition (28) in [1] is satisfied As the title reveals, we prove that no set of points T satisfies both Hypotheses H1 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' The inequalities I will focus on are (22) and (23), namely 2ℓ − Lϵ0 > 0 (22) and ϵ0 > ℓϵ0 + � ℓ2ϵ2 0 − 2ℓLϵ0 + L2ϵ2 0 2ℓ − Lϵ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (23) The central idea of our main proof is showing that these two inequalities contradict each other when given Hypothesis H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' However, we first need to prove the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let a set of points T satisfy Hypothesis H1, then ϵ0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume to the contrary that there exists a ℓ, L and c such that ϵ0 < 1 and Hyopthesis H1 is also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Recall that we can write L = ℓ(1+δ) where δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus we can rewrite (65) as ϵ0 = ℓ + √ ℓ2 − 2Lℓc + L2 2ℓc − L = ℓ + � ℓ2 − 2ℓ2(1 + δ)c + ℓ2(1 + δ)2 2ℓc − ℓ(1 + δ) = 1 + � 1 − 2(1 + δ)c + (1 + δ)2 2c − (1 + δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus we now have 26 1 > 1 + � 1 − 2(1 + δ)c + (1 + δ)2 2c − (1 + δ) Hypotheses H1 tells us that 2ℓc − L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus 2c − (1 + δ) is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We now have that 2c − (1 + δ) > 1 + � 1 − 2(1 + δ)c + (1 + δ)2 ⇒ 2(c − 1) − δ > � 2(1 − c) + 2δ(1 − c) + δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Since 1 ≥ c > 0, we have that 0 ≥ c − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Since δ ≥ 0, we have that the LHS of the last inequality is non-positive and the RHS is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' A contradiction since ϵ0 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Now for the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let a set of points T satisfy Hypothesis H1, then T does not satisfy Hypothesis H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let us rewrite L = ℓ(1 + δ) when δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus we can rewrite (22) as 2 − (1 + δ)ϵ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' and we can rewrite (23) as ϵ0 > ϵ0 + � ϵ2 0 − 2(1 + δ)ϵ0 + (1 + δ)2ϵ2 0 2 − (1 + δ)ϵ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume to the contrary that there exists a set T such that H2 can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Then there exists an ϵ0 and a δ that satisfies both inequalities (22) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' If T satisfies inequality (22) thus we can obtain an equivalent inequality to (23): ϵ0(2 − (1 + δ)ϵ0) − ϵ0 > � ϵ2 0 − 2(1 + δ)ϵ0 + (1 + δ)2ϵ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' With a little bit of algebra on each side, we obtain −δϵ2 0 − (ϵ2 0 − ϵ0) > � (1 + 2δ)(ϵ2 0 − ϵ0) + ϵ2 0δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 27 By the above Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, ϵ0 ≥ 1, thus ϵ2 0 ≥ ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' That combined with the fact that δ ≥ 0, we have that the LHS is non-positive and the RHS is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' This would make the above inequality incorrect, thus we have a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus when given H1, the inequality (23) is not valid when given inequal- ity (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Therefore the inequalities in H2 are contradictory, so H2 cannot be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Discussion In this section, we proved existence theorems for the rotationally symmetric case, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Rotationally symmetric cases are not only practically useful because this case provides a model situation [9], but also because rotationally symmetric solutions can be used to recover nonrotationally symmetric solutions from irradiance distributions without special symmetry assumptions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We also proved an existence theorem for the case where the points in the target set are sufficiently close to each other, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 has the potential to lead to solutions for all kinds of exotic, Lebesgue measurable, target sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' We also proved a uniqueness theorem for the case when the target set is finite, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, and the general case, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Thus, we make significant progress on the original formulation of the refractor problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' For Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1, a possible avenue for further research would be to find precise values for ξ and γ such that the theorem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Another potential avenue for research is to find an explicit algorithm to find proper hyperboloids for the discrete case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' In addition, the author believes that Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1 provide indirect evidence for the following conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Conjecture 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Let T be a target set that satisfies Hypothesis H1 and ϵ0 < limt→K+ 1 t−1 for K = maxx∈T |x| minx∈T |x| when ϵ0 is defined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume that we are given a γ > 0 such that ϵ0 + γ < limt→K+ 1 t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Also, assume we are given a nonnegative g ∈ L1(S2) where g ≡ 0 outside Dδγ T where δγ = 1 ϵ0+γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Assume we are given a nonnegative f ∈ L1(T) and F is the measure defined by (37) such 28 that F(T) = µg(Dδγ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' (66) Then there exists an ϵM ∈ (ϵ0 + γ, limt→K+ 1 t−1) such that there exists a convex refractor R ∈ RϵM convex(T) where R is a convex weak solution to the refractor problem (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Acknowledgements The author would like to especially thank his advisor, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Vladimir Oliker, for introducing him to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' Also, the author would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' David Borthwick for helping him with the presentation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNA0T4oBgHgl3EQfKf_Y/content/2301.02106v1.pdf'} +page_content=' References [1] S.' metadata={'source': 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+Active Learning for Abstractive Text Summarization +Akim Tsvigun1,2, Ivan Lysenko2, Danila Sedashov2, Ivan Lazichny1, +Eldar Damirov2,4, Vladimir Karlov2, Artemy Belousov2, Leonid Sanochkin1,2, +Maxim Panov7, Alexander Panchenko3, Mikhail Burtsev1,5, +Artem Shelmanov1,6,8 +1AIRI, 2HSE, 3Skoltech, 4SberDevices, 5MIPT, 6MBZUAI, 7TII, +8ISP RAS Research Center for Trusted Artificial Intelligence +{tsvigun, shelmanov}@airi.net, artem.shelmanov@mbzuai.ac.ae +Abstract +Construction +of +human-curated +annotated +datasets for abstractive text summarization +(ATS) is very time-consuming and expensive +because creating each instance requires a hu- +man annotator to read a long document and +compose a shorter summary that would pre- +serve the key information relayed by the origi- +nal document. Active Learning (AL) is a tech- +nique developed to reduce the amount of an- +notation required to achieve a certain level of +machine learning model performance. In infor- +mation extraction and text classification, AL +can reduce the amount of labor up to multiple +times. Despite its potential for aiding expen- +sive annotation, as far as we know, there were +no effective AL query strategies for ATS. This +stems from the fact that many AL strategies +rely on uncertainty estimation, while as we +show in our work, uncertain instances are usu- +ally noisy, and selecting them can degrade the +model performance compared to passive anno- +tation. We address this problem by proposing +the first effective query strategy for AL in ATS +based on diversity principles. We show that +given a certain annotation budget, using our +strategy in AL annotation helps to improve the +model performance in terms of ROUGE and +consistency scores. Additionally, we analyze +the effect of self-learning and show that it can +further increase the performance of the model. +1 +Introduction +Abstractive text summarization (ATS) aims to com- +press a document into a brief yet informative and +readable summary, which would retain the key in- +formation of the original document. State-of-the- +art results in this task are achieved by neural seq-to- +seq models (Lewis et al., 2020; Zhang et al., 2020; +Qi et al., 2020; Guo et al., 2021; Liu and Liu, 2021) +based on the Transformer architecture (Vaswani +et al., 2017). Training a model for ATS requires a +dataset that contains pairs of original documents +and their short summaries, which are usually writ- +ten by human annotators. Manually composing +a summary is a very tedious task, which requires +one to read a long original document, select crucial +information, and finally write a small text. Each of +these steps is very time-consuming, resulting in the +fact that constructing each instance in annotated +corpora for text summarization is very expensive. +Active Learning (AL; Cohn et al. (1996)) is a +well-known technique that helps to substantially re- +duce the amount of annotation required to achieve +a certain level of machine learning model perfor- +mance. For example, in tasks related to named +entity recognition, researchers report annotation +reduction by 2-7 times with a loss of only 1% of +F1-score (Settles and Craven, 2008a). This makes +AL especially important when annotation is expen- +sive, which is the case for ATS. +AL works iteratively: on each iteration, (1) a +model is trained on the so far annotated dataset; +(2) the model is used to select some informative +instances from a large unlabeled pool using a query +strategy; (3) informative instances are presented to +human experts, which provide gold-standard anno- +tations; (4) finally, the instances with annotations +are added to the labeled dataset, and a new iteration +begins. Traditional AL query strategies are based +on uncertainty estimation techniques (Lewis and +Gale, 1994; Scheffer et al., 2002). The hypothesis +is that the most uncertain instances for the model +trained on the current iteration are informative for +training the model on the next iteration. We argue +that uncertain predictions of ATS models (uncer- +tain summaries) are not more useful than randomly +selected instances. Moreover, usually, they intro- +duce more noise and detriment to the performance +of summarization models. Therefore, it is not possi- +ble to straightforwardly adapt the uncertainty-based +approach to AL in text summarization. +In this work, we present the first effective query +strategy for AL in ATS, which we call in-domain +diversity sampling (IDDS). It is based on the idea +arXiv:2301.03252v1 [cs.CL] 9 Jan 2023 + +of the selection of diverse instances that are se- +mantically dissimilar from already annotated doc- +uments but at the same time similar to the core +documents of the considered domain. The empiri- +cal investigation shows that while techniques based +on uncertainty cannot overcome the random sam- +pling baseline, IDDS substantially increases the +performance of summarization models. We also +experiment with the self-learning technique that +leverages a training dataset expanded with sum- +maries automatically generated by an ATS model +trained only on the human-annotated dataset. This +approach shows improvements when one needs +to generate short summaries. The code for repro- +ducing the experiments is available online1. The +contributions of this paper are the following: +• We propose the first effective AL query strat- +egy for ATS that beats the random sampling +baseline. +• We conduct a vast empirical investigation and +show that in contrast to such tasks as text clas- +sification and information extraction, in ATS, +uncertainty-based AL query strategies cannot +outperform the random sampling baseline. +• To our knowledge, we are the first to investi- +gate the effect of self-learning in conjunction +with AL for ATS and demonstrate that it can +substantially improve results on the datasets +with short summaries. +2 +Related Work +Abstractive Text Summarization. +The advent +of seq2seq models (Sutskever et al., 2014) along +with the development of the attention mecha- +nism (Bahdanau et al., 2015) consolidated neural +networks as a primary tool for ATS. The attention- +based Transformer (Vaswani et al., 2017) archi- +tecture has formed the basis of many large-scale +pre-trained language models that achieve state-of- +the-art results in ATS (Lewis et al., 2020; Zhang +et al., 2020; Qi et al., 2020; Guo et al., 2021). Re- +cent efforts in this area mostly focus on minor mod- +ifications of the existing architectures (Liu and Liu, +2021; Aghajanyan et al., 2021; Liu et al., 2022). +Active Learning in Natural Language Genera- +tion. +While many recent works leverage AL for +text classification or sequence-tagging tasks (Yuan +et al., 2020; Zhang and Plank, 2021; Shelmanov +1https://github.com/AIRI-Institute/al_ +ats +et al., 2021; Margatina et al., 2021), little atten- +tion has been paid to natural language generation +tasks. Among the works in this area, it is worth +mentioning (Haffari et al., 2009; Ambati, 2012; +Ananthakrishnan et al., 2013). These works focus +on neural machine translation (NMT) and suggest +several uncertainty-based query strategies for AL. +Peris and Casacuberta (2018) successfully apply +AL in the interactive machine translation. Liu et al. +(2018) exploit reinforcement learning to train a +policy-based query strategy for NMT. Although +there is an attempt to apply AL in ATS (Gidiotis +and Tsoumakas, 2021), to the best of our knowl- +edge, there is no published work on this topic yet. +Uncertainty Estimation in Natural Language +Generation. +A simple yet effective approach for +uncertainty estimation in generation is proposed +by Wang et al. (2019). They use a combination of +expected translation probability and variance of the +translation probability, demonstrating that it can +handle noisy instances better and noticeably im- +prove the quality of back-translation. Malinin and +Gales (2021) investigate the ensemble-based mea- +sures of uncertainty for NMT. Their results demon- +strate the superiority of these methods for OOD +detection and for identifying generated translations +of low-quality. Xiao et al. (2020) propose a method +for uncertainty estimation of long sequences of dis- +crete random variables, which they dub “BLEU +Variance”, and apply it for OOD sentence detection +in NMT. It is also shown to be useful for identifying +instances of questionable quality in ATS (Gidiotis +and Tsoumakas, 2022). In this work, we investi- +gate these uncertainty estimation techniques in AL +and show that they do not provide any benefits over +annotating randomly selected instances. +Diversity-based Active Learning. +Along with +the uncertainty-based query strategies, a series of +diversity-based methods have been suggested for +AL (Kim et al., 2006; Sener and Savarese, 2018; +Ash et al., 2019; Citovsky et al., 2021). +The +most relevant work among them is (Kim et al., +2006), where the authors propose to use a Maximal +Marginal Relevance (MMR; Carbonell and Gold- +stein (1998))-based function as a query strategy in +AL for named entity recognition. This function +aims to capture uncertainty and diversity and se- +lects instances for annotation based on these two +perspectives. We adapt this strategy for the ATS +task and compare the proposed method with it. + +3 +Uncertainty-based Active Learning for +Text Generation +In this section, we give a brief formal defini- +tion of the AL procedure for text generation +and uncertainty-based query strategies. Here and +throughout the rest of the paper, we denote an in- +put sequence as x = (x1 . . . xm) and the output +sequence as y = (y1 . . . yn), with m and n being +lengths of x and y respectively. +Let D = {(x(k), y(k))}K +k=1 be a dataset of pairs +(documents, summaries). Consider a probabilis- +tic model pw(y | x) parametrized by a vector w. +Usually, pw(y | x) is a neural network, while the +parameter estimation is done via the maximum like- +lihood approach: +ˆw = arg max +w L(D, w), +(1) +where L(D, w) = �K +k=1 log pw(y(k) | x(k)) is +log-likelihood. +Many AL methods are based on greedy query +strategies that select instances for annotation, op- +timizing a certain criterion A(x | D, ˆw) called an +acquisition function: +x∗ = arg max +x +A(x | D, ˆw). +(2) +The selected instance x∗ is then annotated with a +target value y∗ (document summary) and added to +the training dataset: D := D ∪ (x∗, y∗). Subse- +quently, the model parameters w are updated and +the instance selection process continues until the +desired model quality is achieved or the available +annotation budget is depleted. +The right choice of an acquisition function is +crucial for AL. A common heuristic for acquisition +is selecting instances with high uncertainty. Below, +we consider several measures of uncertainty used +in text generation. +Normalized Sequence Probability (NSP) +was +originally proposed by Ueffing and Ney (2007) and +has been used in many subsequent works (Haffari +et al., 2009; Wang et al., 2019; Xiao et al., 2020; +Lyu et al., 2020). This measure is given by +NSP(x) = 1 − ¯p ˆw(y | x), +(3) +where we define the geometric mean of probabil- +ities of tokens predicted by the model as: ¯p ˆw(y | +x) = exp +� 1 +n log p ˆw(y | x) +� +. +A wide family of uncertainty measures can be +derived using the Bayesian approach to modeling. +Under the Bayesian approach, it is assumed that +model parameters have a prior distribution π(w). +Optimization of the log-likelihood L(D, w) in this +case leads to the optimization of the posterior dis- +tribution of the model parameters: +π(w | D) ∝ exp{L(D, w)} · π(w). +(4) +Usually, the exact computation of the posterior is +intractable, and to perform training and inference, +a family of distributions qθ(w) parameterized by +θ is introduced. The parameter estimate ˆθ mini- +mizes the KL-divergence between the true posterior +π(w | D) and the approximation qˆθ(w). Given +such an approximation, several uncertainty mea- +sures can be constructed. +Expected Normalized Sequence Probability +(ENSP) +is proposed by Wang et al. (2019) and is +also used in (Xiao et al., 2020; Lyu et al., 2020): +ENSP(x) = 1 − Ew∼qˆθ ¯pw(y | x). +(5) +In practice, the expectation is approximated via +Monte Carlo dropout (Gal and Ghahramani, 2016), +i.e. averaging multiple predictions obtained with +activated dropout layers in the network. +Expected +Normalized +Sequence +Variance +(ENSV; Wang et al. (2019)) +measures the +variance of the sequence probabilities obtained via +Monte Carlo dropout: +ENSV(x) = Varw∼qˆθ ¯pw(y | x). +(6) +BLEU Variance (BLEUVar) +is proposed by +Xiao et al. (2020). It treats documents as points in +some high dimensional space and uses the BLEU +metric (Papineni et al., 2002) for measuring the +difference between them. In such a setting, it is +possible to calculate the variance of generated texts +in the following way: +BLEUVar(x) = +(7) += Ew∼qˆθEy,y′∼pw(·|x) +� +1 − BLEU(y, y′) +�2. +The BLEU metric is calculated as a geometric +mean of n-grams overlap up to 4-grams. Conse- +quently, when summaries consist of less than 4 +tokens, the metric is equal to zero since there will +be no common 4-grams. This problem can be mit- +igated with the SacreBLEU metric (Post, 2018), +which smoothes the n-grams with zero counts. +When we use this query strategy with the Sacre- +BLUE metric, we refer to it as SacreBLEUVar. + +Unlabeled Instances +Labeled Instances +Out-of-domain Instances +IDDS Queries +IDDS: far from + labeled, close to + unlabeled on + average + + +Uncertainty-based + methods: far from + both labeled + and unlabeled +Figure 1: The visualization of the idea behind the IDDS +alogrithm on the synthetic data: select instances lo- +cated far from labeled data while close on average to +unlabeled data. +4 +Proposed Methods +4.1 +In-Domain Diversity Sampling +We argue that uncertainty-based query strategies +tend to select noisy instances that have little value +for training ATS models. To alleviate this issue, we +propose a novel query strategy named in-domain +diversity sampling (IDDS). It aims to maximize +the diversity of the annotated instances by select- +ing instances that are dissimilar from the already +annotated ones. At the same time, it avoids se- +lecting noisy outliers. These noisy documents that +are harmful to training an ATS model are usually +semantically dissimilar from the core documents +of the domain represented by the unlabeled pool. +Therefore, IDDS queries instances that are dissimi- +lar to the annotated instances but at the same time +are similar to unannotated ones (Figure 1). +We propose the following acquisition function +that implements the aforementioned idea (the +higher the value – the higher the priority for the +annotation): +IDDS(x) = λ +|U| +� +j=1 +s(x, xj) +|U| +− (1 − λ) +|L| +� +i=1 +s(x, xi) +|L| +, +(8) +where s(x, x′) is a similarity function between +texts, U is the unlabeled set, L is the labeled set, +and λ ∈ [0; 1] is a hyperparameter. +Below, we formalize the resulting algorithm of +the IDDS query strategy. +1. For each document in the unlabeled pool x, +we obtain an embedding vector e(x). For this +purpose, we suggest using the [CLS] pooled +sequence embeddings from BERT. We note +that using a pre-trained checkpoint straightfor- +wardly may lead to unreasonably high sim- +ilarity scores between instances since they +all belong to the same domain, which can be +quite specific. We mitigate this problem by +using the task-adaptive pre-training (TAPT; +Gururangan et al. (2020)) on the unlabeled +pool. TAPT performs several epochs of self- +supervised training of the pre-trained model +on the target dataset to acquaint it with the +peculiarities of the data. +2. Deduplicate the unlabeled pool. Instances +with duplicates will have an overrated sim- +ilarity score with the unlabeled pool. +3. Calculate the informativeness scores using +the IDDS acquisition function (8). As a sim- +ilarity function, we suggest using a scalar +product between document representations: +s(x, x′) = ⟨e(x), e(x′)⟩. +The idea of IDDS is close to the MMR-based +strategy proposed in (Kim et al., 2006). Yet, despite +the resemblance, IDDS differs from it in several +crucial aspects. The MMR-based strategy focuses +on the uncertainty and diversity components. How- +ever, as shown in Section 6.1, selecting instances +by uncertainty leads to worse results compared to +random sampling. Consequently, instead of us- +ing uncertainty, IDDS leverages the unlabeled pool +to capture the representativeness of the instances. +Furthermore, IDDS differs from the MMR-based +strategy in how they calculate the diversity com- +ponent. MMR directly specifies the usage of the +“max” aggregation function for calculating the sim- +ilarity with the already annotated data, while IDDS +uses “average” similarity instead and achieves bet- +ter results as shown in Section 6.2. +We note that IDDS does not require retraining an +acquisition model in contrast to uncertainty-based +strategies since document vector representations +and document similarities can be calculated before +starting the AL annotation process. This results in +the fact that no heavy computations during AL are +required. Consequently, IDDS does not harm the +interactiveness of the annotation process, which is +a common bottleneck (Tsvigun et al., 2022). + +4.2 +Self-learning +Pool-based AL assumes that there is a large unla- +beled pool of data. We propose to use this data +source during AL to improve text summarization +models with the help of self-learning. We train the +model on the labeled data and generate summaries +for the whole unlabeled pool. Then, we concatenate +the generated summaries with the labeled set and +use this data to fine-tune the final model. We note +that generated summaries can be noisy: irrelevant, +grammatically incorrect, contain factual inconsis- +tency, and can harm the model performance. We de- +tect such instances using the uncertainty estimates +obtained via NSP scores and exclude kl% instances +with the lowest scores and kh% of instances with +the highest scores. We choose this uncertainty met- +ric because according to our experiments in Section +6.1, high NSP scores correspond to the noisiest in- +stances. We note that adding the filtration step does +not introduce additional computational overhead, +since the NSP scores are calculated simultaneously +with the summary generation for self-learning. +5 +Experimental Setup +5.1 +Active Learning Setting +We evaluate IDDS and other query strategies using +the conventional scheme of AL annotation emula- +tion applied in many previous works (Settles and +Craven, 2008b; Shen et al., 2017; Siddhant and +Lipton, 2018; Shelmanov et al., 2021; Dor et al., +2020). For uncertainty-based query strategies and +random sampling, we start from a small annotated +seeding set selected randomly. This set is used for +fine-tuning the summarization model on the first it- +eration. For IDDS, the seeding set is not used, since +this query strategy does not require fine-tuning the +model to make a query. On each AL iteration, we +select top-k instances from the unlabeled pool ac- +cording to the informativeness score obtained with +a query strategy. The selected instances with their +gold-standard summaries are added to the so-far +annotated set and are excluded from the unlabeled +pool. On each iteration, we fine-tune a summa- +rization model from scratch and evaluate it on a +held-out test set. We report the performance of +the model on each iteration to demonstrate the dy- +namics of the model performance depending on the +invested annotation effort. +The query size (the number of instances selected +for annotation on each iteration) is set to 10 doc- +uments. We repeat each experiment 9 times with +different random seeds and report the mean and +the standard deviation of the obtained scores. For +the WikiHow and PubMed datasets, on each itera- +tion, we use a random subset from the unlabeled +pool since generating predictions for the whole un- +labeled dataset is too computationally expensive. +In the experiments, the subset size is set to 10,000 +for WikiHow and 1,000 for PubMed. +5.2 +Baselines +We use random sampling as the main baseline. To +our knowledge, in the ATS task, this baseline has +not been outperformed by any other query strategy +yet. In this baseline, an annotator is given randomly +selected instances from the unlabeled pool, which +means that AL is not used at all. We also report +results of uncertainty-based query strategies and an +MMR-based query strategy (Kim et al., 2006) that +is shown to be useful for named entity recognition. +5.3 +Metrics +Quality of Text Summarization. +To measure +the quality of the text summarization model, we use +the commonly adopted ROUGE metric (Lin, 2004). +Following previous works (See et al., 2017; Nal- +lapati et al., 2017; Chen and Bansal, 2018; Lewis +et al., 2020; Zhang et al., 2020), we report ROUGE- +1, ROUGE-2, and ROUGE-L. Since we found the +dynamics of these metrics coinciding, for brevity, +in the main part of the paper, we keep only the +results with the ROUGE-1 metric. The results with +other metrics are presented in the appendix. +Factual Consistency. +Inconsistency (hallucina- +tion) of the generated summaries is one of the +most crucial problems in summarization (Kryscin- +ski et al., 2020; Huang et al., 2021; Nan et al., 2021; +Goyal et al., 2022). Therefore, in addition to the +ROUGE metrics, we measure the factual consis- +tency of the generated summaries with the original +documents. We use the SummaC-ZS (Laban et al., +2022) – a state-of-the-art model for inconsistency +detection. We set granularity = “sentence” and +model_name = “vitc”. +5.4 +Datasets +We experiment with three datasets widely-used for +evaluation of ATS models: AESLC (Zhang and +Tetreault, 2019), PubMed (Cohan et al., 2018), and +WikiHow (Koupaee and Wang, 2018). AESLC con- +sists of emails with their subject lines as summaries. +WikiHow contains articles from WikiHow pages + +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +Random sampling +NSP +ENSP +ENSV +SacreBLEUVar +Num. labeled instances +Performance, Rouge-1 +a) AESLC dataset +20 +40 +60 +80 +100 +120 +140 +25.5 +26 +26.5 +27 +27.5 +28 +28.5 +29 +29.5 +Random sampling +NSP +ENSP +ENSV +BLEUVar +Num. labeled instances +Performance, Rouge-1 +b) WikiHow dataset +20 +40 +60 +80 +100 +120 +140 +20 +22 +24 +26 +28 +30 +32 +Random sampling +NSP +ENSP +ENSV +BLEUVar +Num. labeled instances +Performance, Rouge-1 +c) PubMed dataset +Figure 2: ROUGE-1 scores of BART-base with various uncertainty-based strategies compared with random sam- +pling (baseline) on various datasets. Full results are provided in Figures 6, 8, 9, respectively. +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +30 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-1 +a) AESLC dataset +20 +40 +60 +80 +100 +120 +140 +26 +27 +28 +29 +30 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-1 +b) WikiHow dataset +20 +40 +60 +80 +100 +120 +140 +20 +22 +24 +26 +28 +30 +32 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-1 +c) PubMed dataset +Figure 3: ROUGE-1 scores of BART-base with the IDDS strategy compared with random sampling (baseline) +and NSP (uncertainty-based strategy) on various datasets. Full results are provided in Figures 10, 12 and 13, +respectively. +with their headlines as summaries. PubMed (Co- +han et al., 2018) is a collection of scientific arti- +cles from the PubMed archive with their abstracts. +The choice of datasets is stipulated by the fact that +AESLC contains short documents and summaries, +WikiHow contains medium-sized documents and +summaries, and PubMed contains long documents +and summaries. We also use two non-intersecting +subsets of the Gigaword dataset (Graff et al., 2003; +Rush et al., 2015) of sizes 2,000 and 10,000 for hy- +perparameter optimization of ATS models and addi- +tional experiments with self-learning, respectively. +Gigaword consists of news articles and their head- +lines representing summaries. The dataset statistics +is presented in Table 2 in Appendix A. +5.5 +Models and Hyperparameters +We conduct experiments using the state-of-the-art +text summarization models: BART (Lewis et al., +2020) and PEGASUS (Zhang et al., 2020). In all +experiments, we use the “base” pre-trained version +of BART and the “large” version of PEGASUS. +Most of the experiments are conducted with the +BART model, while PEGASUS is only used for +the AESLC dataset (results are presented in Appen- +dices B, C) since running it on two other datasets +in AL introduces a computational bottleneck. +We tune hyperparameter values of ATS models +using the ROUGE-L score on the subset of the +Gigaword dataset. The hyperparameter values are +provided in Table 3 in Appendix A. +For the IDDS query strategy, we use λ = 0.67. +We analyze the effect of different values of this +parameter in Section 6.2. +6 +Results and Discussion +6.1 +Uncertainty-based Query Strategies +In this series of experiments, we demonstrate that +selected uncertainty-based query strategies are not +suitable for AL in ATS. Figure 2a and Figures 6, 7 +in Appendix B present the results on the AESLC +dataset. As we can see, none of the uncertainty- +based query strategies outperform the random sam- +pling baseline for both BART and PEGASUS mod- +els. NSP and ENSP strategies demonstrate the +worst results with the former having the lowest per- +formance for both ATS models. Similar results are +obtained for the WikiHow and PubMed datasets +(Figures 2b and 2c). +In some previous work on NMT, uncertainty- +based query strategies outperform the random sam- +pling baseline (Haffari et al., 2009; Ambati, 2012; +Ananthakrishnan et al., 2013). Their low results for +ATS compared to NMT might stem from the differ- +ences between these tasks. Both NMT and ATS are + +AL Strat. +Document +Golden Summary +Gener. Summ. +NSP +Aquarius - Horoscope Friday, September 8, 2000 by +Astronet.com. Powerful forces are at work to challenge +you (...) Don’t let hurt feelings prevent you from (...) +These things are +beginning to +scare me... +Invitation – +Aquarius +NSP +Prod Area and Long Haul k# Volume Rec Del 3.6746 +5000 St 62 (...) #6563 PPL (Non NY) should have +this contract tomorrow. (...) 3.5318 6500 Leidy PSE&G +TRCO capacity +for Sep +Prod Area +IDDS +Greg, I wanted to forward this letter to you that I received +from a good friend of mine who is interested in discussing +(...) with Enron. (...) set up a meeting (...) Sincerely, +Meeting with +Enron Networks +n/a +IDDS +Larry, Could I have the price for a 2 day swing peaker +option at NGI Chicago, that can be exercised on any +day in February 2002. Strike is FOM February, (...) +Peaker price for +NGI Chicago +Feb +n/a +Table 1: Examples of instances selected with the NSP and IDDS strategies. Tokens from the source document are +highlighted with green. Tokens that refer to paraphrasing a part of the document and the corresponding part are +highlighted with blue. Tokens that cannot be derived from the document are highlighted with red. +seq2seq tasks and can be solved via similar mod- +els. However, NMT is somewhat easier, since the +output is usually of similar length as the input and +its variability is smaller. It is much easier to train a +model to reproduce an exact translation rather than +make it generate an exact summary. Therefore, +uncertainty estimates of ATS models are way less +reliable than estimates of translation models. These +estimates often select for annotation noisy docu- +ments that are useless or even harmful for training +summarization models. Table 1 reveals several +documents selected by the worst-performing strat- +egy NSP on AESLC. We can see that NSP selects +domain-irrelevant documents or very specific ones. +Their summaries can hardly be restored from the +source documents, which means that they most +likely have little positive impact on the general- +ization ability of the model. More examples of +instances selected by different query strategies are +presented in Table 5 in Appendix E. +6.2 +In-Domain Diversity Sampling +In this series of experiments, we analyze the pro- +posed IDDS query strategy. Figure 3a and Fig- +ures 10, 11 in Appendix C show the performance +of the models with various query strategies on +AESLC. We can see that the proposed strategy +outperforms random sampling on all iterations for +both ATS models and subsequently outperforms +the uncertainty-based strategy NSP. IDDS demon- +strates similar results on the WikiHow and PubMed +datasets, outperforming the baseline with a large +margin as depicted in Figures 3b and 3c. We also +report the improvement of IDDS over random sam- +pling in percentage on several AL iterations in Ta- +ble 4. We can see that IDDS provides an especially +large improvement in the cold-start AL scenario +when the amount of labeled data is very small. +We carry out several ablation studies for the +proposed query strategy. +First, we investigate +the effect of various models for document embed- +dings construction and the necessity of perform- +ing TAPT. Figures 17 and 18 in Appendix F il- +lustrate that TAPT substantially enhances the per- +formance of IDDS. Figure 17 also shows that the +BERT-base encoder appears to be better than Sen- +tenceBERT (Reimers and Gurevych, 2019) and +LongFormer (Beltagy et al., 2020). +Second, we try various functions for calculating +the similarity between instances. Figures 19, 20 in +Appendix F compare the originally used dot prod- +uct with Mahalanobis and Euclidean distances on +AESLC and WikiHow. On AESLC, IDDS with Ma- +halanobis distance persistently demonstrates lower +performance. IDDS with the Euclidean distance +shows a performance drop on the initial AL itera- +tions compared to the dot product. On WikiHow, +however, all the variants perform roughly the same. +Therefore, we suggest keeping the dot product for +computing the document similarity in IDDS since it +provides the most robust results across the datasets. +We also compare the dot product with its nor- +malized version – cosine similarity on AESLC and +PubMed, see Figures 21 and 22 in Appendix F. +On both datasets, adding normalization leads to +substantially worse results on the initial AL itera- +tions. We deem that this happens because normal- +ization may damage the representativeness com- +ponent since the norm of the embedding can be +treated as a measure of the representativeness of +the corresponding document. +Third, we investigate how different values for +the lambda coefficient influence the performance of +IDDS. Table 7 and Figure 23 in Appendix F shows +that smaller values of λ ∈ {0, 0.33, 0.5} substan- +tially deteriorate the performance. Smaller values + +correspond to selecting instances that are highly +dissimilar from the documents in the unlabeled +pool, which leads to picking many outliers. Higher +values lead to the selection of instances from the +core of the unlabeled dataset, but also very similar +to the annotated part. This also results in a lower +quality on the initial AL iterations. The best and +most stable results are obtained with λ = 0.67. +Fourth, we compare IDDS with the MMR-based +strategy suggested in (Kim et al., 2006). Since it +uses uncertainty, it requires a trained model to cal- +culate the scores. Consequently, the initial query +is taken randomly as no trained model is available +on the initial AL iteration. Therefore, we use the +modification, when the initial query is done with +IDDS because it provides substantially better re- +sults on the initial iteration. We also experiment +with different values of the λ hyperparameter of +the MMR-based strategy. Figure 24 illustrates a +large gap in performance of IDDS and the MMR- +based strategy regardless of the initialization / λ +on AESLC. We believe that this is attributed to the +fact that strategies incorporating uncertainty are +harmful to AL in ATS as shown in Section 6.1. +Finally, we compare “aggregation” functions for +estimating the similarity between a document and +a collection of documents (labeled and unlabeled +pools). Following the MMR-based strategy (Kim +et al., 2006), instead of calculating the average +similarity between the embedding of the target doc- +ument and the embeddings of documents from the +labeled set, we calculate the maximum similarity. +We also try replacing the “average” aggregation +function with “maximum” in both IDDS compo- +nents in (8). Figures 25 and 26 in Appendix F +show that average leads to better performance on +both AESLC and WikiHow datasets. +The importance of diversity sampling is illus- +trated in Table 6 in Appendix E. We can see that +NSP-based query batches contain a large number +of overlapping instances. This may partly stipulate +the poor performance of the NSP strategy since al- +most 9% of labeled instances are redundant. IDDS, +on the contrary, does not have instances with over- +lapping summaries inside batches at all. +6.3 +Self-learning +In this section, we investigate the effect of self- +learning in the AL setting. Figures 4a, 4b illus- +trate the effect of self-learning on the AESLC and +Gigaword datasets. For this experiment, we use +kl = 10, kh = 1, filtering out 11% of automati- +cally generated summaries. In both cases: with +AL and without, adding automatically generated +summaries of documents from the unlabeled pool +to the training set improves the performance of the +summarization model. On AESLC, the best results +are obtained with both AL and self-learning: their +combination achieves up to 58% improvement in +all ROUGE metrics compared to using passive an- +notation without self-learning. +The same experiment on the WikiHow dataset is +presented in Figure 4c. To make sure that the qual- +ity is not deteriorated due to the addition of noisy +uncertain instances, we use kl = 38, kh = 2 for +this experiment, filtering out 40% of automatically +generated summaries. On this dataset, self-learning +reduces the performance for both cases (with AL +and without). We deem that the benefit of self- +learning depends on the length of the summaries +in the dataset. AESLC and Gigaword contain very +short summaries (less than 13 tokens on average, +see Table 2). Since the model is capable of gen- +erating short texts that are grammatically correct +and logically consistent, such data augmentation +does not introduce much noise into the dataset, re- +sulting in performance improvement. WikiHow, +on the contrary, contains long summaries (77 to- +kens on average). Generation of long, logically +consistent, and grammatically correct summaries is +still a challenging task even for the state-of-the-art +ATS models. Therefore, the generated summaries +are of low quality, and using them as an additional +training signal deteriorates the model performance. +Consequently, we suggest using self-learning only +if the dataset consists of relatively short texts. We +leave a more detailed investigation of this topic for +future research. +6.4 +Consistency +We analyze how various AL strategies and self- +learning affect the consistency of model output in +two ways. We measure the consistency of the gener- +ated summaries with the original documents on the +test set on each AL iteration. Figure 5 shows that +the model trained on instances queried by IDDS +generates the most consistent summaries across +all considered AL query strategies on AESLC. On +the contrary, the model trained on the instances se- +lected by the uncertainty-based NSP query strategy +generates summaries with the lowest consistency. +Figure 28 in Appendix G demonstrates that on + +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +30 +Random sampling + self-learning +Random sampling +IDDS +IDDS + self-learning +Num. labeled instances +Performance, Rouge-1 +a) AESLC dataset +kl = 10, kh = 1. +20 +40 +60 +80 +100 +120 +140 +23 +24 +25 +26 +27 +28 +29 +30 +31 +Random sampling + self-learning +Random sampling +Num. labeled instances +Performance, Rouge-1 +b) Gigaword dataset +kl = 10, kh = 1 +20 +40 +60 +80 +100 +120 +140 +23 +24 +25 +26 +27 +28 +29 +30 +Random sampling + self-learning +Random sampling +IDDS +IDDS + self-learning +Num. labeled instances +Performance, Rouge-1 +c) WikiHow dataset +kl = 38, kh = 2 +Figure 4: ROUGE-1 scores of the BART-base model with IDDS and random sampling strategies with and without +self-learning on AESLC, Gigaword, and WikiHow. Full results are provided in Figures 14, 15, and 16, respectively. +20 +40 +60 +80 +100 +120 +140 +−0.2 +−0.15 +−0.1 +−0.05 +0 +0.05 +0.1 +0.15 +IDDS +NSP +SacreBLEUVar +Random sampling +Num. labeled instances +Consistency Score +Figure 5: The consistency score calculated via Sum- +maC with BART-base on AESLC with various AL +strategies. +AESLC, self-learning also improves consistency +regardless of the AL strategy. The same trend is +observed on Gigaword (Figure 27 in Appendix G). +However, for WikiHow, there is no clear trend. +Figure 29 in Appendix G shows that all query strate- +gies lead to similar consistency results, with NSP +producing slightly higher consistency, and BLEU- +Var – slightly lower. We deem that this may be due +to the fact that summaries generated by the model +on WikiHow are of lower quality than the golden +summaries regardless of the strategy. Therefore, +this leads to biased scores of the SummaC model +with similar results on average. +6.5 +Query Duration +We compare the average duration of AL iterations +for various query strategies. Figure 30 in the Ap- +pendix H presents the average training time and the +average duration of making a query. We can see +that training a model takes considerably less time +than selecting the instances from the unlabeled pool +for annotation. Therefore, the duration of AL itera- +tions is mostly determined by the efficiency of the +query strategy. The IDDS query strategy does not +require any heavy computations during AL, which +makes it also the best option for keeping the AL +process interactive. +7 +Conclusion +In this work, we convey the first study of AL in +ATS and propose the first active learning query +strategy that outperforms the baseline random sam- +pling. The query strategy aims at selecting for an- +notation the instances with high similarity with the +documents in the unlabeled pool and low similarity +with the already annotated documents. It outper- +forms the random sampling in terms of ROUGE +metrics on all considered datasets. +It also out- +performs random sampling in terms of the con- +sistency score calculated via the SummaC model +on the AESLC dataset. We also demonstrate that +uncertainty-based query strategies fail to outper- +form random sampling, resulting in the same or +even worse performance. Finally, we show that +self-learning can improve the performance of an +ATS model in terms of both the ROUGE metrics +and consistency. This is especially favorable in AL +since there is always a large unlabeled pool of data. +We show that combining AL and self-learning can +give an improvement of up to 58% in terms of +ROUGE metrics. +In future work, we look forward to investigat- +ing IDDS in other sequence generation tasks. This +query strategy might be beneficial for tasks with +the highly variable output when uncertainty es- +timates of model predictions are unreliable and +cannot outperform the random sampling baseline. +IDDS facilitates the representativeness of instances +in the training dataset without leveraging uncer- +tainty scores. + +Limitations +Despite the benefits, the proposed methods require +some conditions to be met to be successfully ap- +plied in practice. IDDS strategy requires making +TAPT of the embeddings-generated model, which +may be computationally consuming for a large +dataset. Self-learning, in turn, may harm the perfor- +mance when the summaries are too long, as shown +in Section 6.3. Consequently, its application re- +quires a detailed analysis of the properties of the +target domain summaries. +Ethical Considerations +It is important to note that active learning is a +method of biased sampling, which can lead to bi- +ased annotated corpora. Therefore, active learning +can be used to deliberately increase the bias in the +datasets. Our research improves the active learning +performance; hence, our contribution would also +make it more efficient for introducing more bias +as well. We also note that our method uses the +pre-trained language models, which usually con- +tain different types of biases by themselves. Since +bias affects all applications of pre-trained models, +this can also unintentionally facilitate the biased +selection of instances for annotation during active +learning. +Acknowledgements +We thank anonymous reviewers for their insight- +ful suggestions to improve this paper. The work +was supported by a grant for research centers in +the field of artificial intelligence (agreement iden- +tifier 000000D730321P5Q0002 dated November +2, 2021 No. 70-2021-00142 with ISP RAS). 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In Findings of the Association for +Computational Linguistics: EMNLP 2021, Virtual +Event / Punta Cana, Dominican Republic, 16-20 +November, 2021, pages 395–406. Association for +Computational Linguistics. +Rui Zhang and Joel R. Tetreault. 2019. +This email +could save your life: Introducing the task of email +subject line generation. In Proceedings of the 57th +Conference of the Association for Computational +Linguistics, ACL 2019, Florence, Italy, July 28- Au- +gust 2, 2019, Volume 1: Long Papers, pages 446– +456. Association for Computational Linguistics. + +A +Dataset Statistics and Model Hyperparameters +Table 2: Dataset statistics. We provide a number of instances for the training and test sets and average lengths of +documents / summaries in terms of tokens. All the datasets are English-language. We filter the WikiHow dataset +since it contains many noisy instances: we exclude instances with documents that have 10 or less tokens and +instances with summaries that have 3 or less tokens. +Dataset +Subset +Num. instances +Av. document len. +Av. summary len. +AESLC +Train +14.4K +142.4 +7.8 +Test +1.9K +143.8 +7.9 +WikiHow +Train +184.6K +377.5 +77.2 +Test +1K +386.9 +77.0 +Pubmed +Train +119.1K +495.4 +263.9 +Test +6.7K +509.5 +268.0 +Gigaword +(self-learning) +Train +10K +38.9 +11.9 +Test +2K +37.1 +12.8 +Gigaword +(hyperparam. optimiz.) +Train +200 +40.8 +13.3 +Test +2K +38.6 +12.5 +Table 3: Hyperparameter values and checkpoints from the HuggingFace repository (Wolf et al., 2019) of the +models. We imitate the low-resource case by randomly selecting 200 instances from Gigaword train dataset as +a train sample, and 2,000 instances from the validation set as a test sample for evaluation consistency. For each +model, we find the optimal hyperparameters according to evaluation scores on the sampled subset. Generation +maximum length is set to the maximum summary length from the available labeled set. +For WikiHow and PubMed datasets, we reduce the batch size and increase gradient accumulation steps by the same +amount due to computational bottleneck. +Hardware configuration: 2 Intel Xeon Platinum 8168, 2.7 GHz, 24 cores CPU; NVIDIA Tesla v100 GPU, 32 Gb +of VRAM. +Hparam +BART +PEGASUS +Checkpoint +facebook/bart-base +google/pegasus-large +# Param. +139M +570M +Number of epochs +6 +4 +Batch size +16 +2 +Gradient accumulation steps +1 +8 +Min. number of training steps +350 +200 +Max. sequence length +1024 +1024 +Optimizer +AdamW +AdamW +Learning rate +2e-5 +5e-4 +Weight decay +0.028 +0.03 +Gradient clipping +0.28 +0.3 +Sheduler +STLR +STLR +% warm-up steps +10 +10 +Num. beams at evaluation +4 +4 +Generation max. length +Adapt. +Adapt. + +B +Full Results for Uncertainty-based Methods +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +Random sampling +NSP +ENSP +ENSV +SacreBLEUVar +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +4 +6 +8 +10 +12 +14 +16 +Random sampling +NSP +ENSP +ENSV +SacreBLEUVar +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +Random sampling +NSP +ENSP +ENSV +SacreBLEUVar +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 6: The performance of the BART-base model with various uncertainty-based strategies compared with +random sampling (baseline) on AESLC. +20 +40 +60 +80 +100 +120 +140 +5 +10 +15 +20 +25 +30 +Random sampling +NSP +ENSP +ENSV +SacreBLEUVar +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +0 +2 +4 +6 +8 +10 +12 +14 +16 +Random sampling +NSP +ENSP +ENSV +SacreBLEUVar +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +5 +10 +15 +20 +25 +Random sampling +NSP +ENSP +ENSV +SacreBLEUVar +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 7: The performance of the PEGASUS-large model with various uncertainty-based strategies compared with +random sampling (baseline) on AESLC. +20 +40 +60 +80 +100 +120 +140 +25.5 +26 +26.5 +27 +27.5 +28 +28.5 +29 +29.5 +Random sampling +NSP +ENSP +ENSV +BLEUVar +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +6.5 +7 +7.5 +8 +8.5 +9 +9.5 +10 +Random sampling +NSP +ENSP +ENSV +BLEUVar +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +17 +18 +19 +20 +21 +22 +Random sampling +NSP +ENSP +ENSV +BLEUVar +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 8: The performance of the BART-base model with various uncertainty-based strategies compared with +random sampling (baseline) on WikiHow. +20 +40 +60 +80 +100 +120 +140 +20 +22 +24 +26 +28 +30 +32 +Random sampling +NSP +ENSP +ENSV +BLEUVar +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +6 +7 +8 +9 +10 +11 +12 +Random sampling +NSP +ENSP +ENSV +BLEUVar +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +13 +14 +15 +16 +17 +18 +19 +20 +Random sampling +NSP +ENSP +ENSV +BLEUVar +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 9: The performance of the BART-base model with various uncertainty-based strategies compared with +random sampling (baseline) on PubMed. + +C +Full Results for IDDS +Iter. 0 +Iter. 5 +Iter. 10 +Iter. 15 +Average +R-1/R-2/R-L +R-1/R-2/R-L +R-1/R-2/R-L +R-1/R-2/R-L +R-1/R-2/R-L +AESLC + BART-base +48.8 / 52.5 / 48.4 +11.2 / 14.9 / 11.4 +5.2 / 5.4 / 5.0 +4.1 / 2.6 / 3.8 +10.2 / 11.9 / 10.0 +AESLC + PEGASUS-large +-24.8 / -19.7 / -24.5 +6.9 / 7.3 / 7.4 +1.6 / 0.4 / 2.0 +4.8 / 3.5 / 4.7 +7.6 / 6.7 / 8.0 +WikiHow + BART-base +6.3 / 12.5 / 5.4 +1.9 / 2.7 / 1.3 +3.0 / 4.2 / 2.5 +2.6 / 2.9 / 1.8 +2.3 / 3.2 / 1.5 +PubMed + BART-base +8.0 / 10.4 / 5.8 +12.0 / 11.7 / 8.0 +8.1 / 6.4 / 4.9 +9.5 / 6.7 / 5.1 +8.9 / 7.7 / 5.5 +Table 4: Percentage increase in ROUGE F-scores of IDDS over the baseline on different AL iterations. Average +refers to the average increase throughout the whole AL cycle. +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +30 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +4 +6 +8 +10 +12 +14 +16 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 10: The performance of the BART-base model with the IDDS strategy compared with random sampling +(baseline) and NSP (uncertainty-based strategy) on AESLC. +20 +40 +60 +80 +100 +120 +140 +5 +10 +15 +20 +25 +30 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +2 +4 +6 +8 +10 +12 +14 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +5 +10 +15 +20 +25 +30 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 11: The performance of the PEGASUS-large model with the IDDS strategy compared with random sam- +pling (baseline) and NSP (uncertainty-based strategy) on AESLC. + +20 +40 +60 +80 +100 +120 +140 +26 +27 +28 +29 +30 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +6.5 +7 +7.5 +8 +8.5 +9 +9.5 +10 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +18 +19 +20 +21 +22 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 12: The performance of the BART-base model with the IDDS strategy compared with random sampling +(baseline) and NSP (uncertainty-based strategy) and NSP (uncertainty-based strategy) on WikiHow. +20 +40 +60 +80 +100 +120 +140 +20 +22 +24 +26 +28 +30 +32 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +6 +7 +8 +9 +10 +11 +12 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +13 +14 +15 +16 +17 +18 +19 +20 +Random sampling +IDDS +NSP +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 13: The performance of the BART-base model with the IDDS strategy compared with random sampling +(baseline) and NSP (uncertainty-based strategy) on PubMed. + +D +Full Results for Self-learning +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +30 +Random sampling + self-learning +Random sampling +IDDS +IDDS + self-learning +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +4 +6 +8 +10 +12 +14 +16 +18 +Random sampling + self-learning +Random sampling +IDDS +IDDS + self-learning +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +30 +Random sampling + self-learning +Random sampling +IDDS +IDDS + self-learning +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 14: The performance of the BART-base model with the IDDS and random sampling strategies with and +without pseudo-labeling of the unlabeled data on AESLC (kl = 0.1, kh = 0.01). +20 +40 +60 +80 +100 +120 +140 +23 +24 +25 +26 +27 +28 +29 +30 +31 +Random sampling + self-learning +Random sampling +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +8 +9 +10 +11 +12 +13 +Random sampling + self-learning +Random sampling +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +21 +22 +23 +24 +25 +26 +27 +28 +29 +Random sampling + self-learning +Random sampling +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 15: The performance of the BART-base model without AL (random sampling) with and without pseudo- +labeling of the unlabeled data on the randomly sampled subset of Gigaword (kl = 0.1, kh = 0.01). +20 +40 +60 +80 +100 +120 +140 +23 +24 +25 +26 +27 +28 +29 +30 +Random sampling + self-learning +Random sampling +IDDS +IDDS + self-learning +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +6 +7 +8 +9 +10 +Random sampling + self-learning +Random sampling +IDDS +IDDS + self-learning +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +16 +17 +18 +19 +20 +21 +22 +Random sampling + self-learning +Random sampling +IDDS +IDDS + self-learning +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 16: The performance of the BART-base model with the IDDS and random sampling strategies with and +without self-supervised learning on WikiHow (kl = 0.38, kh = 0.02). + +E +Diversity Statistics and Query Examples +AL Strat. +Document +Golden summary +Gen. summary +IDDS +"Here’s the latest info. regarding Bloomberg’s ability to +accept deals with Pinnacle West (formerly Arizona +Public Service Co.) (...) +Bloomberg- +Pinnacle/APS deals +n/a +IDDS +Hi. Nice to see you in Houston. I’m giving a +presentation on gas issues on Tuesday. +I’ve got a draft of (...) +Gas Presentation +n/a +IDDS +Kelley, I am writing to you to (...) Can you give +me the name and contact information for the person within +your company that would work with us to put a +Confidentiality Agreement in place (...) +confidentiality agreement +n/a +NSP +tantivy (tan-TIV-ee) adverb At full gallop; at full speed. +noun A fast gallop; rush.adjective Swift.interjection A +hunting cry by a hunter riding a horse at full speed(...) +A.Word.A.Day–tantivy +tricky +(tan-TIV-ee) +adjective +NSP +Prod Area and Long Haul k# Volume Rec +Del 3.6746 5000 St 62 Con Ed 3.4358 +15000 St 65 Con Ed 3.5049 10000 St (...) +TRCO capacity for Sep +Prod Area +and Long +Haul k# +Volume +NSP +This is a list of RisktRAC book-ids corresponding to +what has been created in ERMS. Let me know if the +book-id naming is ok with you. +Regards +Book2.xls +RisktRAC +ENSP +Fred, I suggest a phone call among the team today +to make sure we are all on the same wave length. +What is your schedule? +Thanks +PSEG +Firm +schedule +ENSP +Stephanie - When you get a chance, could you finalize the +attached (also found in Tana’s O drive). I am not sure where +the originals need to go after signed by Enron, but I have a +request for that information currently out to Hess. Thanks. +Hess NDA +Enron +O Drive +ENSP +Current Notes User: To ensure that you experience +a successful migration from Notes to Outlook, it is +necessary to gather individual user information prior +to your date of migration. Please take a few +minutes to completely fill out the following survey (...) +2- SURVEY +/INFORMATION +EMAIL 5-17-01 +Office 2000 +Migration +Survey +Sacre- +BleuVAR +Sheri, We are going to NO for JazzFest at the end of April. +April 27th-29th to be exact. +Let me know if you’re going. +DG +southwest.com weekly +specials +JazzFest +Sacre- +BleuVAR +This warning is sent automatically to inform you that your +mailbox is approaching the maximum size limit. Your +mailbox size is currently 78515 KB. Mailbox size limits (...) +WARNING: Your +mailbox is approaching +the size limit +Mailbox +size limit +Table 5: Examples of the instances queried with different AL strategies. Tokens overlapping with the source docu- +ment are highlighted with green. Tokens that refer to paraphrasing the part of the document and the corresponding +part are highlighted with blue. Tokens that cannot be derived from the document are highlighted with red. Tokens, +the usage of which depends on the peculiarities of the dataset, are not highlighted. Summaries for IDDS are not +presented, because IDDS does not require model inference. + +AL Iter. +SP +ESP +SacreBleuVAR +Random +IDDS +1 +33.3% / 0% +30.0% / 4.4% +0% / 0% +0% / 0% +0% / 0% +6 +15.6% / 0% +0% / 1.1% +0% / 2.2% +0% / 0% +0% / 0% +15 +3.3% / 0% +0% / 0% +0% / 0% +0% / 2.2% +0% / 0% +Mean +7.8% / 1.0% +2.1% / 0.8% +0.1% / 0.3% +0% / 0.7% +0% / 0% +Table 6: Share of fully / partly overlapping summaries among batches of instances, queried with various AL +strategies during AL using BART-base model on AESLC. We consider two summaries to be partly overlapping if +their ROUGE-1 score > 0.66. The results are averaged across 9 seeds for all the strategies except for IDDS, which +has constant seed-independent queries. + +F +Ablation Studies of IDDS +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +30 +IDDS w. BERT-base + TAPT (orig.) +IDDS w. BERT-base +IDDS w. LongFormer +IDDS w. SentBERT +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +4 +6 +8 +10 +12 +14 +16 +IDDS w. BERT-base + TAPT (orig.) +IDDS w. BERT-base +IDDS w. LongFormer +IDDS w. SentBERT +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +IDDS w. BERT-base + TAPT (orig.) +IDDS w. BERT-base +IDDS w. LongFormer +IDDS w. SentBERT +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 17: Ablation study of the document embeddings model & the necessity of performing TAPT for it in the +IDDS strategy with BART-base on AESLC. +20 +40 +60 +80 +100 +120 +140 +26 +26.5 +27 +27.5 +28 +28.5 +29 +29.5 +IDDS w. BERT-base + TAPT (orig.) +IDDS w. BERT-base +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +7 +7.5 +8 +8.5 +9 +9.5 +10 +IDDS w. BERT-base + TAPT (orig.) +IDDS w. BERT-base +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +18 +18.5 +19 +19.5 +20 +20.5 +21 +21.5 +22 +IDDS w. BERT-base + TAPT (orig.) +IDDS w. BERT-base +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 18: Ablation study of the necessity of performing TAPT for the model, which generates embeddings in the +IDDS strategy with BART-base on WikiHow. +20 +40 +60 +80 +100 +120 +140 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +IDDS w. dot product (orig.) +IDDS w. Mahalanobis dist. +IDDS w. Euclidean dist. +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +6 +8 +10 +12 +14 +16 +IDDS w. dot product (orig.) +IDDS w. Mahalanobis dist. +IDDS w. Euclidean dist. +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +12 +14 +16 +18 +20 +22 +24 +26 +28 +IDDS w. dot product (orig.) +IDDS w. Mahalanobis dist. +IDDS w. Euclidean dist. +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 19: The performance of IDDS with different similarity functions with BART-base on AESLC. +20 +40 +60 +80 +100 +120 +140 +24 +25 +26 +27 +28 +29 +30 +IDDS w. dot product (orig.) +IDDS w. Mahalanobis dist. +IDDS w. Euclidean dist. +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +6.5 +7 +7.5 +8 +8.5 +9 +9.5 +10 +IDDS w. dot product (orig.) +IDDS w. Mahalanobis dist. +IDDS w. Euclidean dist. +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +18 +19 +20 +21 +22 +IDDS w. dot product (orig.) +IDDS w. Mahalanobis dist. +IDDS w. Euclidean dist. +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 20: The performance of IDDS with different similarity functions with BART-base on WikiHow. + +20 +40 +60 +80 +100 +120 +140 +14 +16 +18 +20 +22 +24 +26 +28 +IDDS +IDDS w. embeddings normalization +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +IDDS +IDDS w. embeddings normalization +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +14 +16 +18 +20 +22 +24 +26 +28 +IDDS +IDDS w. embeddings normalization +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 21: The performance of the BART-base model with the standard IDDS strategy compared with its modifi- +cation when embeddings are normalized on AESLC. +20 +40 +60 +80 +100 +120 +140 +27 +28 +29 +30 +31 +32 +Embeddings similarity +Embeddings similarity w. embeddings normalization +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +8.5 +9 +9.5 +10 +10.5 +11 +Embeddings similarity +Embeddings similarity w. embeddings normalization +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +16.5 +17 +17.5 +18 +18.5 +Embeddings similarity +Embeddings similarity w. embeddings normalization +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 22: The performance of the BART-base model with the standard IDDS strategy compared with its modifi- +cation when embeddings are normalized on PubMed. +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +30 +IDDS, lambda = 0.5 +IDDS, lambda = 0.67 (orig.) +IDDS, lambda = 0.75 +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +4 +6 +8 +10 +12 +14 +16 +IDDS, lambda = 0.5 +IDDS, lambda = 0.67 (orig.) +IDDS, lambda = 0.75 +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +IDDS, lambda = 0.5 +IDDS, lambda = 0.67 (orig.) +IDDS, lambda = 0.75 +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 23: Ablation study for the hyperparameter λ in the IDDS strategy with BART-base on AESLC. +AL Strategy +Iter. 0 +Iter. 5 +Iter. 10 +Iter. 15 +Average +λ = 0. +9.08 / 4.8 / 8.79 +19.6 / 10.87 / 19.29 +22.6 / 12.58 / 22.1 +23.68 / 13.32 / 23.23 +21.25 / 11.72 / 20.88 +λ = 0.33 +15.77 / 7.67 / 15.46 +22.47 / 12.18 / 22.07 +23.98 / 13.54 / 23.51 +24.68 / 13.81 / 24.21 +23.19 / 12.88 / 22.78 +λ = 0.5 +12.15 / 6.07 / 12.03 +23.82 / 12.97 / 23.3 +25.69 / 14.33 / 25.06 +26.81 / 14.77 / 26.17 +23.84 / 12.94 / 23.31 +λ = 0.67 (orig.) +17.8 / 8.97 / 17.52 +26.4 / 14.4 / 25.86 +27.25 / 14.55 / 26.55 +28.72 / 15.56 / 27.97 +26.7 / 14.43 / 26.07 +λ = 0.75 +10.84 / 4.93 / 10.61 +26.7 / 14.62 / 26.26 +27.42 / 14.62 / 26.72 +28.29 / 15.36 / 27.61 +26.54 / 14.31 / 26.0 +λ = 0.83 +16.47 / 7.84 / 16.03 +26.06 / 14.42 / 25.59 +26.57 / 14.12 / 25.92 +28.7 / 15.22 / 28.0 +26.0 / 13.93 / 25.46 +λ = 1. +16.41 / 8.66 / 16.23 +25.2 / 13.66 / 24.72 +26.74 / 14.4 / 26.04 +27.44 / 14.66 / 26.67 +25.73 / 13.81 / 25.16 +Table 7: ROUGE scores on AL iterations for different values of the lambda hyperparameter in IDDS. We select +with bold the largest values w.r.t. the confidence intervals. + +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +30 +IDDS +MMR, lambda = 0.1 +MMR, lambda = 0.33 +MMR, lambda = 0.67 +MMR, lambda = 0.9 +MMR, lambda = 0.67 + IDDS init. +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +4 +6 +8 +10 +12 +14 +16 +IDDS +MMR, lambda = 0.1 +MMR, lambda = 0.33 +MMR, lambda = 0.67 +MMR, lambda = 0.9 +MMR, lambda = 0.67 + IDDS init. +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +10 +15 +20 +25 +IDDS +MMR, lambda = 0.1 +MMR, lambda = 0.33 +MMR, lambda = 0.67 +MMR, lambda = 0.9 +MMR, lambda = 0.67 + IDDS init. +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 24: Comparison of IDDS with the MMR-based strategy suggested in (Kim et al., 2006) with BART-base +on AESLC. We experiment with different λ values in MMR and the initialization schemes. +20 +40 +60 +80 +100 +120 +140 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +IDDS w. average agg. (orig.) +IDDS w. max agg. for sim. w. labeled data +IDDS w. max agg. for both parts +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +6 +8 +10 +12 +14 +16 +IDDS w. average agg. (orig.) +IDDS w. max agg. for sim. w. labeled data +IDDS w. max agg. for both parts +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +12 +14 +16 +18 +20 +22 +24 +26 +28 +30 +IDDS w. average agg. (orig.) +IDDS w. max agg. for sim. w. labeled data +IDDS w. max agg. for both parts +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 25: Comparison of the average and maximum aggregation functions in IDDS with BART-base on AESLC. +20 +40 +60 +80 +100 +120 +140 +25.5 +26 +26.5 +27 +27.5 +28 +28.5 +29 +29.5 +IDDS w. average agg. (orig.) +IDDS w. max agg. for sim. w. labeled data +IDDS w. max agg. for both parts +Num. labeled instances +Performance, Rouge-1 +a) ROUGE-1 +20 +40 +60 +80 +100 +120 +140 +7 +7.5 +8 +8.5 +9 +9.5 +10 +IDDS w. average agg. (orig.) +IDDS w. max agg. for sim. w. labeled data +IDDS w. max agg. for both parts +Num. labeled instances +Performance, Rouge-2 +b) ROUGE-2 +20 +40 +60 +80 +100 +120 +140 +18 +18.5 +19 +19.5 +20 +20.5 +21 +21.5 +22 +IDDS w. average agg. (orig.) +IDDS w. max agg. for sim. w. labeled data +IDDS w. max agg. for both parts +Num. labeled instances +Performance, Rouge-L +c) ROUGE-L +Figure 26: Comparison of the average and maximum aggregation functions in IDDS with BART-base on WikiHow. + +G +Additional Experiments with Consistency Analysis +20 +40 +60 +80 +100 +120 +140 +0.45 +0.5 +0.55 +0.6 +0.65 +0.7 +0.75 +0.8 +0.85 +Random sampling + self-learning +Random sampling +Num. labeled instances +Consistency Score +Figure 27: The consistency score calculated via SummaC with BART-base on Gigaword without AL (random +sampling) with and without self-learning. +20 +40 +60 +80 +100 +120 +140 +−0.2 +−0.15 +−0.1 +−0.05 +0 +0.05 +0.1 +0.15 +IDDS + self-learning +Random sampling + self-learning +IDDS +Random sampling +Labeled Data, % +Consistency Score +Figure 28: The consistency score calculated via SummaC on the test sample on AESLC for BART-base with the +IDDS and random sampling strategies with and without self-learning. +20 +40 +60 +80 +100 +120 +140 +−0.55 +−0.5 +−0.45 +−0.4 +−0.35 +IDDS +NSP +BLEUVar +Random sampling +Num. labeled instances +Consistency Score +Figure 29: The consistency score calculated via SummaC on the test subset of WikiHow for the BART-base model +with various AL strategies. + +H +Query Duration +Fit +IDDS +Random +NSP +SacreBLEUVar +ENSP +ENSV +0 +500 +1000 +1500 +2000 +2500 +3000 +3500 +Query Strategy +Av. Query Time (seconds) +Figure 30: Average duration in seconds of one AL query of 10 instances with different strategies on the AESLC +dataset with BART-base as an acquisition model. Train refers to the average time required for training the model +throughout the AL cycle. Hardware configuration: 2 Intel Xeon Platinum 8168, 2.7 GHz, 24 cores CPU; NVIDIA +Tesla v100 GPU, 32 Gb of VRAM. + diff --git a/qNE1T4oBgHgl3EQfigRe/content/tmp_files/load_file.txt b/qNE1T4oBgHgl3EQfigRe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f7c0ad6e79bab79cb698269b49bbec5d777ad134 --- /dev/null +++ b/qNE1T4oBgHgl3EQfigRe/content/tmp_files/load_file.txt @@ -0,0 +1,1458 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf,len=1457 +page_content='Active Learning for Abstractive Text Summarization Akim Tsvigun1,2, Ivan Lysenko2, Danila Sedashov2, Ivan Lazichny1, Eldar Damirov2,4, Vladimir Karlov2, Artemy Belousov2, Leonid Sanochkin1,2, Maxim Panov7, Alexander Panchenko3, Mikhail Burtsev1,5, Artem Shelmanov1,6,8 1AIRI, 2HSE, 3Skoltech, 4SberDevices, 5MIPT, 6MBZUAI, 7TII, 8ISP RAS Research Center for Trusted Artificial Intelligence {tsvigun, shelmanov}@airi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='net, artem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='shelmanov@mbzuai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='ae Abstract Construction of human-curated annotated datasets for abstractive text summarization (ATS) is very time-consuming and expensive because creating each instance requires a hu- man annotator to read a long document and compose a shorter summary that would pre- serve the key information relayed by the origi- nal document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Active Learning (AL) is a tech- nique developed to reduce the amount of an- notation required to achieve a certain level of machine learning model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In infor- mation extraction and text classification, AL can reduce the amount of labor up to multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Despite its potential for aiding expen- sive annotation, as far as we know, there were no effective AL query strategies for ATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This stems from the fact that many AL strategies rely on uncertainty estimation, while as we show in our work, uncertain instances are usu- ally noisy, and selecting them can degrade the model performance compared to passive anno- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We address this problem by proposing the first effective query strategy for AL in ATS based on diversity principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We show that given a certain annotation budget, using our strategy in AL annotation helps to improve the model performance in terms of ROUGE and consistency scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Additionally, we analyze the effect of self-learning and show that it can further increase the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 1 Introduction Abstractive text summarization (ATS) aims to com- press a document into a brief yet informative and readable summary, which would retain the key in- formation of the original document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' State-of-the- art results in this task are achieved by neural seq-to- seq models (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Liu and Liu, 2021) based on the Transformer architecture (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Training a model for ATS requires a dataset that contains pairs of original documents and their short summaries, which are usually writ- ten by human annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Manually composing a summary is a very tedious task, which requires one to read a long original document, select crucial information, and finally write a small text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Each of these steps is very time-consuming, resulting in the fact that constructing each instance in annotated corpora for text summarization is very expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Active Learning (AL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Cohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (1996)) is a well-known technique that helps to substantially re- duce the amount of annotation required to achieve a certain level of machine learning model perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For example, in tasks related to named entity recognition, researchers report annotation reduction by 2-7 times with a loss of only 1% of F1-score (Settles and Craven, 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This makes AL especially important when annotation is expen- sive, which is the case for ATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' AL works iteratively: on each iteration, (1) a model is trained on the so far annotated dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2) the model is used to select some informative instances from a large unlabeled pool using a query strategy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (3) informative instances are presented to human experts, which provide gold-standard anno- tations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (4) finally, the instances with annotations are added to the labeled dataset, and a new iteration begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Traditional AL query strategies are based on uncertainty estimation techniques (Lewis and Gale, 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Scheffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The hypothesis is that the most uncertain instances for the model trained on the current iteration are informative for training the model on the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We argue that uncertain predictions of ATS models (uncer- tain summaries) are not more useful than randomly selected instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Moreover, usually, they intro- duce more noise and detriment to the performance of summarization models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, it is not possi- ble to straightforwardly adapt the uncertainty-based approach to AL in text summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In this work, we present the first effective query strategy for AL in ATS, which we call in-domain diversity sampling (IDDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' It is based on the idea arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='03252v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='CL] 9 Jan 2023 of the selection of diverse instances that are se- mantically dissimilar from already annotated doc- uments but at the same time similar to the core documents of the considered domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The empiri- cal investigation shows that while techniques based on uncertainty cannot overcome the random sam- pling baseline, IDDS substantially increases the performance of summarization models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also experiment with the self-learning technique that leverages a training dataset expanded with sum- maries automatically generated by an ATS model trained only on the human-annotated dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This approach shows improvements when one needs to generate short summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The code for repro- ducing the experiments is available online1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The contributions of this paper are the following: We propose the first effective AL query strat- egy for ATS that beats the random sampling baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We conduct a vast empirical investigation and show that in contrast to such tasks as text clas- sification and information extraction, in ATS, uncertainty-based AL query strategies cannot outperform the random sampling baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' To our knowledge, we are the first to investi- gate the effect of self-learning in conjunction with AL for ATS and demonstrate that it can substantially improve results on the datasets with short summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2 Related Work Abstractive Text Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The advent of seq2seq models (Sutskever et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2014) along with the development of the attention mecha- nism (Bahdanau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2015) consolidated neural networks as a primary tool for ATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The attention- based Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2017) archi- tecture has formed the basis of many large-scale pre-trained language models that achieve state-of- the-art results in ATS (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Qi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Re- cent efforts in this area mostly focus on minor mod- ifications of the existing architectures (Liu and Liu, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Aghajanyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Active Learning in Natural Language Genera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' While many recent works leverage AL for text classification or sequence-tagging tasks (Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Zhang and Plank, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Shelmanov 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='com/AIRI-Institute/al_ ats et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Margatina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021), little atten- tion has been paid to natural language generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Among the works in this area, it is worth mentioning (Haffari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Ambati, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Ananthakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' These works focus on neural machine translation (NMT) and suggest several uncertainty-based query strategies for AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Peris and Casacuberta (2018) successfully apply AL in the interactive machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2018) exploit reinforcement learning to train a policy-based query strategy for NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Although there is an attempt to apply AL in ATS (Gidiotis and Tsoumakas, 2021), to the best of our knowl- edge, there is no published work on this topic yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Uncertainty Estimation in Natural Language Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' A simple yet effective approach for uncertainty estimation in generation is proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' They use a combination of expected translation probability and variance of the translation probability, demonstrating that it can handle noisy instances better and noticeably im- prove the quality of back-translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Malinin and Gales (2021) investigate the ensemble-based mea- sures of uncertainty for NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Their results demon- strate the superiority of these methods for OOD detection and for identifying generated translations of low-quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2020) propose a method for uncertainty estimation of long sequences of dis- crete random variables, which they dub “BLEU Variance”, and apply it for OOD sentence detection in NMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' It is also shown to be useful for identifying instances of questionable quality in ATS (Gidiotis and Tsoumakas, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In this work, we investi- gate these uncertainty estimation techniques in AL and show that they do not provide any benefits over annotating randomly selected instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Diversity-based Active Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Along with the uncertainty-based query strategies, a series of diversity-based methods have been suggested for AL (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Sener and Savarese, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Ash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Citovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The most relevant work among them is (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2006), where the authors propose to use a Maximal Marginal Relevance (MMR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Carbonell and Gold- stein (1998))-based function as a query strategy in AL for named entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This function aims to capture uncertainty and diversity and se- lects instances for annotation based on these two perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We adapt this strategy for the ATS task and compare the proposed method with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 3 Uncertainty-based Active Learning for Text Generation In this section, we give a brief formal defini- tion of the AL procedure for text generation and uncertainty-based query strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Here and throughout the rest of the paper, we denote an in- put sequence as x = (x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' xm) and the output sequence as y = (y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' yn), with m and n being lengths of x and y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Let D = {(x(k), y(k))}K k=1 be a dataset of pairs (documents, summaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Consider a probabilis- tic model pw(y | x) parametrized by a vector w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Usually, pw(y | x) is a neural network, while the parameter estimation is done via the maximum like- lihood approach: ˆw = arg max w L(D, w), (1) where L(D, w) = �K k=1 log pw(y(k) | x(k)) is log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Many AL methods are based on greedy query strategies that select instances for annotation, op- timizing a certain criterion A(x | D, ˆw) called an acquisition function: x∗ = arg max x A(x | D, ˆw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2) The selected instance x∗ is then annotated with a target value y∗ (document summary) and added to the training dataset: D := D ∪ (x∗, y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Subse- quently, the model parameters w are updated and the instance selection process continues until the desired model quality is achieved or the available annotation budget is depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The right choice of an acquisition function is crucial for AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' A common heuristic for acquisition is selecting instances with high uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Below, we consider several measures of uncertainty used in text generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Normalized Sequence Probability (NSP) was originally proposed by Ueffing and Ney (2007) and has been used in many subsequent works (Haffari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This measure is given by NSP(x) = 1 − ¯p ˆw(y | x), (3) where we define the geometric mean of probabil- ities of tokens predicted by the model as: ¯p ˆw(y | x) = exp � 1 n log p ˆw(y | x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' A wide family of uncertainty measures can be derived using the Bayesian approach to modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Under the Bayesian approach, it is assumed that model parameters have a prior distribution π(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Optimization of the log-likelihood L(D, w) in this case leads to the optimization of the posterior dis- tribution of the model parameters: π(w | D) ∝ exp{L(D, w)} · π(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (4) Usually, the exact computation of the posterior is intractable, and to perform training and inference, a family of distributions qθ(w) parameterized by θ is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The parameter estimate ˆθ mini- mizes the KL-divergence between the true posterior π(w | D) and the approximation qˆθ(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Given such an approximation, several uncertainty mea- sures can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Expected Normalized Sequence Probability (ENSP) is proposed by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2019) and is also used in (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020): ENSP(x) = 1 − Ew∼qˆθ ¯pw(y | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (5) In practice, the expectation is approximated via Monte Carlo dropout (Gal and Ghahramani, 2016), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' averaging multiple predictions obtained with activated dropout layers in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Expected Normalized Sequence Variance (ENSV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2019)) measures the variance of the sequence probabilities obtained via Monte Carlo dropout: ENSV(x) = Varw∼qˆθ ¯pw(y | x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (6) BLEU Variance (BLEUVar) is proposed by Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' It treats documents as points in some high dimensional space and uses the BLEU metric (Papineni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2002) for measuring the difference between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In such a setting, it is possible to calculate the variance of generated texts in the following way: BLEUVar(x) = (7) = Ew∼qˆθEy,y′∼pw(·|x) � 1 − BLEU(y, y′) �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The BLEU metric is calculated as a geometric mean of n-grams overlap up to 4-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Conse- quently, when summaries consist of less than 4 tokens, the metric is equal to zero since there will be no common 4-grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This problem can be mit- igated with the SacreBLEU metric (Post, 2018), which smoothes the n-grams with zero counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' When we use this query strategy with the Sacre- BLUE metric, we refer to it as SacreBLEUVar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Unlabeled Instances Labeled Instances Out-of-domain Instances IDDS Queries IDDS: far from labeled, close to unlabeled on average Uncertainty-based methods: far from both labeled and unlabeled Figure 1: The visualization of the idea behind the IDDS alogrithm on the synthetic data: select instances lo- cated far from labeled data while close on average to unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 4 Proposed Methods 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 In-Domain Diversity Sampling We argue that uncertainty-based query strategies tend to select noisy instances that have little value for training ATS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' To alleviate this issue, we propose a novel query strategy named in-domain diversity sampling (IDDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' It aims to maximize the diversity of the annotated instances by select- ing instances that are dissimilar from the already annotated ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' At the same time, it avoids se- lecting noisy outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' These noisy documents that are harmful to training an ATS model are usually semantically dissimilar from the core documents of the domain represented by the unlabeled pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, IDDS queries instances that are dissimi- lar to the annotated instances but at the same time are similar to unannotated ones (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We propose the following acquisition function that implements the aforementioned idea (the higher the value – the higher the priority for the annotation): IDDS(x) = λ |U| � j=1 s(x, xj) |U| − (1 − λ) |L| � i=1 s(x, xi) |L| , (8) where s(x, x′) is a similarity function between texts, U is the unlabeled set, L is the labeled set, and λ ∈ [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 1] is a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Below, we formalize the resulting algorithm of the IDDS query strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For each document in the unlabeled pool x, we obtain an embedding vector e(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For this purpose, we suggest using the [CLS] pooled sequence embeddings from BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We note that using a pre-trained checkpoint straightfor- wardly may lead to unreasonably high sim- ilarity scores between instances since they all belong to the same domain, which can be quite specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We mitigate this problem by using the task-adaptive pre-training (TAPT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Gururangan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (2020)) on the unlabeled pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' TAPT performs several epochs of self- supervised training of the pre-trained model on the target dataset to acquaint it with the peculiarities of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Deduplicate the unlabeled pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Instances with duplicates will have an overrated sim- ilarity score with the unlabeled pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Calculate the informativeness scores using the IDDS acquisition function (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' As a sim- ilarity function, we suggest using a scalar product between document representations: s(x, x′) = ⟨e(x), e(x′)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The idea of IDDS is close to the MMR-based strategy proposed in (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Yet, despite the resemblance, IDDS differs from it in several crucial aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The MMR-based strategy focuses on the uncertainty and diversity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' How- ever, as shown in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1, selecting instances by uncertainty leads to worse results compared to random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Consequently, instead of us- ing uncertainty, IDDS leverages the unlabeled pool to capture the representativeness of the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Furthermore, IDDS differs from the MMR-based strategy in how they calculate the diversity com- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' MMR directly specifies the usage of the “max” aggregation function for calculating the sim- ilarity with the already annotated data, while IDDS uses “average” similarity instead and achieves bet- ter results as shown in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We note that IDDS does not require retraining an acquisition model in contrast to uncertainty-based strategies since document vector representations and document similarities can be calculated before starting the AL annotation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This results in the fact that no heavy computations during AL are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Consequently, IDDS does not harm the interactiveness of the annotation process, which is a common bottleneck (Tsvigun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2 Self-learning Pool-based AL assumes that there is a large unla- beled pool of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We propose to use this data source during AL to improve text summarization models with the help of self-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We train the model on the labeled data and generate summaries for the whole unlabeled pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Then, we concatenate the generated summaries with the labeled set and use this data to fine-tune the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We note that generated summaries can be noisy: irrelevant, grammatically incorrect, contain factual inconsis- tency, and can harm the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We de- tect such instances using the uncertainty estimates obtained via NSP scores and exclude kl% instances with the lowest scores and kh% of instances with the highest scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We choose this uncertainty met- ric because according to our experiments in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1, high NSP scores correspond to the noisiest in- stances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We note that adding the filtration step does not introduce additional computational overhead, since the NSP scores are calculated simultaneously with the summary generation for self-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 5 Experimental Setup 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 Active Learning Setting We evaluate IDDS and other query strategies using the conventional scheme of AL annotation emula- tion applied in many previous works (Settles and Craven, 2008b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Siddhant and Lipton, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Shelmanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Dor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For uncertainty-based query strategies and random sampling, we start from a small annotated seeding set selected randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This set is used for fine-tuning the summarization model on the first it- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For IDDS, the seeding set is not used, since this query strategy does not require fine-tuning the model to make a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' On each AL iteration, we select top-k instances from the unlabeled pool ac- cording to the informativeness score obtained with a query strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The selected instances with their gold-standard summaries are added to the so-far annotated set and are excluded from the unlabeled pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' On each iteration, we fine-tune a summa- rization model from scratch and evaluate it on a held-out test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We report the performance of the model on each iteration to demonstrate the dy- namics of the model performance depending on the invested annotation effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The query size (the number of instances selected for annotation on each iteration) is set to 10 doc- uments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We repeat each experiment 9 times with different random seeds and report the mean and the standard deviation of the obtained scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For the WikiHow and PubMed datasets, on each itera- tion, we use a random subset from the unlabeled pool since generating predictions for the whole un- labeled dataset is too computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In the experiments, the subset size is set to 10,000 for WikiHow and 1,000 for PubMed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2 Baselines We use random sampling as the main baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' To our knowledge, in the ATS task, this baseline has not been outperformed by any other query strategy yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In this baseline, an annotator is given randomly selected instances from the unlabeled pool, which means that AL is not used at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also report results of uncertainty-based query strategies and an MMR-based query strategy (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2006) that is shown to be useful for named entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3 Metrics Quality of Text Summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' To measure the quality of the text summarization model, we use the commonly adopted ROUGE metric (Lin, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Following previous works (See et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Nal- lapati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Chen and Bansal, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020), we report ROUGE- 1, ROUGE-2, and ROUGE-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Since we found the dynamics of these metrics coinciding, for brevity, in the main part of the paper, we keep only the results with the ROUGE-1 metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The results with other metrics are presented in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Factual Consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Inconsistency (hallucina- tion) of the generated summaries is one of the most crucial problems in summarization (Kryscin- ski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Nan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, in addition to the ROUGE metrics, we measure the factual consis- tency of the generated summaries with the original documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We use the SummaC-ZS (Laban et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2022) – a state-of-the-art model for inconsistency detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We set granularity = “sentence” and model_name = “vitc”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 Datasets We experiment with three datasets widely-used for evaluation of ATS models: AESLC (Zhang and Tetreault, 2019), PubMed (Cohan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2018), and WikiHow (Koupaee and Wang, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' AESLC con- sists of emails with their subject lines as summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' WikiHow contains articles from WikiHow pages 20 40 60 80 100 120 140 10 15 20 25 Random sampling NSP ENSP ENSV SacreBLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) AESLC dataset 20 40 60 80 100 120 140 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 26 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 27 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 28 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 29 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 Random sampling NSP ENSP ENSV BLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 b) WikiHow dataset 20 40 60 80 100 120 140 20 22 24 26 28 30 32 Random sampling NSP ENSP ENSV BLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 c) PubMed dataset Figure 2: ROUGE-1 scores of BART-base with various uncertainty-based strategies compared with random sam- pling (baseline) on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Full results are provided in Figures 6, 8, 9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 10 15 20 25 30 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) AESLC dataset 20 40 60 80 100 120 140 26 27 28 29 30 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 b) WikiHow dataset 20 40 60 80 100 120 140 20 22 24 26 28 30 32 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 c) PubMed dataset Figure 3: ROUGE-1 scores of BART-base with the IDDS strategy compared with random sampling (baseline) and NSP (uncertainty-based strategy) on various datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Full results are provided in Figures 10, 12 and 13, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' with their headlines as summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' PubMed (Co- han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2018) is a collection of scientific arti- cles from the PubMed archive with their abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The choice of datasets is stipulated by the fact that AESLC contains short documents and summaries, WikiHow contains medium-sized documents and summaries, and PubMed contains long documents and summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also use two non-intersecting subsets of the Gigaword dataset (Graff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Rush et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2015) of sizes 2,000 and 10,000 for hy- perparameter optimization of ATS models and addi- tional experiments with self-learning, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Gigaword consists of news articles and their head- lines representing summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The dataset statistics is presented in Table 2 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 Models and Hyperparameters We conduct experiments using the state-of-the-art text summarization models: BART (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020) and PEGASUS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In all experiments, we use the “base” pre-trained version of BART and the “large” version of PEGASUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Most of the experiments are conducted with the BART model, while PEGASUS is only used for the AESLC dataset (results are presented in Appen- dices B, C) since running it on two other datasets in AL introduces a computational bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We tune hyperparameter values of ATS models using the ROUGE-L score on the subset of the Gigaword dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The hyperparameter values are provided in Table 3 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For the IDDS query strategy, we use λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We analyze the effect of different values of this parameter in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 6 Results and Discussion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 Uncertainty-based Query Strategies In this series of experiments, we demonstrate that selected uncertainty-based query strategies are not suitable for AL in ATS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figure 2a and Figures 6, 7 in Appendix B present the results on the AESLC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' As we can see, none of the uncertainty- based query strategies outperform the random sam- pling baseline for both BART and PEGASUS mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' NSP and ENSP strategies demonstrate the worst results with the former having the lowest per- formance for both ATS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Similar results are obtained for the WikiHow and PubMed datasets (Figures 2b and 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In some previous work on NMT, uncertainty- based query strategies outperform the random sam- pling baseline (Haffari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Ambati, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Ananthakrishnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Their low results for ATS compared to NMT might stem from the differ- ences between these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Both NMT and ATS are AL Strat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Document Golden Summary Gener.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Summ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' NSP Aquarius - Horoscope Friday, September 8, 2000 by Astronet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Powerful forces are at work to challenge you (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') Don’t let hurt feelings prevent you from (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') These things are beginning to scare me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Invitation – Aquarius NSP Prod Area and Long Haul k# Volume Rec Del 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6746 5000 St 62 (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') #6563 PPL (Non NY) should have this contract tomorrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5318 6500 Leidy PSE&G TRCO capacity for Sep Prod Area IDDS Greg, I wanted to forward this letter to you that I received from a good friend of mine who is interested in discussing (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') with Enron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') set up a meeting (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') Sincerely, Meeting with Enron Networks n/a IDDS Larry, Could I have the price for a 2 day swing peaker option at NGI Chicago, that can be exercised on any day in February 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Strike is FOM February, (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') Peaker price for NGI Chicago Feb n/a Table 1: Examples of instances selected with the NSP and IDDS strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Tokens from the source document are highlighted with green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Tokens that refer to paraphrasing a part of the document and the corresponding part are highlighted with blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Tokens that cannot be derived from the document are highlighted with red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' seq2seq tasks and can be solved via similar mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' However, NMT is somewhat easier, since the output is usually of similar length as the input and its variability is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' It is much easier to train a model to reproduce an exact translation rather than make it generate an exact summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, uncertainty estimates of ATS models are way less reliable than estimates of translation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' These estimates often select for annotation noisy docu- ments that are useless or even harmful for training summarization models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Table 1 reveals several documents selected by the worst-performing strat- egy NSP on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We can see that NSP selects domain-irrelevant documents or very specific ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Their summaries can hardly be restored from the source documents, which means that they most likely have little positive impact on the general- ization ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' More examples of instances selected by different query strategies are presented in Table 5 in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2 In-Domain Diversity Sampling In this series of experiments, we analyze the pro- posed IDDS query strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figure 3a and Fig- ures 10, 11 in Appendix C show the performance of the models with various query strategies on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We can see that the proposed strategy outperforms random sampling on all iterations for both ATS models and subsequently outperforms the uncertainty-based strategy NSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS demon- strates similar results on the WikiHow and PubMed datasets, outperforming the baseline with a large margin as depicted in Figures 3b and 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also report the improvement of IDDS over random sam- pling in percentage on several AL iterations in Ta- ble 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We can see that IDDS provides an especially large improvement in the cold-start AL scenario when the amount of labeled data is very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We carry out several ablation studies for the proposed query strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' First, we investigate the effect of various models for document embed- dings construction and the necessity of perform- ing TAPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figures 17 and 18 in Appendix F il- lustrate that TAPT substantially enhances the per- formance of IDDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figure 17 also shows that the BERT-base encoder appears to be better than Sen- tenceBERT (Reimers and Gurevych, 2019) and LongFormer (Beltagy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Second, we try various functions for calculating the similarity between instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figures 19, 20 in Appendix F compare the originally used dot prod- uct with Mahalanobis and Euclidean distances on AESLC and WikiHow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' On AESLC, IDDS with Ma- halanobis distance persistently demonstrates lower performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS with the Euclidean distance shows a performance drop on the initial AL itera- tions compared to the dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' On WikiHow, however, all the variants perform roughly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, we suggest keeping the dot product for computing the document similarity in IDDS since it provides the most robust results across the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also compare the dot product with its nor- malized version – cosine similarity on AESLC and PubMed, see Figures 21 and 22 in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' On both datasets, adding normalization leads to substantially worse results on the initial AL itera- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We deem that this happens because normal- ization may damage the representativeness com- ponent since the norm of the embedding can be treated as a measure of the representativeness of the corresponding document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Third, we investigate how different values for the lambda coefficient influence the performance of IDDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Table 7 and Figure 23 in Appendix F shows that smaller values of λ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5} substan- tially deteriorate the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Smaller values correspond to selecting instances that are highly dissimilar from the documents in the unlabeled pool, which leads to picking many outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Higher values lead to the selection of instances from the core of the unlabeled dataset, but also very similar to the annotated part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This also results in a lower quality on the initial AL iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The best and most stable results are obtained with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Fourth, we compare IDDS with the MMR-based strategy suggested in (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Since it uses uncertainty, it requires a trained model to cal- culate the scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Consequently, the initial query is taken randomly as no trained model is available on the initial AL iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, we use the modification, when the initial query is done with IDDS because it provides substantially better re- sults on the initial iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also experiment with different values of the λ hyperparameter of the MMR-based strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figure 24 illustrates a large gap in performance of IDDS and the MMR- based strategy regardless of the initialization / λ on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We believe that this is attributed to the fact that strategies incorporating uncertainty are harmful to AL in ATS as shown in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Finally, we compare “aggregation” functions for estimating the similarity between a document and a collection of documents (labeled and unlabeled pools).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Following the MMR-based strategy (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2006), instead of calculating the average similarity between the embedding of the target doc- ument and the embeddings of documents from the labeled set, we calculate the maximum similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also try replacing the “average” aggregation function with “maximum” in both IDDS compo- nents in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figures 25 and 26 in Appendix F show that average leads to better performance on both AESLC and WikiHow datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The importance of diversity sampling is illus- trated in Table 6 in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We can see that NSP-based query batches contain a large number of overlapping instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This may partly stipulate the poor performance of the NSP strategy since al- most 9% of labeled instances are redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS, on the contrary, does not have instances with over- lapping summaries inside batches at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3 Self-learning In this section, we investigate the effect of self- learning in the AL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figures 4a, 4b illus- trate the effect of self-learning on the AESLC and Gigaword datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For this experiment, we use kl = 10, kh = 1, filtering out 11% of automati- cally generated summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In both cases: with AL and without, adding automatically generated summaries of documents from the unlabeled pool to the training set improves the performance of the summarization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' On AESLC, the best results are obtained with both AL and self-learning: their combination achieves up to 58% improvement in all ROUGE metrics compared to using passive an- notation without self-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The same experiment on the WikiHow dataset is presented in Figure 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' To make sure that the qual- ity is not deteriorated due to the addition of noisy uncertain instances, we use kl = 38, kh = 2 for this experiment, filtering out 40% of automatically generated summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' On this dataset, self-learning reduces the performance for both cases (with AL and without).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We deem that the benefit of self- learning depends on the length of the summaries in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' AESLC and Gigaword contain very short summaries (less than 13 tokens on average, see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Since the model is capable of gen- erating short texts that are grammatically correct and logically consistent, such data augmentation does not introduce much noise into the dataset, re- sulting in performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' WikiHow, on the contrary, contains long summaries (77 to- kens on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Generation of long, logically consistent, and grammatically correct summaries is still a challenging task even for the state-of-the-art ATS models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, the generated summaries are of low quality, and using them as an additional training signal deteriorates the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Consequently, we suggest using self-learning only if the dataset consists of relatively short texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We leave a more detailed investigation of this topic for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 Consistency We analyze how various AL strategies and self- learning affect the consistency of model output in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We measure the consistency of the gener- ated summaries with the original documents on the test set on each AL iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figure 5 shows that the model trained on instances queried by IDDS generates the most consistent summaries across all considered AL query strategies on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' On the contrary, the model trained on the instances se- lected by the uncertainty-based NSP query strategy generates summaries with the lowest consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figure 28 in Appendix G demonstrates that on 20 40 60 80 100 120 140 10 15 20 25 30 Random sampling + self-learning Random sampling IDDS IDDS + self-learning Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) AESLC dataset kl = 10, kh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 23 24 25 26 27 28 29 30 31 Random sampling + self-learning Random sampling Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 b) Gigaword dataset kl = 10, kh = 1 20 40 60 80 100 120 140 23 24 25 26 27 28 29 30 Random sampling + self-learning Random sampling IDDS IDDS + self-learning Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 c) WikiHow dataset kl = 38, kh = 2 Figure 4: ROUGE-1 scores of the BART-base model with IDDS and random sampling strategies with and without self-learning on AESLC, Gigaword, and WikiHow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Full results are provided in Figures 14, 15, and 16, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='15 IDDS NSP SacreBLEUVar Random sampling Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Consistency Score Figure 5: The consistency score calculated via Sum- maC with BART-base on AESLC with various AL strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' AESLC, self-learning also improves consistency regardless of the AL strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The same trend is observed on Gigaword (Figure 27 in Appendix G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' However, for WikiHow, there is no clear trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figure 29 in Appendix G shows that all query strate- gies lead to similar consistency results, with NSP producing slightly higher consistency, and BLEU- Var – slightly lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We deem that this may be due to the fact that summaries generated by the model on WikiHow are of lower quality than the golden summaries regardless of the strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, this leads to biased scores of the SummaC model with similar results on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 Query Duration We compare the average duration of AL iterations for various query strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Figure 30 in the Ap- pendix H presents the average training time and the average duration of making a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We can see that training a model takes considerably less time than selecting the instances from the unlabeled pool for annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, the duration of AL itera- tions is mostly determined by the efficiency of the query strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The IDDS query strategy does not require any heavy computations during AL, which makes it also the best option for keeping the AL process interactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 7 Conclusion In this work, we convey the first study of AL in ATS and propose the first active learning query strategy that outperforms the baseline random sam- pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The query strategy aims at selecting for an- notation the instances with high similarity with the documents in the unlabeled pool and low similarity with the already annotated documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' It outper- forms the random sampling in terms of ROUGE metrics on all considered datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' It also out- performs random sampling in terms of the con- sistency score calculated via the SummaC model on the AESLC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also demonstrate that uncertainty-based query strategies fail to outper- form random sampling, resulting in the same or even worse performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Finally, we show that self-learning can improve the performance of an ATS model in terms of both the ROUGE metrics and consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This is especially favorable in AL since there is always a large unlabeled pool of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We show that combining AL and self-learning can give an improvement of up to 58% in terms of ROUGE metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In future work, we look forward to investigat- ing IDDS in other sequence generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This query strategy might be beneficial for tasks with the highly variable output when uncertainty es- timates of model predictions are unreliable and cannot outperform the random sampling baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS facilitates the representativeness of instances in the training dataset without leveraging uncer- tainty scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Limitations Despite the benefits, the proposed methods require some conditions to be met to be successfully ap- plied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS strategy requires making TAPT of the embeddings-generated model, which may be computationally consuming for a large dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Self-learning, in turn, may harm the perfor- mance when the summaries are too long, as shown in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Consequently, its application re- quires a detailed analysis of the properties of the target domain summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Ethical Considerations It is important to note that active learning is a method of biased sampling, which can lead to bi- ased annotated corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Therefore, active learning can be used to deliberately increase the bias in the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Our research improves the active learning performance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' hence, our contribution would also make it more efficient for introducing more bias as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We also note that our method uses the pre-trained language models, which usually con- tain different types of biases by themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Since bias affects all applications of pre-trained models, this can also unintentionally facilitate the biased selection of instances for annotation during active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Acknowledgements We thank anonymous reviewers for their insight- ful suggestions to improve this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The work was supported by a grant for research centers in the field of artificial intelligence (agreement iden- tifier 000000D730321P5Q0002 dated November 2, 2021 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 70-2021-00142 with ISP RAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This research was supported in part by computational re- sources of the HPC facilities at the HSE University (Kostenetskiy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2021).' metadata={'source': 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Conference of the European Chapter of the Association for Computational Linguistics: Main Volume, pages 1698–1712, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Yanyao Shen, Hyokun Yun, Zachary Lipton, Yakov Kronrod, and Animashree Anandkumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Deep active learning for named entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In Proceedings of the 2nd Workshop on Representa- tion Learning for NLP, pages 252–256, Vancouver, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Aditya Siddhant and Zachary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Lipton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Deep bayesian active learning for natural language pro- cessing: Results of a large-scale empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, Brussels, Belgium, October 31 - November 4, 2018, pages 2904–2909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Association for Computational Linguis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Ilya Sutskever, Oriol Vinyals, and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2014.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' To- wards computationally feasible deep active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: NAACL 2022, pages 1198–1218, Seat- tle, United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Nicola Ueffing and Hermann Ney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Word-level confidence estimation for machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Com- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Linguistics, 33(1):9–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Gomez, Lukasz Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In Advances in Neural Information Pro- cessing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4- 9, 2017, Long Beach, CA, USA, pages 5998–6008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Shuo Wang, Yang Liu, Chao Wang, Huanbo Luan, and Maosong Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Improving back-translation 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Wat zei je?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' detecting out-of-distribution translations with variational transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' CoRR, abs/2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='08344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Michelle Yuan, Hsuan-Tien Lin, and Jordan Boyd- Graber.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' PEGASUS: pre-training with extracted gap-sentences for abstractive summariza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In Proceedings of the 37th International Con- ference on Machine Learning, ICML 2020, 13-18 July 2020, Virtual Event, volume 119 of Proceedings of Machine Learning Research, pages 11328–11339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Mike Zhang and Barbara Plank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Cartography active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: EMNLP 2021, Virtual Event / Punta Cana, Dominican Republic, 16-20 November, 2021, pages 395–406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Rui Zhang and Joel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Tetreault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' This email could save your life: Introducing the task of email subject line generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' In Proceedings of the 57th Conference of the Association for Computational Linguistics, ACL 2019, Florence, Italy, July 28- Au- gust 2, 2019, Volume 1: Long Papers, pages 446– 456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' A Dataset Statistics and Model Hyperparameters Table 2: Dataset statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We provide a number of instances for the training and test sets and average lengths of documents / summaries in terms of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' All the datasets are English-language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We filter the WikiHow dataset since it contains many noisy instances: we exclude instances with documents that have 10 or less tokens and instances with summaries that have 3 or less tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Dataset Subset Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' instances Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' document len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' summary len.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' AESLC Train 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4K 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 Test 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9K 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 WikiHow Train 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6K 377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2 Test 1K 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 Pubmed Train 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1K 495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 Test 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7K 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 Gigaword (self-learning) Train 10K 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 Test 2K 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 Gigaword (hyperparam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' optimiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') Train 200 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3 Test 2K 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 Table 3: Hyperparameter values and checkpoints from the HuggingFace repository (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2019) of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We imitate the low-resource case by randomly selecting 200 instances from Gigaword train dataset as a train sample, and 2,000 instances from the validation set as a test sample for evaluation consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For each model, we find the optimal hyperparameters according to evaluation scores on the sampled subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Generation maximum length is set to the maximum summary length from the available labeled set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' For WikiHow and PubMed datasets, we reduce the batch size and increase gradient accumulation steps by the same amount due to computational bottleneck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Hardware configuration: 2 Intel Xeon Platinum 8168, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 GHz, 24 cores CPU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' NVIDIA Tesla v100 GPU, 32 Gb of VRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Hparam BART PEGASUS Checkpoint facebook/bart-base google/pegasus-large # Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 139M 570M Number of epochs 6 4 Batch size 16 2 Gradient accumulation steps 1 8 Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' number of training steps 350 200 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' sequence length 1024 1024 Optimizer AdamW AdamW Learning rate 2e-5 5e-4 Weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='03 Gradient clipping 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3 Sheduler STLR STLR % warm-up steps 10 10 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' beams at evaluation 4 4 Generation max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' length Adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Adapt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' B Full Results for Uncertainty-based Methods 20 40 60 80 100 120 140 10 15 20 25 Random sampling NSP ENSP ENSV SacreBLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 4 6 8 10 12 14 16 Random sampling NSP ENSP ENSV SacreBLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 10 15 20 25 Random sampling NSP ENSP ENSV SacreBLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 6: The performance of the BART-base model with various uncertainty-based strategies compared with random sampling (baseline) on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 5 10 15 20 25 30 Random sampling NSP ENSP ENSV SacreBLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 0 2 4 6 8 10 12 14 16 Random sampling NSP ENSP ENSV SacreBLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 5 10 15 20 25 Random sampling NSP ENSP ENSV SacreBLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 7: The performance of the PEGASUS-large model with various uncertainty-based strategies compared with random sampling (baseline) on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 26 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 27 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 28 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 29 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 Random sampling NSP ENSP ENSV BLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 10 Random sampling NSP ENSP ENSV BLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 17 18 19 20 21 22 Random sampling NSP ENSP ENSV BLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 8: The performance of the BART-base model with various uncertainty-based strategies compared with random sampling (baseline) on WikiHow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 20 22 24 26 28 30 32 Random sampling NSP ENSP ENSV BLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 6 7 8 9 10 11 12 Random sampling NSP ENSP ENSV BLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 13 14 15 16 17 18 19 20 Random sampling NSP ENSP ENSV BLEUVar Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 9: The performance of the BART-base model with various uncertainty-based strategies compared with random sampling (baseline) on PubMed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' C Full Results for IDDS Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 0 Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 5 Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 10 Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 15 Average R-1/R-2/R-L R-1/R-2/R-L R-1/R-2/R-L R-1/R-2/R-L R-1/R-2/R-L AESLC + BART-base 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 / 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 / 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 11.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6 / 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2 / 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 AESLC + PEGASUS-large 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 / -19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 / -24.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 / 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6 / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 / 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 WikiHow + BART-base 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3 / 12.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 / 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 / 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 / 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 / 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 / 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 / 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 Table 4: Percentage increase in ROUGE F-scores of IDDS over the baseline on different AL iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Average refers to the average increase throughout the whole AL cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 10 15 20 25 30 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 4 6 8 10 12 14 16 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 10 15 20 25 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 10: The performance of the BART-base model with the IDDS strategy compared with random sampling (baseline) and NSP (uncertainty-based strategy) on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 5 10 15 20 25 30 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 2 4 6 8 10 12 14 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 5 10 15 20 25 30 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 11: The performance of the PEGASUS-large model with the IDDS strategy compared with random sam- pling (baseline) and NSP (uncertainty-based strategy) on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 26 27 28 29 30 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 10 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 18 19 20 21 22 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 12: The performance of the BART-base model with the IDDS strategy compared with random sampling (baseline) and NSP (uncertainty-based strategy) and NSP (uncertainty-based strategy) on WikiHow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 20 22 24 26 28 30 32 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 6 7 8 9 10 11 12 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 13 14 15 16 17 18 19 20 Random sampling IDDS NSP Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 13: The performance of the BART-base model with the IDDS strategy compared with random sampling (baseline) and NSP (uncertainty-based strategy) on PubMed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' D Full Results for Self-learning 20 40 60 80 100 120 140 10 15 20 25 30 Random sampling + self-learning Random sampling IDDS IDDS + self-learning Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 4 6 8 10 12 14 16 18 Random sampling + self-learning Random sampling IDDS IDDS + self-learning Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 10 15 20 25 30 Random sampling + self-learning Random sampling IDDS IDDS + self-learning Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 14: The performance of the BART-base model with the IDDS and random sampling strategies with and without pseudo-labeling of the unlabeled data on AESLC (kl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1, kh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 23 24 25 26 27 28 29 30 31 Random sampling + self-learning Random sampling Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 8 9 10 11 12 13 Random sampling + self-learning Random sampling Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 21 22 23 24 25 26 27 28 29 Random sampling + self-learning Random sampling Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 15: The performance of the BART-base model without AL (random sampling) with and without pseudo- labeling of the unlabeled data on the randomly sampled subset of Gigaword (kl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1, kh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='01).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 23 24 25 26 27 28 29 30 Random sampling + self-learning Random sampling IDDS IDDS + self-learning Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 6 7 8 9 10 Random sampling + self-learning Random sampling IDDS IDDS + self-learning Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 16 17 18 19 20 21 22 Random sampling + self-learning Random sampling IDDS IDDS + self-learning Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 16: The performance of the BART-base model with the IDDS and random sampling strategies with and without self-supervised learning on WikiHow (kl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='38, kh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='02).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' E Diversity Statistics and Query Examples AL Strat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Document Golden summary Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' summary IDDS "Here’s the latest info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' regarding Bloomberg’s ability to accept deals with Pinnacle West (formerly Arizona Public Service Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') Bloomberg- Pinnacle/APS deals n/a IDDS Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Nice to see you in Houston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' I’m giving a presentation on gas issues on Tuesday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' I’ve got a draft of (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') Gas Presentation n/a IDDS Kelley, I am writing to you to (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') Can you give me the name and contact information for the person within your company that would work with us to put a Confidentiality Agreement in place (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') confidentiality agreement n/a NSP tantivy (tan-TIV-ee) adverb At full gallop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' at full speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' noun A fast gallop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='adjective Swift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='interjection A hunting cry by a hunter riding a horse at full speed(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='Word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='Day–tantivy tricky (tan-TIV-ee) adjective NSP Prod Area and Long Haul k# Volume Rec Del 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6746 5000 St 62 Con Ed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4358 15000 St 65 Con Ed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5049 10000 St (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') TRCO capacity for Sep Prod Area and Long Haul k# Volume NSP This is a list of RisktRAC book-ids corresponding to what has been created in ERMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Let me know if the book-id naming is ok with you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Regards Book2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='xls RisktRAC ENSP Fred, I suggest a phone call among the team today to make sure we are all on the same wave length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' What is your schedule?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Thanks PSEG Firm schedule ENSP Stephanie - When you get a chance, could you finalize the attached (also found in Tana’s O drive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' I am not sure where the originals need to go after signed by Enron, but I have a request for that information currently out to Hess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Thanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Hess NDA Enron O Drive ENSP Current Notes User: To ensure that you experience a successful migration from Notes to Outlook, it is necessary to gather individual user information prior to your date of migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Please take a few minutes to completely fill out the following survey (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') 2- SURVEY /INFORMATION EMAIL 5-17-01 Office 2000 Migration Survey Sacre- BleuVAR Sheri, We are going to NO for JazzFest at the end of April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' April 27th-29th to be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Let me know if you’re going.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' DG southwest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='com weekly specials JazzFest Sacre- BleuVAR This warning is sent automatically to inform you that your mailbox is approaching the maximum size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Your mailbox size is currently 78515 KB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Mailbox size limits (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') WARNING: Your mailbox is approaching the size limit Mailbox size limit Table 5: Examples of the instances queried with different AL strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Tokens overlapping with the source docu- ment are highlighted with green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Tokens that refer to paraphrasing the part of the document and the corresponding part are highlighted with blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Tokens that cannot be derived from the document are highlighted with red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Tokens, the usage of which depends on the peculiarities of the dataset, are not highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Summaries for IDDS are not presented, because IDDS does not require model inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' AL Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' SP ESP SacreBleuVAR Random IDDS 1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3% / 0% 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0% / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4% 0% / 0% 0% / 0% 0% / 0% 6 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6% / 0% 0% / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1% 0% / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2% 0% / 0% 0% / 0% 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3% / 0% 0% / 0% 0% / 0% 0% / 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2% 0% / 0% Mean 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8% / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0% 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1% / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1% / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3% 0% / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7% 0% / 0% Table 6: Share of fully / partly overlapping summaries among batches of instances, queried with various AL strategies during AL using BART-base model on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We consider two summaries to be partly overlapping if their ROUGE-1 score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' The results are averaged across 9 seeds for all the strategies except for IDDS, which has constant seed-independent queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' F Ablation Studies of IDDS 20 40 60 80 100 120 140 10 15 20 25 30 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base + TAPT (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' LongFormer IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' SentBERT Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 4 6 8 10 12 14 16 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base + TAPT (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' LongFormer IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' SentBERT Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 10 15 20 25 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base + TAPT (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' LongFormer IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' SentBERT Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 17: Ablation study of the document embeddings model & the necessity of performing TAPT for it in the IDDS strategy with BART-base on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 26 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 27 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 28 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 29 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base + TAPT (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 10 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base + TAPT (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 19 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 20 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 21 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 22 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base + TAPT (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' BERT-base Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 18: Ablation study of the necessity of performing TAPT for the model, which generates embeddings in the IDDS strategy with BART-base on WikiHow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 12 14 16 18 20 22 24 26 28 30 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' dot product (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Mahalanobis dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Euclidean dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 6 8 10 12 14 16 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' dot product (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Mahalanobis dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Euclidean dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 12 14 16 18 20 22 24 26 28 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' dot product (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Mahalanobis dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Euclidean dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 19: The performance of IDDS with different similarity functions with BART-base on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 24 25 26 27 28 29 30 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' dot product (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Mahalanobis dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Euclidean dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 10 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' dot product (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Mahalanobis dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Euclidean dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 18 19 20 21 22 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' dot product (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Mahalanobis dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Euclidean dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 20: The performance of IDDS with different similarity functions with BART-base on WikiHow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 14 16 18 20 22 24 26 28 IDDS IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' embeddings normalization Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 7 8 9 10 11 12 13 14 15 16 IDDS IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' embeddings normalization Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 14 16 18 20 22 24 26 28 IDDS IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' embeddings normalization Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 21: The performance of the BART-base model with the standard IDDS strategy compared with its modifi- cation when embeddings are normalized on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 27 28 29 30 31 32 Embeddings similarity Embeddings similarity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' embeddings normalization Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 11 Embeddings similarity Embeddings similarity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' embeddings normalization Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 17 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 18 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 Embeddings similarity Embeddings similarity w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' embeddings normalization Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 22: The performance of the BART-base model with the standard IDDS strategy compared with its modifi- cation when embeddings are normalized on PubMed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 10 15 20 25 30 IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='75 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 4 6 8 10 12 14 16 IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='75 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 10 15 20 25 IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='75 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 23: Ablation study for the hyperparameter λ in the IDDS strategy with BART-base on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' AL Strategy Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 0 Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 5 Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 10 Iter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 15 Average λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='08 / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 / 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='79 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6 / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='87 / 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='29 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6 / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='58 / 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='68 / 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='32 / 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='23 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='25 / 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='72 / 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='88 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='33 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='77 / 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 / 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='46 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='47 / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='18 / 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='07 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='98 / 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='54 / 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='51 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='68 / 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='81 / 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='21 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='19 / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='88 / 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='78 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='15 / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='07 / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='03 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='82 / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='97 / 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='3 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='69 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='33 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='06 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='81 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='77 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='17 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='84 / 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='94 / 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='31 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 / 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='97 / 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='52 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='86 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='25 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='55 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='55 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='72 / 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='56 / 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='97 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='43 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='07 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='75 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='84 / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='93 / 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='61 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='62 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='26 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='42 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='62 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='72 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='29 / 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='36 / 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='61 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='54 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='31 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='83 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='47 / 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='84 / 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='03 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='06 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='42 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='59 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='57 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='12 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='92 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 / 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='22 / 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='0 / 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='93 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='46 λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='41 / 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='66 / 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='23 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2 / 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='66 / 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='72 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='74 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='04 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='44 / 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='66 / 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='73 / 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='81 / 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='16 Table 7: ROUGE scores on AL iterations for different values of the lambda hyperparameter in IDDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We select with bold the largest values w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' the confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 10 15 20 25 30 IDDS MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='33 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 + IDDS init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 4 6 8 10 12 14 16 IDDS MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='33 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 + IDDS init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 10 15 20 25 IDDS MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='33 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='9 MMR, lambda = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='67 + IDDS init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 24: Comparison of IDDS with the MMR-based strategy suggested in (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=', 2006) with BART-base on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' We experiment with different λ values in MMR and the initialization schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 12 14 16 18 20 22 24 26 28 30 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' average agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' max agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' for sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled data IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' max agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' for both parts Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-1 a) ROUGE-1 20 40 60 80 100 120 140 6 8 10 12 14 16 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' average agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' max agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' for sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled data IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' max agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' for both parts Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-2 b) ROUGE-2 20 40 60 80 100 120 140 12 14 16 18 20 22 24 26 28 30 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' average agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' max agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' for sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled data IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' max agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' for both parts Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 25: Comparison of the average and maximum aggregation functions in IDDS with BART-base on AESLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 26 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 27 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 28 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 29 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 IDDS w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' average agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' (orig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=') IDDS w.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Performance, Rouge-L c) ROUGE-L Figure 26: Comparison of the average and maximum aggregation functions in IDDS with BART-base on WikiHow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' G Additional Experiments with Consistency Analysis 20 40 60 80 100 120 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='85 Random sampling + self-learning Random sampling Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Consistency Score Figure 27: The consistency score calculated via SummaC with BART-base on Gigaword without AL (random sampling) with and without self-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='15 IDDS + self-learning Random sampling + self-learning IDDS Random sampling Labeled Data, % Consistency Score Figure 28: The consistency score calculated via SummaC on the test sample on AESLC for BART-base with the IDDS and random sampling strategies with and without self-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' 20 40 60 80 100 120 140 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='35 IDDS NSP BLEUVar Random sampling Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' labeled instances Consistency Score Figure 29: The consistency score calculated via SummaC on the test subset of WikiHow for the BART-base model with various AL strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' H Query Duration Fit IDDS Random NSP SacreBLEUVar ENSP ENSV 0 500 1000 1500 2000 2500 3000 3500 Query Strategy Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Query Time (seconds) Figure 30: Average duration in seconds of one AL query of 10 instances with different strategies on the AESLC dataset with BART-base as an acquisition model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Train refers to the average time required for training the model throughout the AL cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' Hardware configuration: 2 Intel Xeon Platinum 8168, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content='7 GHz, 24 cores CPU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} +page_content=' NVIDIA Tesla v100 GPU, 32 Gb of VRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qNE1T4oBgHgl3EQfigRe/content/2301.03252v1.pdf'} diff --git a/r9AzT4oBgHgl3EQfBPok/content/tmp_files/2301.00938v1.pdf.txt b/r9AzT4oBgHgl3EQfBPok/content/tmp_files/2301.00938v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..da9275148336a649084404c0394c022b9d6ef9af --- /dev/null +++ b/r9AzT4oBgHgl3EQfBPok/content/tmp_files/2301.00938v1.pdf.txt @@ -0,0 +1,856 @@ +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 1 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +Measuring Physical and Electrical Parameters in Free-Living Subjects: +Motivating an Instrument to Characterize Analytes of Clinical Importance in +Blood Samples +Barry K. Gilbert, Clifton R. Haider, Daniel J. Schwab, Gary S. Delp1 +Special Purpose Processor Development Group, Mayo Clinic, Rochester, MN 55905, USA +ABSTRACT +Significance: A path is described to increase the sensitivity and accuracy of body-worn devices +used to monitor patient health. This path supports improved health management. A wavelength- +choice algorithm developed at Mayo demonstrates that critical biochemical analytes can be +assessed using accurate optical absorption curves over a wide range of wavelengths. +Aim: Combine the requirements for monitoring cardio/electrical, movement, activity, gait, +tremor, and critical biochemical analytes including hemoglobin makeup in the context of body- +worn sensors. Use the data needed to characterize clinically important analytes in blood samples +to drive instrument requirements. +Approach: Using data and knowledge gained over previously separate research threads, some +providing currently usable results from more than eighty years back, determine analyte +characteristics needed to design sensitive and accurate multiuse measurement and recording +units. +Results: Strategies for wavelength selection are detailed. Fine-grained, broad-spectrum +measurement of multiple analytes’ transmission, absorption, and anisotropic scattering are +needed. Post-Beer-Lambert, using the propagation of error from small variations, and utility +functions that include costs and systemic error sources, improved measurements can be +performed. +Conclusions: The Mayo Double-Integrating Sphere Spectrophotometer (referred hereafter as +MDISS), as described in the companion report [1], produces the data necessary for optimal +component choice. These data can provide for robust enhancement of the sensitivity, cost, and +accuracy of body-worn medical sensors. +Keywords: Bio-Analyte, Spectrophotometry, Body-worn monitor, Propagation of error, Double- +Integrating Sphere, Mt. Everest medical measurements, O2SAT +1. INTRODUCTION +This paper describes a bio-analyte characterization process and the associated instrumentation +that was developed to support that characterization. These instruments are used to provide the +parameters to monitor clinically relevant medical data with body-worn devices. Improving +body-worn clinical-grade health monitoring units has been a major end goal of our lab since the +early 2000s. We began by implementing features on these units such as ECG and physical + +1 Corresponding Author: Delp.Gary at Mayo.edu or Gary.Delp at SilverLoon.Systems +This work is licensed under Attribution 4.0 International + + + +ccMeasuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 2 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +activity, but always with the goal of incorporating additional measurement parameters into the +units over time, e.g., blood oxygen saturation and carbon monoxide measurements, detection of +methemoglobins, and other physiological parameters as feasible. This monitoring awaits +appropriate reference data becoming available from laboratory-grade measurements. Discussion +of measuring these additional analytes appears in [1]. +Our intent in this manuscript is to highlight several separate research areas that coalesced in +our laboratory over decades. These related threads led to our recognition of the need for a new +state-of-art spectrophotometer that would yield the previously unavailable baseline data in +support of the design of analyte-measuring untethered body-worn units. Although Section 3 of +this paper may appear in part to be an historical review, that is not our primary intent; dedicated +reviews are in the published literature. Rather, we wish to illustrate the way in which eight +decades of prior work, much of it conducted in the authors’ home institution, resulted in our most +recent efforts, as described below. These fields have been continually active for decades; there +are hundreds of references, some of which we cite herein. +The initial commercial development in the early 2000s of consumer-grade, non-analyte, +body-worn units, and our development in the 2010s of clinical-grade, non-analyte, body-worn +units, are described, followed by a brief review of Mayo Clinic’s optically based analyte +measurements originally made in the 1940s. Thereafter, our development of a high-performance +research spectrophotometer system to collect data to support the design of battery-powered body- +worn analyte measurement units is introduced. The companion report, [1], describes further +spectrophotometer engineering details and presents example analyte measurement results from +the MDISS. +2. INITIAL DEVELOPMENT OF CLINICAL-QUALITY-GRADE, BODY- +WORN, PHYSIOLOGICAL MEASUREMENT UNITS +In the early 2000s, several consumer products companies began to market devices catering to the +burgeoning field of self-help health-and wellbeing techniques, in particular, small, battery- +operated monitoring devices, e.g., a generic class of “step counters” and physical activity +monitors [2-10]. These consumer units were not intended to be used in monitored clinical +settings. However, we and our clinical colleagues at the Mayo Clinic needed similar units to +measure the health and progress of patients, whose collected data would be of clinical grade. +The outcome of these requirements was a multi-year project to develop high-quality, ruggedized, +wearable sensing and recording devices, which could perform and record long-term motion +tracking [11-18], as well as high fidelity monitoring of a free-living individual’s +electrocardiogram [19, 20]. Using miniaturized electronic components and microprocessors, and +small high-energy-density batteries that became available in the first half of the 2000s, we also +created ruggedized versions of these units for extreme-activity athletes and mountaineers [21] +with several-week run times (24/7), as depicted in Figure 1. These studies were conducted under +full written, informed consent according to Mayo Clinic Institutional Review Board (IRB) study +IDs 11-006747, 12-001512 and 14-001445. + +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 3 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +Before embarking on the development for clinical-quality wearable units, and to broaden our +knowledge base, we began by purchasing, reviewing, and documenting the characteristics of + + Figure 1. The small battery-powered body-worn units developed for continuous measurement of +motion and ECG, worn by the mountain climbers on a Mayo-sponsored expedition to Mount +Everest in 2012 [20]. The unit was 38.9 mm wide, 70.2 mm long, 8.9 mm thick; 3 3-axis +accelerometers; 2 or 3-electrode, 400 samples/second ECG at 12-bit sampling resolution; 2-week +run time. Our goal was to incorporate the analyte measurement capabilities into a physical form +factor like these units (this figure also appears in U.S. Patent 8,849,387). (43333) +several consumer self-help devices that were available in the open market [19] (as was also done +by [2, 3]). Guided by these initial reviews of consumer-grade devices, and to ensure clinical- +quality data, we incorporated features into our design such as: 1) Very high sampling rates of the +measured analog signals, initially up to 400 samples/second, and later, up to 1000 +samples/second, 8 or 12 bits/sample; 2) ECG waveform monitoring; 3) rigorous static and +dynamic calibration of the accelerometers in every unit to NIST-traceable standards; 4) +accelerometer slope and offset correction measurements on NIST traceable platforms, with data +values stored to allow for post processed compensation for minor manufacturing differences +(including a unique serial number and manufacturing date for each device for complete +traceability); 5) autonomous multi-day operation without battery change or charging or any other +intervention by the wearer; and 6) a stable time-of-day clock allowing the synchronization of +data from multiple units worn by a single subject. Body-worn units designed, fabricated, tested, +and deployed in this manner resulted in a reliable physiological measurement capability, in a + +5Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 4 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +small form factor, representing a set of potentially useful clinical tools for eventual monitoring of +patients in their free-living environments. We also demonstrated the ability to monitor patients, +via short- and long-haul wireless and wired connections, from their home environments back to a +medical center, where the collected raw data could be analyzed in near real-time [20]. +3. EVOLUTION TO CLINICAL-QUALITY-GRADE, BODY-WORN +BIOLOGICAL ANALYTE MEASUREMENT UNITS +The next request from our clinical colleagues was for an ability to measure and record, +noninvasively and over a duration of days to a few weeks, the blood oxygen saturation levels in +free-living patients (rather than in a hospital or clinic setting); we were also asked if it would be +possible to measure the concentrations of other naturally occurring analytes as well. The tiny, +long-lasting body-worn units represented the target form factors into which our clinical +collaborators asked us to incorporate these additional measurement modalities. +Regarding the measurement of blood oxygen saturation, we relied upon prior research in this +field as a starting point. We began by reviewing the significant body of work, beginning in 1935 +and progressing steadily thereafter, on techniques for measuring blood oxygen saturation +noninvasively. This capability, using optical techniques based on two wavelengths of light, was +first demonstrated in 1935 by Matthes [22], then extended by Milliken [23] and by Goldie [24] in +the early 1940s. Also, in the early 1940s, significant contributions to this field were made by +Wood and colleagues [25-35]. However, the Wood team was unable to publish their results until +1947 and thereafter [36] because of wartime restrictions on the release of “sensitive” information +since this work was conducted under the auspices of the U.S. Army Air Corps [though funded by +Mayo Clinic as a contribution to the WW2 scientific efforts]. As with the work described in [22- +24], the Wood earpiece oximeter employed two optical wavelengths, but it also incorporated a +pressure-activated plunger to expel blood from the upper edge of the pinna (i.e., the upper +portion of the outer ear) to achieve a hemoglobin-free tissue baseline that could be incorporated +into the blood O2 calculations (Figure 2). By present standards, the units were heavy and bulky, +and had to be taped to the subject’s head to provide mechanical support (Figure 3). The Mayo- +developed units were used in studies of G-induced loss of consciousness (G-LOC) in human +subjects during World War II on a full-sized human centrifuge (partially visible in Figure 3) +installed at the Mayo Clinic. The earpiece oximeters continued in use without major changes +until the early 1960s, in centrifuge studies of the Project Mercury astronaut couches. +The on-body oximetry technology continued to evolve. The next significant advance, +referred to as pulse oximetry (a variant on the original continuous oximetry) was first described +in 1972 by Aoyagi and Kishi [37-39], with additional refinements in the subsequent decades. +The pulse oximetry approach effectively supplanted the original non-pulsatile approach in +clinical implementations (see below). Over the decades, commercial industry extended the +implementation of pulse oximetry through hardware refinements using newer components and +with algorithmic and software extensions to improve the usefulness and accuracy of the collected +data, e.g., [40]. In 2000 Masimo introduced a technology approach referred to as Signal +Extraction Technology (SET) [41], in which five proprietary algorithms were developed to + +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 5 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +remove the extraneous variability in the arterial oxygenation waveform caused by changes in the +venous circulation, thereby providing a more accurate arterial oxygenation signal. + +Figure 2: Earpiece oximeter developed at Mayo Clinic, illustrating light and plunger fully +depressed against photodetector, ca. 1943. (42175) + + +Figure 3: Volunteer in the cockpit of Mayo Clinic’s human centrifuge, wearing earpiece oximeter +on right ear, as indicated by white arrow, ca. 1962 (Photo courtesy of Don Hagland). (1954)2 +Others have continued to document and refine the understanding of the physiological and +optical processes underlying blood oximetry; see Severinghaus’s excellent review of the early + +2 The (numbers) trailing the figure caption denote the figure’s location in the SPPDG image archive. + +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 6 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +years of oximetry [39], and an exposition by Mannheimer of the optical physics and +hemodynamics of the process [42]. By 2010, “finger-tip” oximeters, cable-powered by a desktop +unit at the patient’s bedside, were coming into use in hospital settings. These wired, clip-on +devices are placed on the patient’s index finger, and use “transmission oximetry”, i.e., where +light from two or more light-emitting diodes (LEDs) of different wavelengths passes through the +finger, with the residual light then detected by one or more solid-state photodiodes. +To incorporate noninvasive analyte detection and measurement into our planned free- +standing body-worn units we needed to identify optical components that would be compatible +with the physical form factor constraints of the ECG/motion units. Our motion-and-ECG units +were powered by small batteries. We viewed the analyte detection capability as an addition to +the original baseline functionality. Thus, we needed to remain within the size and power +constraints of those units, with the battery limitation being the most important. The consumer- +grade LEDs used in the wired units require more power than we could support with small +batteries. LEDs also have relatively wide emission bandwidths (20-70 nm full-width half-max +[FWHM]) and uncertain center frequencies, which, as we later demonstrated, degrade the quality +of data generated from them (discussed below). We turned to a family of small solid-state lasers, +referred to as vertical cavity surface emitting lasers (VCSELs), which, in addition to their +narrow-banded emission characteristics (FWHM optical bandwidths of 2-5 nm), were available +over a wide range of optical center frequencies. +The first practical room temperature non-pulsatile (i.e., continuous-wave or CW) VCSEL +was reported by Koyama et al. in 1988 [43]. Development of these tiny optical sources +accelerated in the late 1990s and early 2000s by DARPA funding [44]. Our intent was to pair a +small number of frequency-selected VCSELs with equally tiny solid-state photodetectors, either +avalanche photodiodes (APDs) or P-I-N photodiodes (PIN diodes). APDs exhibit more signal +gain than PIN diodes, but also have more intrinsic noise, so we concentrated on PIN diodes for +our application. PIN diodes can be selected to cover a range of optical wavelengths. By +combining narrow-spectrum VCSELs with PIN diodes having wide wavelength sensitivity, we +could allow the light from VCSELS of different wavelengths to impinge on a single PIN diode. +If in addition the light output from each VCSEL was modulated with a unique on-off keyed +(OOK) pseudo-random pulsatile sequence (code division, Sig/Noise gain), and the PIN diode +were integrated with an on-unit multi-channel correlator (i.e., in a small microcontroller or a +custom correlator chip), accurate through-the-skin measurement of several analytes of clinical +importance could be accomplished, simultaneously, in a small form factor, unlike in +conventional pulse oximetry, where the measurements from the different LEDs must be made +sequentially (thus introducing time skew between the two measured values in each pair). +The VCSELs’ narrow emission lines indicated that considerable analyte measurement +specificity could be achieved, far better than the LED-based wire-tethered fingertip pulse +oximeters, based on Aoyogi’s implementation [37-39], that entered hospital use in the early +2000s. However, to select the appropriate VCSEL wavelengths, we needed more frequency- +accurate data than was identifiable in the published literature. It was this need for improved +wavelength resolution, a larger wavelength measurement span, and additional wavelength +measurement parameters that led to our decision to design and fabricate a spectrophotometer + +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 7 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +with extended functionality, as discussed below and in [1]. With this type of higher resolution +and more comprehensive data available, it appeared feasible to design a wearable analyte sensor +with considerably extended in vivo measurement capabilities compared to the then-commercial +offerings. We undertook that effort. The goal of achieving an extended-capability autonomous +on-body analyte measurement unit, underpinned by the historical evolution of such a capability +as described above, represents the end goal of the entire sequence of projects described here and +in [1]. +Because we did not wish to be bound to through-the-finger transmission measurements, we +investigated an alternate approach, referred to as “reflection oximetry”, in which light is reflected +from underlying bone, e.g., at the forehead, back to the photodetectors. The requirement for +underlying bone also constrained placement options for the body-worn system. Therefore, we +investigated a third approach, “scatter” oximetry, in which photons from the optical sources are +directed into the skin and are sensed by one or more photodetectors placed several cm from the +sources. Scattered photons entering the inputs of the photodetectors acquire and carry with them +the optical information required to calculate the concentrations of the analytes of interest. Using +scatter oximetry, the battery-powered measurement device can be placed on a wrist, on an arm, +or on the torso, i.e., less intrusively than on a finger. +4. APPROACHES FOR SELECTING THE OPTIMUM MEASUREMENT +WAVELENGTHS FOR ON-BODY OXIMETRY +Next, with this conceptual optical measurement chain sketched out, we needed a method to select +the optimum wavelengths to measure analyte concentrations in vivo, so that the correct VCSELs +could be incorporated into body-worn units. To address this problem one of us (CRH) developed +a robust algorithm [45, 46] for the selection of optimal excitation wavelengths. Using the +wavelengths selected by the algorithm allowed for the measurement of relative and absolute +concentrations of a set of analytes in a sample. +The Beer-Lambert molar extinction coefficients of homogeneous materials can be measured +for selected frequencies. Measurements in “the wild,” however, need to incorporate many more +factors, e.g., diffusion; heterogeneous paths; reflection; and the propagation of potential error +from the inputs, through the measurement system, and continuing to a consideration of the +variations and non-linearities of receivers. This system-level approach required more accurate +and higher-resolution measurements of analyte characteristics, including diffusion, reflection, +mean-free-path variation, florescence, phosphorescence, and various anisotropies. Our view of +the importance of this information was informed by the guidance provided by [45, 46]. +However, we did not have this data. Thus, we instituted a literature search for absorption +curves for various forms of hemoglobin over a wide range of wavelengths. Such curves were +first published in 1987 by Barker and Tremper [47], and again in 1989 by Tremper and Barker +[48] (the authors credit Susan Manson, Biox/Ohmeda as their source of this data, as do many +subsequent authors, which is widely accepted in the field), and are reproduced and duplicated in +both the left and right panels of Figure 4; Figure 7 is from the same published sources. (Note: +these curves might or might not be “correct” in some absolute sense, but they were the data + +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 8 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +available to us; “errors” in the shapes of these curves would of course have deleterious effects on +the results that we present below but were unknowable.) Next, we searched the published +literature for a selection of wavelengths that would yield the most accurate measures of the +concentrations of four forms of hemoglobin (COHb, MetHb, OxyHb, and DeoxyHb). Lamego et +al. [40] describes eight “optimum” wavelengths (610, 620, 630, 655, 700, 720, 800 and 905 nm). +The wavelengths in the rightmost panel of Figure 4, illustrated as vertical lines, are similar +though not identical to those documented in Lamego et al. +The left panel of that figure depicts four wavelengths calculated from the algorithm of [45, +46] also provides a means of computing a quality factor, referred to as the “condition number” +for a selected set of wavelengths, in which a lower number is “better”. A condition number for +the eight wavelengths in the right panel of Figure 4, taken from [40] as the outcome of a +literature search, was calculated as 31.2, whereas the condition number for the four wavelengths +in the left panel of that figure was calculated to be 19.8, i.e., a better condition number for the +use of fewer wavelengths. The algorithm of [45, 46] teaches the following conditions, all of +which must be satisfied simultaneously as best possible: 1) each selected measurement +wavelength should be maximally separated from other wavelengths; 2) the measured curves +should be maximally separated in the vertical, or extinction coefficient, axis; 3) wavelengths +should not be selected in regions where an extinction curve is “steep”, since slight differences in +the shapes of the curves and/or in their left-to-right or vertical positions due to component +variations or other differences can introduce errors in the final resulting measurements (this +constraint is often difficult to obey). In the rightmost panel, the wavelengths 610, 620, and 630 +nm are “too close together”. The wavelengths at 660, 730, 805 and 905 nm are “good” choices, +according to our algorithm, although the overall condition number is degraded by the +wavelengths at 610, 620, and 630 nm. +The four wavelengths selected by our algorithm, in the left panel of Figure 4, are in partial +violation of the ground rules described above; the wavelengths are constrained to some extent by +the shapes of the curves themselves. In addition, there is a fifth curve, labeled “Penalty”, which +we incorporated to account for the fact that some VCSELs are more difficult and costly to +manufacture than others; the shape of this curve changes over time as manufacturing processes +evolve [courtesy of M. Hibbs-Brenner, Vixar Corporation, ca. 2010]. Were the penalty curve +not incorporated into the calculations, the selection of wavelengths would have been somewhat +different, and might yield results which obey the above-described guidelines more closely. +Finally, in Figure 4, note the addition of small, bell-shaped curves at the bottom of the +rightmost panel, which presents the wavelengths published by a commercial vendor [40]. These +bell-shaped curves illustrate the wavelength spread generated by LEDs on either side of their +center frequencies, which, as commented above, can be 20-70 nm (FWHM) wide. The LEDs +thus generate wavelength components which overlap one another, thereby degrading the +measured signal-to-noise ratios at the photodetector circuitry, because each light source will be +carrying “information” that optimally would be out-of-band for those light sources. VCSELs +generating FWHM wavelengths only a few nm in width will not create or will at least minimize +these artifactual components, thereby motivating their use in place of LEDs. + +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 9 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 + +Figure 4: Oxyhemoglobin, Deoxyhemoglobin, Carboxyhemoglobin, and methemoglobin +extinction coefficients as a function of wavelength (four algorithmically selected maximally +linearly independent wavelengths in the left panel, versus eight wavelengths selected by a +commercial vendor in the right panel). Narrow-band VCSEL sources are assumed, though +commercial vendors typically employ LEDs [40], whose wider bandwidths and overlaps are +illustrated by the bell-shaped curves at the bottom of the rightmost panel. Condition numbers: +left panel: 19.8; right panel: 31.2. (42003) +As is clear from this chart, the optical wavelengths at which meaningful hemoglobin +measurements can be performed are in the range of approximately 450 nm to 1000 nm. As will +be noted below, several other analytes can also be measured using a similar implementation, but +the wavelength ranges needed to be extended down to 200 nm and up to 2000 nm. +5. THE NEED FOR, AND OUR APPROACH TO OBTAINING, AN IMPROVED +SPECTROPHOTOMETER +The results described above demonstrated that with good continuous optical absorption data over +a broad wavelength span, the algorithm described here could select the optimum wavelengths, +and hence the correct VCSELs, to detect one or more naturally occurring analytes present in +sufficient concentration in the blood of an individual as measured by an autonomous battery- +operated (i.e., untethered!) body-worn unit. We also recognized that the more optical parameters +from laboratory samples that we had, the more specifically we could select the optimum VCSEL +wavelengths. To gather the requisite data, we needed a spectrophotometer capable of measuring +many optical parameters simultaneously, in a wide variety of physiological samples (e.g., whole +blood, plasma, lymph, etc.). +A review of available commercial spectrophotometers confirmed that such a machine did not +exist. The available models for purchase measured only two parameters, i.e., transmission and +backscatter, and were lacking most of the desired operational features. We also conducted a + +Algorithmically-Selected Wavelengths +Wavelengths + Used by Commercial Vendor +OxyHB +610 +OxyHB +10 +DeoxyHB +DeoxyHB +COHB +620 +COHB +MetHB +MetHB +Penalty +730 +0 +10 +cm +mM-1 +805 +700 +857 +905 +673 +630 +2 +10 +650 +611 +600 +700 +800 +900 +1000 +600 +700 +800 +900 +1000 +(A) +(B) +Wavelength, nm +Wavelength, nmMeasuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 10 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +literature search against the possibility that other research groups had developed very capable +spectrophotometers, from which they might have published high accuracy absorption curves of +one or more naturally occurring analytes. We discovered developments, both theoretically and +in some cases through the implementation of actual hardware [49-52] that addressed some of the +parameters that appeared to us to be important, but none that were sensitive to as many +parameters as our studies indicated might be physically implementable and clinically relevant. +In several of those previously published papers, descriptions of the characteristics of those +machines were limited. Therefore, we elected to design and fabricate an optical instrument +capable of measuring more of these parameters, over a wide spectral range, in very short +durations. Our intent was to create a set of baseline data that would enable the next step, that of +designing the desired autonomous body-worn units. Following a brief introduction to the +spectrophotometer developed here, we will return to a discussion of the value of the extended +measurement parameters that we believed to be important. +5.1. The Mayo Double-Integrating Sphere Spectrophotometer (MDISS): Lessons-Learned +Regarding the Selection of Optical Wavelengths for Body-Worn Analyte +Measurement Units +Only a summary of the features of the MDISS is presented here; a complete description appears +in [1]. From a functional perspective, we consider the MDISS to be a multi-generational +successor to the Beckman DU spectrophotometer [53-55]; also, we wished to extend the +capabilities of the experimental machines previously referenced with a combination of +quantitative functionality features including: a broad wavelength measurement span (190-2750 +nm); high wavelength accuracy and high wavelength precision (FWHM on the order of 1 nm); +fine wavelength resolution (0.1-5 nm wavelength step sizes); high amplitude sensitivity; a rapid- +throughput automated measurement capability; simultaneous acquisition of diffuse reflected +(DR), diffuse transmitted (DT), and unscattered transmitted (UT) energy; input energy +monitoring for high accuracy energy values on a pulse by pulse basis (with options for +fluorescent and opto-acoustic acquisition); and the flexibility to accommodate a wide variety +sample holder types and volumes, including high-pressure (up to 30 atmosphere [450 psi]) cells +for measurements of blood oxygen saturation characteristics in hyperbaric environments). We +also attempted to mitigate potential sources of measurement error of parameters, as will be noted +below. Figure 5 is a photo of an early version of the unit. + +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 11 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 + +Figure 5: Double-integrating sphere spectrophotometer system (narrow-linewidth pump laser +coupled to a tunable optical parametric oscillator, allowing precise optical characterization of +biological analytes over a 192-2750 nm wavelength span. (44414) + +Figure 6: A set of spectroscopic transmission curves measured with the MDISS at each of six +oxygen saturation levels, at wavelength separations of 2-nm, over a range of 800 nm. The +desaturation gas in this example was carbon monoxide (note the isosbestic point at 650 nm, +rather than at 800 nm if the desaturation gas had been carbon dioxide). (46584) +5.2. MDISS Design Tradeoffs +The design of the MDISS system required a recognition of the various cost tradeoffs in terms of +money, time (development time and actual test time), and specific performance parameters. A +single example of a specific performance parameter trade-off was prioritizing wavelength span +rather than absolute energy sensitivity of the detectors. This selection was accepted to address +the numerous unknowns regarding the analyte characteristics, since we had no prior knowledge + +BeamPath(ShowninOrange) +Fixed Turning +Mirror +401-2750nm +Neodymium-DopedYttrium +Transmitted Light +OutputPort +AluminumGarnet (Nd:YAG) Pump +(ForwardScattered) +Energy Meters (4) +Input Energy +Laser,withHarmonicGenerators +CollectionSphere +Detector +192-400nm +(1064nm,532nm,355nm) +(EnergyDetectorOnTop) +OutputPort +Optical +Parametric +Oscillator,oPo +(192-2750nm) +Sample +Thickness +System +Display +Control +Computer +Switchable +Turning Mirror +(Selects +DesiredOPO +Output Port) +Detector +for +Unscattered +Light +Aperture +to Filter +Unwanted +4By10Foot +ReflectedLight +Light +Air Table +(BackwardScattered) +CollectionSphere +(Energy Detector On Top) +Input Energy +Beamsplitter +PeristalticPump +AdjustableBiological +(0.5%/99.5%) +SampleReservoir +(ToCirculateSample) +AnalyteSampleHolder +(1-10mmThickness)0.40 +Carbon Monoxide +9 +0.30 +(Lo) +0.20 +lized +0.10 +0.08 +0.06 +Oxygenation Saturation in % +rma +1.0 +42.0 +e +0.04 +79.0 +83.7 +Pul +88.0 +92.3 +0.02 +100.0 +600 +650 +700 +750 +800 +850 +900 +950 +1000 +1050 +1100 +1150 +1200 +1250 +1300 +1350 +1400 +Wavelength (nm)Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 12 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +of those subregions in the entire wavelength span in which a given analyte would display +maximum amplitude differences. This requirement for sensors with wide optical bandwidths +justified the use of pyroelectric sensors, which have broad wavelengths spanning 190–12,000 +nm, and the use of an optical parametric oscillator (OPO) which produced energy from 192 to +2750 nm. If it had been determined that, even with the broadband capability, there was a need +for more sensitivity, then alternate detectors could have been implemented, albeit at the expense +of reduced bandwidth. As was discovered during the blood hemoglobin and oxygenation +characterizations with different desaturation gases, most of the variations that we needed to +detect occurred in the 600-1200 nm span. +In a mutually reinforcing fashion, the underlying theory was subsequently corroborated by +the measured results from the MDISS when it became operational. Without the machine, which +was designed and constructed under the assumption that the guidance gleaned from [45, 46] was +correct, many if not most of the results demonstrated by the machine to are valid, and necessary +for the design of accurate body-worn analyte measurement units, would have been unavailable. +Noninvasive optical sensing can be used with other clinically important physiological and +biochemical variables besides hemoglobin species. As one example, Figure 7 displays +absorption curves for blood glucose, blood protein, and blood lipids in the near-IR range of 1350 +nm to 1850 nm. As in Figure 4, these waveforms were identified and employed following a +search of the open literature. An absorption curve for water is provided for comparison. With +the appropriate sensing wavelengths as determined by the algorithm and indicated by the vertical +lines in this figure, detection and measurements of glucose, lipids, proteins, and even a measure +of either total body water or central blood volume could be made. + + +Measuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 13 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 + +Figure 7: Water, protein, and glucose extinction coefficients as a function of wavelength, +assuming ideal 1-nm bandwidth light sources using maximally independent wavelengths. +Condition number: 10.9. (41964) +6. THE LONG-TERM GOAL OF THIS EFFORT +The long-term goal of this multiple-step project was the development of a sufficient technical +base so that small, battery-powered, microcontroller-enhanced body-worn units could be +designed to measure and record, with high accuracy and in real-time, blood oxygen saturation, +blood levels of carbon monoxide, and concentrations of other clinically relevant analytes. The +first developmental stage was the MDISS instrument, which was to yield the type of clinically +actionable data described above. A next step, described in [55], was the prototyping of the +battery-powered electrical and optical components to conduct these measurements continually, in +a sufficiently small form factor that they could be incorporated into versions of the body-worn +units depicted in Figure 1, that could be fielded into the clinical practice. In addition, although +we did not pursue this path, commercial versions of the MDISS system could be used as research + +Water +Glucose +Protein +Lipid +101 +Extinction, +1679 nm +1707 nm +100 +1371nm +1597nm +1350 +1400 +1450 +1500 +1550 +1600 +1650 +1700 +1750 +1800 +1850 +Wavelength, nmMeasuring Physical and Electrical Parameters in Free-Living Subjects: + 22 December 2022 +Page 14 of 18 +Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples +Report# Mayo-SPPDG-R22-15-V01 +instruments; or, reduced-wavelength-span and/or lower cost versions targeted for a subset of +analytes could be used for higher throughput routine clinical diagnostic purposes. +7. SUMMARY +We have described the evolution of consumer-grade body-worn physiological measurement +units. We then introduced an evolutionary thread from early work in the development of +research-grade body-worn blood oxygen saturation units conducted in the first half of the 1940s. +Next, we reviewed our recognition in the 2010s of the requirements for higher-quality laboratory +measurements of the optical characteristics of medically relevant blood analytes (e.g., oxygen +saturation-versus-wavelength behavior of critical blood analytes). We extended that line of +investigation with the design of a spectrophotometer, the MDISS, with the needed sensitivity and +specificity to help us gather new optically based blood characterization data [1]. We also +initiated efforts to use the results reported here and in [1] to develop a new-technology body +worn unit that operates on a somewhat different principle from classical pulse oximetry. +Although body-worn units were our major end goal, we have also underscored the benefits of +using optical analyte data over a wide wavelength range with high wavelength resolution. An +instrument with capabilities beyond those available is required; these new data support a future +effort to simplify and optimize body-worn monitors. +Disclosures +The authors declare no conflicts of interest. Although various patents cover various aspects of +this work, none of these patents are licensed for gain or profit. +Acknowledgements +We wish to acknowledge the years-long contributions of the following individuals to this project: +Charles Burfield, Anthony Ebert, Theresa Funk, Nicholas Klitzke, Steven Polzer, Jason Prairie, +and Steven Schuster; and Drs. Franklyn Cockerill, Kendall Cradic, Graham Cameron, E. Rolland +Dickson, Stefan Grebe, Ravinder Singh, and Nathan Harff. The clinical materials used in the +study were produced by the Mayo Clinic Department of Laboratory Medicine and Pathology and +processed by them for us using their standard clinical laboratory tools. +References +[1] Schwab, DJ, CR Haider, G Delp, SK Grebe, and BK Gilbert, “An Experimental Double- +Integrating Sphere Spectrophotometer for in Vitro Optical Analysis of Blood and Tissue +Samples, Including Examples of Analyte Measurement Results,” MDISS Curves Technical +Reports, SPPDG, Mayo Clinic, Rochester, MN, Technical Report, Mayo-SPPDG-R22-16- +R0, 21 Dec 2022. doi:10.48550/arXiv.2212.08763 +[2] Crouter, SE, PL Schneider, M Karabulut, and DRJ Bassett: “Validity of 10 Electronic +Pedometers for Measuring Steps, Distance, and Energy Cost,” Medicine & Science in Sports +& Exercise, 35(8), 2003. doi:10.1249/01.MSS.00000789 32.61440.A2 +[3] Yang, C-C and Y-L Hsu: “A Review of Accelerometry-Based Wearable Motion Detectors +for Physical Activity Monitoring,” Sensors in Biomechanics and 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Beckman,” Journal of Biological Chemistry, +278(49):79-81, 5 Dec 2003. doi:10.1016/S0021-9258(20)75750-9 +[55] Hewlett Packard: “Compound Identification with Hp 8450 A Uv Visible +Spectrophotometer,” Analytical Chemistry, 51(12):1188A-1189A, Oct 1979. doi:10.1021 +/ac50048a728 +[56] White, C, CR Haider, DJ Schwab, and BK Gilbert: “Demonstration of Compact Free-Space +Optical MIMO Transmitter and Receiver Using Tri-Wavelength Vcsels and Silicon Apds,” +International Journal of Optics and Applications, 8(1):12-20, 2019. doi:10.5923 +/j.optics.20190801.03 + diff --git a/r9AzT4oBgHgl3EQfBPok/content/tmp_files/load_file.txt b/r9AzT4oBgHgl3EQfBPok/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..633b065d082907663cc18eb8e8b3bce649fe7160 --- /dev/null +++ b/r9AzT4oBgHgl3EQfBPok/content/tmp_files/load_file.txt @@ -0,0 +1,510 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf,len=509 +page_content='Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 1 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 Measuring Physical and Electrical Parameters in Free-Living Subjects: Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Barry K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Gilbert, Clifton R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Haider, Daniel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Schwab, Gary S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Delp1 Special Purpose Processor Development Group, Mayo Clinic, Rochester, MN 55905, USA ABSTRACT Significance: A path is described to increase the sensitivity and accuracy of body-worn devices used to monitor patient health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' This path supports improved health management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' A wavelength- choice algorithm developed at Mayo demonstrates that critical biochemical analytes can be assessed using accurate optical absorption curves over a wide range of wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Aim: Combine the requirements for monitoring cardio/electrical, movement, activity, gait, tremor, and critical biochemical analytes including hemoglobin makeup in the context of body- worn sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Use the data needed to characterize clinically important analytes in blood samples to drive instrument requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Approach: Using data and knowledge gained over previously separate research threads, some providing currently usable results from more than eighty years back, determine analyte characteristics needed to design sensitive and accurate multiuse measurement and recording units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Results: Strategies for wavelength selection are detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Fine-grained, broad-spectrum measurement of multiple analytes’ transmission, absorption, and anisotropic scattering are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Post-Beer-Lambert, using the propagation of error from small variations, and utility functions that include costs and systemic error sources, improved measurements can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Conclusions: The Mayo Double-Integrating Sphere Spectrophotometer (referred hereafter as MDISS), as described in the companion report [1], produces the data necessary for optimal component choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' These data can provide for robust enhancement of the sensitivity, cost, and accuracy of body-worn medical sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Keywords: Bio-Analyte, Spectrophotometry, Body-worn monitor, Propagation of error, Double- Integrating Sphere, Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Everest medical measurements, O2SAT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' INTRODUCTION This paper describes a bio-analyte characterization process and the associated instrumentation that was developed to support that characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' These instruments are used to provide the parameters to monitor clinically relevant medical data with body-worn devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Improving body-worn clinical-grade health monitoring units has been a major end goal of our lab since the early 2000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We began by implementing features on these units such as ECG and physical 1 Corresponding Author: Delp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='Gary at Mayo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='edu or Gary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='Delp at SilverLoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='Systems This work is licensed under Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='0 International ccMeasuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 2 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 activity, but always with the goal of incorporating additional measurement parameters into the units over time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', blood oxygen saturation and carbon monoxide measurements, detection of methemoglobins, and other physiological parameters as feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' This monitoring awaits appropriate reference data becoming available from laboratory-grade measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Discussion of measuring these additional analytes appears in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Our intent in this manuscript is to highlight several separate research areas that coalesced in our laboratory over decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' These related threads led to our recognition of the need for a new state-of-art spectrophotometer that would yield the previously unavailable baseline data in support of the design of analyte-measuring untethered body-worn units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Although Section 3 of this paper may appear in part to be an historical review, that is not our primary intent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' dedicated reviews are in the published literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Rather, we wish to illustrate the way in which eight decades of prior work, much of it conducted in the authors’ home institution, resulted in our most recent efforts, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' These fields have been continually active for decades;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' there are hundreds of references, some of which we cite herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The initial commercial development in the early 2000s of consumer-grade, non-analyte, body-worn units, and our development in the 2010s of clinical-grade, non-analyte, body-worn units, are described, followed by a brief review of Mayo Clinic’s optically based analyte measurements originally made in the 1940s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Thereafter, our development of a high-performance research spectrophotometer system to collect data to support the design of battery-powered body- worn analyte measurement units is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The companion report, [1], describes further spectrophotometer engineering details and presents example analyte measurement results from the MDISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' INITIAL DEVELOPMENT OF CLINICAL-QUALITY-GRADE, BODY- WORN, PHYSIOLOGICAL MEASUREMENT UNITS In the early 2000s, several consumer products companies began to market devices catering to the burgeoning field of self-help health-and wellbeing techniques, in particular, small, battery- operated monitoring devices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', a generic class of “step counters” and physical activity monitors [2-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' These consumer units were not intended to be used in monitored clinical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' However, we and our clinical colleagues at the Mayo Clinic needed similar units to measure the health and progress of patients, whose collected data would be of clinical grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The outcome of these requirements was a multi-year project to develop high-quality, ruggedized, wearable sensing and recording devices, which could perform and record long-term motion tracking [11-18], as well as high fidelity monitoring of a free-living individual’s electrocardiogram [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Using miniaturized electronic components and microprocessors, and small high-energy-density batteries that became available in the first half of the 2000s, we also created ruggedized versions of these units for extreme-activity athletes and mountaineers [21] with several-week run times (24/7), as depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' These studies were conducted under full written, informed consent according to Mayo Clinic Institutional Review Board (IRB) study IDs 11-006747, 12-001512 and 14-001445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 3 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 Before embarking on the development for clinical-quality wearable units, and to broaden our knowledge base, we began by purchasing, reviewing, and documenting the characteristics of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The small battery-powered body-worn units developed for continuous measurement of motion and ECG, worn by the mountain climbers on a Mayo-sponsored expedition to Mount Everest in 2012 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The unit was 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='9 mm wide, 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='2 mm long, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='9 mm thick;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 3 3-axis accelerometers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 2 or 3-electrode, 400 samples/second ECG at 12-bit sampling resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 2-week run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Our goal was to incorporate the analyte measurement capabilities into a physical form factor like these units (this figure also appears in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Patent 8,849,387).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' (43333) several consumer self-help devices that were available in the open market [19] (as was also done by [2, 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Guided by these initial reviews of consumer-grade devices, and to ensure clinical- quality data, we incorporated features into our design such as: 1) Very high sampling rates of the measured analog signals, initially up to 400 samples/second, and later, up to 1000 samples/second, 8 or 12 bits/sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 2) ECG waveform monitoring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 3) rigorous static and dynamic calibration of the accelerometers in every unit to NIST-traceable standards;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 4) accelerometer slope and offset correction measurements on NIST traceable platforms, with data values stored to allow for post processed compensation for minor manufacturing differences (including a unique serial number and manufacturing date for each device for complete traceability);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 5) autonomous multi-day operation without battery change or charging or any other intervention by the wearer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' and 6) a stable time-of-day clock allowing the synchronization of data from multiple units worn by a single subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Body-worn units designed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' fabricated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' tested,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' and deployed in this manner resulted in a reliable physiological measurement capability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' in a 5Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 4 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 small form factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' representing a set of potentially useful clinical tools for eventual monitoring of patients in their free-living environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We also demonstrated the ability to monitor patients, via short- and long-haul wireless and wired connections, from their home environments back to a medical center, where the collected raw data could be analyzed in near real-time [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' EVOLUTION TO CLINICAL-QUALITY-GRADE, BODY-WORN BIOLOGICAL ANALYTE MEASUREMENT UNITS The next request from our clinical colleagues was for an ability to measure and record, noninvasively and over a duration of days to a few weeks, the blood oxygen saturation levels in free-living patients (rather than in a hospital or clinic setting);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' we were also asked if it would be possible to measure the concentrations of other naturally occurring analytes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The tiny, long-lasting body-worn units represented the target form factors into which our clinical collaborators asked us to incorporate these additional measurement modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Regarding the measurement of blood oxygen saturation, we relied upon prior research in this field as a starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We began by reviewing the significant body of work, beginning in 1935 and progressing steadily thereafter, on techniques for measuring blood oxygen saturation noninvasively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' This capability, using optical techniques based on two wavelengths of light, was first demonstrated in 1935 by Matthes [22], then extended by Milliken [23] and by Goldie [24] in the early 1940s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Also, in the early 1940s, significant contributions to this field were made by Wood and colleagues [25-35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' However, the Wood team was unable to publish their results until 1947 and thereafter [36] because of wartime restrictions on the release of “sensitive” information since this work was conducted under the auspices of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Army Air Corps [though funded by Mayo Clinic as a contribution to the WW2 scientific efforts].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' As with the work described in [22- 24], the Wood earpiece oximeter employed two optical wavelengths, but it also incorporated a pressure-activated plunger to expel blood from the upper edge of the pinna (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', the upper portion of the outer ear) to achieve a hemoglobin-free tissue baseline that could be incorporated into the blood O2 calculations (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' By present standards, the units were heavy and bulky, and had to be taped to the subject’s head to provide mechanical support (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The Mayo- developed units were used in studies of G-induced loss of consciousness (G-LOC) in human subjects during World War II on a full-sized human centrifuge (partially visible in Figure 3) installed at the Mayo Clinic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The earpiece oximeters continued in use without major changes until the early 1960s, in centrifuge studies of the Project Mercury astronaut couches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The on-body oximetry technology continued to evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The next significant advance, referred to as pulse oximetry (a variant on the original continuous oximetry) was first described in 1972 by Aoyagi and Kishi [37-39], with additional refinements in the subsequent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The pulse oximetry approach effectively supplanted the original non-pulsatile approach in clinical implementations (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Over the decades, commercial industry extended the implementation of pulse oximetry through hardware refinements using newer components and with algorithmic and software extensions to improve the usefulness and accuracy of the collected data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' In 2000 Masimo introduced a technology approach referred to as Signal Extraction Technology (SET) [41],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' in which five proprietary algorithms were developed to Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 5 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 remove the extraneous variability in the arterial oxygenation waveform caused by changes in the venous circulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' thereby providing a more accurate arterial oxygenation signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Figure 2: Earpiece oximeter developed at Mayo Clinic, illustrating light and plunger fully depressed against photodetector, ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 1943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' (42175) Figure 3: Volunteer in the cockpit of Mayo Clinic’s human centrifuge, wearing earpiece oximeter on right ear, as indicated by white arrow, ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 1962 (Photo courtesy of Don Hagland).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' (1954)2 Others have continued to document and refine the understanding of the physiological and optical processes underlying blood oximetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' see Severinghaus’s excellent review of the early 2 The (numbers) trailing the figure caption denote the figure’s location in the SPPDG image archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 6 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 years of oximetry [39], and an exposition by Mannheimer of the optical physics and hemodynamics of the process [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' By 2010, “finger-tip” oximeters, cable-powered by a desktop unit at the patient’s bedside, were coming into use in hospital settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' These wired, clip-on devices are placed on the patient’s index finger, and use “transmission oximetry”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', where light from two or more light-emitting diodes (LEDs) of different wavelengths passes through the finger, with the residual light then detected by one or more solid-state photodiodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' To incorporate noninvasive analyte detection and measurement into our planned free- standing body-worn units we needed to identify optical components that would be compatible with the physical form factor constraints of the ECG/motion units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Our motion-and-ECG units were powered by small batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We viewed the analyte detection capability as an addition to the original baseline functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Thus, we needed to remain within the size and power constraints of those units, with the battery limitation being the most important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The consumer- grade LEDs used in the wired units require more power than we could support with small batteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' LEDs also have relatively wide emission bandwidths (20-70 nm full-width half-max [FWHM]) and uncertain center frequencies, which, as we later demonstrated, degrade the quality of data generated from them (discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We turned to a family of small solid-state lasers, referred to as vertical cavity surface emitting lasers (VCSELs), which, in addition to their narrow-banded emission characteristics (FWHM optical bandwidths of 2-5 nm), were available over a wide range of optical center frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The first practical room temperature non-pulsatile (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', continuous-wave or CW) VCSEL was reported by Koyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' in 1988 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Development of these tiny optical sources accelerated in the late 1990s and early 2000s by DARPA funding [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Our intent was to pair a small number of frequency-selected VCSELs with equally tiny solid-state photodetectors, either avalanche photodiodes (APDs) or P-I-N photodiodes (PIN diodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' APDs exhibit more signal gain than PIN diodes, but also have more intrinsic noise, so we concentrated on PIN diodes for our application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' PIN diodes can be selected to cover a range of optical wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' By combining narrow-spectrum VCSELs with PIN diodes having wide wavelength sensitivity, we could allow the light from VCSELS of different wavelengths to impinge on a single PIN diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' If in addition the light output from each VCSEL was modulated with a unique on-off keyed (OOK) pseudo-random pulsatile sequence (code division, Sig/Noise gain), and the PIN diode were integrated with an on-unit multi-channel correlator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', in a small microcontroller or a custom correlator chip), accurate through-the-skin measurement of several analytes of clinical importance could be accomplished, simultaneously, in a small form factor, unlike in conventional pulse oximetry, where the measurements from the different LEDs must be made sequentially (thus introducing time skew between the two measured values in each pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The VCSELs’ narrow emission lines indicated that considerable analyte measurement specificity could be achieved, far better than the LED-based wire-tethered fingertip pulse oximeters, based on Aoyogi’s implementation [37-39], that entered hospital use in the early 2000s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' However, to select the appropriate VCSEL wavelengths, we needed more frequency- accurate data than was identifiable in the published literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' It was this need for improved wavelength resolution, a larger wavelength measurement span, and additional wavelength measurement parameters that led to our decision to design and fabricate a spectrophotometer Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 7 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 with extended functionality, as discussed below and in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' With this type of higher resolution and more comprehensive data available, it appeared feasible to design a wearable analyte sensor with considerably extended in vivo measurement capabilities compared to the then-commercial offerings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We undertook that effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The goal of achieving an extended-capability autonomous on-body analyte measurement unit, underpinned by the historical evolution of such a capability as described above, represents the end goal of the entire sequence of projects described here and in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Because we did not wish to be bound to through-the-finger transmission measurements, we investigated an alternate approach, referred to as “reflection oximetry”, in which light is reflected from underlying bone, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', at the forehead, back to the photodetectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The requirement for underlying bone also constrained placement options for the body-worn system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Therefore, we investigated a third approach, “scatter” oximetry, in which photons from the optical sources are directed into the skin and are sensed by one or more photodetectors placed several cm from the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Scattered photons entering the inputs of the photodetectors acquire and carry with them the optical information required to calculate the concentrations of the analytes of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Using scatter oximetry, the battery-powered measurement device can be placed on a wrist, on an arm, or on the torso, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', less intrusively than on a finger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' APPROACHES FOR SELECTING THE OPTIMUM MEASUREMENT WAVELENGTHS FOR ON-BODY OXIMETRY Next, with this conceptual optical measurement chain sketched out, we needed a method to select the optimum wavelengths to measure analyte concentrations in vivo, so that the correct VCSELs could be incorporated into body-worn units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' To address this problem one of us (CRH) developed a robust algorithm [45, 46] for the selection of optimal excitation wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Using the wavelengths selected by the algorithm allowed for the measurement of relative and absolute concentrations of a set of analytes in a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The Beer-Lambert molar extinction coefficients of homogeneous materials can be measured for selected frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Measurements in “the wild,” however, need to incorporate many more factors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' heterogeneous paths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' reflection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' and the propagation of potential error from the inputs, through the measurement system, and continuing to a consideration of the variations and non-linearities of receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' This system-level approach required more accurate and higher-resolution measurements of analyte characteristics, including diffusion, reflection, mean-free-path variation, florescence, phosphorescence, and various anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Our view of the importance of this information was informed by the guidance provided by [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' However, we did not have this data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Thus, we instituted a literature search for absorption curves for various forms of hemoglobin over a wide range of wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Such curves were first published in 1987 by Barker and Tremper [47], and again in 1989 by Tremper and Barker [48] (the authors credit Susan Manson, Biox/Ohmeda as their source of this data, as do many subsequent authors, which is widely accepted in the field), and are reproduced and duplicated in both the left and right panels of Figure 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Figure 7 is from the same published sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' (Note: these curves might or might not be “correct” in some absolute sense, but they were the data Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 8 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 available to us;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' “errors” in the shapes of these curves would of course have deleterious effects on the results that we present below but were unknowable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=') Next, we searched the published literature for a selection of wavelengths that would yield the most accurate measures of the concentrations of four forms of hemoglobin (COHb, MetHb, OxyHb, and DeoxyHb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Lamego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' [40] describes eight “optimum” wavelengths (610, 620, 630, 655, 700, 720, 800 and 905 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The wavelengths in the rightmost panel of Figure 4, illustrated as vertical lines, are similar though not identical to those documented in Lamego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The left panel of that figure depicts four wavelengths calculated from the algorithm of [45, 46] also provides a means of computing a quality factor, referred to as the “condition number” for a selected set of wavelengths, in which a lower number is “better”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' A condition number for the eight wavelengths in the right panel of Figure 4, taken from [40] as the outcome of a literature search, was calculated as 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='2, whereas the condition number for the four wavelengths in the left panel of that figure was calculated to be 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', a better condition number for the use of fewer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The algorithm of [45, 46] teaches the following conditions, all of which must be satisfied simultaneously as best possible: 1) each selected measurement wavelength should be maximally separated from other wavelengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 2) the measured curves should be maximally separated in the vertical, or extinction coefficient, axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 3) wavelengths should not be selected in regions where an extinction curve is “steep”, since slight differences in the shapes of the curves and/or in their left-to-right or vertical positions due to component variations or other differences can introduce errors in the final resulting measurements (this constraint is often difficult to obey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' In the rightmost panel, the wavelengths 610, 620, and 630 nm are “too close together”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The wavelengths at 660, 730, 805 and 905 nm are “good” choices, according to our algorithm, although the overall condition number is degraded by the wavelengths at 610, 620, and 630 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The four wavelengths selected by our algorithm, in the left panel of Figure 4, are in partial violation of the ground rules described above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' the wavelengths are constrained to some extent by the shapes of the curves themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' In addition, there is a fifth curve, labeled “Penalty”, which we incorporated to account for the fact that some VCSELs are more difficult and costly to manufacture than others;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' the shape of this curve changes over time as manufacturing processes evolve [courtesy of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Hibbs-Brenner, Vixar Corporation, ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Were the penalty curve not incorporated into the calculations, the selection of wavelengths would have been somewhat different, and might yield results which obey the above-described guidelines more closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Finally, in Figure 4, note the addition of small, bell-shaped curves at the bottom of the rightmost panel, which presents the wavelengths published by a commercial vendor [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' These bell-shaped curves illustrate the wavelength spread generated by LEDs on either side of their center frequencies, which, as commented above, can be 20-70 nm (FWHM) wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The LEDs thus generate wavelength components which overlap one another, thereby degrading the measured signal-to-noise ratios at the photodetector circuitry, because each light source will be carrying “information” that optimally would be out-of-band for those light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' VCSELs generating FWHM wavelengths only a few nm in width will not create or will at least minimize these artifactual components, thereby motivating their use in place of LEDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 9 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 Figure 4: Oxyhemoglobin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Deoxyhemoglobin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Carboxyhemoglobin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' and methemoglobin extinction coefficients as a function of wavelength (four algorithmically selected maximally linearly independent wavelengths in the left panel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' versus eight wavelengths selected by a commercial vendor in the right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Narrow-band VCSEL sources are assumed, though commercial vendors typically employ LEDs [40], whose wider bandwidths and overlaps are illustrated by the bell-shaped curves at the bottom of the rightmost panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Condition numbers: left panel: 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' right panel: 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' (42003) As is clear from this chart, the optical wavelengths at which meaningful hemoglobin measurements can be performed are in the range of approximately 450 nm to 1000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' As will be noted below, several other analytes can also be measured using a similar implementation, but the wavelength ranges needed to be extended down to 200 nm and up to 2000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' THE NEED FOR, AND OUR APPROACH TO OBTAINING, AN IMPROVED SPECTROPHOTOMETER The results described above demonstrated that with good continuous optical absorption data over a broad wavelength span, the algorithm described here could select the optimum wavelengths, and hence the correct VCSELs, to detect one or more naturally occurring analytes present in sufficient concentration in the blood of an individual as measured by an autonomous battery- operated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', untethered!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=') body-worn unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We also recognized that the more optical parameters from laboratory samples that we had, the more specifically we could select the optimum VCSEL wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' To gather the requisite data, we needed a spectrophotometer capable of measuring many optical parameters simultaneously, in a wide variety of physiological samples (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', whole blood, plasma, lymph, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' A review of available commercial spectrophotometers confirmed that such a machine did not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The available models for purchase measured only two parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', transmission and backscatter, and were lacking most of the desired operational features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We also conducted a Algorithmically-Selected Wavelengths Wavelengths Used by Commercial Vendor OxyHB 610 OxyHB 10 DeoxyHB DeoxyHB COHB 620 COHB MetHB MetHB Penalty 730 0 10 cm mM-1 805 700 857 905 673 630 2 10 650 611 600 700 800 900 1000 600 700 800 900 1000 (A) (B) Wavelength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' nm Wavelength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' nmMeasuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 10 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 literature search against the possibility that other research groups had developed very capable spectrophotometers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' from which they might have published high accuracy absorption curves of one or more naturally occurring analytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We discovered developments, both theoretically and in some cases through the implementation of actual hardware [49-52] that addressed some of the parameters that appeared to us to be important, but none that were sensitive to as many parameters as our studies indicated might be physically implementable and clinically relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' In several of those previously published papers, descriptions of the characteristics of those machines were limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Therefore, we elected to design and fabricate an optical instrument capable of measuring more of these parameters, over a wide spectral range, in very short durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Our intent was to create a set of baseline data that would enable the next step, that of designing the desired autonomous body-worn units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Following a brief introduction to the spectrophotometer developed here, we will return to a discussion of the value of the extended measurement parameters that we believed to be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The Mayo Double-Integrating Sphere Spectrophotometer (MDISS): Lessons-Learned Regarding the Selection of Optical Wavelengths for Body-Worn Analyte Measurement Units Only a summary of the features of the MDISS is presented here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' a complete description appears in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' From a functional perspective, we consider the MDISS to be a multi-generational successor to the Beckman DU spectrophotometer [53-55];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' also, we wished to extend the capabilities of the experimental machines previously referenced with a combination of quantitative functionality features including: a broad wavelength measurement span (190-2750 nm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' high wavelength accuracy and high wavelength precision (FWHM on the order of 1 nm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' fine wavelength resolution (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='1-5 nm wavelength step sizes);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' high amplitude sensitivity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' a rapid- throughput automated measurement capability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' simultaneous acquisition of diffuse reflected (DR), diffuse transmitted (DT), and unscattered transmitted (UT) energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' input energy monitoring for high accuracy energy values on a pulse by pulse basis (with options for fluorescent and opto-acoustic acquisition);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' and the flexibility to accommodate a wide variety sample holder types and volumes, including high-pressure (up to 30 atmosphere [450 psi]) cells for measurements of blood oxygen saturation characteristics in hyperbaric environments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We also attempted to mitigate potential sources of measurement error of parameters, as will be noted below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Figure 5 is a photo of an early version of the unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 11 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 Figure 5: Double-integrating sphere spectrophotometer system (narrow-linewidth pump laser coupled to a tunable optical parametric oscillator, allowing precise optical characterization of biological analytes over a 192-2750 nm wavelength span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' (44414) Figure 6: A set of spectroscopic transmission curves measured with the MDISS at each of six oxygen saturation levels, at wavelength separations of 2-nm, over a range of 800 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The desaturation gas in this example was carbon monoxide (note the isosbestic point at 650 nm, rather than at 800 nm if the desaturation gas had been carbon dioxide).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' (46584) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' MDISS Design Tradeoffs The design of the MDISS system required a recognition of the various cost tradeoffs in terms of money, time (development time and actual test time), and specific performance parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' A single example of a specific performance parameter trade-off was prioritizing wavelength span rather than absolute energy sensitivity of the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' This selection was accepted to address the numerous unknowns regarding the analyte characteristics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' since we had no prior knowledge BeamPath(ShowninOrange) Fixed Turning Mirror 401-2750nm Neodymium-DopedYttrium Transmitted Light OutputPort AluminumGarnet (Nd:YAG) Pump (ForwardScattered) Energy Meters (4) Input Energy Laser,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='withHarmonicGenerators CollectionSphere Detector 192-400nm (1064nm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='532nm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='355nm) (EnergyDetectorOnTop) OutputPort Optical Parametric Oscillator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='oPo (192-2750nm) Sample Thickness System Display Control Computer Switchable Turning Mirror (Selects DesiredOPO Output Port) Detector for Unscattered Light Aperture to Filter Unwanted 4By10Foot ReflectedLight Light Air Table (BackwardScattered) CollectionSphere (Energy Detector On Top) Input Energy Beamsplitter PeristalticPump AdjustableBiological (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='5%/99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='5%) SampleReservoir (ToCirculateSample) AnalyteSampleHolder (1-10mmThickness)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='40 Carbon Monoxide 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='30 (Lo) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='20 lized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='06 Oxygenation Saturation in % rma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='0 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='04 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='7 Pul 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='0 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='02 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='0 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 Wavelength (nm)Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 12 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 of those subregions in the entire wavelength span in which a given analyte would display maximum amplitude differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' This requirement for sensors with wide optical bandwidths justified the use of pyroelectric sensors, which have broad wavelengths spanning 190–12,000 nm, and the use of an optical parametric oscillator (OPO) which produced energy from 192 to 2750 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' If it had been determined that, even with the broadband capability, there was a need for more sensitivity, then alternate detectors could have been implemented, albeit at the expense of reduced bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' As was discovered during the blood hemoglobin and oxygenation characterizations with different desaturation gases, most of the variations that we needed to detect occurred in the 600-1200 nm span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' In a mutually reinforcing fashion, the underlying theory was subsequently corroborated by the measured results from the MDISS when it became operational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Without the machine, which was designed and constructed under the assumption that the guidance gleaned from [45, 46] was correct, many if not most of the results demonstrated by the machine to are valid, and necessary for the design of accurate body-worn analyte measurement units, would have been unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Noninvasive optical sensing can be used with other clinically important physiological and biochemical variables besides hemoglobin species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' As one example, Figure 7 displays absorption curves for blood glucose, blood protein, and blood lipids in the near-IR range of 1350 nm to 1850 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' As in Figure 4, these waveforms were identified and employed following a search of the open literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' An absorption curve for water is provided for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' With the appropriate sensing wavelengths as determined by the algorithm and indicated by the vertical lines in this figure, detection and measurements of glucose, lipids, proteins, and even a measure of either total body water or central blood volume could be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Measuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 13 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 Figure 7: Water, protein, and glucose extinction coefficients as a function of wavelength, assuming ideal 1-nm bandwidth light sources using maximally independent wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Condition number: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' (41964) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' THE LONG-TERM GOAL OF THIS EFFORT The long-term goal of this multiple-step project was the development of a sufficient technical base so that small, battery-powered, microcontroller-enhanced body-worn units could be designed to measure and record, with high accuracy and in real-time, blood oxygen saturation, blood levels of carbon monoxide, and concentrations of other clinically relevant analytes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The first developmental stage was the MDISS instrument, which was to yield the type of clinically actionable data described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' A next step, described in [55], was the prototyping of the battery-powered electrical and optical components to conduct these measurements continually, in a sufficiently small form factor that they could be incorporated into versions of the body-worn units depicted in Figure 1, that could be fielded into the clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' although we did not pursue this path,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' commercial versions of the MDISS system could be used as research Water Glucose Protein Lipid 101 Extinction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 1679 nm 1707 nm 100 1371nm 1597nm 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 Wavelength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' nmMeasuring Physical and Electrical Parameters in Free-Living Subjects: 22 December 2022 Page 14 of 18 Motivating an Instrument to Characterize Analytes of Clinical Importance in Blood Samples Report# Mayo-SPPDG-R22-15-V01 instruments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' or, reduced-wavelength-span and/or lower cost versions targeted for a subset of analytes could be used for higher throughput routine clinical diagnostic purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' SUMMARY We have described the evolution of consumer-grade body-worn physiological measurement units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We then introduced an evolutionary thread from early work in the development of research-grade body-worn blood oxygen saturation units conducted in the first half of the 1940s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Next, we reviewed our recognition in the 2010s of the requirements for higher-quality laboratory measurements of the optical characteristics of medically relevant blood analytes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=', oxygen saturation-versus-wavelength behavior of critical blood analytes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We extended that line of investigation with the design of a spectrophotometer, the MDISS, with the needed sensitivity and specificity to help us gather new optically based blood characterization data [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' We also initiated efforts to use the results reported here and in [1] to develop a new-technology body worn unit that operates on a somewhat different principle from classical pulse oximetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Although body-worn units were our major end goal, we have also underscored the benefits of using optical analyte data over a wide wavelength range with high wavelength resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' An instrument with capabilities beyond those available is required;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' these new data support a future effort to simplify and optimize body-worn monitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Disclosures The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Although various patents cover various aspects of this work, none of these patents are licensed for gain or profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Acknowledgements We wish to acknowledge the years-long contributions of the following individuals to this project: Charles Burfield, Anthony Ebert, Theresa Funk, Nicholas Klitzke, Steven Polzer, Jason Prairie, and Steven Schuster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' and Drs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Franklyn Cockerill, Kendall Cradic, Graham Cameron, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' Rolland Dickson, Stefan Grebe, Ravinder Singh, and Nathan Harff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' The clinical materials used in the study were produced by the Mayo Clinic Department of Laboratory Medicine and Pathology and processed by them for us using their standard clinical laboratory tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content=' References [1] Schwab, DJ, CR Haider, G Delp, SK Grebe, and BK Gilbert, “An Experimental Double- Integrating Sphere Spectrophotometer for in Vitro Optical Analysis of 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='20190801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} +page_content='03' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9AzT4oBgHgl3EQfBPok/content/2301.00938v1.pdf'} diff --git a/rNE4T4oBgHgl3EQfwA2r/content/tmp_files/2301.05247v1.pdf.txt b/rNE4T4oBgHgl3EQfwA2r/content/tmp_files/2301.05247v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ecd5644cd100a1dd95d6519a1e100f11888db97e --- /dev/null +++ b/rNE4T4oBgHgl3EQfwA2r/content/tmp_files/2301.05247v1.pdf.txt @@ -0,0 +1,1689 @@ +Draft version January 16, 2023 +Typeset using LATEX twocolumn style in AASTeX63 +Co-evolution of Dust and Chemistry in Galaxy Simulations with a Resolved Interstellar Medium +Chia-Yu Hu (胡家瑜 ),1, 2 Amiel Sternberg,3, 4, 1 and Ewine F. van Dishoeck1, 5 +1Max-Planck-Institut f¨ur Extraterrestrische Physik, Giessenbachstrasse 1, D-85748 Garching, Germany +2Department of Astronomy, University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611, USA +3School of Physics & Astronomy, Tel Aviv University, Ramat Aviv 69978, Israel +4Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA +5Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, the Netherlands +ABSTRACT +Nearby dwarf irregular galaxies are ideal laboratories for studying the interstellar medium (ISM) at +low metallicity, which is expected to be common for galaxies at very high redshift that will be observed +by the James Webb Space Telescope. We present the first high-resolution (∼ 0.2 pc) hydrodynam- +ical simulations of an isolated low-metallicity (0.1 Z⊙) dwarf galaxy coupled with a time-dependent +chemistry network and a dust evolution model where dust is locally produced and destroyed by var- +ious processes. To accurately model carbon monoxide (CO), we post-process the simulations with +a detailed chemistry network including the time-dependent effect of molecular hydrogen (H2). Our +model successfully reproduces the observed star formation rate and CO(1–0) luminosity (LCO). We +find that dust growth in dense gas is required to reproduce the observed LCO as otherwise CO would +be completely photodissociated. +In contrast, the H2 abundance is extremely small and is insensi- +tive to dust growth, leading to a CO-to-H2 conversion factor similar to the Milky Way value despite +the low metallicity. Observationally inferred dust-to-gas ratio is thus underestimated if adopting the +metallicity-dependent CO-to-H2 conversion factor. The newly-produced dust in dense gas mixes with +the ISM through supernova feedback without being completely destroyed by sputtering, which leads to +galactic outflows 20% – 50% dustier than the ISM, providing a possible source for intergalactic dust. +Keywords: Interstellar medium (847); Astrochemistry (75); Hydrodynamical simulations (767) +1. INTRODUCTION +The formation and evolution of galaxies are critically +controlled by how stars form and how they affect the +gas cycle in and around galaxies via stellar feedback +(Somerville & Dav´e 2015; Naab & Ostriker 2017). Over +the last decade, the cold, star-forming gas (dominated +by molecular hydrogen, H2) in galaxies from the lo- +cal Universe to “cosmic noon” at redshift z ∼ 2 has +been systematically quantified by submillimeter and far- +infrared (FIR) telescopes, leading to a physical picture +where galaxies grow primarily by gas accretion onto the +rotationally-supported disks, which fuels star formation +in their interstellar medium (ISM) regulated by feed- +back across cosmic time (see Tacconi et al. 2020 and +references therein). The molecular gas mass inferred by +Corresponding author: Chia-Yu Hu +cyhu.astro@gmail.com +dust-based methods has found to be broadly consistent +with the conventional method based on carbon monox- +ide (CO), strengthening the robustness of the results. +However, while significant progress has been made +in understanding the evolution of star-forming gas in +galaxies of solar or slightly sub-solar metallicity, little +is known in galaxies of low metallicity (Z ≲ 0.1Z⊙). In +the era of James Webb Space Telescope (JWST), a deep +understanding of the ISM chemistry at low metallicity +is urgently needed as we begin to observe galaxies in the +very early Universe. Indeed, Curtis-Lake et al. (2022) +recently reported four galaxies at extremely high red- +shift (z ∼ 10 − 13) discovered by JWST, all of which +have metallicity of Z < 0.1Z⊙. +On the other hand, nearby dwarf irregular galaxies +provide a unique laboratory to study the chemical prop- +erties and observational signatures of the star-forming +gas at comparably low metallicity in great detail thanks +to their proximity, even though their galaxy proper- +ties (such as mass, size, surface density, etc.) may be +arXiv:2301.05247v1 [astro-ph.GA] 12 Jan 2023 + +2 +Hu et al. +different from their high-redshift counterparts. It has +long been recognized that CO emission in these galax- +ies tends to be extremely faint and becomes undetected +when the metallicity drops below 0.2Z⊙ (Leroy et al. +2005; Schruba et al. 2012; Madden et al. 2020). This +has changed with the advent of the Atacama Large Mil- +limeter/submillimeter Array (ALMA) thanks to its ex- +tremely high sensitivity. Rubio et al. (2015) observed +the Wolf-Lundmark-Melotte (WLM) dwarf galaxy and +detected the first CO emission at a metallicity of 0.1Z⊙. +However, the molecular gas mass still cannot be robustly +determined due to the highly uncertain CO-to-H2 con- +version factor. Similarly, the dust-based method suffers +from the uncertainty in the assumed dust-to-gas ratio +(DGR) which has been shown to scale super-linearly +with metallicity in this regime, but the exact scaling +relation is still uncertain (see, e.g., R´emy-Ruyer et al. +2014; De Vis et al. 2019). In fact, the detection of CO +in the WLM galaxy is rather surprising given the ex- +tremely low DGR expected at this metallicity. +Recent hydrodynamical simulations coupled with +time-dependent chemistry have achieved the required +numerical resolution of a few parsecs (pc) to directly +resolve feedback from individual supernova (SN) explo- +sions in isolated dwarf galaxies (Hu et al. 2016, 2017; +Hu 2019; Emerick et al. 2019; Lah´en et al. 2020; His- +lop et al. 2022; Whitworth et al. 2022b,a; Steinwandel +et al. 2022a,b; Katz et al. 2022; Lah´en et al. 2022). This +is an important milestone as it avoids the need of sub- +grid prescriptions for SN feedback which is one of the +major uncertainties in cosmological simulations. Coin- +cidentally, H2 can be resolved at a similar resolution +(Gong et al. 2018), at least at solar metallicity. How- +ever, proper modeling of CO requires a significantly +higher resolution of ∼ 0.1 pc (Seifried et al. 2017; Hu +et al. 2021) as CO typically exists in dense, spatially +compact gas. Lagrangian codes are therefore particu- +larly suitable for such a task thanks to their built-in +adaptive spatial resolution that can more easily reach +sub-pc scales. Meanwhile, the carbon network responsi- +ble for CO formation is much larger than the hydrogen +network (Sternberg & Dalgarno 1995), leading to a sig- +nificant computational overhead if coupled with simu- +lations, compromising the achievable resolution. To ad- +dress this dilemma, Hu et al. (2021) introduced a hybrid +approach where the hydrogen network is solved on-the- +fly to capture the time-dependent (non-steady-state) ef- +fect of H2 while an accurate chemistry network including +carbon chemistry is solved in post-processing. +Dust plays an important role in ISM chemistry. The +surfaces of dust grains are the main sites where H2 for- +mation occurs, which then initiates the formation of +other important molecules such as CO. Furthermore, +dust provides shielding against the Lyman-Werner ra- +diation that can photodissociate both H2 and CO. Ob- +servations of the Small and Large Magellanic Clouds +(SMC and LMC) have demonstrated that the spatially- +resolved DGR can vary by almost an order of magnitude +from region to region in low-metallicity galaxies, indicat- +ing significant dust evolution (Roman-Duval et al. 2017, +2022). However, simulations that resolves the ISM to +date have all assumed a constant DGR in both space +and time, which is an over-simplification. +Cosmological simulations and isolated galaxy simula- +tions with comparable resolution have started to include +dust evolution models at different levels of sophistication +(McKinnon et al. 2017, 2018; Aoyama et al. 2017, 2018; +Li et al. 2019; Parente et al. 2022; Lewis et al. 2022; Ro- +mano et al. 2022; Lower et al. 2022). However, these sim- +ulations do not resolve either SN feedback or the density +structure of the ISM, forcing them to adopt dust evo- +lution models in a sub-grid fashion. For example, dust +destruction by SNe is generally based on results from +1D calculations in plane-parallel shocks, which does not +apply for more complex situations such as inhomoge- +neous ISM or clustered SNe. While this simplification +is perhaps justified as SN feedback in these simulations +is unresolved anyway, a more accurate model is clearly +desirable in resolved simulations. Hu et al. (2019) intro- +duced a numerical method to directly simulate thermal +and nonthermal sputtering of dust designed for simula- +tions with pc- or sub-pc resolutions. However, the model +did not include any dust production processes, nor was +it coupled with chemistry. +In this work, we extend the dust evolution model in +Hu et al. (2019) by including dust growth in cold gas +and dust production in asymptotic giant branch (AGB) +stars. We couple this with a time-dependent chemistry +network and the ISM model developed in Hu et al. (2016, +2017, 2021) and perform hydrodynamical simulations of +an isolated dwarf galaxy similar to the WLM galaxy at +a mass resolution of 1 M⊙ (spatial resolution ∼ 0.2 pc). +To our knowledge, this is the first ISM-resolved simula- +tion coupled with chemistry and dust evolution (but see +Romano et al. 2022 for a model for significantly coarser +resolutions). This paper is organized as follows. In Sec- +tion 2, we describe our numerical model. In Section 3, +we study how dust evolution affects the ISM chemistry, +how the DGR is distributed in the ISM and galactic +outflows, and how the projected DGR varies with the +gas surface density at different telescope beam sizes. In +Section 4, we discuss the implications of our results for +the observationally inferred DGR and the intergalactic +dust. In Section 5, we summarize our work. + +Dust and chemistry co-evolution in dwarf galaxies +3 +2. NUMERICAL METHODS +2.1. Gravity and hydrodynamics +We use the public version of Gizmo (Hopkins 2015), a +multi-solver code for hydrodynamics that is built on the +massively parallel TreeSPH code Gadget-3 (Springel +2005). We adopt its meshless finite-mass (MFM) solver +for hydrodynamics (Hopkins 2015) which is a variation +of the meshless Godunov method (Gaburov & Nita- +dori 2011). Gravity is calculated using the Barnes–Hut +method (“treecode”). +2.2. The ISM model +In this section, we summarize the physical processes +in the ISM in our simulations excluding dust evolution, +which will be described in the Sec. 2.3. The methods and +implementations are largely based on Hu et al. (2016, +2017, 2021) where more details can be found. +2.2.1. Time-dependent cooling and chemistry +We adopt a time-dependent chemistry network devel- +oped in Glover & Mac Low (2007) and Glover & Clark +(2012a) that is widely used in ISM simulations. +The +abundances of H2, H+, H i, and the free electron frac- +tion are integrated based on the chemistry reactions in +the network. The hydrogen network includes H2 forma- +tion on dust, H2 destruction by photodissociation, colli- +sional dissociation and cosmic ray ionization, and recom- +bination in the gas phase and on dust grains. Individ- +ual cooling and heating processes are calculated based +on the time-dependent chemical abundances, which in- +cludes cooling from fine structure metal lines, molecu- +lar lines, Lyman alpha, H2 collisional dissociation, colli- +sional ionization of H, and recombination of H+ in the +gas phase and on grains. Heating includes photoelectric +effect, cosmic ray ionization, H2 photodissociation, UV +pumping of H2 and the formation of H2. Following Clark +et al. (2012), shielding against FUV radiation uses the +HEALPix algorithm (G´orski & Hivon 2011) in combi- +nation with the “treecode” approximation to integrate +the relevant column densities along 12 sightlines up to +a pre-defined radius of 100 pc. +2.2.2. Star formation +We adopt the stochastic star formation recipe com- +monly used in the field of galaxy formation. +A gas +particle eligible for star formation is converted into a +star particle of the same mass on a timescale of tff/ϵsf +stochastically, where ϵsf is the star formation efficiency +and tff = +� +3π/(32Gρ) is the gas free-fall time where +G is the gravitational constant and ρ is the gas den- +sity. We adopt ϵsf = 0.5 in this work. Such a high effi- +ciency is justified by our resolution as we can follow the +gravitational collapse down to the scales of individual +molecular cores. Gas is eligible for star formation when +its local Jeans mass MJ = (π2.5c3 +s)/(6G1.5ρ0.5), drops +below the kernel mass Mker = Nngbmg, where cs is the +sound speed, mg is the gas particle mass, and Nngb = 32 +is the number of neighboring particles in a kernel. +2.2.3. Sampling individual stars from an IMF +Massive stars (initial mass > 8 M⊙) inject energy and +momentum into their surrounding gas commonly termed +as “stellar feedback”. +At our resolution of 1 M⊙ per +star particle, it is unphysical to assume that each parti- +cle represents a star cluster with a fully sampled stellar +initial mass function (IMF), as is commonly the case +in cosmological simulations with much coarser resolu- +tions. Following Hu et al. (2021), we adopt the tech- +nique of “importance sampling” to stochastically sam- +ple stellar masses from a Kroupa IMF (Kroupa 2001). +The sampled stellar masses are used to determine the +stellar lifetime (Ekstr¨om et al. 2012) and UV luminos- +ity from the BaSeL stellar library (Lejeune et al. 1997, +1998) and they do not affect the dynamics of gravity, as +the gravitational mass of the star particles (m∗) remains +unchanged. +2.2.4. Stellar feedback +We include stellar feedback from supernovae (SNe) +and photoionization. +SN feedback is done by inject- +ing thermal energy of 1051 erg per SN into its near- +est Nngb gas particles in a kernel-weighted fashion. As +our resolution is able to resolve the Sedov-Taylor phase +in each SN event, a simple thermal feedback is able +to achieve numerical convergence (Hu 2019). Feedback +from photoionization follows Hu et al. (2017), where +each massive star searches for its ionization front it- +eratively by balancing recombination and photoioniza- +tion and heats up the interior gas to 104 K. This ap- +proach reproduces the dynamics of an expanding H ii +region predicted by radiative transfer codes in a uni- +form medium. More importantly, it captures the cor- +rect behaviors in overlapping H ii regions where a naive +Str¨omgren-sphere method would suffer from the numer- +ical artifact of double-counting. +2.2.5. Spatially variable FUV radiation +Following Hu et al. (2017), the unattenuated FUV ra- +diation field is both spatially and temporally variable +and is calculated directly from the star particles. For +a given gas particle, every star particle contributes a +radiation flux of LFUV/(4πr2) where LFUV is the FUV +luminosity based on the sampled stellar mass and r is +the distance between the gas and star particle. +The +summation is over all star particles and is done via the + +4 +Hu et al. +“treecode” approximation to avoid the O(n2) operation +and speed up the calculation. The FUV radiation af- +fects both the thermal balance via photoelectric heating +and the chemistry via photodissociation. +2.3. Dust evolution model +We adopt the “one-fluid” approach where dust is as- +sumed to be spatially coupled with the gas. +This is +justified as dust is expected to be charged and gyrates +around the magnetic fields in the ISM with a small gyro- +radius. Each gas particle is associated with a dust mass +md = md(Sil) + md(C), where md(Sil) and md(C) are, +respectively, the masses of silicate dust and carbona- +ceous dust which evolve separately in the simulations +due to the production and destruction processes as we +will describe below. +Dust grains are assumed to be +spherical with a radius of a and a material density of sd, +which leads to a grain mass of mgr = (4πa3/3)sd. The +formation or destruction rate of dust can be expressed +as +dmd +dt += Ngr +dmgr +dt += 3 ˙a +amd +(1) +where Ngr = md/mgr is number of dust grains in a gas +cell. Time integration is done via sub-cycling in order +to resolve the timescales of dust dynamics and sputter- +ing which can be orders of magnitude smaller than the +hydrodynamical timesteps. +We include physical processes that directly modify +md: +sputtering, dust growth, and dust formation in +AGB ejecta. Processes that modify the grain size while +keeping md fixed, such as shattering and coagulation, +are not included. This is because we are only interested +in the evolution of dust mass rather than other dust +properties such as the extinction law. We adopt a fixed +grain-size distribution that follows Mathis et al. (1977) +(the “MRN” distribution). +2.3.1. Sputtering +In shocks or in hot gas, dust can be destroyed via +sputtering which returns metals locked up in dust grains +back to the ISM. We adopt the sputtering model in Hu +et al. (2019) that includes thermal and nonthermal sput- +tering, which we brief summarize as follow. The dust +destruction rate due to sputtering can be expressed as +dmd +dt +��� +sput = − md +tsput +(2) +where +tsput ≡ +a +3nYtot += 10 kyr +� +a +0.03µm +�� +n +cm−3 +�−1� +106 Ytot +µm yr−1cm3 +�−1 +, +(3) +where n is the hydrogen number density and Ytot is the +erosion rate that includes thermal and nonthermal sput- +tering, which we adopt from Nozawa et al. (2006). The +thermal erosion rate is a function of gas temperature +while the nonthermal erosion rate is a function of the +relative bulk velocity between dust and gas, which we +obtain by integrating the equation of motion for dust ac- +counting for direct collision, plasma drag, and betatron +acceleration. +2.3.2. Dust growth +Dust can grow in the cold gas when gas-phase metals +interact with dust and stick onto the surfaces of dust +grains, which can be viewed as the reverse process of +sputtering. The exact mechanism is still poorly under- +stood, though its feasibility has been supported by lab- +oratory experiments (Krasnokutski et al. 2014; Henning +et al. 2018; Rouill´e et al. 2020). The dust production +rate due to dust growth can be expressed as +dmd +dt +��� +grow = (1 − f) md +tgrow +(4) +where tgrow is the dust growth timescale (see below) and +f is the fraction of metals locked in dust grains. For +element A (where A = Si or C), this can be written as +fA = mA,d +mA,tot += md(A)ξA +mgXAZ′ , +(5) +where mA,d is the mass of element A in the dust phase, +mA,tot is the total mass of element A (dust + gas), XA +is the solar abundance of element A, and ξA is the mass +fraction of element A in the assumed grain material. For +carbonaceous dust, ξC = 1. For silicate dust, we adopt +MgFeSiO4 as the grain material which leads to ξSi = +0.165. +The dust growth timescale takes the following +form +tgrow = 1.5 Gyr +� +n +cm−3 +�−1� +T +100K +�−0.5 +� +a3 +0.03µm +� +(Z′αsDeff)−1. +(6) +Here, αs is the sticking coefficient, a3 ≡ ⟨a3⟩/⟨a2⟩ is +the average grain size where the bracket ⟨. . .⟩ refers +to integration over the grain size distribution, and +Deff ≡ ⟨a2D(a)⟩/⟨a2⟩ is the enhancement factor D(a) +due to Coulomb focusing (Weingartner & Draine 1999) +weighted by the surface area of grains. The sticking co- +efficient encompasses the complex physical and chemical +processes on the surfaces of grains which is still uncer- +tain (see, e.g., Zhukovska et al. 2018 for a discussion). +To first order, it is expected to be close to unity at low +temperatures and decrease substantially at high temper- +atures. We follow Zhukovska et al. (2016) and assume + +Dust and chemistry co-evolution in dwarf galaxies +5 +that αs = 1 at T < 300 K while αs = 0 at T ≥ 300 K. +The area-weighted enhancement factor Deff is also un- +certain and is expected to depend on the a number of +properties such as density, temperature, free electron +fraction, grain size distribution, and grain charge. Wein- +gartner & Draine (1999) found that Coulomb focusing +shortens the growth timescale by more than an order +of magnitude. However, Priestley et al. (2021) found a +much weaker effect if the evolution of grain size is taken +into account. We adopt Deff = 10 for simplicity, but +noting that the uncertainties in both αs and Deff are +potential caveats of our model. +2.3.3. Dust production from AGB stars +We adopt the mass-dependent dust yields from +Zhukovska et al. (2008) at Z′ = 0.1 based on the individ- +ual stellar masses sampled from the IMF to account for +dust produced in the ejecta of AGB stars. The produced +dust mass is injected in the neighboring gas particles in +a kernel-weighted fashion. We do so only for carbona- +ceous dust as the silicate dust yields at this metallicity +is essentially zero. Dust produced from AGB stars is +expected to play a sub-dominant effect on the spatial +variation of the DGR as AGB stars are more uniformly +distributed in the ISM compared to sputtering in SN +shocks and dust growth in dense clouds that are highly +clustered. +2.4. Chemistry network in post-processing +In order to accurately model the transitions of +C+/C i/CO (the “carbon cycle”), a detailed carbon +chemistry is required, which is much more complicated +and computationally costly to solve compared to the hy- +drogen chemistry. We therefore post-process the simu- +lation snapshots using AstroChemistry1, a chemistry +network code developed in Hu et al. (2021) which con- +sists of 31 species: H, H−, H2, H+, H+ +2 , H+ +3 , e−, He, +He+, HeH+, C, C+, CO, HCO+, O, O+, OH, OH+, +H2O+, H3O+, H2O, O2, CO+, O+ +2 , CH2, CH+ +2 , CH, +CH+, CH+ +3 , Si+ and Si. All chemical reactions in the +UMIST database (McElroy et al. 2013) that exclusively +involve the above-mentioned species are included in the +network, which leads to 286 reactions in total. +The +time-dependent abundances of H2 and H+ in simula- +tions are taken as input parameters when solving the +network, which is of crucial importance as H2 can be +out of steady state significantly, especially at low metal- +licity. +This approach has been applied to ISM-patch +simulations which successfully reproduced the observed +relationship between the column densities of CO and +1 Available at https://github.com/huchiayu/AstroChemistry.jl. +H2 in Galactic clouds (Hu et al. 2021) as well as the +Milky Way CO-to-H2 conversion factor (XCO) (Hu et al. +2022a). +Hu et al. (2021) found that CO(1–0) is almost always +optically thin in their simulated ISM at Z′ = 0.1. This +is primarily a direct consequence of the low CO abun- +dance. Furthermore, CO(1–0) can remain optically thin +even at the highest column densities as the population +of CO is distributed over more excited levels. Assuming +optically thin conditions, the CO luminosity from each +gas particle can be expressed by +lCO = 0.5λ3 +10T10A10nCOf1V +(7) +where nCO is the CO number density, V = mg/ρ is +the volume of the gas particle, and λ10 = 0.26 cm, +T10 += +5.53 K, A10 += +7.2 × 10−8 s−1 are the +wavelength, energy level, and the Einstein A coeffi- +cient for the CO(1–0) line, respectively. +f1(Tex) = +3 exp(−5.53/Tex)/ +� +1 + (Tex/2.77)2 is the fraction of +CO in the level J = 1 (Draine 2011). Note that f1 only +varies within a factor of 2 in the range of 3 < Tex/K < 30 +where most CO exists. +2.5. Simulation setup +The initial conditions consist of a rotating disk +galaxy embedded in a dark matter halo with prop- +erties resembling the WLM galaxy, generated by the +MakeDiskGalaxy code (Springel 2005). The halo has +a virial radius Rvir = 45 kpc and a virial mass Mvir = +1010 M⊙, and it follows a Hernquist profile (Hernquist +1990) matching an NFW (Navarro et al. 1997) profile +at small radii with the concentration parameter c = 15 +and the spin parameter λ = 0.035. The baryonic mass +fraction is 0.8%, with a stellar disk of 107 M⊙ and a +gaseous disk of 7 × 107 M⊙, both following an exponen- +tial profile a with scale-length of 1 kpc. The central gas +surface density is Σgas ∼ 10 M⊙ pc−2. The stellar disk +follows an exponential vertical profile with a scale-height +of 1 kpc. The vertical density profile of the gaseous disk +is set up to maintain hydrostatic equilibrium. The ini- +tial gas temperature is set to be 104 K. The particle +mass for gas, stars, and dark matter are mg = 1 M⊙, +m∗ = 1 M⊙, and mdm = 103 M⊙, respectively. The cor- +responding spatial resolution for gas is ∼ 0.2 pc, defined +as the minimum kernel radius where the Jeans length +can be resolved (Hu et al. 2021). Such a high resolution +is needed in order to resolve the dense and compact cores +where CO is observed in the WLM galaxy. The gravi- +tational softening length is 0.2 pc for gas and 100 pc for +the dark matter. +The metallicity is Z′ ≡ Z/Z⊙ = 0.1 throughout +the simulations. The initial dust-to-gas ratio is set to + +6 +Hu et al. +Figure 1. Face-on images of the surface densities of H i, H2, and H+, gas temperature (T), projected DGR (Z′proj +d +), and FUV +radiation field (G0, in units of the Habing 1968 field) at simulation time t = 230 Myr. +be 1% of the Milky Way value, Z′ +d ≡ Zd/Zd,MW = +0.01, motivated by observations of low-metallicity galax- +ies (R´emy-Ruyer et al. 2014). +We adopt the Milky +Way dust abundances as Zd,MW(C) = 1.9 × 10−3 and +Zd,MW(Sil) = 3.5×10−3 (Dwek 2005) which corresponds +to a carbonaceous-to-silicate ratio ∼ 0.54 and a total +dust-to-gas ratio of Zd,MW = 5.4 × 10−3. +Galaxy scale simulations with a setup like ours are +known to undergo an artificial burst of star formation +during the initial collapse, which in turn leads to overly- +energetic SN feedback that blows out the entire gaseous +disk, substantially reducing the gas surface density. Fol- +lowing Hu et al. (2022b), we minimize this artifact by +first running a simulation without dust evolution for +100 Myr and with the SN delay time set to zero, which +reduces the dynamical impact of feedback due to sup- +pressed SN clustering. The simulation snapshot at the +end of this “pre-simulation” is used as the new initial +conditions with a relaxed configuration. +We run three simulations: (1) a fiducial model with +fully coupled chemistry and dust evolution, (2) a model +without dust evolution (i.e., Z′ +d = 0.01 throughout the +simulation), and (3) a model with dust evolution where +dust growth is switched off. Each simulation is run for +0.5 Gyr. +3. RESULTS +3.1. Overview +Fig. 1 shows the face-on images of the surface densities +of H i, H2, and H+, gas temperature, projected DGR, +and FUV radiation field at simulation time t = 230 Myr. +The ISM has a complex structure, with dense clouds +where star formation occurs and holes driven by stellar +feedback (“SN bubbles”). The majority of gas is in the +form of H i while H+ traces the young massive stars + +time = 230 Myr +0 +0 +log1oZHi [M。pc-2] +log10ZH2 [M。pc-2] +log1oZH+ [M。 pc-2] +2 +3 +4 +5 +6 +2.4-2.2-2.0-1.8-1.6 +-2 +-1 +0 +log1oT [K] +log10 Zd +log1o GoDust and chemistry co-evolution in dwarf galaxies +7 +WLM +WLM +aCO,MW +Figure 2. +Time evolution of the following global properties integrated over the simulated galaxy: the H2 mass (MH2, top +left), the star formation rate (SFR, top right), the luminosity of the CO(1–0) emission (LCO, bottom left), and the CO-to-H2 +conversion factor (αCO ≡ MH2/LCO, bottom right). The solid blue lines represent our fiducial run including dust evolution +while the dashed orange lines represent a controlled run without dust evolution. The red dotted lines indicate the observed LCO +and SFR in the WLM galaxy as well as the αCO in the Milky Way. Dust evolution strongly enhances LCO, but it has little +effect on MH2 and SFR. +photoionizing the ambient gas (the H ii regions). The +FUV radiation also traces young stars but it is more +spatially extended. The H2 abundance is extremely low +everywhere besides the densest part of clouds. These pc- +scale dense cores are also where CO exists (cf. Fig. 3). +Most of the gas is in the warm phase with a temperature +of T ∼ 104 K, while the hot gas (T ∼ 106 K) is found in +the interior of the SN bubbles. Cold gas with T ∼ 100 K +only exists in dense and compact clouds. +The projected DGR image demonstrates that DGR is +not spatially uniform. The DGR is elevated in dense +gas where dust growth is most efficient. Interestingly, +the DGR is not suppressed in the interior of SN bubbles +where dust is expected to be destroyed via sputtering. +Instead, the DGR seems to be elevated inside the SN +bubbles. More quantitative analysis will be provided in +Sec. 3.3. +3.2. Effect on ISM chemistry +Fig. 2 shows the following global quantities of the sim- +ulated galaxy as a function of time: the H2 mass (MH2, +top left), the star formation rate (SFR, top right), the +luminosity of the CO(1–0) emission (LCO, bottom left), +and the CO-to-H2 conversion factor (αCO ≡ MH2/LCO, +bottom right). +Two simulations are shown, one with +dust evolution (solid blue lines, our fiducial model) and +one without dust evolution where the DGR is constant +everywhere (dashed orange lines). +The WLM galaxy +has an observed CO luminosity of 1229 K km s−1 pc2 +(Rubio et al. 2015) and an observed SFR of 1.74 × +10−3 M⊙ yr−1 (Hunter et al. 2010), as indicated by the +red dotted lines. +The CO-to-H2 conversion factor in +the Milky Way αCO,MW = 3.2 M⊙ pc−2 (K km s−1)−1 +(excluding helium) is also overplotted as the red dotted +line. +Table 1 summarizes the median values of these +global quantities. +Our fiducial model successfully reproduces the ob- +served LCO and SFR in the WLM galaxy. On the other +hand, MH2 is extremely low and it contributes to a mass +fraction of ∼ 10−4 in the ISM. This is due to the long H2 +formation time compared to the dynamical time in the +highly turbulent ISM and is consistent with previous + +8 +Hu et al. +Table 1. Median values of global properties over 100 < t < 500 Myr. +model +SFR +MH2 +MCO +LCO +αCO +SFR/LCO +(1) +(2) +(3) +(4) +(5) +(6) +w/ dust evolution +1.59 × 10−3 +2435 +1.64 × 10−1 +469 +4.63 +3.41 × 10−6 +no dust evolution +2.02 × 10−3 +2095 +1.97 × 10−4 +0.72 +3156 +2.19 × 10−3 +Note— (1) Star formation rate [M⊙ yr−1]. (2) Total H2 mass [M⊙] (3) Total CO +mass [M⊙] (4) Total CO(1–0) luminosity [K km s−1 pc2] (5) CO-to-H2 conver- +sion factor αCO ≡ MH2/LCO [M⊙ pc−2 (K km s−1)−1] (6) Ratio of SFR/LCO +[M⊙ yr−1 pc−2 (K km s−1)−1] +simulations of isolated dwarf galaxies (Hu et al. 2016, +2017; Whitworth et al. 2022b). The low MH2 leads to a +conversion factor close to αCO,MW despite the low metal- +licity. Dust evolution has a substantial impact on CO +luminosity. Without dust evolution, LCO is about three +orders of magnitude lower than the observed value. On +the other hand, MH2 and the SFR are both insensitive +to dust evolution. +Our low αCO may seem to be in conflict with Hu et al. +(2022a) where the kpc-scale αCO was found to scales +with Z′−0.71, implying αCO ∼ 5αCO,MW at Z′ = 0.1. +This is because we adopt Z′ +d = 0.01 in this work (mo- +tivated by the observed super-linear metallicity–DGR +relation), which is ten times lower than what Hu et al. +(2022a) assumed. As a result, the molecular mass frac- +tion FH2 is also about ten times lower, which explains +the difference in αCO. +We now take a closer look at the local chemical abun- +dances and DGR as a function of hydrogen number den- +sity n as shown in Fig. 3. The top panels show the DGR +and abundances of H2 and H i while the bottom panels +show the abundances of C+, C i, and CO. The right and +left panels are models with and without dust evolution, +respectively. +Dust growth enhances Z′ +d only at high enough densi- +ties where n > 103 cm−3. This is a result of the short +dynamical time in the ISM (tdyn) which limits the avail- +able time for dust growth to operate before the dense +clouds are destroyed. Indeed, the dust growth rate in +Eq. 4 in a static medium has the following analytic so- +lution (Zhukovska et al. 2008): +f(t) = f(0) +exp(t/tgrow) +1 − f(0) + f(0) exp(t/tgrow), +(8) +where f(0) is the initial dust depletion fraction and tgrow +is given in Eq. 6 which is density-dependent. For a given +tdyn, we can therefore construct the analytic solution as +a function of n, as shown in black lines in the upper +right panel in Fig. 3 for tdyn = 0.1 (dashed), 1 (dotted), +and 10 Myr (dash-dotted), respectively. The median Z′ +d +from our fiducial simulation can be reproduced remark- +ably well with tdyn = 1 Myr. This Myr-scale dynamical +time in the ISM is consistent with Hu et al. (2021). +The fact that dust growth only operates at high den- +sities (n > 103 cm−3) means that it only modestly in- +creases the H2 formation rate on dust grains. +Mean- +while, dust growth can also enhance radiation shielding +in dense gas. However, this does not affect the H2 abun- +dance very much either as (1) H2 can self-shield against +the FUV radiation and (2) the limiting factor for H2 +is the available time for it to form while shielding only +plays a secondary role (Glover & Mac Low 2011; Hu +et al. 2021). Indeed, the H2 abundance is only notably +enhanced at n > 104 cm−3 with dust evolution. Simi- +larly, the SFR is insensitive to dust growth because the +thermal balance of the ISM is mostly unaffected except +for the densest gas. +In contrast, dust evolution has a very significant ef- +fect on the C+/C i/CO transitions. Without dust evo- +lution, both C i and CO are completely destroyed by +the FUV radiation. This is a natural consequence of the +adopted low Z′ +d as motivated by observations. On the +other hand, with dust evolution, C+/C i/CO transitions +take place at very high densities (n ≳ 105 cm−3) due to +enhanced dust shielding. +This is consistent with the +cloud simulations in Glover & Clark (2012b) where they +found that the CO luminosity of low-metallicity clouds +is dominated by emission from gravitationally collapsing +dense gas of similar densities. Note that the high-Z′ +d gas +occupies a very small mass fraction in the ISM such that +the global galaxy-integrated Z′ +d is only ∼ 20% higher +than in the constant-DGR run. +Due to the long H2 formation time, gas with n ∼ +100 cm−3, the typical density for molecular clouds in +the Milky Way, is completely dominated by H i. In other +words, there is very little CO-dark H2 gas in the ISM as +often assumed at low metallicity. This leads to a galaxy- + +Dust and chemistry co-evolution in dwarf galaxies +9 +dust +dust +H2 +H2 +HI +HI +C+ +CI +CO +C+ +no dust evolution +with dust evolution +CI +Figure 3. +Top panels: the DGR (green) and chemical abundances of H i (blue) and H2 (red) as a function of the hydrogen +number density n. Bottom panels: chemical abundances of C+ (blue), C i (green), and CO (red) as a function of n. The right +and left panels are the runs with and without dust evolution, respectively. The solid lines show the median value in a given +n bin while the shaded area brackets the 16 and 84 percentiles. The black lines in the upper right panel indicate the analytic +solution in Eq. 4 with the dynamical time tdyn = 0.1 (dashed), 1 (dotted), and 10 Myr (dash-dotted), respectively. Without +dust evolution, CO is completely photo-dissociated due to insufficient shielding while H2 is almost unaffected. +integrated αCO factor only 50% higher than the Milky +Way value as shown in Fig. 2. +3.3. Spatial variation of DGR +We now turn our attention to how the DGR spatial +variation comes about. We compare our fiducial simula- +tion with the simulation where dust growth is switched +off to investigate the relative importance between dust +growth and sputtering. +Fig. 4 shows the time-averaged phase diagram (density +vs. temperature) color-coded by Z′ +d. The right and left +panels are for runs with (right panel) and without (left +panel) dust growth. The gas distribution on the phase +diagram broadly follows the classical “S-shape” curve +as determined by the thermal balance between radiative +cooling and heating. Hot gas with T > 105 K is gener- +ated by SN feedback while the narrow line at T = 104 K +is the signature of photoionization from massive stars. +Without dust growth, Z′ +d in hot gas decreases due to +sputtering. However, the highly-sputtered gas (shown +in red) concentrates in the relatively high-density gas in +the hot phase (n = 0.1−1 cm−3), while the more diffuse +hot gas is only weakly sputtered. This results from the +density dependence in the sputtering rate (see Eq. 3). +The situation becomes quite different once dust +growth is included. Firstly, as already shown in Fig. 3, +Z′ +d in high-density gas (n > 103 cm−3) is strongly en- +hanced as this is the place where dust growth occurs. +Furthermore, Z′ +d in the hot gas is slightly enhanced ex- +cept for the densest and hottest region where sputtering +is most efficient. This indicates that the high-Z′ +d dense +gas where star formation occurs is dispersed by the sub- +sequent stellar feedback and mixes with the ISM. As +sputtering only slightly decreases Z′ +d, the net effect is +that the hot gas is “dust-enriched”. +This is qualita- +tively similar to metal enrichment in supernova rem- +nants, where the SN ejecta of high metallicity mix with +the ISM and increase its metallicity. +Observationally, it is more straightforward to measure +the projected DGR Zd,proj = Σd/Σg rather than the lo- +cal Z′ +d. +Fig. 5 shows the normalized projected DGR +Z′ +d,proj ≡ Zd,proj/Zd,MW as a function of the gas surface +density (Σg) with a pixel size lp = 3 pc for our fidu- + +10 +Hu et al. +no dust growth +with dust growth +Figure 4. +The time-averaged phase diagram (density vs. temperature) color-coded by the DGR. The right and left panels +are for runs with and without dust growth, respectively. Dust growth enhances the DGR in dense gas which mixes with the hot +gas generated by SN feedback. +cial model (blue solid line) and the model without dust +growth (orange dashed line). +Without dust growth, the projected DGR is fairly ho- +mogeneous everywhere. Even in the SN bubbles (Σg ≲ +1 M⊙ pc−2) where sputtering occurs, Z′proj +d +is only 20% +lower. Therefore, sputtering alone is insufficient to gen- +erate DGR variation. +If dust growth is included, we +see significant DGR variation which can be broadly di- +vided into three regimes: (1) the compact gas clumps +(Σg ≳ 100 M⊙ pc−2) where Z′proj +d +increases sharply with +Σg as a direct consequence of the density-dependent dust +growth. (2) the diffuse ISM (Σg ≈ 1 − 100 M⊙ pc−2) +where Z′proj +d +increases slowly with Σg, reflecting the +large-scale DGR variation (e.g., radial gradient) as the +high-DGR gas clumps mix with the diffuse ISM and (3) +the SN bubbles (Σg ≲ 1 M⊙ pc−2) where Z′proj +d +is en- +hanced by roughly a factor of two. +3.3.1. Dust in galactic outflows +Observations suggest that dust exists in the inter- +galactic medium far away from galaxies (M´enard et al. +2010; Peek et al. 2015). In this section, we quantify the +SN-driven outflow rates from our simulated galaxy. +We define the mass outflow rate for gas as +˙M out +g += +� +S +ρv · ˆndA, +(9) +where ρ is the gas density, v is the gas velocity, ˆn is the +outward unit normal vector of the area dA and S is the +surface where we measure the outflow rate. Similarly, +the mass outflow rate for dust is defined as +˙M out +d += +� +S +Zdρv · ˆndA. +(10) +10 +2 +10 +1 +100 +101 +102 +103 +104 +g [M +pc +2] +10 +2 +10 +1 +Z′proj +d +with dust growth +no dust growth +Figure 5. +The time-averaged projected DGR as a function +of the gas surface density with a pixel size of 3 pc. The blue +solid line shows our fiducial model while the orange dashed +line shows the model without dust growth. The lines show +the median in each bin while the shaded area brackets the +16th and 84th percentiles. Dust growth is the main driver of +the DGR variation in the ISM. +In this work, we measure the outflow rates at |z| = +zout kpc parallel to the mid-plane of the galactic disk. +We adopt two choices of zout: 1 kpc and 10 kpc. Follow- +ing Hu (2019), the discretized outflow rate for gas and +dust can be expressed as +˙M out +g += +� +(zvz)i>0 +(mgvz)i +dz +, +(11) +˙M out +d += +� +(zvz)i>0 +(mgvzZd)i +dz +, +(12) + +8 +-1.00 +-1.25 +6 +-1.50 +5 +-1.75 +N +4 +601 +-2.25 +3 +-2.50 +2 +-2.75 +1 +-3.00 +4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +5 +6 +log1on [cm-3]8 +-1.00 +7 +-1.25 +6 +-1.50 +[K] +5 +-1.75 +N +4 +601 +-2.25 +3 +-2.50 +2 +-2.75 +1 +-3.00 +4 +-2 +-1 +0 +1 +2 +3 +4 +5 +6 +log1on [cm-3]Dust and chemistry co-evolution in dwarf galaxies +11 +0 +100 +200 +300 +400 +500 +time [Myr] +100 +101 +mass loading factor ( +m) +|z| = 1 kpc +|z| = 1 kpc +0 +100 +200 +300 +400 +500 +time [Myr] +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +dust enrichment factor (yd) +|z| = 1 kpc +|z| = 1 kpc +0 +100 +200 +300 +400 +500 +time [Myr] +10 +1 +100 +mass loading factor ( +m) +|z| = 10 kpc +|z| = 10 kpc +0 +100 +200 +300 +400 +500 +time [Myr] +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +dust enrichment factor (yd) +|z| = 10 kpc +|z| = 10 kpc +with dust growth +no dust growth +Figure 6. +Time evolution of the mass loading factor (left panels) and the dust enrichment factor (right panels) of the galactic +outflows measured at |z| = 1 kpc (top panels) and |z| = 10 kpc (bottom panels). Dust growth leads to dust-enriched outflows. +where the subscript i represents the particle index, vz is +gas velocity in the vertical direction, and dz = 0.1zout is +the thickness of the measuring plane. The summation is +over particles with zvz > 0 (i.e., outflowing gas) within +z = zout ± 0.5dz and z = −zout ± 0.5dz. +We define the mass loading factor as +ηm(t) ≡ +˙M out +g +(t) +SFR +(13) +where SFR is the time-averaged star formation rate. We +take the averaged SFR instead of the instantaneous SFR +for normalization as there is a time delay between the +star formation events and the associated outflowing gas +arriving at the measuring planes. Given the burstiness +of the SFR, if we took the instantaneous SFR for normal- +ization, ηm can be misleadingly high when the outflow +rate is modest but the SFR is very low. +To quantify whether dust is preferentially expelled out +of the galaxy, we define the dust enrichment factor +yd(t) ≡ +˙M out +d +(t) +˙M out +g +(t)ZISM +d +(t) +(14) +where ZISM +d +is the DGR in the ISM defined as |z| < +0.5 kpc. Here we take the instantaneous DGR in the +ISM, ZISM +d +(t), for normalization. +This is because, in +contrast to the SFR, the DGR in the ISM varies very +slowly over time. +Note that the ratio +˙M out +d +/ ˙M out +g +is +essentially the DGR of outflows weighted by the mass +flux (mgvz). +Fig. 6 shows the time evolution of ηm (left panels) and +yd (right panels) measured at |z| = 1 kpc (top panels) +and |z| = 10 kpc (bottom panels). The blue solid line +shows our fiducial model while the orange dashed line +shows the model without dust growth. +We first discuss the strength of outflows. +At |z| = +1 kpc, ηm fluctuates strongly with time between 1 and +30 as a result of the bursty star formation and SN feed- +back. At |z| = 10 kpc, it drops by an order of magni- +tude to ηm = 0.3 – 1, suggesting that a large fraction of +outflows measured at |z| = 1 kpc is balanced by inflow- +ing gas that falls back to the disk (i.e., fountain flows), +which is consistent with Hu (2019). Dust growth has +a negligible effect on ηm as it does not affect the ther- +mal balance and the dynamics in the ISM significantly. +The difference between two models is likely due to the +intrinsic stochasticity of star formation and feedback. +We now examine the dust content in outflows. With- +out dust growth, the dust enrichment factor is less than + +12 +Hu et al. +Figure 7. +Images of the gas surface density (upper panels) and the projected DGR (lower panels) at t = 240 Myr with lb = 24, +48, 96, 192, and 384 pc from left to right. +unity: yd ∼ 0.8. This is because dust is sputtered in +the shocked-heated gas in SNRs (see Fig. 4) which is +then launched as outflows. +Sputtering only destroys +∼ 20% of dust even when the outflows have traveled +to |z| = 10 kpc. The slightly lower yd at |z| = 1 kpc +does not mean that dust is created during its journey +from |z| = 1 kpc to |z| = 10 kpc, which is unlikely given +the low gas densities. Instead, it is likely due to dilution +by the entrained ISM which has Zd = ZISM +d +by defini- +tion and is expected to fall back as fountain flows rather +than travel to |z| = 10 kpc. +The situation becomes very different once dust growth +is included. +The outflows are now more dusty than +the ISM, with yd ∼ 1.2 – 1.5 at |z| = 10 kpc. as the +shocked-heated gas in SNRs is “dust-enriched” due to +dust growth (see Fig. 4). Again, this is analogous to +metal enrichment from SN ejecta which leads to metal- +enriched outflows. +3.3.2. The effect of beam size +Extragalactic observations often have a telescope +beam size significantly coarser than 3 pc as adopted in +Fig. 5. To understand the effect of beam size, we con- +struct images of Σg and Σd for our fiducial model at +systematically coarser beam sizes of lb = 3, 6, 12, 24, +48, 96, 192, and 384 pc. For example, +Σg (6 pc) = +� +Σg dA +� +dA += 1 +22 +4 +� +i=1 +Σg,i (3 pc) , +(15) +Σd (6 pc) = +� +Σd dA +� +dA += 1 +22 +4 +� +i=1 +Σd,i (3 pc) , +(16) +and, correspondingly, +Z′proj +d +(6 pc) = +Σd (6 pc) +Σg (6 pc)Zd,MW +. +(17) +Fig. 7 shows the coarsened images of Σg (upper pan- +els) and Z′proj +d +(lower panels) at t = 240 Myr with +lb = 24, 48, 96, 192, and 384 pc from left to right. +The compact dense gas with very high Z′ +d can be re- +solved reasonably well at lb = 24 pc. As lb increases, +this high-Z′ +d gas is gradually smoothed out and the DGR +is significantly diluted by the diffuse ISM which has a +much lower DGR. At lb = 384 pc, Z′proj +d +becomes very +uniform. There is a slight radial gradient of Z′proj +d +which +reflects the large-scale radial distribution of the gas sur- +face density, similar to the metallicity gradient in galax- +ies caused by metal enrichment. +To be more quantitative, Fig. 8 shows the relation- +ship between Σg and Z′proj +d +at various beam sizes for +all snapshots between t = 100 – 500 Myr. +As lb in- +creases, beam averaging smooths out the dense gas such +that both Σg and Σd decrease. However, Σd decreases +less significantly because Z′proj +d +is significantly higher at +high Σg, shifting the relationship leftward. As a result, +the gas surface density above which Z′proj +d +rises sharply +decreases at larger lb. The observational implication is +that measurements of the Σg – Z′proj +d +relationship must +be compared at a similar beam size. +At even larger beam sizes (lb ≥ 92 pc), the sharply ris- +ing part disappears completely, and the Σg – Z′proj +d +rela- +tionship becomes insensitive to lb. This happens when +the high-Z′ +d dense gas is completely diluted away by the +diffuse ISM at large enough lb. The slowly-rising part + +-1 +0 +1 +log10Zg [Mo pc-2]-2.0 -1.8 -i.6 -1.4 -i1.2Dust and chemistry co-evolution in dwarf galaxies +13 +10 +1 +100 +101 +102 +103 +104 +g [M +pc +2] +10 +2 +10 +1 +Z′ proj +d +lb = 3 pc +lb = 6 pc +lb = 12 pc +lb = 24 pc +lb = 48 pc +lb = 96 pc +lb = 192 pc +lb = 384 pc +Figure 8. +Same with Fig. 5 but only for the fiducial model and with systematically coarser beam sizes (lb). The scatters +are not shown for clarity. Beam averaging shifts the relationship to lower gas surface densities as it smooths out the high-DGR +dense gas. +corresponds to the large-scale radial gradient of Z′proj +d +as can be seen in Fig. 7. +The Σg – Z′proj +d +relation is likely to depend on the +metallicity and the large-scale gas surface density. We +plan to conduct simulations of SMC- and LMC-like +galaxies in the future for direct comparison with high- +resolution FIR and UV observations such as Roman- +Duval et al. (2017, 2022). +4. DISCUSSION +4.1. Implications for the observationally derived DGR +Observationally, the galaxy-integrated DGR is often +estimated using +Zd = +Md +MH2 + MH i += +Md +LCOαCO + MH i +, +(18) +where MH i is the total H i mass from the 21-cm line and +Md is the total dust mas from the FIR continuum. A +major uncertainty is in the adopted αCO. As a result, +R´emy-Ruyer et al. (2014) reported two versions of the +metallicity–DGR relation based on two different choices +of αCO, one is a constant Milky Way value αCO,MW and +the other is metallicity-dependent that scales as Z′−2. +While the latter one is more frequently adopted in the +literature, our results suggest that the one based on the +Milky Way conversion factor is actually more appropri- +ate at low metallicity. In fact, the metallicity-dependent +αCO strongly overestimates MH2 which in turn underes- +timates Zd. +Furthermore, our simulations showed, consistent with +previous studies in Hu et al. (2016, 2017), that the +molecular mass fraction is extremely small (FH2 ∼ 10−4) +in dwarf galaxies with Z′ = 0.1 such that MH2 con- +tributes negligibly to the total gas mass. This is a nat- +ural consequence of the long H2 formation time tH2 ∼ +1 Gyr(nZ′ +d)−1. +With our adopted Z′ +d = 0.01, gas at +n = 100 cm−2 takes tH2 ∼ 1 Gyr to form H2, which is +orders of magnitude longer than the Myr-scale dynam- +ical time in the ISM. Therefore, the H2 abundance is +primarily limited by the dynamical time (Glover & Mac +Low 2011; Hu et al. 2021). The situation remains quali- +tatively similar even if we adopted Z′ +d = 0.1 (i.e., a linear +metallicity–DGR relation) as shown in Hu et al. (2016) +who found FH2 ∼ 10−3. Consequently, the DGR should +be observationally estimated simply by Zd = Md/MH i +at low metallicity, and the uncertainty in αCO is irrele- +vant. +4.2. Implications for the intergalactic dust +Observations of reddening effect suggest that dust ex- +ists in the intergalactic medium 20 kpc to several Mpc +away from galaxies (M´enard et al. 2010; Peek et al. +2015), whose origin is still poorly understood. One of +the possible scenarios is dust entrained in galactic out- +flows (Aguirre 1999; Bianchi & Ferrara 2005). Kannan +et al. (2021) conducted simulations of isolated galaxies +with similar properties of the Milky Way and LMC cou- +pled with a dust evolution model. They found that out- +flows are able to entrain dust in the fountain flows that +circulate around the galaxies within a few kpc. How- +ever, they found that dust cannot be expelled to the +outer part of the halos, which is in contrast to our case +where outflows at 10 kpc are still dust-enriched (rather +than depleted) and are expected to eventually escape + +14 +Hu et al. +the halo (Hu 2019). +This might indicate that dwarf +galaxies are preferable sites to pollute the intergalactic +medium with dust as they have low gravitational po- +tential wells and the lack of hot gaseous halos around +them prevents further destruction via thermal sputter- +ing. On the other hand, the difference could also arise +from numerics as the resolution in Kannan et al. (2021) +is 103 M⊙ which was presumably too coarse to resolve +the dense gas where dust growth is most efficient. Re- +solved simulations of LMC-like galaxies have recently +been conducted by Steinwandel et al. (2022b), and a +systematic study of outflows across a range of galaxy +masses will be valuable to shed light on this topic. +4.3. Neglected physics +Our dust evolution model does not include dust pro- +duction in SN ejecta, which has been observed (Wooden +et al. 1993; Indebetouw et al. 2014) and might be major +source of dust production in the early Universe when +there was not enough time for AGB stars to kick in. It +is still an open question about how much dust will even- +tually survive once the reversed shock hits back which +depends sensitively on local gas properties (Bianchi & +Schneider 2007; Micelotta et al. 2016; Kirchschlager +et al. 2019; Priestley et al. 2022). Including dust pro- +duction in SNe is unlikely to affect the DGR in dense +gas and the ISM chemistry, but it could make the shock- +heated hot gas and galactic outflows even dustier. +In addition, our dust model assumes a fixed “MRN” +grain-size distribution and neglects processes that can +vary the grain size such as shattering and coagula- +tion. As the timescales for sputtering and dust growth +both depend linearly on grain size, the dust produc- +tion/destruction rate would be affected if the actual +grain-size distribution deviates significantly from MRN. +In addition, the grain-size distribution may also affect +the H2 formation rate on dust as well as dust shield- +ing (Jonkheid et al. 2006; Romano et al. 2022). Includ- +ing the evolution of grain-size distribution is therefore a +highly desirable extension for future work. +5. SUMMARY +We have presented the first ISM-resolved galaxy +scale simulations coupled with time-dependent hydrogen +chemistry and dust evolution. Our aim is to understand +the ISM chemistry and dust properties at low metallic- +ity, a condition expected to be common for galaxies in +the very early Universe observed by JWST. Our simu- +lated galaxy is similar to the WLM dwarf galaxy, which +has a metallicity of 0.1 Z⊙ and has detected CO(1–0) +emission. +We adopt a mass resolution of 1 M⊙ per +gas particle which corresponds to a spatial resolution +∼ 0.2 pc to properly resolve the compact CO cores. We +post-process the simulation snapshots with a detailed +chemistry network to accurately model the C+/C i/CO +transitions, taking the time-dependent abundances of +H2 and H+ from the simulations as input parameters. +Our main findings can be summarized as follows. +1. Our fiducial simulation successfully reproduces +both the observed SFR and CO(1–0) luminosity +(LCO) in the WLM dwarf galaxy (Fig. 2). LCO +can only be reproduced if dust growth in the ISM +is included, as otherwise dust shielding would be +insufficient to protect CO from being photodissoci- +ated by FUV radiation, suppressing LCO by more +than two orders of magnitude (Fig. 3). +2. The predicted total H2 fraction is extremely low +(∼ 10−4) either with or without dust evolution +due to the long H2 formation time. +This leads +to very little CO-dark H2 gas and a CO-to- +H2 conversion factor (excluding helium) αCO = +4.63 M⊙ pc−2 (K km s−1)−1 which is very close +to the Milky Way value despite the low metallic- +ity (Table 1). +Observationally inferred dust-to- +gas ratio (DGR) is underestimated if assuming a +metallicity-dependent αCO at low metallicity (Sec- +tion 4.1). +3. Dust growth is the primary driver of the spatial +variation of DGR in the ISM. Dust growth signif- +icantly increases the DGR in cold, star-forming +clouds. +Subsequent SN feedback disperses the +high-DGR gas without significantly destroying the +dust, leading to elevated DGR in the diffuse gas +associated with supernova remnants, qualitatively +similar to metal enrichment (Figs. 4 and 5). As a +result, galactic outflows are about 20 – 50% dustier +than the ISM (Fig. 6). Dwarf galaxies therefore +could be a source of the intergalactic dust (Sec- +tion 4.2). +4. The projected DGR increases with gas surface +density due to dust growth. The relationship be- +tween these two quantities varies with the tele- +scope beam size as coarse beams smooth out the +high-DGR gas clumps (Figs. 7 and 8). Observa- +tional measurements should therefore be compared +at a similar beam size. +ACKNOWLEDGMENTS +C.Y.H. acknowledges support from the DFG via +German-Israel Project Cooperation grant STE1869/2- +1 GE625/17-1 and NASA ATP grant 80NSSC22K0716. +A.S. thanks the Center for Computational Astrophysics + +Dust and chemistry co-evolution in dwarf galaxies +15 +(CCA) of the Flatiron Institute, and the Mathemat- +ics and Physical Science (MPS) division of the Simons +Foundation for support. 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P., & Trieloff, M. 2008, A&A, 479, +453, doi: 10.1051/0004-6361:20077789 +Zhukovska, S., Henning, T., & Dobbs, C. 2018, ApJ, 857, +94, doi: 10.3847/1538-4357/aab438 + diff --git a/rNE4T4oBgHgl3EQfwA2r/content/tmp_files/load_file.txt b/rNE4T4oBgHgl3EQfwA2r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..429ec5c58b53197779f15279484197b4a1cf7e72 --- /dev/null +++ b/rNE4T4oBgHgl3EQfwA2r/content/tmp_files/load_file.txt @@ -0,0 +1,1232 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf,len=1231 +page_content='Draft version January 16, 2023 Typeset using LATEX twocolumn style in AASTeX63 Co-evolution of Dust and Chemistry in Galaxy Simulations with a Resolved Interstellar Medium Chia-Yu Hu (胡家瑜 ),1, 2 Amiel Sternberg,3, 4, 1 and Ewine F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' van Dishoeck1, 5 1Max-Planck-Institut f¨ur Extraterrestrische Physik, Giessenbachstrasse 1, D-85748 Garching, Germany 2Department of Astronomy, University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611, USA 3School of Physics & Astronomy, Tel Aviv University, Ramat Aviv 69978, Israel 4Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA 5Leiden Observatory, Leiden University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Box 9513, NL-2300 RA Leiden, the Netherlands ABSTRACT Nearby dwarf irregular galaxies are ideal laboratories for studying the interstellar medium (ISM) at low metallicity, which is expected to be common for galaxies at very high redshift that will be observed by the James Webb Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We present the first high-resolution (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 pc) hydrodynam- ical simulations of an isolated low-metallicity (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 Z⊙) dwarf galaxy coupled with a time-dependent chemistry network and a dust evolution model where dust is locally produced and destroyed by var- ious processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' To accurately model carbon monoxide (CO), we post-process the simulations with a detailed chemistry network including the time-dependent effect of molecular hydrogen (H2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Our model successfully reproduces the observed star formation rate and CO(1–0) luminosity (LCO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We find that dust growth in dense gas is required to reproduce the observed LCO as otherwise CO would be completely photodissociated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In contrast, the H2 abundance is extremely small and is insensi- tive to dust growth, leading to a CO-to-H2 conversion factor similar to the Milky Way value despite the low metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Observationally inferred dust-to-gas ratio is thus underestimated if adopting the metallicity-dependent CO-to-H2 conversion factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The newly-produced dust in dense gas mixes with the ISM through supernova feedback without being completely destroyed by sputtering, which leads to galactic outflows 20% – 50% dustier than the ISM, providing a possible source for intergalactic dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Keywords: Interstellar medium (847);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Astrochemistry (75);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Hydrodynamical simulations (767) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' INTRODUCTION The formation and evolution of galaxies are critically controlled by how stars form and how they affect the gas cycle in and around galaxies via stellar feedback (Somerville & Dav´e 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Naab & Ostriker 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Over the last decade, the cold, star-forming gas (dominated by molecular hydrogen, H2) in galaxies from the lo- cal Universe to “cosmic noon” at redshift z ∼ 2 has been systematically quantified by submillimeter and far- infrared (FIR) telescopes, leading to a physical picture where galaxies grow primarily by gas accretion onto the rotationally-supported disks, which fuels star formation in their interstellar medium (ISM) regulated by feed- back across cosmic time (see Tacconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2020 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The molecular gas mass inferred by Corresponding author: Chia-Yu Hu cyhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='astro@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='com dust-based methods has found to be broadly consistent with the conventional method based on carbon monox- ide (CO), strengthening the robustness of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, while significant progress has been made in understanding the evolution of star-forming gas in galaxies of solar or slightly sub-solar metallicity, little is known in galaxies of low metallicity (Z ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1Z⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In the era of James Webb Space Telescope (JWST), a deep understanding of the ISM chemistry at low metallicity is urgently needed as we begin to observe galaxies in the very early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Indeed, Curtis-Lake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2022) recently reported four galaxies at extremely high red- shift (z ∼ 10 − 13) discovered by JWST, all of which have metallicity of Z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' On the other hand, nearby dwarf irregular galaxies provide a unique laboratory to study the chemical prop- erties and observational signatures of the star-forming gas at comparably low metallicity in great detail thanks to their proximity, even though their galaxy proper- ties (such as mass, size, surface density, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=') may be arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='05247v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='GA] 12 Jan 2023 2 Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' different from their high-redshift counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' It has long been recognized that CO emission in these galax- ies tends to be extremely faint and becomes undetected when the metallicity drops below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2Z⊙ (Leroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Schruba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Madden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This has changed with the advent of the Atacama Large Mil- limeter/submillimeter Array (ALMA) thanks to its ex- tremely high sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Rubio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2015) observed the Wolf-Lundmark-Melotte (WLM) dwarf galaxy and detected the first CO emission at a metallicity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1Z⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, the molecular gas mass still cannot be robustly determined due to the highly uncertain CO-to-H2 con- version factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Similarly, the dust-based method suffers from the uncertainty in the assumed dust-to-gas ratio (DGR) which has been shown to scale super-linearly with metallicity in this regime, but the exact scaling relation is still uncertain (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=', R´emy-Ruyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' De Vis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In fact, the detection of CO in the WLM galaxy is rather surprising given the ex- tremely low DGR expected at this metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Recent hydrodynamical simulations coupled with time-dependent chemistry have achieved the required numerical resolution of a few parsecs (pc) to directly resolve feedback from individual supernova (SN) explo- sions in isolated dwarf galaxies (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Hu 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Emerick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Lah´en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' His- lop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Whitworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022b,a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Steinwandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Lah´en et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is an important milestone as it avoids the need of sub- grid prescriptions for SN feedback which is one of the major uncertainties in cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Coin- cidentally, H2 can be resolved at a similar resolution (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2018), at least at solar metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' How- ever, proper modeling of CO requires a significantly higher resolution of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 pc (Seifried et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2021) as CO typically exists in dense, spatially compact gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Lagrangian codes are therefore particu- larly suitable for such a task thanks to their built-in adaptive spatial resolution that can more easily reach sub-pc scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Meanwhile, the carbon network responsi- ble for CO formation is much larger than the hydrogen network (Sternberg & Dalgarno 1995), leading to a sig- nificant computational overhead if coupled with simu- lations, compromising the achievable resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' To ad- dress this dilemma, Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2021) introduced a hybrid approach where the hydrogen network is solved on-the- fly to capture the time-dependent (non-steady-state) ef- fect of H2 while an accurate chemistry network including carbon chemistry is solved in post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust plays an important role in ISM chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The surfaces of dust grains are the main sites where H2 for- mation occurs, which then initiates the formation of other important molecules such as CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Furthermore, dust provides shielding against the Lyman-Werner ra- diation that can photodissociate both H2 and CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Ob- servations of the Small and Large Magellanic Clouds (SMC and LMC) have demonstrated that the spatially- resolved DGR can vary by almost an order of magnitude from region to region in low-metallicity galaxies, indicat- ing significant dust evolution (Roman-Duval et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2017, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, simulations that resolves the ISM to date have all assumed a constant DGR in both space and time, which is an over-simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Cosmological simulations and isolated galaxy simula- tions with comparable resolution have started to include dust evolution models at different levels of sophistication (McKinnon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Aoyama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Parente et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Ro- mano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Lower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, these sim- ulations do not resolve either SN feedback or the density structure of the ISM, forcing them to adopt dust evo- lution models in a sub-grid fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' For example, dust destruction by SNe is generally based on results from 1D calculations in plane-parallel shocks, which does not apply for more complex situations such as inhomoge- neous ISM or clustered SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' While this simplification is perhaps justified as SN feedback in these simulations is unresolved anyway, a more accurate model is clearly desirable in resolved simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2019) intro- duced a numerical method to directly simulate thermal and nonthermal sputtering of dust designed for simula- tions with pc- or sub-pc resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, the model did not include any dust production processes, nor was it coupled with chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In this work, we extend the dust evolution model in Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2019) by including dust growth in cold gas and dust production in asymptotic giant branch (AGB) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We couple this with a time-dependent chemistry network and the ISM model developed in Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2016, 2017, 2021) and perform hydrodynamical simulations of an isolated dwarf galaxy similar to the WLM galaxy at a mass resolution of 1 M⊙ (spatial resolution ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' To our knowledge, this is the first ISM-resolved simula- tion coupled with chemistry and dust evolution (but see Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022 for a model for significantly coarser resolutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In Sec- tion 2, we describe our numerical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In Section 3, we study how dust evolution affects the ISM chemistry, how the DGR is distributed in the ISM and galactic outflows, and how the projected DGR varies with the gas surface density at different telescope beam sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In Section 4, we discuss the implications of our results for the observationally inferred DGR and the intergalactic dust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In Section 5, we summarize our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust and chemistry co-evolution in dwarf galaxies 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' NUMERICAL METHODS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Gravity and hydrodynamics We use the public version of Gizmo (Hopkins 2015), a multi-solver code for hydrodynamics that is built on the massively parallel TreeSPH code Gadget-3 (Springel 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We adopt its meshless finite-mass (MFM) solver for hydrodynamics (Hopkins 2015) which is a variation of the meshless Godunov method (Gaburov & Nita- dori 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Gravity is calculated using the Barnes–Hut method (“treecode”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The ISM model In this section, we summarize the physical processes in the ISM in our simulations excluding dust evolution, which will be described in the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The methods and implementations are largely based on Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2016, 2017, 2021) where more details can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Time-dependent cooling and chemistry We adopt a time-dependent chemistry network devel- oped in Glover & Mac Low (2007) and Glover & Clark (2012a) that is widely used in ISM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The abundances of H2, H+, H i, and the free electron frac- tion are integrated based on the chemistry reactions in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The hydrogen network includes H2 forma- tion on dust, H2 destruction by photodissociation, colli- sional dissociation and cosmic ray ionization, and recom- bination in the gas phase and on dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Individ- ual cooling and heating processes are calculated based on the time-dependent chemical abundances, which in- cludes cooling from fine structure metal lines, molecu- lar lines, Lyman alpha, H2 collisional dissociation, colli- sional ionization of H, and recombination of H+ in the gas phase and on grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Heating includes photoelectric effect, cosmic ray ionization, H2 photodissociation, UV pumping of H2 and the formation of H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Following Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2012), shielding against FUV radiation uses the HEALPix algorithm (G´orski & Hivon 2011) in combi- nation with the “treecode” approximation to integrate the relevant column densities along 12 sightlines up to a pre-defined radius of 100 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Star formation We adopt the stochastic star formation recipe com- monly used in the field of galaxy formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' A gas particle eligible for star formation is converted into a star particle of the same mass on a timescale of tff/ϵsf stochastically, where ϵsf is the star formation efficiency and tff = � 3π/(32Gρ) is the gas free-fall time where G is the gravitational constant and ρ is the gas den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We adopt ϵsf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5 in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Such a high effi- ciency is justified by our resolution as we can follow the gravitational collapse down to the scales of individual molecular cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Gas is eligible for star formation when its local Jeans mass MJ = (π2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5c3 s)/(6G1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5), drops below the kernel mass Mker = Nngbmg, where cs is the sound speed, mg is the gas particle mass, and Nngb = 32 is the number of neighboring particles in a kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Sampling individual stars from an IMF Massive stars (initial mass > 8 M⊙) inject energy and momentum into their surrounding gas commonly termed as “stellar feedback”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' At our resolution of 1 M⊙ per star particle, it is unphysical to assume that each parti- cle represents a star cluster with a fully sampled stellar initial mass function (IMF), as is commonly the case in cosmological simulations with much coarser resolu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Following Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2021), we adopt the tech- nique of “importance sampling” to stochastically sam- ple stellar masses from a Kroupa IMF (Kroupa 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The sampled stellar masses are used to determine the stellar lifetime (Ekstr¨om et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2012) and UV luminos- ity from the BaSeL stellar library (Lejeune et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 1997, 1998) and they do not affect the dynamics of gravity, as the gravitational mass of the star particles (m∗) remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Stellar feedback We include stellar feedback from supernovae (SNe) and photoionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' SN feedback is done by inject- ing thermal energy of 1051 erg per SN into its near- est Nngb gas particles in a kernel-weighted fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As our resolution is able to resolve the Sedov-Taylor phase in each SN event, a simple thermal feedback is able to achieve numerical convergence (Hu 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Feedback from photoionization follows Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2017), where each massive star searches for its ionization front it- eratively by balancing recombination and photoioniza- tion and heats up the interior gas to 104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This ap- proach reproduces the dynamics of an expanding H ii region predicted by radiative transfer codes in a uni- form medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' More importantly, it captures the cor- rect behaviors in overlapping H ii regions where a naive Str¨omgren-sphere method would suffer from the numer- ical artifact of double-counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Spatially variable FUV radiation Following Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2017), the unattenuated FUV ra- diation field is both spatially and temporally variable and is calculated directly from the star particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' For a given gas particle, every star particle contributes a radiation flux of LFUV/(4πr2) where LFUV is the FUV luminosity based on the sampled stellar mass and r is the distance between the gas and star particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The summation is over all star particles and is done via the 4 Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' “treecode” approximation to avoid the O(n2) operation and speed up the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The FUV radiation af- fects both the thermal balance via photoelectric heating and the chemistry via photodissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust evolution model We adopt the “one-fluid” approach where dust is as- sumed to be spatially coupled with the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is justified as dust is expected to be charged and gyrates around the magnetic fields in the ISM with a small gyro- radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Each gas particle is associated with a dust mass md = md(Sil) + md(C), where md(Sil) and md(C) are, respectively, the masses of silicate dust and carbona- ceous dust which evolve separately in the simulations due to the production and destruction processes as we will describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust grains are assumed to be spherical with a radius of a and a material density of sd, which leads to a grain mass of mgr = (4πa3/3)sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The formation or destruction rate of dust can be expressed as dmd dt = Ngr dmgr dt = 3 ˙a amd (1) where Ngr = md/mgr is number of dust grains in a gas cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Time integration is done via sub-cycling in order to resolve the timescales of dust dynamics and sputter- ing which can be orders of magnitude smaller than the hydrodynamical timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We include physical processes that directly modify md: sputtering, dust growth, and dust formation in AGB ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Processes that modify the grain size while keeping md fixed, such as shattering and coagulation, are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is because we are only interested in the evolution of dust mass rather than other dust properties such as the extinction law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We adopt a fixed grain-size distribution that follows Mathis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (1977) (the “MRN” distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Sputtering In shocks or in hot gas, dust can be destroyed via sputtering which returns metals locked up in dust grains back to the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We adopt the sputtering model in Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2019) that includes thermal and nonthermal sput- tering, which we brief summarize as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The dust destruction rate due to sputtering can be expressed as dmd dt ��� sput = − md tsput (2) where tsput ≡ a 3nYtot = 10 kyr � a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='03µm �� n cm−3 �−1� 106 Ytot µm yr−1cm3 �−1 , (3) where n is the hydrogen number density and Ytot is the erosion rate that includes thermal and nonthermal sput- tering, which we adopt from Nozawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The thermal erosion rate is a function of gas temperature while the nonthermal erosion rate is a function of the relative bulk velocity between dust and gas, which we obtain by integrating the equation of motion for dust ac- counting for direct collision, plasma drag, and betatron acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust growth Dust can grow in the cold gas when gas-phase metals interact with dust and stick onto the surfaces of dust grains, which can be viewed as the reverse process of sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The exact mechanism is still poorly under- stood, though its feasibility has been supported by lab- oratory experiments (Krasnokutski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Henning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Rouill´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The dust production rate due to dust growth can be expressed as dmd dt ��� grow = (1 − f) md tgrow (4) where tgrow is the dust growth timescale (see below) and f is the fraction of metals locked in dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' For element A (where A = Si or C), this can be written as fA = mA,d mA,tot = md(A)ξA mgXAZ′ , (5) where mA,d is the mass of element A in the dust phase, mA,tot is the total mass of element A (dust + gas), XA is the solar abundance of element A, and ξA is the mass fraction of element A in the assumed grain material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' For carbonaceous dust, ξC = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' For silicate dust, we adopt MgFeSiO4 as the grain material which leads to ξSi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The dust growth timescale takes the following form tgrow = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5 Gyr � n cm−3 �−1� T 100K �−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5 � a3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='03µm � (Z′αsDeff)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (6) Here, αs is the sticking coefficient, a3 ≡ ⟨a3⟩/⟨a2⟩ is the average grain size where the bracket ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='⟩ refers to integration over the grain size distribution, and Deff ≡ ⟨a2D(a)⟩/⟨a2⟩ is the enhancement factor D(a) due to Coulomb focusing (Weingartner & Draine 1999) weighted by the surface area of grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The sticking co- efficient encompasses the complex physical and chemical processes on the surfaces of grains which is still uncer- tain (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=', Zhukovska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2018 for a discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' To first order, it is expected to be close to unity at low temperatures and decrease substantially at high temper- atures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We follow Zhukovska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2016) and assume Dust and chemistry co-evolution in dwarf galaxies 5 that αs = 1 at T < 300 K while αs = 0 at T ≥ 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The area-weighted enhancement factor Deff is also un- certain and is expected to depend on the a number of properties such as density, temperature, free electron fraction, grain size distribution, and grain charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Wein- gartner & Draine (1999) found that Coulomb focusing shortens the growth timescale by more than an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, Priestley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2021) found a much weaker effect if the evolution of grain size is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We adopt Deff = 10 for simplicity, but noting that the uncertainties in both αs and Deff are potential caveats of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust production from AGB stars We adopt the mass-dependent dust yields from Zhukovska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2008) at Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 based on the individ- ual stellar masses sampled from the IMF to account for dust produced in the ejecta of AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The produced dust mass is injected in the neighboring gas particles in a kernel-weighted fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We do so only for carbona- ceous dust as the silicate dust yields at this metallicity is essentially zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust produced from AGB stars is expected to play a sub-dominant effect on the spatial variation of the DGR as AGB stars are more uniformly distributed in the ISM compared to sputtering in SN shocks and dust growth in dense clouds that are highly clustered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Chemistry network in post-processing In order to accurately model the transitions of C+/C i/CO (the “carbon cycle”), a detailed carbon chemistry is required, which is much more complicated and computationally costly to solve compared to the hy- drogen chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We therefore post-process the simu- lation snapshots using AstroChemistry1, a chemistry network code developed in Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2021) which con- sists of 31 species: H, H−, H2, H+, H+ 2 , H+ 3 , e−, He, He+, HeH+, C, C+, CO, HCO+, O, O+, OH, OH+, H2O+, H3O+, H2O, O2, CO+, O+ 2 , CH2, CH+ 2 , CH, CH+, CH+ 3 , Si+ and Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' All chemical reactions in the UMIST database (McElroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2013) that exclusively involve the above-mentioned species are included in the network, which leads to 286 reactions in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The time-dependent abundances of H2 and H+ in simula- tions are taken as input parameters when solving the network, which is of crucial importance as H2 can be out of steady state significantly, especially at low metal- licity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This approach has been applied to ISM-patch simulations which successfully reproduced the observed relationship between the column densities of CO and 1 Available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='com/huchiayu/AstroChemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='jl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' H2 in Galactic clouds (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2021) as well as the Milky Way CO-to-H2 conversion factor (XCO) (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2021) found that CO(1–0) is almost always optically thin in their simulated ISM at Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is primarily a direct consequence of the low CO abun- dance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Furthermore, CO(1–0) can remain optically thin even at the highest column densities as the population of CO is distributed over more excited levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Assuming optically thin conditions, the CO luminosity from each gas particle can be expressed by lCO = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5λ3 10T10A10nCOf1V (7) where nCO is the CO number density, V = mg/ρ is the volume of the gas particle, and λ10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='26 cm, T10 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='53 K, A10 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 × 10−8 s−1 are the wavelength, energy level, and the Einstein A coeffi- cient for the CO(1–0) line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' f1(Tex) = 3 exp(−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='53/Tex)/ � 1 + (Tex/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='77)2 is the fraction of CO in the level J = 1 (Draine 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Note that f1 only varies within a factor of 2 in the range of 3 < Tex/K < 30 where most CO exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Simulation setup The initial conditions consist of a rotating disk galaxy embedded in a dark matter halo with prop- erties resembling the WLM galaxy, generated by the MakeDiskGalaxy code (Springel 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The halo has a virial radius Rvir = 45 kpc and a virial mass Mvir = 1010 M⊙, and it follows a Hernquist profile (Hernquist 1990) matching an NFW (Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 1997) profile at small radii with the concentration parameter c = 15 and the spin parameter λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The baryonic mass fraction is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='8%, with a stellar disk of 107 M⊙ and a gaseous disk of 7 × 107 M⊙, both following an exponen- tial profile a with scale-length of 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The central gas surface density is Σgas ∼ 10 M⊙ pc−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The stellar disk follows an exponential vertical profile with a scale-height of 1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The vertical density profile of the gaseous disk is set up to maintain hydrostatic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The ini- tial gas temperature is set to be 104 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The particle mass for gas, stars, and dark matter are mg = 1 M⊙, m∗ = 1 M⊙, and mdm = 103 M⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The cor- responding spatial resolution for gas is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 pc, defined as the minimum kernel radius where the Jeans length can be resolved (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Such a high resolution is needed in order to resolve the dense and compact cores where CO is observed in the WLM galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The gravi- tational softening length is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 pc for gas and 100 pc for the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The metallicity is Z′ ≡ Z/Z⊙ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 throughout the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The initial dust-to-gas ratio is set to 6 Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Face-on images of the surface densities of H i, H2, and H+, gas temperature (T), projected DGR (Z′proj d ), and FUV radiation field (G0, in units of the Habing 1968 field) at simulation time t = 230 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' be 1% of the Milky Way value, Z′ d ≡ Zd/Zd,MW = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='01, motivated by observations of low-metallicity galax- ies (R´emy-Ruyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We adopt the Milky Way dust abundances as Zd,MW(C) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='9 × 10−3 and Zd,MW(Sil) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5×10−3 (Dwek 2005) which corresponds to a carbonaceous-to-silicate ratio ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='54 and a total dust-to-gas ratio of Zd,MW = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='4 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Galaxy scale simulations with a setup like ours are known to undergo an artificial burst of star formation during the initial collapse, which in turn leads to overly- energetic SN feedback that blows out the entire gaseous disk, substantially reducing the gas surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Fol- lowing Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2022b), we minimize this artifact by first running a simulation without dust evolution for 100 Myr and with the SN delay time set to zero, which reduces the dynamical impact of feedback due to sup- pressed SN clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The simulation snapshot at the end of this “pre-simulation” is used as the new initial conditions with a relaxed configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We run three simulations: (1) a fiducial model with fully coupled chemistry and dust evolution, (2) a model without dust evolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=', Z′ d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='01 throughout the simulation), and (3) a model with dust evolution where dust growth is switched off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Each simulation is run for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' RESULTS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Overview Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 1 shows the face-on images of the surface densities of H i, H2, and H+, gas temperature, projected DGR, and FUV radiation field at simulation time t = 230 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The ISM has a complex structure, with dense clouds where star formation occurs and holes driven by stellar feedback (“SN bubbles”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The majority of gas is in the form of H i while H+ traces the young massive stars time = 230 Myr 0 0 log1oZHi [M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='pc-2] log10ZH2 [M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='pc-2] log1oZH+ [M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' pc-2] 2 3 4 5 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='4-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='0-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='8-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='6 2 1 0 log1oT [K] log10 Zd log1o GoDust and chemistry co-evolution in dwarf galaxies 7 WLM WLM aCO,MW Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Time evolution of the following global properties integrated over the simulated galaxy: the H2 mass (MH2, top left), the star formation rate (SFR, top right), the luminosity of the CO(1–0) emission (LCO, bottom left), and the CO-to-H2 conversion factor (αCO ≡ MH2/LCO, bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The solid blue lines represent our fiducial run including dust evolution while the dashed orange lines represent a controlled run without dust evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The red dotted lines indicate the observed LCO and SFR in the WLM galaxy as well as the αCO in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust evolution strongly enhances LCO, but it has little effect on MH2 and SFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' photoionizing the ambient gas (the H ii regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The FUV radiation also traces young stars but it is more spatially extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The H2 abundance is extremely low everywhere besides the densest part of clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' These pc- scale dense cores are also where CO exists (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Most of the gas is in the warm phase with a temperature of T ∼ 104 K, while the hot gas (T ∼ 106 K) is found in the interior of the SN bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Cold gas with T ∼ 100 K only exists in dense and compact clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The projected DGR image demonstrates that DGR is not spatially uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The DGR is elevated in dense gas where dust growth is most efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Interestingly, the DGR is not suppressed in the interior of SN bubbles where dust is expected to be destroyed via sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Instead, the DGR seems to be elevated inside the SN bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' More quantitative analysis will be provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Effect on ISM chemistry Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2 shows the following global quantities of the sim- ulated galaxy as a function of time: the H2 mass (MH2, top left), the star formation rate (SFR, top right), the luminosity of the CO(1–0) emission (LCO, bottom left), and the CO-to-H2 conversion factor (αCO ≡ MH2/LCO, bottom right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Two simulations are shown, one with dust evolution (solid blue lines, our fiducial model) and one without dust evolution where the DGR is constant everywhere (dashed orange lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The WLM galaxy has an observed CO luminosity of 1229 K km s−1 pc2 (Rubio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2015) and an observed SFR of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='74 × 10−3 M⊙ yr−1 (Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2010), as indicated by the red dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The CO-to-H2 conversion factor in the Milky Way αCO,MW = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 M⊙ pc−2 (K km s−1)−1 (excluding helium) is also overplotted as the red dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Table 1 summarizes the median values of these global quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Our fiducial model successfully reproduces the ob- served LCO and SFR in the WLM galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' On the other hand, MH2 is extremely low and it contributes to a mass fraction of ∼ 10−4 in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is due to the long H2 formation time compared to the dynamical time in the highly turbulent ISM and is consistent with previous 8 Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Median values of global properties over 100 < t < 500 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' model SFR MH2 MCO LCO αCO SFR/LCO (1) (2) (3) (4) (5) (6) w/ dust evolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='59 × 10−3 2435 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='64 × 10−1 469 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='41 × 10−6 no dust evolution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='02 × 10−3 2095 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='97 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='72 3156 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='19 × 10−3 Note— (1) Star formation rate [M⊙ yr−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2) Total H2 mass [M⊙] (3) Total CO mass [M⊙] (4) Total CO(1–0) luminosity [K km s−1 pc2] (5) CO-to-H2 conver- sion factor αCO ≡ MH2/LCO [M⊙ pc−2 (K km s−1)−1] (6) Ratio of SFR/LCO [M⊙ yr−1 pc−2 (K km s−1)−1] simulations of isolated dwarf galaxies (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Whitworth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The low MH2 leads to a conversion factor close to αCO,MW despite the low metal- licity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust evolution has a substantial impact on CO luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Without dust evolution, LCO is about three orders of magnitude lower than the observed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' On the other hand, MH2 and the SFR are both insensitive to dust evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Our low αCO may seem to be in conflict with Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2022a) where the kpc-scale αCO was found to scales with Z′−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='71, implying αCO ∼ 5αCO,MW at Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is because we adopt Z′ d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='01 in this work (mo- tivated by the observed super-linear metallicity–DGR relation), which is ten times lower than what Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2022a) assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As a result, the molecular mass frac- tion FH2 is also about ten times lower, which explains the difference in αCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We now take a closer look at the local chemical abun- dances and DGR as a function of hydrogen number den- sity n as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The top panels show the DGR and abundances of H2 and H i while the bottom panels show the abundances of C+, C i, and CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The right and left panels are models with and without dust evolution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust growth enhances Z′ d only at high enough densi- ties where n > 103 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is a result of the short dynamical time in the ISM (tdyn) which limits the avail- able time for dust growth to operate before the dense clouds are destroyed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Indeed, the dust growth rate in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4 in a static medium has the following analytic so- lution (Zhukovska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2008): f(t) = f(0) exp(t/tgrow) 1 − f(0) + f(0) exp(t/tgrow), (8) where f(0) is the initial dust depletion fraction and tgrow is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 6 which is density-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' For a given tdyn, we can therefore construct the analytic solution as a function of n, as shown in black lines in the upper right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3 for tdyn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 (dashed), 1 (dotted), and 10 Myr (dash-dotted), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The median Z′ d from our fiducial simulation can be reproduced remark- ably well with tdyn = 1 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This Myr-scale dynamical time in the ISM is consistent with Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The fact that dust growth only operates at high den- sities (n > 103 cm−3) means that it only modestly in- creases the H2 formation rate on dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Mean- while, dust growth can also enhance radiation shielding in dense gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, this does not affect the H2 abun- dance very much either as (1) H2 can self-shield against the FUV radiation and (2) the limiting factor for H2 is the available time for it to form while shielding only plays a secondary role (Glover & Mac Low 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Indeed, the H2 abundance is only notably enhanced at n > 104 cm−3 with dust evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Simi- larly, the SFR is insensitive to dust growth because the thermal balance of the ISM is mostly unaffected except for the densest gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In contrast, dust evolution has a very significant ef- fect on the C+/C i/CO transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Without dust evo- lution, both C i and CO are completely destroyed by the FUV radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is a natural consequence of the adopted low Z′ d as motivated by observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' On the other hand, with dust evolution, C+/C i/CO transitions take place at very high densities (n ≳ 105 cm−3) due to enhanced dust shielding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is consistent with the cloud simulations in Glover & Clark (2012b) where they found that the CO luminosity of low-metallicity clouds is dominated by emission from gravitationally collapsing dense gas of similar densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Note that the high-Z′ d gas occupies a very small mass fraction in the ISM such that the global galaxy-integrated Z′ d is only ∼ 20% higher than in the constant-DGR run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Due to the long H2 formation time, gas with n ∼ 100 cm−3, the typical density for molecular clouds in the Milky Way, is completely dominated by H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In other words, there is very little CO-dark H2 gas in the ISM as often assumed at low metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This leads to a galaxy- Dust and chemistry co-evolution in dwarf galaxies 9 dust dust H2 H2 HI HI C+ CI CO C+ no dust evolution with dust evolution CI Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Top panels: the DGR (green) and chemical abundances of H i (blue) and H2 (red) as a function of the hydrogen number density n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Bottom panels: chemical abundances of C+ (blue), C i (green), and CO (red) as a function of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The right and left panels are the runs with and without dust evolution, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The solid lines show the median value in a given n bin while the shaded area brackets the 16 and 84 percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The black lines in the upper right panel indicate the analytic solution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4 with the dynamical time tdyn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 (dashed), 1 (dotted), and 10 Myr (dash-dotted), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Without dust evolution, CO is completely photo-dissociated due to insufficient shielding while H2 is almost unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' integrated αCO factor only 50% higher than the Milky Way value as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Spatial variation of DGR We now turn our attention to how the DGR spatial variation comes about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We compare our fiducial simula- tion with the simulation where dust growth is switched off to investigate the relative importance between dust growth and sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4 shows the time-averaged phase diagram (density vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' temperature) color-coded by Z′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The right and left panels are for runs with (right panel) and without (left panel) dust growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The gas distribution on the phase diagram broadly follows the classical “S-shape” curve as determined by the thermal balance between radiative cooling and heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Hot gas with T > 105 K is gener- ated by SN feedback while the narrow line at T = 104 K is the signature of photoionization from massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Without dust growth, Z′ d in hot gas decreases due to sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, the highly-sputtered gas (shown in red) concentrates in the relatively high-density gas in the hot phase (n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1−1 cm−3), while the more diffuse hot gas is only weakly sputtered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This results from the density dependence in the sputtering rate (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The situation becomes quite different once dust growth is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Firstly, as already shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3, Z′ d in high-density gas (n > 103 cm−3) is strongly en- hanced as this is the place where dust growth occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Furthermore, Z′ d in the hot gas is slightly enhanced ex- cept for the densest and hottest region where sputtering is most efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This indicates that the high-Z′ d dense gas where star formation occurs is dispersed by the sub- sequent stellar feedback and mixes with the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As sputtering only slightly decreases Z′ d, the net effect is that the hot gas is “dust-enriched”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is qualita- tively similar to metal enrichment in supernova rem- nants, where the SN ejecta of high metallicity mix with the ISM and increase its metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Observationally, it is more straightforward to measure the projected DGR Zd,proj = Σd/Σg rather than the lo- cal Z′ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 5 shows the normalized projected DGR Z′ d,proj ≡ Zd,proj/Zd,MW as a function of the gas surface density (Σg) with a pixel size lp = 3 pc for our fidu- 10 Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' no dust growth with dust growth Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The time-averaged phase diagram (density vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' temperature) color-coded by the DGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The right and left panels are for runs with and without dust growth, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust growth enhances the DGR in dense gas which mixes with the hot gas generated by SN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' cial model (blue solid line) and the model without dust growth (orange dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Without dust growth, the projected DGR is fairly ho- mogeneous everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Even in the SN bubbles (Σg ≲ 1 M⊙ pc−2) where sputtering occurs, Z′proj d is only 20% lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Therefore, sputtering alone is insufficient to gen- erate DGR variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' If dust growth is included, we see significant DGR variation which can be broadly di- vided into three regimes: (1) the compact gas clumps (Σg ≳ 100 M⊙ pc−2) where Z′proj d increases sharply with Σg as a direct consequence of the density-dependent dust growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2) the diffuse ISM (Σg ≈ 1 − 100 M⊙ pc−2) where Z′proj d increases slowly with Σg, reflecting the large-scale DGR variation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=', radial gradient) as the high-DGR gas clumps mix with the diffuse ISM and (3) the SN bubbles (Σg ≲ 1 M⊙ pc−2) where Z′proj d is en- hanced by roughly a factor of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust in galactic outflows Observations suggest that dust exists in the inter- galactic medium far away from galaxies (M´enard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Peek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In this section, we quantify the SN-driven outflow rates from our simulated galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We define the mass outflow rate for gas as ˙M out g = � S ρv · ˆndA, (9) where ρ is the gas density, v is the gas velocity, ˆn is the outward unit normal vector of the area dA and S is the surface where we measure the outflow rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Similarly, the mass outflow rate for dust is defined as ˙M out d = � S Zdρv · ˆndA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (10) 10 2 10 1 100 101 102 103 104 g [M pc 2] 10 2 10 1 Z′proj d with dust growth no dust growth Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The time-averaged projected DGR as a function of the gas surface density with a pixel size of 3 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The blue solid line shows our fiducial model while the orange dashed line shows the model without dust growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The lines show the median in each bin while the shaded area brackets the 16th and 84th percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust growth is the main driver of the DGR variation in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In this work, we measure the outflow rates at |z| = zout kpc parallel to the mid-plane of the galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We adopt two choices of zout: 1 kpc and 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Follow- ing Hu (2019), the discretized outflow rate for gas and dust can be expressed as ˙M out g = � (zvz)i>0 (mgvz)i dz , (11) ˙M out d = � (zvz)i>0 (mgvzZd)i dz , (12) 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='25 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='50 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='75 N 4 601 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='25 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='50 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='75 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='00 4 3 2 1 0 1 2 3 4 5 6 log1on [cm-3]8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='00 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='25 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='50 [K] 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='75 N 4 601 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='25 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='50 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='75 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='00 4 2 1 0 1 2 3 4 5 6 log1on [cm-3]Dust and chemistry co-evolution in dwarf galaxies 11 0 100 200 300 400 500 time [Myr] 100 101 mass loading factor ( m) |z| = 1 kpc |z| = 1 kpc 0 100 200 300 400 500 time [Myr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='8 dust enrichment factor (yd) |z| = 1 kpc |z| = 1 kpc 0 100 200 300 400 500 time [Myr] 10 1 100 mass loading factor ( m) |z| = 10 kpc |z| = 10 kpc 0 100 200 300 400 500 time [Myr] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='8 dust enrichment factor (yd) |z| = 10 kpc |z| = 10 kpc with dust growth no dust growth Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Time evolution of the mass loading factor (left panels) and the dust enrichment factor (right panels) of the galactic outflows measured at |z| = 1 kpc (top panels) and |z| = 10 kpc (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust growth leads to dust-enriched outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' where the subscript i represents the particle index, vz is gas velocity in the vertical direction, and dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1zout is the thickness of the measuring plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The summation is over particles with zvz > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=', outflowing gas) within z = zout ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5dz and z = −zout ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We define the mass loading factor as ηm(t) ≡ ˙M out g (t) SFR (13) where SFR is the time-averaged star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We take the averaged SFR instead of the instantaneous SFR for normalization as there is a time delay between the star formation events and the associated outflowing gas arriving at the measuring planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Given the burstiness of the SFR, if we took the instantaneous SFR for normal- ization, ηm can be misleadingly high when the outflow rate is modest but the SFR is very low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' To quantify whether dust is preferentially expelled out of the galaxy, we define the dust enrichment factor yd(t) ≡ ˙M out d (t) ˙M out g (t)ZISM d (t) (14) where ZISM d is the DGR in the ISM defined as |z| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Here we take the instantaneous DGR in the ISM, ZISM d (t), for normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is because, in contrast to the SFR, the DGR in the ISM varies very slowly over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Note that the ratio ˙M out d / ˙M out g is essentially the DGR of outflows weighted by the mass flux (mgvz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 6 shows the time evolution of ηm (left panels) and yd (right panels) measured at |z| = 1 kpc (top panels) and |z| = 10 kpc (bottom panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The blue solid line shows our fiducial model while the orange dashed line shows the model without dust growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We first discuss the strength of outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' At |z| = 1 kpc, ηm fluctuates strongly with time between 1 and 30 as a result of the bursty star formation and SN feed- back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' At |z| = 10 kpc, it drops by an order of magni- tude to ηm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3 – 1, suggesting that a large fraction of outflows measured at |z| = 1 kpc is balanced by inflow- ing gas that falls back to the disk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=', fountain flows), which is consistent with Hu (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust growth has a negligible effect on ηm as it does not affect the ther- mal balance and the dynamics in the ISM significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The difference between two models is likely due to the intrinsic stochasticity of star formation and feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We now examine the dust content in outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' With- out dust growth, the dust enrichment factor is less than 12 Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Images of the gas surface density (upper panels) and the projected DGR (lower panels) at t = 240 Myr with lb = 24, 48, 96, 192, and 384 pc from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' unity: yd ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is because dust is sputtered in the shocked-heated gas in SNRs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4) which is then launched as outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Sputtering only destroys ∼ 20% of dust even when the outflows have traveled to |z| = 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The slightly lower yd at |z| = 1 kpc does not mean that dust is created during its journey from |z| = 1 kpc to |z| = 10 kpc, which is unlikely given the low gas densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Instead, it is likely due to dilution by the entrained ISM which has Zd = ZISM d by defini- tion and is expected to fall back as fountain flows rather than travel to |z| = 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The situation becomes very different once dust growth is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The outflows are now more dusty than the ISM, with yd ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='5 at |z| = 10 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' as the shocked-heated gas in SNRs is “dust-enriched” due to dust growth (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Again, this is analogous to metal enrichment from SN ejecta which leads to metal- enriched outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The effect of beam size Extragalactic observations often have a telescope beam size significantly coarser than 3 pc as adopted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' To understand the effect of beam size, we con- struct images of Σg and Σd for our fiducial model at systematically coarser beam sizes of lb = 3, 6, 12, 24, 48, 96, 192, and 384 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' For example, Σg (6 pc) = � Σg dA � dA = 1 22 4 � i=1 Σg,i (3 pc) , (15) Σd (6 pc) = � Σd dA � dA = 1 22 4 � i=1 Σd,i (3 pc) , (16) and, correspondingly, Z′proj d (6 pc) = Σd (6 pc) Σg (6 pc)Zd,MW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (17) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 7 shows the coarsened images of Σg (upper pan- els) and Z′proj d (lower panels) at t = 240 Myr with lb = 24, 48, 96, 192, and 384 pc from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The compact dense gas with very high Z′ d can be re- solved reasonably well at lb = 24 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As lb increases, this high-Z′ d gas is gradually smoothed out and the DGR is significantly diluted by the diffuse ISM which has a much lower DGR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' At lb = 384 pc, Z′proj d becomes very uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' There is a slight radial gradient of Z′proj d which reflects the large-scale radial distribution of the gas sur- face density, similar to the metallicity gradient in galax- ies caused by metal enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' To be more quantitative, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 8 shows the relation- ship between Σg and Z′proj d at various beam sizes for all snapshots between t = 100 – 500 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As lb in- creases, beam averaging smooths out the dense gas such that both Σg and Σd decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' However, Σd decreases less significantly because Z′proj d is significantly higher at high Σg, shifting the relationship leftward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As a result, the gas surface density above which Z′proj d rises sharply decreases at larger lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The observational implication is that measurements of the Σg – Z′proj d relationship must be compared at a similar beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' At even larger beam sizes (lb ≥ 92 pc), the sharply ris- ing part disappears completely, and the Σg – Z′proj d rela- tionship becomes insensitive to lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This happens when the high-Z′ d dense gas is completely diluted away by the diffuse ISM at large enough lb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The slowly-rising part 1 0 1 log10Zg [Mo pc-2]-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='0 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='8 -i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='6 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='4 -i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2Dust and chemistry co-evolution in dwarf galaxies 13 10 1 100 101 102 103 104 g [M pc 2] 10 2 10 1 Z′ proj d lb = 3 pc lb = 6 pc lb = 12 pc lb = 24 pc lb = 48 pc lb = 96 pc lb = 192 pc lb = 384 pc Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Same with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 5 but only for the fiducial model and with systematically coarser beam sizes (lb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The scatters are not shown for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Beam averaging shifts the relationship to lower gas surface densities as it smooths out the high-DGR dense gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' corresponds to the large-scale radial gradient of Z′proj d as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The Σg – Z′proj d relation is likely to depend on the metallicity and the large-scale gas surface density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We plan to conduct simulations of SMC- and LMC-like galaxies in the future for direct comparison with high- resolution FIR and UV observations such as Roman- Duval et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2017, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' DISCUSSION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Implications for the observationally derived DGR Observationally, the galaxy-integrated DGR is often estimated using Zd = Md MH2 + MH i = Md LCOαCO + MH i , (18) where MH i is the total H i mass from the 21-cm line and Md is the total dust mas from the FIR continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' A major uncertainty is in the adopted αCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As a result, R´emy-Ruyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2014) reported two versions of the metallicity–DGR relation based on two different choices of αCO, one is a constant Milky Way value αCO,MW and the other is metallicity-dependent that scales as Z′−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' While the latter one is more frequently adopted in the literature, our results suggest that the one based on the Milky Way conversion factor is actually more appropri- ate at low metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In fact, the metallicity-dependent αCO strongly overestimates MH2 which in turn underes- timates Zd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Furthermore, our simulations showed, consistent with previous studies in Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2016, 2017), that the molecular mass fraction is extremely small (FH2 ∼ 10−4) in dwarf galaxies with Z′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 such that MH2 con- tributes negligibly to the total gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This is a nat- ural consequence of the long H2 formation time tH2 ∼ 1 Gyr(nZ′ d)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' With our adopted Z′ d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='01, gas at n = 100 cm−2 takes tH2 ∼ 1 Gyr to form H2, which is orders of magnitude longer than the Myr-scale dynam- ical time in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Therefore, the H2 abundance is primarily limited by the dynamical time (Glover & Mac Low 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The situation remains quali- tatively similar even if we adopted Z′ d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=', a linear metallicity–DGR relation) as shown in Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2016) who found FH2 ∼ 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Consequently, the DGR should be observationally estimated simply by Zd = Md/MH i at low metallicity, and the uncertainty in αCO is irrele- vant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Implications for the intergalactic dust Observations of reddening effect suggest that dust ex- ists in the intergalactic medium 20 kpc to several Mpc away from galaxies (M´enard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Peek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2015), whose origin is still poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' One of the possible scenarios is dust entrained in galactic out- flows (Aguirre 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Bianchi & Ferrara 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Kannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2021) conducted simulations of isolated galaxies with similar properties of the Milky Way and LMC cou- pled with a dust evolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' They found that out- flows are able to entrain dust in the fountain flows that circulate around the galaxies within a few kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' How- ever, they found that dust cannot be expelled to the outer part of the halos, which is in contrast to our case where outflows at 10 kpc are still dust-enriched (rather than depleted) and are expected to eventually escape 14 Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' the halo (Hu 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This might indicate that dwarf galaxies are preferable sites to pollute the intergalactic medium with dust as they have low gravitational po- tential wells and the lack of hot gaseous halos around them prevents further destruction via thermal sputter- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' On the other hand, the difference could also arise from numerics as the resolution in Kannan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2021) is 103 M⊙ which was presumably too coarse to resolve the dense gas where dust growth is most efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Re- solved simulations of LMC-like galaxies have recently been conducted by Steinwandel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' (2022b), and a systematic study of outflows across a range of galaxy masses will be valuable to shed light on this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Neglected physics Our dust evolution model does not include dust pro- duction in SN ejecta, which has been observed (Wooden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Indebetouw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2014) and might be major source of dust production in the early Universe when there was not enough time for AGB stars to kick in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' It is still an open question about how much dust will even- tually survive once the reversed shock hits back which depends sensitively on local gas properties (Bianchi & Schneider 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Micelotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Kirchschlager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Priestley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Including dust pro- duction in SNe is unlikely to affect the DGR in dense gas and the ISM chemistry, but it could make the shock- heated hot gas and galactic outflows even dustier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In addition, our dust model assumes a fixed “MRN” grain-size distribution and neglects processes that can vary the grain size such as shattering and coagula- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As the timescales for sputtering and dust growth both depend linearly on grain size, the dust produc- tion/destruction rate would be affected if the actual grain-size distribution deviates significantly from MRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' In addition, the grain-size distribution may also affect the H2 formation rate on dust as well as dust shield- ing (Jonkheid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Romano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Includ- ing the evolution of grain-size distribution is therefore a highly desirable extension for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' SUMMARY We have presented the first ISM-resolved galaxy scale simulations coupled with time-dependent hydrogen chemistry and dust evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Our aim is to understand the ISM chemistry and dust properties at low metallic- ity, a condition expected to be common for galaxies in the very early Universe observed by JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Our simu- lated galaxy is similar to the WLM dwarf galaxy, which has a metallicity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1 Z⊙ and has detected CO(1–0) emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We adopt a mass resolution of 1 M⊙ per gas particle which corresponds to a spatial resolution ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2 pc to properly resolve the compact CO cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' We post-process the simulation snapshots with a detailed chemistry network to accurately model the C+/C i/CO transitions, taking the time-dependent abundances of H2 and H+ from the simulations as input parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Our main findings can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Our fiducial simulation successfully reproduces both the observed SFR and CO(1–0) luminosity (LCO) in the WLM dwarf galaxy (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' LCO can only be reproduced if dust growth in the ISM is included, as otherwise dust shielding would be insufficient to protect CO from being photodissoci- ated by FUV radiation, suppressing LCO by more than two orders of magnitude (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The predicted total H2 fraction is extremely low (∼ 10−4) either with or without dust evolution due to the long H2 formation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' This leads to very little CO-dark H2 gas and a CO-to- H2 conversion factor (excluding helium) αCO = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='63 M⊙ pc−2 (K km s−1)−1 which is very close to the Milky Way value despite the low metallic- ity (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Observationally inferred dust-to- gas ratio (DGR) is underestimated if assuming a metallicity-dependent αCO at low metallicity (Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust growth is the primary driver of the spatial variation of DGR in the ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dust growth signif- icantly increases the DGR in cold, star-forming clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Subsequent SN feedback disperses the high-DGR gas without significantly destroying the dust, leading to elevated DGR in the diffuse gas associated with supernova remnants, qualitatively similar to metal enrichment (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4 and 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' As a result, galactic outflows are about 20 – 50% dustier than the ISM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Dwarf galaxies therefore could be a source of the intergalactic dust (Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The projected DGR increases with gas surface density due to dust growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' The relationship be- tween these two quantities varies with the tele- scope beam size as coarse beams smooth out the high-DGR gas clumps (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' Observa- tional measurements should therefore be compared at a similar beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' ACKNOWLEDGMENTS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' acknowledges support from the DFG via German-Israel Project Cooperation grant STE1869/2- 1 GE625/17-1 and NASA ATP grant 80NSSC22K0716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' thanks the Center for Computational Astrophysics Dust and chemistry co-evolution in dwarf galaxies 15 (CCA) of the Flatiron Institute, and the Mathemat- ics and Physical Science (MPS) division of the Simons Foundation for support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' All simulations were run on the Raven and Cobra supercomputers at the Max 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content=' 2018, ApJ, 857, 94, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} +page_content='3847/1538-4357/aab438' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rNE4T4oBgHgl3EQfwA2r/content/2301.05247v1.pdf'} diff --git a/sdFKT4oBgHgl3EQfJy1E/content/tmp_files/2301.11739v1.pdf.txt b/sdFKT4oBgHgl3EQfJy1E/content/tmp_files/2301.11739v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f10170eca71d67c7c86b79994e437c24405299d9 --- /dev/null +++ b/sdFKT4oBgHgl3EQfJy1E/content/tmp_files/2301.11739v1.pdf.txt @@ -0,0 +1,1407 @@ +Special Session: Noisy Intermediate-Scale +Quantum (NISQ) Computers—How They Work, +How They Fail, How to Test Them? +Sebastian Brandhofer1 +Simon Devitt2 +Thomas Wellens3 +Ilia Polian1 +1University of Stuttgart and Center for +Integrated Quantum Science and Technology +Stuttgart, Germany +{sebastian.brandhofer | ilia.polian} +@informatik.uni-stuttgart.de +2University of Technology Sydney +Sydney, Australia +simon.devitt@uts.edu.au +3Fraunhofer Institute for +Applied Solid State Physics IAF +Freiburg, Germany +thomas.wellens@iaf.fraunhofer.de +Abstract—First quantum computers very recently have demon- +strated “quantum supremacy” or “quantum advantage”: Execut- +ing a computation that would have been impossible on a classical +machine. Today’s quantum computers follow the NISQ paradigm: +They exhibit error rates that are much higher than in conventional +electronics and have insufficient quantum resources to support +powerful error correction protocols. This raises questions which +relevant computations are within the reach of NISQ architectures. +Several “NISQ-era algorithms” are assumed to match the specifics +of such computers; for instance, variational optimisers are based +on intertwining relatively short quantum and classical computa- +tions, thus maximizing the chances of success. This paper will +critically assess the promise and challenge of NISQ computing. +What has this field achieved so far, what are we likely to achieve +soon, where do we have to be skeptical and wait for the advent +of larger-scale fully error-corrected architectures? +Index Terms—Quantum Computing, NISQ Computing, Error +Simulation, Error Tolerance Analysis, Error Characterisation +I. INTRODUCTION +In spite of classical computing technology being on an expo- +nential trajectory since 1960s, quantum computing (QC) contin- +ues to fuel the fantasies of scientists and practitioners alike. QC +promises asymptotic, and in some cases exponential, speedups +for hard problems from domains such as cryptography, compu- +tational chemistry, simulation, or machine learning. Theoretical +results on quantum algorithms have been complemented by a +development of actual quantum hardware potentially capable +of practically executing quantum computations, even though +the demonstrated systems were of limited size and could be +simulated on a classical computer within a meaningful time. +The “quantum supremacy” experiment [1] showed compu- +tations on a 53-qubit (quantum bit) machine that the authors +argued would be intractable on a classical compute server. Even +though the latter argument is currently disputed [2], [3], as +©2021 IEEE. Personal use of this material is permitted. Permission from +IEEE must be obtained for all other uses, in any current or future media, +including reprinting/republishing this material for advertising or promotional +purposes, creating new collective works, for resale or redistribution to servers +or lists, or reuse of any copyrighted component of this work in other works. +DOI: 10.1109/VTS50974.2021.9441047. +other authors are proposing more efficient classical solutions +for the problem of [1], this problem serves no practical or sci- +entific purpose other than demonstrating “quantum supremacy” +(or “quantum advantage”). Using QC for solving a practical +problem faster than any classical computer is still open at the +time of writing. +One key challenge on the road to quantum advantage is +increasing the number of qubits that a quantum computer can +calculate on, as the asymptotic speedups naturally unfold their +effects on larger problem instances. Less obvious but not less +important, the quality of the available qubits and operations on +them (quantum gates) is essential for meaningful computations. +Quantum states are fragile, and even error rates reported for the +hardware in [1] (which are among the best implementations +that exist today) were orders of magnitude larger compared +to conventional electronics. These error rates are expected to +improve in the future, but not come close to the nearly error- +free operation of classical computers. +When it comes to the near future of quantum computing +given the non-trivial error rates of its hardware, two schools +of thought exist with. The first is to deploy quantum error +correction (QEC), where several poor-quality physical qubits +together constitute a logical qubit with a more reasonable +robustness. The idea is not unsimilar to processing encoded +information in self-checking design; however, state-of-the-art +QEC schemes [4] have an overhead of hundreds or thousands +physical qubits per logical qubit. The second approach, known +under the name “NISQ”, suggests to accept the inherent noise +as given and try to find applications that can survive it without +resorting to fully-fledged QEC. Some classes of algorithms +are being investigated specifically with NISQ computers in +mind [5]. Near-term NISQ computers are also the focus of this +paper. +Since “noise” is an integral part of NISQ, understanding its +origins, characterising its properties, quantifying its magnitude, +and assessing its application-level impact are of utmost impor- +tance. To this end, this paper will be organised as follows. After +some background information about QC in Section II, Section +arXiv:2301.11739v1 [quant-ph] 27 Jan 2023 + +III will introduce the basic NISQ concepts and revisit the +expectations raised since its inception. Section IV will discuss +future developments in QC, going beyond NISQ. +Section V will put today’s successful demonstrations of +the NISQ principle for small instances of practically relevant +problems into perspective. For example, certain computational +chemistry tasks have been solved for small molecules, such as +the hydrogen atom, for which classical solving methods work as +well. What would be needed to extend such approaches to more +complex tasks? Section VI will explain what role characterisa- +tion play in making NISQ computers useful. The approaches +go far beyond the pass-fail tests used for conventional CMOS +circuits. Individual qubits and ensembles of entangled qubits +must be characterised using techniques such as randomised +benchmarking [6] or gate set tomography [7], [8]. Section VII +will conclude the paper. +II. QUANTUM COMPUTING BACKGROUND +A quantum computer can perform computations on n qubits, +where one qubit might be, for instance, an ion, a photon, +or a solid-state transmon circuit. An individual qubit can be +set to two basis states |0⟩ and |1⟩, which correspond to two- +dimensional column vectors (1, 0)T and (0, 1)T, respectively. +A single qubit assumes a state, which can be |0⟩, |1⟩, or their +superposition α0|0⟩+α1|1⟩ = (α0, α1)T, where α0 and α1 are +complex numbers and |α0|2 + |α1|2 = 1. An ensemble of n +qubits assumes a state described by a 2n-dimensional complex +vector of amplitudes (α0...00, α0...01, . . . α1...11)T, where the +indices of α are all n-bit combinations of 0s and 1s and again +|α0...00|2 + |α0...01|2 + · · · + |α1...11|2 = 1. +A quantum gate G acting on k qubits is described by a +2k × 2k complex unitary matrix UG. Examples of quantum +gates (some of which play a key role in modeling errors) along +with their symbols and matrices are shown in Fig. 1. Note +that all quantum gates are invertible and therefore have the +same number of inputs and outputs. Gates acting in parallel on +different qubits are combined using tensor product; gates acting +sequentially can be combined using matrix multiplication. +Fig. 1 includes an example circuit; let us follow its operation +if the first and the qubit are initialised in states |0⟩ and |1⟩, +respectively: +|ϕ1⟩ += +|01⟩ = +� +1 +0 +� +⊗ +� +0 +1 +� += +� +� +� +� +0 +1 +0 +0 +� +� +� +� +|ϕ2⟩ += +1 +√ +2 +� +1 +1 +1 +−1 +� +⊗ +� +0 +1 +1 +0 +� +|ϕ1⟩ += +1 +√ +2 +� +� +� +� +0 +1 +0 +1 +1 +0 +1 +0 +0 +1 +0 +−1 +1 +0 +−1 +0 +� +� +� +� +� +� +� +� +0 +1 +0 +0 +� +� +� +� = +� +� +� +� +1/ +√ +2 +0 +1/ +√ +2 +0 +� +� +� +� +|ϕ3⟩ += +� +� +� +� +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +1 +0 +� +� +� +� +� +� +� +� +1/ +√ +2 +0 +1/ +√ +2 +0 +� +� +� +� = +� +� +� +� +1/ +√ +2 +0 +0 +1/ +√ +2 +� +� +� +� +Hadamard +1 +√ +2 +� +1 +1 +1 +−1 +� +Pauli-X +� +0 +1 +1 +0 +� +Pauli-Y +� +0 +−i +i +0 +� +Pauli-Z +� +1 +0 +0 +−1 +� +Controlled-NOT +� +� +� +� +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +1 +0 +0 +1 +0 +� +� +� +� +Example circuit: +Fig. 1. Some important quantum gates and an example quantum circuit. +Note that state |ϕ2⟩ corresponds to the system being in superpo- +sition of |00⟩ and |10⟩, which can be attributed to its individual +qubits: qubit 1 is in the superposition state (0⟩ + |1⟩)/ +√ +2, and +qubit 2 is in state |0⟩ (the Pauli-X gate acts as an inverter). +In contrast, state |ϕ3⟩ is a superposition of |00⟩ and |11⟩ and +cannot be expressed by the individual qubits; this phenomenon +is known as entanglement. +It sounds quite unspectacular that most of a quantum com- +puter’s operations are multiplying matrices (of its gates) by the +state vector. It seems a lot more spectacular that these vectors +and matrices have an exponential size in the number n of actual +physical objects, i.e. qubits. One would expect radical speed- +ups from a system that is seemingly able to hold 2n intermedi- +ate results in the state vector (α0...00, α0...01, . . . α1...11)T and +applying computations to all these results at once. The reality +is more intricate: we cannot read out the values of the state +vector directly, but can only perform a measurement on it. +Measuring a qubit will result in either outcome 0 or 1. If a +qubit’s state was |0⟩ or |1⟩, the measurement will deterministi- +cally yield 0 or 1. However, if a qubit is in a superposition +α0|0⟩ + α1|1⟩ = (α0, α1)T, the outcome will be |0⟩ with +probability |α0|2 and |0⟩ with probability |α1|2. For example, +measuring the second qubit in state |ϕ2⟩ in the example above +will yield |0⟩ with certainty, and measuring the first qubit +will result in |0⟩ or |1⟩ with probability |1/ +√ +2|2 = 1/2 +each. The measurement is destructive: after the measurement, +the state of this qubit will change to the state corresponding +to the measurement outcome, and all non-trivial information +about α1 and α2 will be lost. An interesting effect happens in +entangled states like |ϕ3⟩: measuring one qubit can determine +the value of the other. For example, if measuring the first qubit +in |ϕ3⟩ = (|00⟩ + |11⟩)/ +√ +2 yields |0⟩ (this happens with 50% +probability), then the 2-qubit state collapses to |00⟩; the next +measurement of qubit 2 will result in |0⟩ with certainty. +The destructive character of measurements prevents the +seemingly obvious parallelism. It is not possible to obtain +the exact values αj by copying the end-state of the circuit +and measuring it multiple times. It is possible to re-run the +circuit multiple times from the beginning, but the required + +number of repetitions would offset any gain from the paral- +lelism. Instead, quantum algorithms employ steps to transform +(entangled) quantum states such that measurements yield useful +information. +The description so far had assumed a perfectly working +quantum computer. Real-world hardware is prone to distur- +bances due to noise, interaction with the environment, and +imperfections of the physical apparatus [9]. One can distinguish +different types of failures. Pauli noise is modeled by Pauli-X, +Y and Z gates (Fig. 1) randomly added to different locations +within the circuit. Somewhat counter-intuitive, this discrete +fault model captures parametric (large and small) errors in the +individual qubit states; this is because a qubit is measured +at some point and even a small error either manifests itself +(this is captured by Pauli errors) or not (then it has no effect). +Frequently used special cases of Pauli noise are depolarizing +noise, where Pauli-X, Y or Z gates are applied with uniform +error rates independently to each qubit, and dephasing noise, +where only Pauli-Z gates are randomly applied within the +circuit. +Independent Pauli errors do not cover correlated errors that +can be generated by effects such as physical control line +crosstalk between qubits, but these effects can be bounded +as joint Pauli errors occurring at the same time across the +relevant qubits - the error is often bounded to weight two Pauli +errors as the underlying physics of qubit systems is limited to +pairwise Hamiltonian interactions. Quantum error correction +(QEC) protocols make use of heavily entangled states to form +highly redundant “logical qubits”, such that errors affecting one +or few physical qubits are compensated by error-unaffected +redundant physical qubits within the same entangled state. +However, NISQ computers follow a different route. +III. DEFINING NISQ, AND HOW USEFUL WILL THE NISQ +ERA BE? +In terms of quantum hardware, the necessary components to +build a scalable quantum computer (in a variety of different +systems, including superconductors) are well known [10]–[13]. +At a minimum, a hardware device requires a regular array of +qubits arranged in a square 2D grid. Each qubit needs the ability +to initialise in a known state, perform arbitrary single qubit +operations, be able to be measured and be able to couple to the +four neighbouring qubits in the array. All of these fundamental +operations need to occur with an error rate of 0.1% or lower +[4]. This is the bare minimum for a scalable system. +As anticipated by many, the first realisation of quantum +computing technology has occurred over the cloud, with users +logging onto dedicated hardware over the classical internet. +These types of ‘quantum in the cloud’ systems began with +the connection of a two-qubit photonic chip to the classical +internet by the University of Bristol in 2013 and accelerated +significantly in 2016 with the introduction of IBM of their +quantum experience platform. We now see both free and paid +services offered by IBM, Microsoft, Amazon, Xanadu and +Rigetti, across a variety of hardware modalities for small scale +quantum computing chipsets up to 65 physical qubits. This has +spurred the so-called era of noisy intermediate-scale quantum +(NISQ) [14] research. +Noisy intermediate-scale quantum also refers to quantum +algorithms that are small enough to be faithfully executed on +near term, low qubit count, high error rate, quantum hardware +without the need for resource-costly error correction protocols. +While NISQ algorithms exist in principle—quantum supremacy +is a quintessential example [1]—an added caveat is that the +algorithm needs to be either scientifically and/or commercially +viable—quantum supremacy is not. Consequently NISQ al- +gorithms must be highly compact but still either reach the +quantum supremacy regime—where the problem simply cannot +be solved on any classical computer—or reach the regime +where it is more cost efficient—either in terms of actual dollars +or in terms of computational time—to run the algorithm on a +quantum computer. NISQ algorithms satisfying the commercial +or scientific viability condition are under active research, but +do not currently exist. +The actual definition of NISQ computation is somewhat +nebulous and depends on who you ask. However, to a first +approximation we can examine NISQ compatibility by calcu- +lating for a given algorithm the value A = n · d, called area, +where n is the number of qubits the algorithm needs and d is the +number of time-steps in the algorithm. If the physical error rate +of the qubits and gates in a physical quantum computing chip +is p < 1/A, the algorithm is likely NISQ compatible. This is +due to the error sensitivity of quantum algorithms. A quantum +computation is akin to an optical interferometer and its ability +to perform efficient computation comes about through a delicate +interference effect of the computational wavefunction as gate +operations are applied. As with physical interferometers, where +a minor misalignment can completely disrupt the interference +effect and make the device non-functional, even one error in +a quantum algorithm can disrupt the required computational +interference effect, resulting in incorrect output. It has been +demonstrated for a variety of quantum algorithms that single +errors can dramatically reduce the probability of success of +generating the correct output [15], [16]. This makes p < 1/A— +which represents the error rate needed such that one error +occurs during execution on average—a good bound for the error +tolerance of an algorithm. +The difficulty is finding a quantum algorithm that is comfort- +ably in the regime where classical algorithms simply cannot be +used at the same time as still being small enough to satisfy the +p < 1/A bound which defines when error correction or other +mitigation techniques are needed. The quantum supremacy +formalism that was used by the Google Sycamore processor +[1] are, by definition, the smallest quantum circuits known that +are provably difficult to simulate on a classical computer. They +were constructed explicitly for this purpose—this is why they +are not practically useful beyond demonstrations of quantum +supremacy. The Google team simulated a 53-qubit circuit over +a depth of approximately 40, giving A ≈ 2100. Even this +circuit was ‘border-line’ in terms of being implementable on +classical hardware. A preprint paper in early 2021 has in +fact presented results that replicated the Google experiment +using new techniques in classical algorithms based on tensor + +contraction [3]. +A circuit with A ≈ 2100 would therefore require physical +errors in the quantum processor of p < 5×10−4 to occasionally +run without errors. Not even the Google Sycamore processor +had this level of accuracy. Single and two-qubit gates in +the experiment had, on average, error rates of approximately +0.1-1% while measurement gates had errors up to 5% for +simultaneous readout [1]. +Hardware errors for a variety of quantum processors have +decreased from of order 10-50% when basic qubit operations +were first demonstrated in the late 1990’s and early 2000’s +to, of order, 1-0.1% today (over two decades later). This +is extremely impressive from an experimental point of view, +and the ability to replicate the fabrication of low error rate +quantum chips is getting better and better. However, given +that the smallest quantum circuit that has provable quantum +advantage—that is commercially and scientifically useless— +requires physical error rates at least another two orders of +magnitude lower to be unambiguously achievable in a quantum +processor, it is difficult to accept the prospect that an algorithm +that can be implemented with higher physical error rate that +can simultaneously revolutionise a scientific discipline or a +commercial industry is possible. The only other option is +that experimental progress in reducing physical error rates in +chip-sets accelerates rapidly, such that the next two orders of +magnitude in error improvement occurs over the next two or +three years rather than the next two or three decades. +Consequently, NISQ currently has no identified, commer- +cially relevant application at the 50-1000 qubit level. There are +also good reasons to believe that finding such applications (in +an era where error rates on quantum chips will be of the order +of 0.1% – 1% at best) will not be possible. However, there are +no results that prove that the existence of these algorithms are +impossible, so there is still strong motivation to keep searching. +This search should be done in a way that addresses these +fundamental issues rather than sidestepping them, or in some +unfortunate cases, simply obfuscating the problems. +IV. WHAT HAPPENS BEYOND NISQ? +Beyond the NISQ era is one of two possibilities. Either +physical error rates in quantum computing chips are drastically +reduced—via a combination of better fabrication physics, better +quantum control and passive error mitigation techniques [17]— +or active techniques are employed to reduce error rates which +fall under the umbrella of Quantum Error Correction (QEC) [9]. +QEC uses redundant qubit encoding, active measurement and +feedback to stabilise quantum information over long periods +of computational time. It is a highly successful theoretical +discipline and is unarguably one of the main pillars of quantum +information science. QEC is commonly combined with the +method of fault-tolerant circuit design, where error correction +protocols and logic operations are performed in a manner +such that physical errors do not ‘cascade’ out of control. The +combination of these two techniques lead to one of the most +important results in quantum computing, the threshold theorem. +The threshold theorem for fault-tolerant quantum computing +[18] states that with a polylogarithmic overhead of physi- +cal resources—qubits and time—an arbitrarily long quantum +circuit can be implemented provided the physical errors of +the hardware are less than a critical value, pth, known as +the fault-tolerant threshold. Currently, the threshold for the +most commonly used QEC scheme for large-scale architecture +design, the surface code, is pth ≈ 0.6%. This means that several +hardware platforms already have reached physical error rates +satisfying the fault-tolerant threshold for QEC in a subset of +operations, although it should be noted that no hardware system +has yet demonstrated error rates below threshold for a universal +set of gate operations on a single device. +The unfortunate issue with QEC is the drastic increase in +physical qubit resources once it is even implemented to a +minimal degree. Once error correction is incorporated into an +algorithm, the total number of physical qubits required in the +quantum chipset can easily jump from 50-100 qubits to 50,000- +100,000. This includes both the raw overhead in performing +the qubit encoding and ancillary physical resources to maintain +fault-tolerance in operational logic. This would only be for a +small amount of error correction, enough to reduce error rates +from approximately 0.1% on the physical level to the order of +0.0001% at the encoded level. This number of physical qubits is +just not possible to engineer in the near future, in any hardware +system. +The error correction overhead for large-scale algorithm, such +as Shor’s factoring algorithm or quantum simulation is very +high. However, two points should be strongly emphasised. +• Factoring or quantum simulation are extremely large al- +gorithms, even if they scale polynomially with problem +size. +Given the error sensitivity of quantum algorithms, this +means that the effective error rate required for encoded +qubits can be of the order of 10−13% or lower. Hence, +QEC needs to provide at least 12 orders of magnitude of +error improvement. This massive amount of suppression +does not come for free. +• Classical techniques for circuit optimisation both at the +algorithmic level and at the QEC level has already done an +extraordinary job at reducing the overhead. For example, +in 2012, an explicitly compiled, error-corrected analysis of +Shor’s algorithm estimated that of the order of 100 billion +qubits would be needed to break RSA-2048 encryption +[19]. By 2019, this number had been reduced to 20 million +[20], a reduction by a factor of 5,000 without changing any +of the operating assumptions of the underlying hardware. +Ultimately, the demands required on hardware accuracy for +large-scale algorithms will necessitate active error correction +in the long term. How far experimental improvements in +fabrication, control and passive error mitigation can go before +active techniques need to be implemented is up for debate, but +there will be (and may already be) decisions made to the most +cost effective and expedient way to move in the future. Either +invest in reducing physical error rates by orders of magnitude +or invest in being able to make chip-sets containing very large +numbers of integrated qubits quickly and cheaply. + +V. ANALYSING NISQ COMPATIBILITY OF ALGORITHMS +Recent progress in quantum computing technologies [1], [5], +[21], [22] offers quantum computational resources that have not +been available before. Although only noisy and intermediate- +sized quantum computational resources can be exploited by +current NISQ computers, recent experiments yield results that +are challenging to obtain using classical resources [1], [22]. In +the light of these experiments, increasing quantum resources of +NISQ computers and growing business adaptation, it is crucial +to determine the resource requirements of a computation. +Assessing these requirements determines the set of suitable +NISQ computers, can help to improve the implementation of a +computation, and informs whether quantum error correction is +necessary. +The resource requirements of a computation consist of the +number of qubits, the depth and the tolerable error rate. The +first two requirements can be computed by inspection [23]. +The tolerable error rate can be determined through approaches +ranging from guidelines on the size of the quantum computa- +tion, simulations subject to errors, and experiments on quantum +computers to reliability models of quantum computers based +on machine learning. The determination of the tolerable error +rate must be able to consider noise sources ubiquitous in NISQ +computers, arbitrary quantum computations, flexible definitions +of success and must be able to make statements about a wide +range of quantum computing technologies. +In [14] and [24] bounds on the quantum computation size +and error rate p for successful computations are described. +The work in [14] states that the error rate may not be much +larger than G−1, with G denoting the number of quantum gates +constituting the quantum computation. The work in [24] states +the observation +p < (n · d)−1 +(1) +with n qubits used in a computation that has depth d. These +guidelines provide a rough estimation and are not suitable +to distinguish between different success criteria or quantum +computations that have the same size but exhibit a different +error behaviour. +A more accurate approach to determine the tolerable error +rate of a quantum computation is to conduct simulations subject +to errors [15], [16], [25]–[31]. Using Monte Carlo or exhaus- +tive error simulations, the error rate at which the quantum +computation starts failing the target success criteria can be +determined. Here, the employed error model significantly con- +tributes to the accuracy, simulation effort and generality of the +determined tolerable error rate. If an error model is employed +that is fitted to the dynamic adverse physical processes (noise +processes) in a specific quantum computer, the determined +error rate requirement can often not be generalized to other +quantum computers [32] or quantum computing technologies. +Alternatively, if the employed error model is too general, it may +not represent noise processes sufficiently well and thus lead to +incorrect predictions. While simulations subject to errors are +a suitable choice to determine the tolerable error rate, they +also incur an exponential runtime and space overhead in the +number of qubits in general. It is therefore crucial to reduce +the simulation effort by a suitable choice of error model and +simulation techniques [15]. +Another option is to probe the success probability of a +quantum computation on a target quantum computer. While this +can be performed efficiently, it is challenging to generalise the +results to other quantum computers of the same generation due +to the dynamic error rates of NISQ computers, and the results +are not valid for other quantum computing technologies. In +addition, a computation is performed on a quantum computer +with respect to a specific error rate that can only be increased +in a limited way [33]. Thus, the error rate requirement of a +quantum computation can not be determined in general since +only a small range of error rates can be probed. +Reliability models for quantum computers based on machine +learning have also been proposed [34]. These models are trained +on a specific quantum computer and uses features such as quan- +tum computation size or structure, auxiliary experiments and +success probabilities of previous quantum computations to yield +a prediction about the success probability of future quantum +computations. While these methods exhibit a high prediction +accuracy for some quantum computers and computations, these +approaches have not been demonstrated to generalise well over +multiple quantum computers or different quantum computing +technologies. +Thus, for determining the error rate requirement of a quan- +tum computation in the NISQ era, simulations subject to +errors is a suitable choice. Arbitrary noise sources, quantum +computations and success criteria can be considered with +simulations. While the runtime overhead is exponential in the +number of qubits, there currently is only a small number of +publicly available NISQ computers that can not be simulated +in reasonable time. We will now give details about commonly +used success criteria, simulation methods, error modeling and +results for a set of small-scale quantum algorithms. +A. Success Criteria +The success of a quantum circuit computation is determined +depending on the available reference information of the ideal +result. In the simplest case, when the target state is known +and exact probability amplitudes are available upon simulation, +the fidelity measure F(|ψt⟩ , |ψe⟩) is often used to quantify +how much an erroneous state |ψe⟩ deviates from a target +state |ψt⟩ [35]. However, this is often not applicable as quan- +tum circuit computations on a quantum computer only return +measurement results and the target state is often not known. +Therefore, a number of success criteria were proposed and used +in previous works such as [36]: +• The measurement probability of the correct result in single +outcome quantum circuits is above a certain threshold. +• Measurements are in a set of acceptable results with high +probability larger than a threshold. Such a set can e.g. be +defined by the Hamming distance or the heavy output [37]. +• The deviation between the measurement output distri- +bution and the expected distribution is smaller than a +threshold according to some measure of distance [36]. +• The measurement result probabilities are consistent with +predictions, i.e. the cross-entropy of results is low [38]. + +Since quantum circuit computations on a quantum computer are +subject to errors and sampling noise, above probabilities can +not be determined with certainty [36]. It is therefore essential +to also consider a metric of confidence for reaching a defined +success criteria. In addition, applying a success criterion to +a quantum computation may not make the results classically +tractable. +B. Simulation Methods +Simulating a quantum circuit incurs an exponential runtime +and space overhead in general [35]. However, specific quantum +circuit structures and properties can be exploited by simulators +based on tensor networks [39], matrix product states [40], de- +cision diagrams [41] or stabilizers [42], [43]. These simulators +exhibit a better runtime or memory requirement for specific +quantum circuits than general simulators based on the state +vector or density matrix. +In general, simulating a n-qubit state requires a state vector +to store and manipulate 2n complex amplitudes whereas the +density matrix simulator has to store and manipulate 2n n- +qubit states. Simulating 53-qubit quantum circuits such as the +quantum circuit in Google’s quantum supremacy experiment +using a state vector or density matrix simulator imposes a large +memory requirement: 72.1 petabytes and 6.49 · 1017 petabytes +would be required for the state vector and density matrix +simulator respectively, if one complex number is stored using +two double-precision floating-point numbers. +C. Error Modeling +Noise in NISQ computers is ubiquitous and affect the compu- +tation of a quantum circuit in various ways. In general, a qubit +can be subject to coherent or incoherent noise, be lost, leak +out of the computational space, or be erroneously initialised +or read out [9]. These effects can be modeled by device- +oriented or device-agnostic error models. In a device-oriented +error model, the adverse physical processes in a quantum +computing device are replicated to model the errors affecting +a computation. Device-agnostic error models, however, consist +of a set of operators that cover the effects of relevant noise +sources. The choice of error model has consequences for the +prediction accuracy, applicability of results, quantifiability of +the model, and simulation effort of simulations subject to these +error models. +A device-oriented error model fitted to one specific quan- +tum computer would be expected to yield the best prediction +accuracy. However, the results of simulations subject to that +error model would not be applicable to different quantum +computers or quantum computing technologies. Furthermore, +estimating each error parameter of such an error model through +characterisation protocols can be complex, as described in more +detail in Section VI. Device-oriented error models also incur +a large simulation effort since they require the density matrix +formalism in general. +A device-agnostic error model is applicable to diverse quan- +tum computers, if the noise parameters are quantified on them +using characterisation protocols. Thus, if it is known under +which noise parameters the success criterion of a quantum +circuit computation is met, we can determine suitable NISQ +computers that do not exceed the determined noise parameter. +Recently, quantum supremacy experiments confirmed that the +device-agnostic Pauli error model is sufficiently accurate to +make predictions about the success probability [1]. The Pauli +error model also only incurs a low simulation effort for arbitrary +quantum circuits since the error operators constituting the error +model are in the Clifford set and can be simulated using the +state vector or stabilizer simulator. +D. Results +In this section, we show results on various quantum circuit +with up to 16 qubits and 214 gates subject to the Pauli +error model with uniform error rates (depolarizing noise). The +tolerable Pauli error rate given a success probability of 66% +and the success probability given a Pauli error rate of 0.15% +are reported. The tolerable Pauli error rate specifies the error +rate at which a quantum computation can still be performed +successfully given a target success criterion and probability. +The Pauli error rate of 0.15% was chosen as this is currently +the lowest (single-qubit gate) error rate exhibited by one of +the largest NISQ computer [1]. The evaluated quantum circuits +implement the Grover algorithm [44], arithmetic functions [45], +Bernstein-Vazirani (BV) algorithm [46] (using CNOTs), the +quantum Fourier transform (QFT) [47] and the hidden linear +function (HLF) [48]. Furthermore, and quantum circuits for +chemistry applications (RYRZ [5], UCCSD [49]) were evalu- +ated. +The evaluated quantum circuits were generated using Qiskit +[50] for a general quantum computer with single-qubit rota- +tions, the controlled-NOT two-qubit gate and without geometric +constraints. Geometric constraints imposed by many quan- +tum computing technologies such superconducting qubits [1] +decrease the herein reported tolerable Pauli error rate and +success probability since satisfying these constraints require the +insertion of additional error-prone operations. +The evaluated success criterion was the measurement prob- +ability of the correct result for single outcome algorithms such +as BV, and fidelity for all other. +The quantum circuits were simulated subject to the Pauli +error model using a state vector simulator [16]. For combina- +tions of quantum circuits and Pauli error rate where less than +one Pauli error occurs during the quantum circuit computation +on average, an exhaustive error simulation was employed. For +this exhaustive error simulation, one Pauli X, Z, or Y error +was simulated successively at every space-time location in the +quantum circuit and the impact of these errors on the target +success criterion was scaled according to the specified Pauli +error rates [15]. For combinations of quantum circuits and Pauli +error rates that incur more than one Pauli error per quantum +circuit computation on average, Monte Carlo simulations were +conducted to obtain the success probability and tolerable Pauli +error rate. +1) Success Probability: Fig. 2 shows the success probability +of the evaluated quantum circuits at a Pauli error rate of 0.15%. +On the x-axis, the area of the evaluated quantum circuits is + +0 +100 +200 +300 +400 +500 +600 +Quantum Circuit Area +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Success Probability +P = 0.66 +Arithmetic +BV +Grover +HLF +QFT +RYRZ +UCCSD +Fig. 2. Success probability of quantum circuits subject to a Pauli error rate of +0.15% and no geometric constraints. +shown. All quantum circuits with an area smaller than 322 +could be executed with a success probability of at least 66%. +All evaluated BV and UCCSD quantum circuits could be +computed successfully. For other quantum circuits and mod- +erate quantum circuit sizes, the probability of successfully +computing the quantum circuit quickly falls below 66%. These +results show that the quantum circuit area is a good indicator for +determining the success probability for many of the evaluated +quantum circuits. +2) Tolerable Pauli Error Rate: Fig. 3 depicts the tolerable +Pauli error rate of the evaluated quantum circuits for a success +probability of 66%. The x-axis shows the quantum circuit area. +Both axes are in log-scale. The bound of the error rate on the +quantum circuit area (n · d)−1 stated in the literature [24] is +also shown as a black line. +All evaluated quantum circuits only tolerate a Pauli error +rate that is smaller than the inverse of their quantum circuit +area. There are small difference in the tolerable Pauli error +rate between quantum circuits implementing different quantum +algorithms. Quantum circuits for arithmetic functions tolerate a +Pauli error rate that is mostly the average of evaluated quantum +circuits with the same quantum circuit area. BV, RYRZ and +QFT quantum circuits tolerate a slightly larger Pauli error rate +and HLF, UCCSD and Grover quantum circuits only tolerate +a slightly smaller Pauli error rate. The tolerable Pauli error +rate of HLF quantum circuits constitute a lower bound for the +evaluated quantum circuits. +These results confirm the bound given in [24] and indicate +that the evaluated quantum circuits do not tolerate one Pauli +error on average. The tolerable Pauli error rate is mainly +dominated by the area of a quantum circuit; large difference +due the structure of a quantum circuit or other properties were +not observed. +VI. CHARACTERISING THE UNDERLYING HARDWARE +Above, we have seen the important impact of physical error +rates onto the applicability of quantum error correction proto- +cols and NISQ algorithms. In this section, we will therefore +102 +Quantum Circuit Area +10 +3 +10 +2 +Tolerable Pauli Error Rate +(area) +1 +Arithmetic +BV +Grover +HLF +QFT +RYRZ +UCCSD +Fig. 3. Tolerable Pauli error rate of the evaluated quantum circuits for a success +probability of 66%. +present some of the most commonly used methods for actu- +ally measuring the error rates. These methods do not require +any knowledge about how the quantum gates are physically +realised. Instead, the basic idea is the following: we execute +suitably chosen quantum circuits on the given hardware and +count the measurement outcomes. The desired information con- +cerning the errors is then extracted by comparing the measured +outcomes with the outcomes expected in the absence of errors. +A. Modeling Errors in Quantum Circuits +To introduce these methods, we first explain some basic +concepts concerning the mathematical modelling of errors. In +section II, we have introduced the state of a qubit as a two- +dimensional complex vector |φ⟩ = (α0, α1)T. Such a state is +called a pure quantum state. More generally, however—and +especially so in the presence of errors—the qubit may be in a +mixed state, which is described by a 2 × 2 density matrix: +ρ = +� +ρ00 +ρ01 +ρ10 +ρ11 +� +with ρ = ρ† (self-adjoint; the symbol † denotes transposition +and complex conjugation), Tr(ρ) = 1 (normalised) and ρ ≥ 0 +(positive semi-definite). In the special case of a pure state φ, +we have +ρ = |φ⟩⟨φ| = +� +|α0|2 +α0α∗ +1 +α∗ +0α1 +|α1|2 +� +, +where ⟨φ| = (α∗ +0, α∗ +1). In a similar way, states of n qubits are +described by 2n × 2n dimensional density matrices. +If a unitary operator (or quantum gate) U is applied to the +qubits, their density matrix changes according to +ρ′ = EU(ρ) := UρU †, +where U †U = 1 +If ρ is a pure state, then ρ′ is also a pure state. For this reason, +there is no need to consider density matrices if the quantum +computer exhibits no errors and all gates are perfectly unitary. + +In the presence of errors, however, the resulting quantum oper- +ation assumes the following general form (completely positive, +trace preserving map) [51]: +ρ′ = E(ρ) = +� +j +AjρA† +j, +where +� +j +A† +jAj = 1 +This can be interpreted as follows: instead of a single well- +defined unitary operation U, one of several possible operators +Aj is applied to the qubits with probability pj = tr(AjρA† +j). +As an example, the depolarising channel for a single qubit is +defined by A0 = √1 − p 1, A1 = +� p +3X, A2 = +� p +3Y and +A3 = +� p +3Y . In other words, with probability 1−p (in this case +independent of ρ), the state ρ remains unchanged (no error), +whereas an error amounting to the application of Pauli-X, Y +or Z occurs with probability p/3 each. +It can be shown [51] that the sum over j in the above general +representation of a quantum operation can be restricted such as +to contain at most 4n different terms. Thereby, a noisy quantum +operation is defined by up to 4n different 2n × 2n matrices Ai +(called Kraus operators), or—equivalently—by a single 4n×4n +matrix E (sometimes called superoperator). +There are several ways for defining the error rate of a +quantum operation. Suppose we want to implement the unitary +quantum gate U, but realise E instead. Then, we define the +corresponding error operator by Λ = (EU)−1 ◦ E (i.e. the +inverse of the desired operation EU is applied after E, such that +Λ = 1 in the absence of errors). For a given initial pure state +ρ = |φ⟩⟨φ|), we can determine the fidelity F(φ) = ⟨φ| Λ(ρ) |φ⟩ +quantifying how well the state is preserved in spite of the error. +A uniform average over all initial states yields the average +fidelity F = +� +dφ F(φ) with corresponding average error rate +r = 1 − F +As an example, in case of the n-qubit depolarising channel +Λdep(ρ) = (1 − p)ρ + p +2n 1 +the average error rate r is related to the Pauli error rate p by +r = +� +1 − 1 +2n +� +p +The difference between r and p can be traced back to the fact +that, even if the state is perturbed by the undesired application +of a Pauli operator, it may still retain some overlap with the +original state. Apart from the average error rate, also other error +measures are used, e.g. the diamond norm in connection with +quantum error correction threshold theorems [18]. +B. Randomised Benchmarking +A robust and scalable way for measuring the average error +rate is provided by the method of randomised benchmarking +[6], [52]. Here, random sequences of quantum gates are ex- +ecuted on the noisy quantum hardware and the measurement +results are compared to the ideal result expected in the absence +of errors. This method is proven to be scalable (i.e. applicable +for a large number of qubits), which is a remarkable property +given the fact that, in general, quantum circuits with a large +number of qubits cannot be simulated on classical computers. +In randomised benchmarking, however, the quantum circuits +are chosen to consist of so-called Clifford gates, which can be +efficiently simulated classically (according to the Gottesman- +Knill theorem [51]). This makes it possible to determine the +ideal (i.e. error-free) result as a benchmark. +These +Clifford +gates +form +a +finite +group +{C1, C2, . . . , C|Clif|n} (of size |Clif|n = 22n2n2 �n +j=1(22j −1) +for n qubits) generated by the single-qubit Hadamard gate +H, the two-qubit controlled NOT-gate (see Fig. 1) and the +single-qubit phase gate S = +� +1 +0 +0 +i +� +. Each element of +the Clifford group can be decomposed into O(n2) of these +elementary gates. Moreover, a random twirl over the Clifford +group turns any quantum operation Λ into the depolarising +channel +Λdep = +1 +|Clif|n +|Clif|n +� +j=1 +C† +j ◦ Λ ◦ Cj +with the same average error rate r. This makes it possible to +characterise the error of Λ by a single number r: the average +error per Clifford gate. +More precisely, the randomised benchmarking protocol +works as follows: the quantum computer starts in the standard +initial state |00 . . . 0⟩ (all qubits in state |0⟩). Then, a sequence +Ckm . . . Ck2Ck1 of m random Clifford gates (uniformly chosen +from the above group) is applied. At the end, another Clifford +gate Ckm+1 = (Ckm . . . Ck2Ck1)−1 is applied, which—in the +absence of errors—reverses the effect of the previous sequence. +Therefore, the expected ideal measurement results is 00 . . . 0. +With errors, however, the probability p00...0 to measure this +state will be smaller than 1. Repeating this procedure N times +with k = 1..N for various lengths m, the measured results are +fitted according to +p00...0 = A0(1 − p)m + B0 +see Fig. 4. This fit determines the Pauli error rate p of the +corresponding depolarising channel, from which the average +error per Clifford gate results in r = +� +1 − +1 +2n +� +p. The remaining +parameters A0 and B0 take into account so-called SPAM errors, +i.e. errors in the preparation of the initial state and the mea- +surement of the final state, as well as an edge effect from the +error on the final gate. The above fitting formula is strictly valid +in the case where the errors Λ of each gate are identical and +time-independent. However, a weak dependence of the errors +on gates and on time can be included, resulting in a slightly +more complicated fitting formula. Moreover, a modification of +the above described protocol, called interleaved randomised +benchmarking [53] can be used to determine different error +rates for individual Clifford gates. +C. Gate Set Tomography (GST) +As already mentioned above, the main advantage of ran- +domised benchmarking is its efficiency and scalability with +respect to the number of qubits. On the other hand, it only +yields a rough characterisation of errors in terms of a single +number (the average error rate). Although this provides useful +information concerning the quality of the quantum hardware, + +Fig. 4. Result of a two-qubit randomised benchmarking experiment performed +on the recently installed Fraunhofer-IBM quantum computer ‘ibmq ehningen’. +The average error rate per Clifford gate, extracted from an exponential fit +(blue line) to experimental data (red dashed line, obtained as mean value over +5 different sequences of Clifford gates with various lengths) results as r = +0.01966. Since the dominant error originates from CNOT gates (rather than +from the single-qubits gates H and S), and the average number of CNOT gates +per Clifford gate used in this experiment is 1.485, the resulting error rate per +CNOT gate turns out as 0.01966/1.48 = 0.0132. +it would be desirable to get a more detailed description. The +average error rate only tells us how frequently errors occur, +but nothing about what kind of errors these are. More detailed +knowledge can be useful, e.g. to devise specific quantum error +correction schemes that correct certain errors better than others, +possibly at lower cost. +Such a detailed characterisation is provided by the method of +gate set tomography (GST) [7], [8]. Here, we consider a certain +set of quantum gates {G1, G2, . . . } and seek to determine +their actual implementation on the noisy quantum hardware. +For this purpose, the effect of each gate is modelled as a +general quantum operation, i.e. each gate Gi corresponds to +a 4n × 4n dimensional superoperator (see above). In general, +the complete characterisation of a quantum operation (known +as quantum process tomography [54]) requires the preparation +of at least 4n different (linearly independent) initial states, +subsequent application of the respective quantum operation on +all these states and finally, measurements with respect to 4n +different final states. When working with a gate-based quantum +computer, however, we usually have only one initial state at +our disposal (e.g. all qubits in state |0⟩), and measurements +are only performed with respect to the standard basis (i.e. +the outcome corresponds to state |0⟩ or |1⟩ for each qubit). +Although the required complete set of initial and final states +can be generated by applying suitable quantum gates (e.g. a +Hadamard gate to generate the superposition +1 +√ +2(|0⟩ + |1⟩) as +initial state), these gates themselves suffer from errors, which +have to be distinguished from the errors of the gate one seeks to +characterise. Apart from that, also the preparation of the initial +state |00 . . . 0⟩ as well as the final measurement with respect +to the standard basis exhibit errors (SPAM errors) that must be +taken into account. +The solution provided by GST is to characterise a whole set +of gates simultaneously and self-consistently. For this purpose, +a large number of quantum circuits, each consisting of a specific +sequence of gates chosen from the respective gate set, are +executed on the noisy quantum hardware. Then, the quantum +operation corresponding to each gate, as well as the SPAM +errors mentioned above, are extracted from the measurement +results (More precisely, gates and SPAM errors can only be +characterised up to a global gauge transformation [8] which, +however, is irrelevant for practical purposes, since it does not +affect the observable outcomes of any quantum circuit). In +the simplest implementation of gate set tomography (LGST— +linear gate set tomography), this extraction can be performed +using methods from linear algebra (e.g. matrix inversion), +whereas numerically more expensive techniques like maximum +likelihood estimation are required in more advanced implemen- +tations that yield more accurate results (long-sequence GST) +[8]. Due to the exponentially large number of parameters that +have to be estimated (4n×4n for each quantum gate), however, +a complete characterisation of errors as performed by GST is +realistically feasible only for a small number of qubits (1 or 2, +in practice). +In summary, we have discussed two of the most commonly +used methods for characterising quantum hardware: randomised +benchmarking, which can be applied for many qubits, but +yields only a rough characterisation of errors, and gate set +tomography, which provides a detailed characterisation for a +few qubits. An active area of research consists of finding inter- +mediate methods, which are scalable, but yield more detailed +information than randomised benchmarking. A promising idea +in this direction is to restrict the range of possible correlations +(or crosstalk) between the errors of gates applied to different +qubits by modelling the errors as a Gibbs random field. This +approach has recently been used to characterise errors in a 14- +qubit quantum computer [55]. +VII. CONCLUSIONS +Quantum computing promises spectacular breakthroughs, en- +abling computations that were deemed impossible by classical +computers. This capability stems from fully utilising properties +of physical objects that combine an unprecedented sophistica- +tion with an extreme fragility. Understanding and controlling +errors in quantum circuits is the holy grail on the road to +achieving quantum advantage for practically useful problems. +In this paper, we attempted a realistic view on the current +progress of this journey. We critically reviewed the NISQ +paradigm and identified its realistic capabilities but also its +obvious limitations. We provided an overview about methods +for both: analysing a quantum algorithm using the simplified, +device-agnostic Pauli error model, and establishing far more +detailed knowledge about the error mechanisms of a given +quantum machine by means of randomised benchmarking and +gate set tomography. +ACKNOWLEDGMENTS +Parts of this work are supported by project QORA within the +Competence Center Quantum Computing Baden-W¨urttemberg + +0.9 +alpha: 0.974(7.8e-04) EPC: 1.966e-02(6.0e-04) +lation +0.8 +Popul: +0.7 +Ground +0.5 +0.4 +0.3 +0 +25 +50 +75 +100 +125 +150 +175 +200 +Clifford Length(funded by the Ministerium f¨ur Wirtschaft, Arbeit und Woh- +nungbau Baden-W¨urttemberg), and by a project funded by Carl +Zeiss foundation. +REFERENCES +[1] F. Arute et al., “Quantum supremacy using a programmable supercon- +ducting processor,” Nature, vol. 574, no. 7779, pp. 505–510, Oct. 2019. +[2] E. Pednault, J. A. Gunnels, G. Nannicini, L. Horesh, and R. 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Wallman, “Efficient learning of +quantum noise,” Nature Physics, vol. 16, no. 12, pp. 1184–1188, 2020. + diff --git a/sdFKT4oBgHgl3EQfJy1E/content/tmp_files/load_file.txt b/sdFKT4oBgHgl3EQfJy1E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c09daf7f4620d7e500bb31778223d4e470f23c9c --- /dev/null +++ b/sdFKT4oBgHgl3EQfJy1E/content/tmp_files/load_file.txt @@ -0,0 +1,1157 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf,len=1156 +page_content='Special Session: Noisy Intermediate-Scale Quantum (NISQ) Computers—How They Work, How They Fail, How to Test Them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Sebastian Brandhofer1 Simon Devitt2 Thomas Wellens3 Ilia Polian1 1University of Stuttgart and Center for Integrated Quantum Science and Technology Stuttgart, Germany {sebastian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='brandhofer | ilia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='polian} @informatik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='uni-stuttgart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='de 2University of Technology Sydney Sydney, Australia simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='devitt@uts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='au 3Fraunhofer Institute for Applied Solid State Physics IAF Freiburg, Germany thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='wellens@iaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='fraunhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='de Abstract—First quantum computers very recently have demon- strated “quantum supremacy” or “quantum advantage”: Execut- ing a computation that would have been impossible on a classical machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Today’s quantum computers follow the NISQ paradigm: They exhibit error rates that are much higher than in conventional electronics and have insufficient quantum resources to support powerful error correction protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This raises questions which relevant computations are within the reach of NISQ architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Several “NISQ-era algorithms” are assumed to match the specifics of such computers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' for instance, variational optimisers are based on intertwining relatively short quantum and classical computa- tions, thus maximizing the chances of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This paper will critically assess the promise and challenge of NISQ computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' What has this field achieved so far, what are we likely to achieve soon, where do we have to be skeptical and wait for the advent of larger-scale fully error-corrected architectures?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Index Terms—Quantum Computing, NISQ Computing, Error Simulation, Error Tolerance Analysis, Error Characterisation I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' INTRODUCTION In spite of classical computing technology being on an expo- nential trajectory since 1960s, quantum computing (QC) contin- ues to fuel the fantasies of scientists and practitioners alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' QC promises asymptotic, and in some cases exponential, speedups for hard problems from domains such as cryptography, compu- tational chemistry, simulation, or machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Theoretical results on quantum algorithms have been complemented by a development of actual quantum hardware potentially capable of practically executing quantum computations, even though the demonstrated systems were of limited size and could be simulated on a classical computer within a meaningful time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The “quantum supremacy” experiment [1] showed compu- tations on a 53-qubit (quantum bit) machine that the authors argued would be intractable on a classical compute server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Even though the latter argument is currently disputed [2], [3], as ©2021 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1109/VTS50974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='9441047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' other authors are proposing more efficient classical solutions for the problem of [1], this problem serves no practical or sci- entific purpose other than demonstrating “quantum supremacy” (or “quantum advantage”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Using QC for solving a practical problem faster than any classical computer is still open at the time of writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' One key challenge on the road to quantum advantage is increasing the number of qubits that a quantum computer can calculate on, as the asymptotic speedups naturally unfold their effects on larger problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Less obvious but not less important, the quality of the available qubits and operations on them (quantum gates) is essential for meaningful computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Quantum states are fragile, and even error rates reported for the hardware in [1] (which are among the best implementations that exist today) were orders of magnitude larger compared to conventional electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These error rates are expected to improve in the future, but not come close to the nearly error- free operation of classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' When it comes to the near future of quantum computing given the non-trivial error rates of its hardware, two schools of thought exist with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The first is to deploy quantum error correction (QEC), where several poor-quality physical qubits together constitute a logical qubit with a more reasonable robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The idea is not unsimilar to processing encoded information in self-checking design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' however, state-of-the-art QEC schemes [4] have an overhead of hundreds or thousands physical qubits per logical qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The second approach, known under the name “NISQ”, suggests to accept the inherent noise as given and try to find applications that can survive it without resorting to fully-fledged QEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Some classes of algorithms are being investigated specifically with NISQ computers in mind [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Near-term NISQ computers are also the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Since “noise” is an integral part of NISQ, understanding its origins, characterising its properties, quantifying its magnitude, and assessing its application-level impact are of utmost impor- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' To this end, this paper will be organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' After some background information about QC in Section II, Section arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='11739v1 [quant-ph] 27 Jan 2023 III will introduce the basic NISQ concepts and revisit the expectations raised since its inception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Section IV will discuss future developments in QC, going beyond NISQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Section V will put today’s successful demonstrations of the NISQ principle for small instances of practically relevant problems into perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For example, certain computational chemistry tasks have been solved for small molecules, such as the hydrogen atom, for which classical solving methods work as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' What would be needed to extend such approaches to more complex tasks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Section VI will explain what role characterisa- tion play in making NISQ computers useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The approaches go far beyond the pass-fail tests used for conventional CMOS circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Individual qubits and ensembles of entangled qubits must be characterised using techniques such as randomised benchmarking [6] or gate set tomography [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Section VII will conclude the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' QUANTUM COMPUTING BACKGROUND A quantum computer can perform computations on n qubits, where one qubit might be, for instance, an ion, a photon, or a solid-state transmon circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' An individual qubit can be set to two basis states |0⟩ and |1⟩, which correspond to two- dimensional column vectors (1, 0)T and (0, 1)T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A single qubit assumes a state, which can be |0⟩, |1⟩, or their superposition α0|0⟩+α1|1⟩ = (α0, α1)T, where α0 and α1 are complex numbers and |α0|2 + |α1|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' An ensemble of n qubits assumes a state described by a 2n-dimensional complex vector of amplitudes (α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='00, α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='01, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='11)T, where the indices of α are all n-bit combinations of 0s and 1s and again |α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='00|2 + |α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='01|2 + · · · + |α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='11|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A quantum gate G acting on k qubits is described by a 2k × 2k complex unitary matrix UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Examples of quantum gates (some of which play a key role in modeling errors) along with their symbols and matrices are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Note that all quantum gates are invertible and therefore have the same number of inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Gates acting in parallel on different qubits are combined using tensor product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' gates acting sequentially can be combined using matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 1 includes an example circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' let us follow its operation if the first and the qubit are initialised in states |0⟩ and |1⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='respectively: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='|ϕ1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='|01⟩ = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='|ϕ2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='√ ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='|ϕ1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='Example circuit: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Some important quantum gates and an example quantum circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Note that state |ϕ2⟩ corresponds to the system being in superpo- sition of |00⟩ and |10⟩, which can be attributed to its individual qubits: qubit 1 is in the superposition state (0⟩ + |1⟩)/ √ 2, and qubit 2 is in state |0⟩ (the Pauli-X gate acts as an inverter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In contrast, state |ϕ3⟩ is a superposition of |00⟩ and |11⟩ and cannot be expressed by the individual qubits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' this phenomenon is known as entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It sounds quite unspectacular that most of a quantum com- puter’s operations are multiplying matrices (of its gates) by the state vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It seems a lot more spectacular that these vectors and matrices have an exponential size in the number n of actual physical objects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' One would expect radical speed- ups from a system that is seemingly able to hold 2n intermedi- ate results in the state vector (α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='00, α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='01, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='11)T and applying computations to all these results at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The reality is more intricate: we cannot read out the values of the state vector directly, but can only perform a measurement on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Measuring a qubit will result in either outcome 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' If a qubit’s state was |0⟩ or |1⟩, the measurement will deterministi- cally yield 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, if a qubit is in a superposition α0|0⟩ + α1|1⟩ = (α0, α1)T, the outcome will be |0⟩ with probability |α0|2 and |0⟩ with probability |α1|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For example, measuring the second qubit in state |ϕ2⟩ in the example above will yield |0⟩ with certainty, and measuring the first qubit will result in |0⟩ or |1⟩ with probability |1/ √ 2|2 = 1/2 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The measurement is destructive: after the measurement, the state of this qubit will change to the state corresponding to the measurement outcome, and all non-trivial information about α1 and α2 will be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' An interesting effect happens in entangled states like |ϕ3⟩: measuring one qubit can determine the value of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For example, if measuring the first qubit in |ϕ3⟩ = (|00⟩ + |11⟩)/ √ 2 yields |0⟩ (this happens with 50% probability), then the 2-qubit state collapses to |00⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' the next measurement of qubit 2 will result in |0⟩ with certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The destructive character of measurements prevents the seemingly obvious parallelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It is not possible to obtain the exact values αj by copying the end-state of the circuit and measuring it multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It is possible to re-run the circuit multiple times from the beginning, but the required number of repetitions would offset any gain from the paral- lelism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Instead, quantum algorithms employ steps to transform (entangled) quantum states such that measurements yield useful information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The description so far had assumed a perfectly working quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Real-world hardware is prone to distur- bances due to noise, interaction with the environment, and imperfections of the physical apparatus [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' One can distinguish different types of failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Pauli noise is modeled by Pauli-X, Y and Z gates (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 1) randomly added to different locations within the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Somewhat counter-intuitive, this discrete fault model captures parametric (large and small) errors in the individual qubit states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' this is because a qubit is measured at some point and even a small error either manifests itself (this is captured by Pauli errors) or not (then it has no effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Frequently used special cases of Pauli noise are depolarizing noise, where Pauli-X, Y or Z gates are applied with uniform error rates independently to each qubit, and dephasing noise, where only Pauli-Z gates are randomly applied within the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Independent Pauli errors do not cover correlated errors that can be generated by effects such as physical control line crosstalk between qubits, but these effects can be bounded as joint Pauli errors occurring at the same time across the relevant qubits - the error is often bounded to weight two Pauli errors as the underlying physics of qubit systems is limited to pairwise Hamiltonian interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Quantum error correction (QEC) protocols make use of heavily entangled states to form highly redundant “logical qubits”, such that errors affecting one or few physical qubits are compensated by error-unaffected redundant physical qubits within the same entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, NISQ computers follow a different route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' DEFINING NISQ, AND HOW USEFUL WILL THE NISQ ERA BE?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In terms of quantum hardware, the necessary components to build a scalable quantum computer (in a variety of different systems, including superconductors) are well known [10]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' At a minimum, a hardware device requires a regular array of qubits arranged in a square 2D grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Each qubit needs the ability to initialise in a known state, perform arbitrary single qubit operations, be able to be measured and be able to couple to the four neighbouring qubits in the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' All of these fundamental operations need to occur with an error rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1% or lower [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This is the bare minimum for a scalable system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' As anticipated by many, the first realisation of quantum computing technology has occurred over the cloud, with users logging onto dedicated hardware over the classical internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These types of ‘quantum in the cloud’ systems began with the connection of a two-qubit photonic chip to the classical internet by the University of Bristol in 2013 and accelerated significantly in 2016 with the introduction of IBM of their quantum experience platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' We now see both free and paid services offered by IBM, Microsoft, Amazon, Xanadu and Rigetti, across a variety of hardware modalities for small scale quantum computing chipsets up to 65 physical qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This has spurred the so-called era of noisy intermediate-scale quantum (NISQ) [14] research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Noisy intermediate-scale quantum also refers to quantum algorithms that are small enough to be faithfully executed on near term, low qubit count, high error rate, quantum hardware without the need for resource-costly error correction protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' While NISQ algorithms exist in principle—quantum supremacy is a quintessential example [1]—an added caveat is that the algorithm needs to be either scientifically and/or commercially viable—quantum supremacy is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Consequently NISQ al- gorithms must be highly compact but still either reach the quantum supremacy regime—where the problem simply cannot be solved on any classical computer—or reach the regime where it is more cost efficient—either in terms of actual dollars or in terms of computational time—to run the algorithm on a quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' NISQ algorithms satisfying the commercial or scientific viability condition are under active research, but do not currently exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The actual definition of NISQ computation is somewhat nebulous and depends on who you ask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, to a first approximation we can examine NISQ compatibility by calcu- lating for a given algorithm the value A = n · d, called area, where n is the number of qubits the algorithm needs and d is the number of time-steps in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' If the physical error rate of the qubits and gates in a physical quantum computing chip is p < 1/A, the algorithm is likely NISQ compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This is due to the error sensitivity of quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A quantum computation is akin to an optical interferometer and its ability to perform efficient computation comes about through a delicate interference effect of the computational wavefunction as gate operations are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' As with physical interferometers, where a minor misalignment can completely disrupt the interference effect and make the device non-functional, even one error in a quantum algorithm can disrupt the required computational interference effect, resulting in incorrect output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It has been demonstrated for a variety of quantum algorithms that single errors can dramatically reduce the probability of success of generating the correct output [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This makes p < 1/A— which represents the error rate needed such that one error occurs during execution on average—a good bound for the error tolerance of an algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The difficulty is finding a quantum algorithm that is comfort- ably in the regime where classical algorithms simply cannot be used at the same time as still being small enough to satisfy the p < 1/A bound which defines when error correction or other mitigation techniques are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The quantum supremacy formalism that was used by the Google Sycamore processor [1] are, by definition, the smallest quantum circuits known that are provably difficult to simulate on a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' They were constructed explicitly for this purpose—this is why they are not practically useful beyond demonstrations of quantum supremacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The Google team simulated a 53-qubit circuit over a depth of approximately 40, giving A ≈ 2100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Even this circuit was ‘border-line’ in terms of being implementable on classical hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A preprint paper in early 2021 has in fact presented results that replicated the Google experiment using new techniques in classical algorithms based on tensor contraction [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A circuit with A ≈ 2100 would therefore require physical errors in the quantum processor of p < 5×10−4 to occasionally run without errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Not even the Google Sycamore processor had this level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Single and two-qubit gates in the experiment had, on average, error rates of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1-1% while measurement gates had errors up to 5% for simultaneous readout [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Hardware errors for a variety of quantum processors have decreased from of order 10-50% when basic qubit operations were first demonstrated in the late 1990’s and early 2000’s to, of order, 1-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1% today (over two decades later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This is extremely impressive from an experimental point of view, and the ability to replicate the fabrication of low error rate quantum chips is getting better and better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, given that the smallest quantum circuit that has provable quantum advantage—that is commercially and scientifically useless— requires physical error rates at least another two orders of magnitude lower to be unambiguously achievable in a quantum processor, it is difficult to accept the prospect that an algorithm that can be implemented with higher physical error rate that can simultaneously revolutionise a scientific discipline or a commercial industry is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The only other option is that experimental progress in reducing physical error rates in chip-sets accelerates rapidly, such that the next two orders of magnitude in error improvement occurs over the next two or three years rather than the next two or three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Consequently, NISQ currently has no identified, commer- cially relevant application at the 50-1000 qubit level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' There are also good reasons to believe that finding such applications (in an era where error rates on quantum chips will be of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1% – 1% at best) will not be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, there are no results that prove that the existence of these algorithms are impossible, so there is still strong motivation to keep searching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This search should be done in a way that addresses these fundamental issues rather than sidestepping them, or in some unfortunate cases, simply obfuscating the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' WHAT HAPPENS BEYOND NISQ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Beyond the NISQ era is one of two possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Either physical error rates in quantum computing chips are drastically reduced—via a combination of better fabrication physics, better quantum control and passive error mitigation techniques [17]— or active techniques are employed to reduce error rates which fall under the umbrella of Quantum Error Correction (QEC) [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' QEC uses redundant qubit encoding, active measurement and feedback to stabilise quantum information over long periods of computational time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It is a highly successful theoretical discipline and is unarguably one of the main pillars of quantum information science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' QEC is commonly combined with the method of fault-tolerant circuit design, where error correction protocols and logic operations are performed in a manner such that physical errors do not ‘cascade’ out of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The combination of these two techniques lead to one of the most important results in quantum computing, the threshold theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The threshold theorem for fault-tolerant quantum computing [18] states that with a polylogarithmic overhead of physi- cal resources—qubits and time—an arbitrarily long quantum circuit can be implemented provided the physical errors of the hardware are less than a critical value, pth, known as the fault-tolerant threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Currently, the threshold for the most commonly used QEC scheme for large-scale architecture design, the surface code, is pth ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This means that several hardware platforms already have reached physical error rates satisfying the fault-tolerant threshold for QEC in a subset of operations, although it should be noted that no hardware system has yet demonstrated error rates below threshold for a universal set of gate operations on a single device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The unfortunate issue with QEC is the drastic increase in physical qubit resources once it is even implemented to a minimal degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Once error correction is incorporated into an algorithm, the total number of physical qubits required in the quantum chipset can easily jump from 50-100 qubits to 50,000- 100,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This includes both the raw overhead in performing the qubit encoding and ancillary physical resources to maintain fault-tolerance in operational logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This would only be for a small amount of error correction, enough to reduce error rates from approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1% on the physical level to the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0001% at the encoded level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This number of physical qubits is just not possible to engineer in the near future, in any hardware system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The error correction overhead for large-scale algorithm, such as Shor’s factoring algorithm or quantum simulation is very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, two points should be strongly emphasised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Factoring or quantum simulation are extremely large al- gorithms, even if they scale polynomially with problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Given the error sensitivity of quantum algorithms, this means that the effective error rate required for encoded qubits can be of the order of 10−13% or lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Hence, QEC needs to provide at least 12 orders of magnitude of error improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This massive amount of suppression does not come for free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Classical techniques for circuit optimisation both at the algorithmic level and at the QEC level has already done an extraordinary job at reducing the overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For example, in 2012, an explicitly compiled, error-corrected analysis of Shor’s algorithm estimated that of the order of 100 billion qubits would be needed to break RSA-2048 encryption [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' By 2019, this number had been reduced to 20 million [20], a reduction by a factor of 5,000 without changing any of the operating assumptions of the underlying hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Ultimately, the demands required on hardware accuracy for large-scale algorithms will necessitate active error correction in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' How far experimental improvements in fabrication, control and passive error mitigation can go before active techniques need to be implemented is up for debate, but there will be (and may already be) decisions made to the most cost effective and expedient way to move in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Either invest in reducing physical error rates by orders of magnitude or invest in being able to make chip-sets containing very large numbers of integrated qubits quickly and cheaply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' ANALYSING NISQ COMPATIBILITY OF ALGORITHMS Recent progress in quantum computing technologies [1], [5], [21], [22] offers quantum computational resources that have not been available before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Although only noisy and intermediate- sized quantum computational resources can be exploited by current NISQ computers, recent experiments yield results that are challenging to obtain using classical resources [1], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In the light of these experiments, increasing quantum resources of NISQ computers and growing business adaptation, it is crucial to determine the resource requirements of a computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Assessing these requirements determines the set of suitable NISQ computers, can help to improve the implementation of a computation, and informs whether quantum error correction is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The resource requirements of a computation consist of the number of qubits, the depth and the tolerable error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The first two requirements can be computed by inspection [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The tolerable error rate can be determined through approaches ranging from guidelines on the size of the quantum computa- tion, simulations subject to errors, and experiments on quantum computers to reliability models of quantum computers based on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The determination of the tolerable error rate must be able to consider noise sources ubiquitous in NISQ computers, arbitrary quantum computations, flexible definitions of success and must be able to make statements about a wide range of quantum computing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In [14] and [24] bounds on the quantum computation size and error rate p for successful computations are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The work in [14] states that the error rate may not be much larger than G−1, with G denoting the number of quantum gates constituting the quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The work in [24] states the observation p < (n · d)−1 (1) with n qubits used in a computation that has depth d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These guidelines provide a rough estimation and are not suitable to distinguish between different success criteria or quantum computations that have the same size but exhibit a different error behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A more accurate approach to determine the tolerable error rate of a quantum computation is to conduct simulations subject to errors [15], [16], [25]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Using Monte Carlo or exhaus- tive error simulations, the error rate at which the quantum computation starts failing the target success criteria can be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Here, the employed error model significantly con- tributes to the accuracy, simulation effort and generality of the determined tolerable error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' If an error model is employed that is fitted to the dynamic adverse physical processes (noise processes) in a specific quantum computer, the determined error rate requirement can often not be generalized to other quantum computers [32] or quantum computing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Alternatively, if the employed error model is too general, it may not represent noise processes sufficiently well and thus lead to incorrect predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' While simulations subject to errors are a suitable choice to determine the tolerable error rate, they also incur an exponential runtime and space overhead in the number of qubits in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It is therefore crucial to reduce the simulation effort by a suitable choice of error model and simulation techniques [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Another option is to probe the success probability of a quantum computation on a target quantum computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' While this can be performed efficiently, it is challenging to generalise the results to other quantum computers of the same generation due to the dynamic error rates of NISQ computers, and the results are not valid for other quantum computing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In addition, a computation is performed on a quantum computer with respect to a specific error rate that can only be increased in a limited way [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Thus, the error rate requirement of a quantum computation can not be determined in general since only a small range of error rates can be probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Reliability models for quantum computers based on machine learning have also been proposed [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These models are trained on a specific quantum computer and uses features such as quan- tum computation size or structure, auxiliary experiments and success probabilities of previous quantum computations to yield a prediction about the success probability of future quantum computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' While these methods exhibit a high prediction accuracy for some quantum computers and computations, these approaches have not been demonstrated to generalise well over multiple quantum computers or different quantum computing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Thus, for determining the error rate requirement of a quan- tum computation in the NISQ era, simulations subject to errors is a suitable choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Arbitrary noise sources, quantum computations and success criteria can be considered with simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' While the runtime overhead is exponential in the number of qubits, there currently is only a small number of publicly available NISQ computers that can not be simulated in reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' We will now give details about commonly used success criteria, simulation methods, error modeling and results for a set of small-scale quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Success Criteria The success of a quantum circuit computation is determined depending on the available reference information of the ideal result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In the simplest case, when the target state is known and exact probability amplitudes are available upon simulation, the fidelity measure F(|ψt⟩ , |ψe⟩) is often used to quantify how much an erroneous state |ψe⟩ deviates from a target state |ψt⟩ [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, this is often not applicable as quan- tum circuit computations on a quantum computer only return measurement results and the target state is often not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Therefore, a number of success criteria were proposed and used in previous works such as [36]: The measurement probability of the correct result in single outcome quantum circuits is above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Measurements are in a set of acceptable results with high probability larger than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Such a set can e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' be defined by the Hamming distance or the heavy output [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The deviation between the measurement output distri- bution and the expected distribution is smaller than a threshold according to some measure of distance [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The measurement result probabilities are consistent with predictions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' the cross-entropy of results is low [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Since quantum circuit computations on a quantum computer are subject to errors and sampling noise, above probabilities can not be determined with certainty [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It is therefore essential to also consider a metric of confidence for reaching a defined success criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In addition, applying a success criterion to a quantum computation may not make the results classically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Simulation Methods Simulating a quantum circuit incurs an exponential runtime and space overhead in general [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, specific quantum circuit structures and properties can be exploited by simulators based on tensor networks [39], matrix product states [40], de- cision diagrams [41] or stabilizers [42], [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These simulators exhibit a better runtime or memory requirement for specific quantum circuits than general simulators based on the state vector or density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In general, simulating a n-qubit state requires a state vector to store and manipulate 2n complex amplitudes whereas the density matrix simulator has to store and manipulate 2n n- qubit states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Simulating 53-qubit quantum circuits such as the quantum circuit in Google’s quantum supremacy experiment using a state vector or density matrix simulator imposes a large memory requirement: 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='1 petabytes and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='49 · 1017 petabytes would be required for the state vector and density matrix simulator respectively, if one complex number is stored using two double-precision floating-point numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Error Modeling Noise in NISQ computers is ubiquitous and affect the compu- tation of a quantum circuit in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In general, a qubit can be subject to coherent or incoherent noise, be lost, leak out of the computational space, or be erroneously initialised or read out [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These effects can be modeled by device- oriented or device-agnostic error models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In a device-oriented error model, the adverse physical processes in a quantum computing device are replicated to model the errors affecting a computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Device-agnostic error models, however, consist of a set of operators that cover the effects of relevant noise sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The choice of error model has consequences for the prediction accuracy, applicability of results, quantifiability of the model, and simulation effort of simulations subject to these error models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A device-oriented error model fitted to one specific quan- tum computer would be expected to yield the best prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, the results of simulations subject to that error model would not be applicable to different quantum computers or quantum computing technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Furthermore, estimating each error parameter of such an error model through characterisation protocols can be complex, as described in more detail in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Device-oriented error models also incur a large simulation effort since they require the density matrix formalism in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A device-agnostic error model is applicable to diverse quan- tum computers, if the noise parameters are quantified on them using characterisation protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Thus, if it is known under which noise parameters the success criterion of a quantum circuit computation is met, we can determine suitable NISQ computers that do not exceed the determined noise parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Recently, quantum supremacy experiments confirmed that the device-agnostic Pauli error model is sufficiently accurate to make predictions about the success probability [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The Pauli error model also only incurs a low simulation effort for arbitrary quantum circuits since the error operators constituting the error model are in the Clifford set and can be simulated using the state vector or stabilizer simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Results In this section, we show results on various quantum circuit with up to 16 qubits and 214 gates subject to the Pauli error model with uniform error rates (depolarizing noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The tolerable Pauli error rate given a success probability of 66% and the success probability given a Pauli error rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='15% are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The tolerable Pauli error rate specifies the error rate at which a quantum computation can still be performed successfully given a target success criterion and probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The Pauli error rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='15% was chosen as this is currently the lowest (single-qubit gate) error rate exhibited by one of the largest NISQ computer [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The evaluated quantum circuits implement the Grover algorithm [44], arithmetic functions [45], Bernstein-Vazirani (BV) algorithm [46] (using CNOTs), the quantum Fourier transform (QFT) [47] and the hidden linear function (HLF) [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Furthermore, and quantum circuits for chemistry applications (RYRZ [5], UCCSD [49]) were evalu- ated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The evaluated quantum circuits were generated using Qiskit [50] for a general quantum computer with single-qubit rota- tions, the controlled-NOT two-qubit gate and without geometric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Geometric constraints imposed by many quan- tum computing technologies such superconducting qubits [1] decrease the herein reported tolerable Pauli error rate and success probability since satisfying these constraints require the insertion of additional error-prone operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The evaluated success criterion was the measurement prob- ability of the correct result for single outcome algorithms such as BV, and fidelity for all other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The quantum circuits were simulated subject to the Pauli error model using a state vector simulator [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For combina- tions of quantum circuits and Pauli error rate where less than one Pauli error occurs during the quantum circuit computation on average, an exhaustive error simulation was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For this exhaustive error simulation, one Pauli X, Z, or Y error was simulated successively at every space-time location in the quantum circuit and the impact of these errors on the target success criterion was scaled according to the specified Pauli error rates [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For combinations of quantum circuits and Pauli error rates that incur more than one Pauli error per quantum circuit computation on average, Monte Carlo simulations were conducted to obtain the success probability and tolerable Pauli error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 1) Success Probability: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 2 shows the success probability of the evaluated quantum circuits at a Pauli error rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' On the x-axis, the area of the evaluated quantum circuits is 0 100 200 300 400 500 600 Quantum Circuit Area 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 Success Probability P = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='66 Arithmetic BV Grover HLF QFT RYRZ UCCSD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Success probability of quantum circuits subject to a Pauli error rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='15% and no geometric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' All quantum circuits with an area smaller than 322 could be executed with a success probability of at least 66%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' All evaluated BV and UCCSD quantum circuits could be computed successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For other quantum circuits and mod- erate quantum circuit sizes, the probability of successfully computing the quantum circuit quickly falls below 66%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These results show that the quantum circuit area is a good indicator for determining the success probability for many of the evaluated quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 2) Tolerable Pauli Error Rate: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 3 depicts the tolerable Pauli error rate of the evaluated quantum circuits for a success probability of 66%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The x-axis shows the quantum circuit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Both axes are in log-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The bound of the error rate on the quantum circuit area (n · d)−1 stated in the literature [24] is also shown as a black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' All evaluated quantum circuits only tolerate a Pauli error rate that is smaller than the inverse of their quantum circuit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' There are small difference in the tolerable Pauli error rate between quantum circuits implementing different quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Quantum circuits for arithmetic functions tolerate a Pauli error rate that is mostly the average of evaluated quantum circuits with the same quantum circuit area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' BV, RYRZ and QFT quantum circuits tolerate a slightly larger Pauli error rate and HLF, UCCSD and Grover quantum circuits only tolerate a slightly smaller Pauli error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The tolerable Pauli error rate of HLF quantum circuits constitute a lower bound for the evaluated quantum circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These results confirm the bound given in [24] and indicate that the evaluated quantum circuits do not tolerate one Pauli error on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The tolerable Pauli error rate is mainly dominated by the area of a quantum circuit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' large difference due the structure of a quantum circuit or other properties were not observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' CHARACTERISING THE UNDERLYING HARDWARE Above, we have seen the important impact of physical error rates onto the applicability of quantum error correction proto- cols and NISQ algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In this section, we will therefore 102 Quantum Circuit Area 10 3 10 2 Tolerable Pauli Error Rate (area) 1 Arithmetic BV Grover HLF QFT RYRZ UCCSD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Tolerable Pauli error rate of the evaluated quantum circuits for a success probability of 66%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' present some of the most commonly used methods for actu- ally measuring the error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These methods do not require any knowledge about how the quantum gates are physically realised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Instead, the basic idea is the following: we execute suitably chosen quantum circuits on the given hardware and count the measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The desired information con- cerning the errors is then extracted by comparing the measured outcomes with the outcomes expected in the absence of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Modeling Errors in Quantum Circuits To introduce these methods, we first explain some basic concepts concerning the mathematical modelling of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In section II, we have introduced the state of a qubit as a two- dimensional complex vector |φ⟩ = (α0, α1)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Such a state is called a pure quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' More generally, however—and especially so in the presence of errors—the qubit may be in a mixed state, which is described by a 2 × 2 density matrix: ρ = � ρ00 ρ01 ρ10 ρ11 � with ρ = ρ† (self-adjoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' the symbol † denotes transposition and complex conjugation), Tr(ρ) = 1 (normalised) and ρ ≥ 0 (positive semi-definite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In the special case of a pure state φ, we have ρ = |φ⟩⟨φ| = � |α0|2 α0α∗ 1 α∗ 0α1 |α1|2 � , where ⟨φ| = (α∗ 0, α∗ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In a similar way, states of n qubits are described by 2n × 2n dimensional density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' If a unitary operator (or quantum gate) U is applied to the qubits, their density matrix changes according to ρ′ = EU(ρ) := UρU †, where U †U = 1 If ρ is a pure state, then ρ′ is also a pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For this reason, there is no need to consider density matrices if the quantum computer exhibits no errors and all gates are perfectly unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In the presence of errors, however, the resulting quantum oper- ation assumes the following general form (completely positive, trace preserving map) [51]: ρ′ = E(ρ) = � j AjρA† j, where � j A† jAj = 1 This can be interpreted as follows: instead of a single well- defined unitary operation U, one of several possible operators Aj is applied to the qubits with probability pj = tr(AjρA† j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' As an example, the depolarising channel for a single qubit is defined by A0 = √1 − p 1, A1 = � p 3X, A2 = � p 3Y and A3 = � p 3Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In other words, with probability 1−p (in this case independent of ρ), the state ρ remains unchanged (no error), whereas an error amounting to the application of Pauli-X, Y or Z occurs with probability p/3 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' It can be shown [51] that the sum over j in the above general representation of a quantum operation can be restricted such as to contain at most 4n different terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Thereby, a noisy quantum operation is defined by up to 4n different 2n × 2n matrices Ai (called Kraus operators), or—equivalently—by a single 4n×4n matrix E (sometimes called superoperator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' There are several ways for defining the error rate of a quantum operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Suppose we want to implement the unitary quantum gate U, but realise E instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Then, we define the corresponding error operator by Λ = (EU)−1 ◦ E (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' the inverse of the desired operation EU is applied after E, such that Λ = 1 in the absence of errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For a given initial pure state ρ = |φ⟩⟨φ|), we can determine the fidelity F(φ) = ⟨φ| Λ(ρ) |φ⟩ quantifying how well the state is preserved in spite of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A uniform average over all initial states yields the average fidelity F = � dφ F(φ) with corresponding average error rate r = 1 − F As an example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' in case of the n-qubit depolarising channel Λdep(ρ) = (1 − p)ρ + p 2n 1 the average error rate r is related to the Pauli error rate p by r = � 1 − 1 2n � p The difference between r and p can be traced back to the fact that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' even if the state is perturbed by the undesired application of a Pauli operator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' it may still retain some overlap with the original state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Apart from the average error rate, also other error measures are used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' the diamond norm in connection with quantum error correction threshold theorems [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Randomised Benchmarking A robust and scalable way for measuring the average error rate is provided by the method of randomised benchmarking [6], [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Here, random sequences of quantum gates are ex- ecuted on the noisy quantum hardware and the measurement results are compared to the ideal result expected in the absence of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This method is proven to be scalable (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' applicable for a large number of qubits), which is a remarkable property given the fact that, in general, quantum circuits with a large number of qubits cannot be simulated on classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In randomised benchmarking, however, the quantum circuits are chosen to consist of so-called Clifford gates, which can be efficiently simulated classically (according to the Gottesman- Knill theorem [51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This makes it possible to determine the ideal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' error-free) result as a benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' These Clifford gates form a finite group {C1, C2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' , C|Clif|n} (of size |Clif|n = 22n2n2 �n j=1(22j −1) for n qubits) generated by the single-qubit Hadamard gate H, the two-qubit controlled NOT-gate (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 1) and the single-qubit phase gate S = � 1 0 0 i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Each element of the Clifford group can be decomposed into O(n2) of these elementary gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Moreover, a random twirl over the Clifford group turns any quantum operation Λ into the depolarising channel Λdep = 1 |Clif|n |Clif|n � j=1 C† j ◦ Λ ◦ Cj with the same average error rate r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This makes it possible to characterise the error of Λ by a single number r: the average error per Clifford gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' More precisely, the randomised benchmarking protocol works as follows: the quantum computer starts in the standard initial state |00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 0⟩ (all qubits in state |0⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Then, a sequence Ckm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Ck2Ck1 of m random Clifford gates (uniformly chosen from the above group) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' At the end, another Clifford gate Ckm+1 = (Ckm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Ck2Ck1)−1 is applied, which—in the absence of errors—reverses the effect of the previous sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Therefore, the expected ideal measurement results is 00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' With errors, however, the probability p00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 to measure this state will be smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Repeating this procedure N times with k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='.N for various lengths m, the measured results are fitted according to p00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0 = A0(1 − p)m + B0 see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This fit determines the Pauli error rate p of the corresponding depolarising channel, from which the average error per Clifford gate results in r = � 1 − 1 2n � p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The remaining parameters A0 and B0 take into account so-called SPAM errors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' errors in the preparation of the initial state and the mea- surement of the final state, as well as an edge effect from the error on the final gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The above fitting formula is strictly valid in the case where the errors Λ of each gate are identical and time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' However, a weak dependence of the errors on gates and on time can be included, resulting in a slightly more complicated fitting formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Moreover, a modification of the above described protocol, called interleaved randomised benchmarking [53] can be used to determine different error rates for individual Clifford gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Gate Set Tomography (GST) As already mentioned above, the main advantage of ran- domised benchmarking is its efficiency and scalability with respect to the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' On the other hand, it only yields a rough characterisation of errors in terms of a single number (the average error rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Although this provides useful information concerning the quality of the quantum hardware, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Result of a two-qubit randomised benchmarking experiment performed on the recently installed Fraunhofer-IBM quantum computer ‘ibmq ehningen’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The average error rate per Clifford gate, extracted from an exponential fit (blue line) to experimental data (red dashed line, obtained as mean value over 5 different sequences of Clifford gates with various lengths) results as r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='01966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Since the dominant error originates from CNOT gates (rather than from the single-qubits gates H and S), and the average number of CNOT gates per Clifford gate used in this experiment is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='485, the resulting error rate per CNOT gate turns out as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='01966/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='48 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' it would be desirable to get a more detailed description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The average error rate only tells us how frequently errors occur, but nothing about what kind of errors these are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' More detailed knowledge can be useful, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' to devise specific quantum error correction schemes that correct certain errors better than others, possibly at lower cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Such a detailed characterisation is provided by the method of gate set tomography (GST) [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Here, we consider a certain set of quantum gates {G1, G2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' } and seek to determine their actual implementation on the noisy quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For this purpose, the effect of each gate is modelled as a general quantum operation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' each gate Gi corresponds to a 4n × 4n dimensional superoperator (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In general, the complete characterisation of a quantum operation (known as quantum process tomography [54]) requires the preparation of at least 4n different (linearly independent) initial states, subsequent application of the respective quantum operation on all these states and finally, measurements with respect to 4n different final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' When working with a gate-based quantum computer, however, we usually have only one initial state at our disposal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' all qubits in state |0⟩), and measurements are only performed with respect to the standard basis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' the outcome corresponds to state |0⟩ or |1⟩ for each qubit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Although the required complete set of initial and final states can be generated by applying suitable quantum gates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' a Hadamard gate to generate the superposition 1 √ 2(|0⟩ + |1⟩) as initial state), these gates themselves suffer from errors, which have to be distinguished from the errors of the gate one seeks to characterise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Apart from that, also the preparation of the initial state |00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' 0⟩ as well as the final measurement with respect to the standard basis exhibit errors (SPAM errors) that must be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' The solution provided by GST is to characterise a whole set of gates simultaneously and self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' For this purpose, a large number of quantum circuits, each consisting of a specific sequence of gates chosen from the respective gate set, are executed on the noisy quantum hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Then, the quantum operation corresponding to each gate, as well as the SPAM errors mentioned above, are extracted from the measurement results (More precisely, gates and SPAM errors can only be characterised up to a global gauge transformation [8] which, however, is irrelevant for practical purposes, since it does not affect the observable outcomes of any quantum circuit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In the simplest implementation of gate set tomography (LGST— linear gate set tomography), this extraction can be performed using methods from linear algebra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' matrix inversion), whereas numerically more expensive techniques like maximum likelihood estimation are required in more advanced implemen- tations that yield more accurate results (long-sequence GST) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Due to the exponentially large number of parameters that have to be estimated (4n×4n for each quantum gate), however, a complete characterisation of errors as performed by GST is realistically feasible only for a small number of qubits (1 or 2, in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In summary, we have discussed two of the most commonly used methods for characterising quantum hardware: randomised benchmarking, which can be applied for many qubits, but yields only a rough characterisation of errors, and gate set tomography, which provides a detailed characterisation for a few qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' An active area of research consists of finding inter- mediate methods, which are scalable, but yield more detailed information than randomised benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' A promising idea in this direction is to restrict the range of possible correlations (or crosstalk) between the errors of gates applied to different qubits by modelling the errors as a Gibbs random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This approach has recently been used to characterise errors in a 14- qubit quantum computer [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' CONCLUSIONS Quantum computing promises spectacular breakthroughs, en- abling computations that were deemed impossible by classical computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' This capability stems from fully utilising properties of physical objects that combine an unprecedented sophistica- tion with an extreme fragility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' Understanding and controlling errors in quantum circuits is the holy grail on the road to achieving quantum advantage for practically useful problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' In this paper, we attempted a realistic view on the current progress of this journey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' We critically reviewed the NISQ paradigm and identified its realistic capabilities but also its obvious limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' We provided an overview about methods for both: analysing a quantum algorithm using the simplified, device-agnostic Pauli error model, and establishing far more detailed knowledge about the error mechanisms of a given quantum machine by means of randomised benchmarking and gate set tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content=' ACKNOWLEDGMENTS Parts of this work are supported by project QORA within the Competence Center Quantum Computing Baden-W¨urttemberg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='9 alpha: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='974(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='8e-04) EPC: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='966e-02(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='0e-04) lation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='8 Popul: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='7 Ground 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sdFKT4oBgHgl3EQfJy1E/content/2301.11739v1.pdf'} +page_content='3 0 25 50 75 100 125 150 175 200 Clifford Length(funded by the Ministerium f¨ur Wirtschaft, Arbeit und Woh- nungbau 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Mirasola,b Nathan Musoke,a Mark C. +Neyrinck,c Chanda Prescod-Weinsteina +aDepartment of Physics & Astronomy +University of New Hampshire +Durham, NH 03824, USA +bDepartment of Physics +University of Illinois at Urbana-Champaign +Urbana, IL 6180, USA +cIkerbasque, the Basque Foundation for Science +48009, Bilbao, Spain +E-mail: nglennon@wildcats.unh.edu, aem8@illinois.edu, nathan.musoke@unh.edu, +Mark.Neyrinck@gmail.com, chanda.prescod-weinstein@unh.edu +Abstract. Galaxies and their dark-matter halos are commonly presupposed to spin. But it is +an open question how this spin manifests in halos and soliton cores made of scalar dark matter +(SDM, including fuzzy/wave/ultralight-axion dark matter). One way spin could manifest in +a necessarily irrotational SDM velocity field is with a vortex. But recent results have cast +doubt on this scenario, finding that vortices are generally unstable except with substantial +repulsive self-interaction. In this paper, we introduce an alternative route to stability: in +both (non-relativistic) analytic calculations and simulations, a black hole or other central +mass at least as massive as a soliton can stabilize a vortex within it. This conclusion may +also apply to stellar-scale Bose stars. +Keywords: scalar dark matter, fuzzy dark matter, vortex, black hole +arXiv:2301.13220v1 [astro-ph.CO] 30 Jan 2023 + +Contents +1 +Introduction +1 +2 +Analytic description of vortices in SDM +4 +2.1 +Rotating solutions to Gross–Pitaevskii–Poisson equations +4 +2.2 +Instability of vortex-solitons +5 +3 +Simulations of vortex solitons +8 +4 +Discussion +14 +1 +Introduction +Astronomical objects such as galaxies and stars in the Universe generally have some nonzero +angular momentum. If an object’s angular momentum is drawn from some continuous dis- +tribution about zero, it has formally zero chance of being exactly zero. +It is not trivial to explain cosmological-scale spin in galaxies, because the primordial +velocity field is thought to have had negligible vorticity since it stretched away during and +after inflation, leaving only a gravity-sourced potential flow until collapsed (multistream) ob- +jects form. These then have angular momenta distributed with some nonzero width around +zero. The standard, widely accepted explanation for how a cosmological-scale object comes +to rotate as it forms in a primordially irrotational velocity field is the tidal torque theory [1]: +protogalaxies and protohalos, the regions of the primordial, homogeneous density field that +collapse to form galaxies and the dark-matter halos around them, are never perfectly spher- +ical. Their aspherical protuberances are generally torqued up by gravitational tidal fields. +This can also be understood as the protuberances carrying some nonzero primordial gravity- +sourced velocities, whose contribution to the final collapsed object’s angular momentum does +not cancel out [2]. +Intergalactic-scale torques should spin up both baryons and dark matter similarly, e.g. +with similar angular momentum per unit mass. +One way to understand this is that the +torquing is gravitational, which by the equivalence principle should affect all matter equiva- +lently. But this spin presents a question in a scalar dark matter (SDM) scenario, applicable +to fuzzy, wave, or ultralight-axion dark matter. This is because an SDM velocity field is +irrotational, being defined as a gradient, with zero curl. How can an object that is part of +an irrotational flow be said to spin? +SDM models have seen substantial recent attention as a promising alternative to the +longstanding WIMP cold dark matter (WIMP CDM) model. Simulations of SDM show that +these alternative models behave similarly to WIMP CDM in large-scale structure formation +(where WIMP CDM has been successful), but ultralight scalars radically differ on galactic +scales. +WIMP CDM encounters well-known issues on these scales; there is an apparent +deficit of observed satellite galaxies compared to simple estimates from N-body simulations +of WIMPs, and there is a disagreement between simulated and observed density profiles in +galactic cores [3–5]. Some work has suggested that baryonic effects could possibly account +for these discrepancies [6, 7], but this motivation for SDM or warm dark matter remains. +Some have even argued that a 10−25 eV particle comprising a small fraction of the dark +– 1 – + +matter density would affect the H0 tension [8]. There also are other motivations for SDM, +beyond addressing observational problems such as the missing-satellite problem. They are +well-motivated in string theory [9–12]. +Coming back to the question of spin in SDM, there are three mechanisms for a patch +of an irrotational fluid such as SDM to carry angular momentum. First, it can sport vor- +tices, where the density goes to zero, and the vorticity to infinity. This mechanism is unusual +from an astrophysical standpoint, but it is seen in laboratory superfluid-torquing experiments +(e.g. [13]). A second mechanism to exhibit angular momentum in SDM is in an inhomoge- +neous density field. If the density is spherically symmetric, the angular momentum will be +zero within a sphere. But if we increase the density in a moving lump near the edge of the +sphere, that will generally unbalance the angular momentum integral and make the total +nonzero. A third mechanism, possible even in a homogeneous density field, is if the patch +has an aspherical boundary. In this case, regions outside a maximal sphere about the center +generally will contribute angular momentum (e.g. [2]). An example of such an object is a +Riemann S-ellipsoid [14]. Torqued-up SDM subhalos in cosmological simulations have been +observed to resemble these ellipsoids [15]. +While these latter two mechanisms exist, we imagine them not to be able to carry +arbitrarily large spin except in contrived cases. We are most interested in the first, i.e. ‘spin- +driven vortices’ that would be produced when an object is sufficiently spun up. They are +particularly interesting because of the exotic phenomenology, that has the most plausible +observational relevance. Spin-driven vortices contrast conceptually with ‘random vortices,’ +meandering loci where the wave function happens to go precisely to zero; these are straight- +forward to study with random wave interference [16]. In an interfering jumble far from the +center of a dark-matter halo, it is not completely clear how to distinguish random from +spin-driven vortices apart from their different formation mechanism, i.e., from a single snap- +shot of the wave-function. But if the vortex threads a soliton core, another piece of SDM +phenomenology, it would be hard not to identify it with physical spin. +Solitons are stable structures thought to reside in the centers of SDM halos, from +both simulations and theoretical arguments [17–22]. +They are supported against gravi- +tational collapse by “quantum” pressure [23, 24], and in some models, by repulsive self- +interactions [25] so long as certain soliton mass limits are not exceeded (in the case of at- +tractive self-interactions) [26–28]. These structures are long-lived, localized solutions which +are stable against perturbations, and so are commonly referred to as solitons. These solitons +are expected to form through gravitational thermalization in the centers of SDM halos, and +they have a variety of observational effects which could lead to their identification. +It would be a major change in the standard SDM soliton picture if vortices commonly +inhabited solitons. Some previous studies have argued that this arrangement is unstable (and +therefore uncommon) unless the SDM has sufficiently strong repulsive self-interactions [29, +30]. That is, a soliton with one vortex inside the core transitions to a system with the same +angular momentum and many vortices outside the soliton core. While vortices still exist in +such a system, when they are far away from the soliton core, they exist in regions of much +lower density, or can become lost in the chaotic halo that surrounds the soliton. Thus for all +practical purposes the vortex has decayed out of the system. +In this work, we show that there is another scenario besides strongly repulsive self- +interactions that can stabilize vortices within soliton cores: a background gravitational well +generated not by the SDM itself, but by other matter. +While this does not change the +energy analysis, and the rotating soliton still is not the energy minimum with fixed mass and +– 2 – + +angular momentum, it does suppress the decay channel, causing the rotating soliton to have +an extremely long lifetime. When the soliton lives in such a gravitational potential sourced +by other matter (baryonic, black-hole, or ambient non-solitonic dark matter), the vortex can +have a long lifetime before decaying. This stabilizing mechanism is relevant because all SDM +solitons are expected to form in galactic cores where there is a concentration of other matter. +Particularly, supermassive black holes (mass 106 to 109 M⊙,) are thought to be present in the +centers of almost all galaxies (e.g. [31]), plausibly coincident with soliton centers. Simulations +exist that include SDM, hydrodynamics, and star formation [22, 32], but none as far as we +know that self-consistently include central black holes as well. Thus, even if the situation of +a vortex-soliton supported by a supermassive black hole were common, we would not expect +to find them in previous cosmological simulations. +Interactions between the angular momentum of supermassive black holes and their en- +vironments is important, for example to the merger of such black holes [33]. The literature +contains other studies of the interaction between scalar and ultralight dark matter and black +holes, including: dynamical friction due to black holes moving through scalar dark matter +fields [34–36]; the formation of black holes in scalar dark matter halos [37, 38]; the formation +of vortex-less solitons through accretion of dark matter on black holes and interpretation +as black-hole hair [39–50]; black-hole superradiance and interactions between spinning black +holes and scalar halos [46, 51–59], and the merger of black holes in wave dark matter en- +vironments [60–62]. The last two classes are particularly pertinent; they are scenarios with +significant angular momentum. +Another, recently found example of cosmological-scale rotation is in filaments (com- +prised of gas, dark matter, and small galaxies) that generally connect neighboring galax- +ies [63, 64]. Simulations in a SDM scenario that include hydrodynamics [22] have found that +intergalactic filaments often contain soliton tubes in their centers, although these seem to +fragment into cores on long timescales. It would be quite interesting if spin-driven vortex +tubes often resided in filaments. But in a 2D version of the 3D discussion above about halo +spin, an irrotational SDM velocity field could carry some amount of filamentary angular mo- +mentum without vortex tubes, through velocity or density asymmetry in the filament’s cross +section. +The route to stability we find in this paper does not immediately pertain to vortex +tubes within intergalactic filaments, because there is no known stringlike analog of a black +hole that would gravitationally support a vortex tube. Still, it would be interesting to further +investigate in simulations what form filament spin takes in SDM, and if vortex tubes might +often inhabit filaments. It is possible that they could have long decay/fragmentation lifetimes, +as the soliton tubes themselves have, or could be supported by repulsive self-interactions. +This paper is organised as follows. In section 2 we analyse theoretical considerations +for rotating solitons in the presence of a background potential. In section 2.1 we present an +approximate solution for vortices stabilised by the gravitational potential of a point mass. In +section 2.2 we discuss the energy and decay modes of a rotating soliton in an central gravi- +tational potential, making analytic arguments that support the results found in simulations. +In section 3, we use simulations from UltraDark.jl to demonstrate that vortex solitons can +be long-lived even in the presence of perturbations. We end with a discussion of possible +applications and future directions in section 4. +– 3 – + +2 +Analytic description of vortices in SDM +2.1 +Rotating solutions to Gross–Pitaevskii–Poisson equations +We are interested in scalar particles with mass ma and Lagrangian density +L = 1 +2gµν∂µφ∂νφ − 1 +2m2 +aφ2 − λ +4 φ4 . +(2.1) +In the non-relativistic limit, the field φ can be rewritten as +φ = +1 +√ +2m +� +e−imtψ + e+imtψ∗� +, +(2.2) +and transformed to a field ψ whose equations of motion are the Gross–Pitaevskii–Poisson +equations +i∂ψ +∂t = − 1 +2m∇2ψ + mψ (Φ + Φbg) + +λ +2m2 |ψ|2ψ +(2.3) +∇2Φ(r) = 4πGρ(r) = 4π|ψ|2 . +(2.4) +Here Φ is the gravitational potential sourced by the ULDM density |ψ|2, Φbg is the gravi- +tational potential due to a mass with density ρbg, and λ parametrizes the self-interactions +arising from a quartic term in the SDM Lagrangian, with λ > 0 corresponding to repulsive +self-interactions. +Such a quartic self-interaction is common in axion models arising from +string theory [10]. +We do not consider backreaction on the background density, instead +treating ρbg as a fixed background. +The simplest solutions to the Gross–Pitaevskii equations that carry non-zero velocity +circulation are axially symmetric solutions with a central vortex line, characterized by cir- +culation l, which is an integer defining the winding number of the phase around the vortex +line [29]. These solutions have the form +ψl(r, z)e−iωlt+ilφ, +(2.5) +where a vortex line of circulation l is located at r = 0. The phase-independent amplitude +ψl(r, z) can be explicitly written if self-interactions vanish (λ = 0) and the background po- +tential in equation (2.3) is due to a central point mass with Φbg ≫ |Φ|2. Then equation (2.3) +is approximately +i∂ψ +∂t ≈ − 1 +2m∇2ψ − mψMbg +r +. +(2.6) +The gravitational potential from the point mass is analogous to the Coulomb potential, so +in this limit, the rotating soliton is well-described by hydrogen atom wavefunctions [65–68]. +The lowest energy solution exhibiting a vortex with circulation 1 is +ψ2,1,±1 = +1 +2 +√ +6a−3/2 �r +a +� +e−r/2a +� +− +� 3 +8π +�1/2 +sin θe±iφ +� +(2.7) +where a = 1/Mbg, r is the radial coordinate, θ is the polar angle, and φ is the azimuthal +angle. Due to the nonlinearities in the Gross–Pitaevskii–Poisson equations, the exact solution +exhibits deviations from the hydrogen wavefunctions in its radial profile. Indeed, we observe +– 4 – + +in simulations that starting from a radial profile that deviates from that of the ground state +results in radial-profile oscillations, but not azimuthal perturbations. In the absence of a +background potential, there are significant deviations from these hydrogen-like solutions. +However, qualitative properties such as axial symmetry, and the presence of a central vortex +in the middle of the densest region of the soliton remain [69]. +We refer to all such systems as “vortex solitons” to indicate the essential point that the +vortex is located in the center of the soliton, although we note that these objects may not +satisfy all properties normally associated with solitons, in particular, their stability against +collisions and perturbations. +When the vortex line passes through the soliton core, as it does in these hydrogen-like +wavefunctions, the angular momentum Ltot is quantized to integer multiples of the particle +number N in the soliton, Ltot = ℏNl. However, the system can carry a lower value of angular +momentum if the vortex line departs from the center of the soliton. In the limit that the +vortex is far from the soliton center, the angular momentum asymptotically approaches zero. +When the vortex is far away from the soliton, there is hardly any mass near the vortex. The +mass, and hence angular momentum density, is concentrated in the soliton core. But the +velocity field is highest near the vortex, because the velocity is proportional to the phase +gradient, and the velocity falls off as one over the distance from the vortex. Therefore when +the vortex is displaced from the soliton core, the total angular momentum can take any value +L < Ltot = Nl. +2.2 +Instability of vortex-solitons +Ref. [29] demonstrated that in the absence of repulsive self-interactions, the vortices in ro- +tating solitons of scalar field dark matter are unstable. They showed that the vortex state +is not the lowest energy state; it can attain lower energy by shedding angular momentum. +They did not consider the influence of a central background potential. We show first that +when an background potential Φbg caused by point mass Mbg is included, the vortex state is +still not the lowest energy state, so it cannot truly be stable. We then show that the decay +channel is suppressed, leading to the long lifetimes of the vortices. +Begin with an initial state in the configuration ψ′ +s, with circulation l > 0 and particle +number Ns. We will construct a configuration with the same particle number Ns and total +angular momentum ℏlNs, but lower energy, and without a vortex in the soliton core. This +shows that the vortex is not a global minimum energy state in the sector with angular +momentum l, so it is not stable to sufficiently large perturbations. +To construct the lower-energy configuration, we remove dNs particles from the initial +state, and add dN0 particles to the l = 0 mode and dNl′ to the l′ ≫ l mode. In order to +conserve particle number, we require +dNs = dN0 + dNl′. +(2.8) +In order to conserve angular momentum, we require +ldNs = l′dNl′. +(2.9) +Thus the final wavefunction is +ψ′ +s → ψ = ψs + +� +dN0Ψ0 + +� +dNl′Ψl′, +(2.10) +where the magnitude of the original soliton configuration has decreased, |ψs(x)| < |ψ′ +s(x)|. +– 5 – + +We will show that the energy of the initial rotating soliton is greater than the energy +of this new configuration that has the same angular momentum. The energy of the final +configuration is +Ef = Es + ω0dN0 + ωl′dNl′, +(2.11) +where ωl′ is the energy of a single particle in state l′, and Es is the energy of all particles +remaining in the initial state of the rotating soliton. The initial energy is +Ei = Es + ωsdNs + O(dN2 +s ), +(2.12) +so the change in energy is +∆E = (ω0 − ωs)dN0 + (ωl′ − ωs)dNl′. +(2.13) +To show that this configuration has a lower energy than the vortex soliton, we must estimate +the energy differences ωl′ − ωs, for l′ = 0, and for l′ ≫ 1. +The modes Ψl′ and energies ωl′, are the solutions to the non-interacting Gross–Pitaevskii +equation with background potential generated by the soliton and (in our case) the background +central potential: +ωl′Ψl′ = −∇2Ψl′ +2m ++ m(Φs + Φbg)Ψl′. +(2.14) +For any l′ < l, where l is the angular momentum of the soliton, we must have ωl′ < ωs, +because the centrifugal barrier is weaker, while the other energy terms are the same. So for +l = 0, we have +ω0 − ωs < − +� +d3xl2|Ψl|2 +2mr2 < 0. +(2.15) +The background potential does not depend on l, so it does not affect this estimate of the +energy difference. In fact, it further lowers the energy of the l′ = 0 state, because that state +has more mass closer to the center of the potential. +Now consider the l′ ≫ 1 state. The wavefunctions become more and more spatially +spread out as l′ increases, so we can approximate both the wavefunctions and the energies as +the eigenstates of the hydrogen atom with mass Ms + Mbg (since the soliton behaves like a +point mass when viewed from long distances). The energies are therefore +ωl′ ≈ −m3G2(Ms + Mbg)2 +2(l′ + 1)2 +∼ O(l′−2). +(2.16) +Note that the background potential significantly increases the energy gaps between states +with increasing l′ (by adding a term proportional to M2 +bg, but does not affect the scaling of +the energies with l′). In particular, as as l′ → ∞, the energies asymptotically approach zero. +Returning to Eq. (2.13), we can re-write the energy difference as +∆E = (ω0 − ωs)dN0 + (O(l′−2) − ωs)dNl′ += (ω0 − ωs)dN0 + (O(l′−2) − ωs) l +l′ dNs += (ω0 − ωs)dNs + O(l′−1)dNs < 0, +(2.17) +where in the second line we used Eq. (2.9), which ensures conservation of angular momentum. +Therefore, for sufficiently large l′, the energy gap is dominated by the first term, which we +– 6 – + +estimated in Eq. (2.15) is negative. This shows that the initial vortex state is not a global +energy minimum over l, so it is not stable against perturbations. +While the vortex state is not a global energy minimum, it might still be extraordinarily +long lived. To decay from the initial vortex state into the lower energy configuration con- +structed above, we would require transfers from the initial l = 1 state into an l′ ≫ 1 state, +but these are not strongly coupled together. In simulations of vortex solitons without back- +ground potential (which do show an instability) [29], the decay has been demonstrated to +occur due to pairwise transitions from l state to neighboring l′ = l ± 1 states. In particular, +an initial l = 1 loses occupancy to l′ = 0, 2 states, whose occupation numbers grow. At later +times, the occupation of the l = 2 state slows while l′ = 1, 3 begins to grow. +The transitions from the initial state into a decay state can only take place when there +are finite transition matrix elements between the coupled states, so that there is a decay +channel between the coupled states. These transition matrix elements can be estimated in +a perturbation theory [70]. In the limit where the background mass is significantly greater +than the soliton mass, Mbg/Ms ≫ 1, all transition matrix elements vanish, because the +background potential dominates the self-potential of the soliton and the Gross–Pitaevskii– +Poisson equations become linear equations with exact eigenstates corresponding to hydrogen +atom wavefunctions. Thus in this limit we expect the soliton to be long-lived and stable +against all perturbations. However, the transition matrix elements are suppressed to the first +order in perturbation theory even when the nonlinearities of the system are still significant. In +particular, the decay mode identified by Ref. [29] is suppressed by the background potential. +At the first order in perturbation theory, the stationary states are the eigenstates of the +background potential. Since our background potential is that of a point mass, these are the +hydrogen atom wavefunctions. The self-gravity of the soliton creates a perturbative potential +∆V that introduces transition matrix elements between the unperturbed stationary states. In +our initial state in Eq. (2.7), the mass distribution is axially symmetric, ρ(r, θ, φ) = ρ(r, θ), +and hence the gravitational potential is axially symmetric as well. In this case, the mass +distribution and gravitational potential admits a multipole expansion in terms of Legendre +polynomials Pn(cos θ), +ρ(r, θ) = +� +n +ρn(r)Pn(cos θ) +(2.18) +Φ(r, θ) = +� +n +Φn(r)Pn(cos θ), +(2.19) +where the components Φn of the potential are determined by +Φn(r) = − 2πG +n + 1 +2 +r−n−1 +� r +0 +r′n+2ρn(r′)dr′ − 2πG +n + 1 +2 +rn +� ∞ +r +r′n−1ρn(r′)dr′. +(2.20) +In our case, the initial state has a mass distribution with only P0 and P2 components +nonzero. Therefore these are the only nonvanishing components of the potential caused by +the rotating soliton. Moreover, the background gravitational potential caused by the point +mass is a central potential and so only has nonvanishing P0 component. These monopole +moments conserve angular momentum and do not lead to any transitions between states +of different l. The component proportional to P2 does allow transitions. The transitional +matrix element is +⟨l′m′|∆V |lm⟩ = +� +f′∗(r)Φ2(r)f(r)r2dr +� +Y m′ +l′ +∗Y 0 +2 Y m +l dΩ, +(2.21) +– 7 – + +where f, f′ are the radial wavefunctions of the initial and final states and Y m +l +are spherical +harmonics. The integral over angles is a Wigner 3-j symbol, +⟨l′m′|Φ|lm⟩ ∝ +� l 2 +l′ +m 0 −m′ +� +. +(2.22) +The Wigner 3-j symbols are nonzero only when their selection rules are satisfied. For our +initial rotating soliton state |l = 1, m = 1⟩, we have permitted transitions to |l = 2, m = 1⟩, +|l = 3, m = 1⟩, and no others. In particular, the pairwise transition which leads to the decay +of the rotating soliton without background potential is not permitted. Thus the dominant +instability mode observed by Ref. [29] is not initially present in this system, and is suppressed +by the background potential, at least at first order in perturbation theory. This suppression +of the dominant instability mode of the soliton without background potential leads to the +long lifetime of our vortex solitons compared to those in other studies. +3 +Simulations of vortex solitons +We use UltraDark.jl to simulate these approximate solutions and understand their dynam- +ics and stability. UltraDark.jl is a pseudospectral solver for the Gross–Pitaevskii–Poisson +equations that allows for contributions from background gravitational fields, such as the +potential Φbg in equation (2.4) [71]. +UltraDark.jl uses code units for time, length and mass, +T = +� 3 +8πH2 +0Ωm,0 +�−1/2 +≈ 74 Gyr +(3.1) +L = +� ℏ +m +�1/2� 3 +8πΩm,0H2 +0 +�−1/4 +≈ 38 +�10−22 eV +m +�1/2 +kpc +(3.2) +M = +� ℏ +m +�3/2 1 +G +� 3 +8πΩm,0H2 +0 +�1/4 +≈ 2.2 × 106 +�10−22 eV +m +�3/2 +M⊙ . +(3.3) +In these units, which we write as primed, equations (2.3) and (2.4) become +i∂ψ′ +∂t′ = − 1 +a2 ∇′2ψ′ + ψ′ � +Φ′(r′) + Φ′ +bg(r) +� +(3.4) +∇′2Φ′(r) = 1 +a4π|ψ′|2 . +(3.5) +We use H0 = 70 km/s/Mpc and Ωm,0 = 0.3. In these units, the critical density of the +Universe today is 1 M/L3. +These units are dependent on the mass m of SDM particle +considered, and so the following simulations can be interpreted as corresponding to different +scenarios for different values of m. In the interest of clarity, units in figures assume m = 10−22, +with the understanding that a different choice of m would scale the units as above. We run +simulations with a resolution of 2563 and a box length of lbox = 22/Mbg set by the length +scale in the initial wavefunction equation (2.7). +UltraDark.jl has periodic boundary conditions. The density at the edge of the grid +is ≲ 1% of the maximum density, and so only a small amount of matter passes through +the boundary. The periodic boundary conditions also cause the solutions to see gravitational +fields due to the scalar field in neighbouring boxes; this adds a violation of spherical symmetry +– 8 – + +beyond that of the Cartesian grid. The gravitational potential due to the central mass does +not experience these effects. +Our simulations assume Newtonian gravity, but are sufficient to model black holes on +these scales. The Schwarzschild radius of such a black hole would be ≲ 1 × 10−6 kpc, far +smaller than the grid spacing of ∼ 10−1 kpc. Furthermore, the vortex has vanishing density +at the center so accretion would be minimal. +We consider a vortex to be stable up to a decay time tdecay if the winding number does +not change in this time interval. We measure the winding number by integrating the phase +difference between neighbouring grid points around a loop. The choice of loop around which +to compute the winding number is important but somewhat subjective. When perturbations +are introduced, the vortex orbits the central mass with a small but non-zero radius, even +when has not decayed (see for example the second snapshot in figure 3). If the loop is too +small, the vortex moves outside of it even though it has not decayed. If the loop is too large, +it includes transient vortices in the underdense outskirts of the soliton. These can increase +the winding number if the have the same chirality as the central vortex, or decrease it if they +have opposite chirality. We used loops with radius similar to the radius of a ground state +soliton of the same mass. In practice, this corresponded to a radius of 10 to 12 grid points. +Figures 1 to 3 show a circle with a radius of 10 grid points. +In figure 1 we show the results of evolving the initial conditions of equation (2.7) forward +with Mbg = 3.5×107 M⊙ × +� +10−22 eV/m +�3/2 and Msol = +√ +2Mbg. One can see that although +there are radial oscillations, the vortex persists to the end of the simulation. There are also +perturbations aligned with the simulation grid; these break the axial symmetry of the ansatz, +so should not increase the stability of the vortex. +Verifying stability requires the addition of perturbations that break the cylindrical sym- +metry of the system. We perturb the simulation with 10 small Gaussian overdensities with +random position and velocity. In particular, +δψ = A +Npert +� +j=1 +ρc exp +� +−(rj − r)2 +σ +� +exp (i(rj − r) · vj) , +(3.6) +where σ = lbox/50, and for each perturbation the spherical radius rj is uniformly sampled +from the interval [lbox/2.5, lbox/2 × 0.9], the azimuthal angle from [0, 2π] and the polar an- +gle from [0, π]. Each perturbation is assumed to have an angular velocity ωj, where each +component of ωj is drawn from a normal distribution with mean 0 and standard deviation +1; vj = rj × ωj. The same positions and velocities of random perturbations are used for all +simulations. The overall scaling A is set such that the ratio of the mass of the perturbations +and soliton is |δψ|2/|ψ|2 = 1/100. A realistic galaxy core might be subject to much larger +perturbations, but preliminary simulations suggest that a central mass prolongs the lifetime +of vortex-solitons even in the presence of much larger perturbations. We have not yet fully +investigated this area of parameter space. +Figure 2 shows the result of adding this perturbation to the approximate solution in +equation (2.7) and evolving it forward. +The other initial conditions are the same as in +figure 1: Mbg = 3.5 × 107 M⊙ × +� +10−22 eV/m +�3/2 and Msol = +√ +2Mbg. One may note that +perturbations are not obvious in the initial phase; this is because the perturbation densities +and velocities are small, and many of the perturbations lie outside the plane. The vortex +persists until the end of the simulation, 7.6 Gyr. +– 9 – + +t = 0:0 [Gyr] +t = 3:7 [Gyr] +x [kpc] +-20 +-10 +0 +10 +20 +t = 7:6 [Gyr] +½=½crit +0 +1.0×10⁴ +2.0×10⁴ +3.0×10⁴ +y [kpc] +-20 +-10 +0 +10 +20 +y [kpc] +-20 +-10 +0 +10 +20 +x [kpc] +-20 +-10 +0 +10 +20 +y [kpc] +-20 +-10 +0 +10 +20 +arg(Ã) +¡ ¼ +¡ ¼=2 +0 +¼=2 +¼ +Figure 1. (Stable vortex-soliton, with no initial perturbations.) Snapshots of a simulation with +initial conditions equation (2.7) with Mbg = 3.5 × 107 M⊙ × +� +10−22 eV/m +�3/2 and Msol = +√ +2Mbg. +Time increases from top to bottom. The x- and y-grids are the same in each panel. The left column +shows the density projected into the x-y plane. The right column is the phase in the x-y plane in a +slice through z = 0; the red curve is that used to compute the winding number. This initial condition +is not an exact equilibrium because Msol > 0. One can see that although there are radial fluctuations, +the vortex persists. See https://www.youtube.com/watch?v=dEHL1Io0akY for an animation. +– 10 – + +t = 0:0 [Gyr] +t = 3:7 [Gyr] +x [kpc] +-20 +-10 +0 +10 +20 +t = 7:6 [Gyr] +½=½crit +0 +1.0×10⁴ 2.0×10⁴ 3.0×10⁴ +y [kpc] +-20 +-10 +0 +10 +20 +y [kpc] +-20 +-10 +0 +10 +20 +x [kpc] +-20 +-10 +0 +10 +20 +y [kpc] +-20 +-10 +0 +10 +20 +arg(Ã) +¡ ¼ +¡ ¼=2 +0 +¼=2 +¼ +Figure 2. (Stable vortex-soliton, with initial perturbations.) Snapshots of a simulation with initial +conditions equation (2.7) with Mbg = 3.5 × 107 M⊙ × +� +10−22 eV/m +�3/2, Msol = +√ +2Mbg and pertur- +bations as in equation (3.6). Time increases from top to bottom. The left column is the projected +density. The right column is the phase in the x-y plane in a slice through z = 0; the red curve is that +used to compute the winding number. This scenario does not have axial symmetry, but the central +vortex persists. See https://www.youtube.com/watch?v=DYeL5UHQjdE for an animation. +– 11 – + +.t = 0:0 [Gyr] +t = 3:7 [Gyr] +x [kpc] +-20 +-10 +0 +10 +20 +t = 7:6 [Gyr] +½=½crit +0 +1.0×10⁶ +2.0×10⁶ +3.0×10⁶ +y [kpc] +-20 +-10 +0 +10 +20 +y [kpc] +-20 +-10 +0 +10 +20 +x [kpc] +-20 +-10 +0 +10 +20 +y [kpc] +-20 +-10 +0 +10 +20 +arg(Ã) +¡ ¼ +¡ ¼=2 +0 +¼=2 +¼ +Figure 3. +(Unstable vortex-soliton, with initial perturbations.) +Snapshots of a simulation with +initial conditions equation (2.7) with Mbg = 3.5 × 107 M⊙ × +� +10−22 eV/m +�3/2, Msol = 8Mbg and +perturbations as in equation (3.6). +Time increases from top to bottom. +The left column is the +projected density. +The right column is the phase in the x-y plane in a slice through z = 0; the +red curve is that used to compute the winding number. The ratio of the soliton mass and central +mass is sufficiently large that the vortex is unstable even in the presence of the central mass. See +https://www.youtube.com/watch?v=g9NVU4LK2Lc for an animation. +– 12 – + +Msol=Mbg +2.0 +4.0 +8.0 +tdecay [Gyr] +10⁰⋅⁰ +10⁰⋅⁵ +10¹⋅⁰ +Mbg [M ¯ £ 107] +3.52 +7.04 +Figure 4. Plot of decay time against the mass ratio Msol/Mbg. The solid blue curve with circles has +a central mass of Mbg = 3.5 × 107 M⊙ × +� +10−22 eV/m +�3/2. The dashed orange curve with triangles +has a Mbg = 7.0 × 107 M⊙ × +� +10−22 eV/m +�3/2. In each case, the bold (faint) curve has winding +number computed with an aperture of 10 (12) grid points. In both cases, increasing the mass of the +soliton relative to the central mass decreases the time taken for the vortex to decay. +Figure 3 shows snapshots of a scenario which is unstable because the central mass is +not dominant. As in figures 1 and 2, Mbg = 3.5 × 107 M⊙ × +� +10−22 eV/m +�3/2. However, the +vortex-soliton is much more massive, with Msol = 8Mbg. As the simulation proceeds, the +initial dark matter distribution contracts to form a genus 0 soliton. This soliton falls to the +bottom of the central potential. Even before the initial central vortex dissipates, it orbits +the central potential, instead of remaining centred. +To explore the relation between the mass ratio Msol/Mbg and decay time, we ran a set +of simulations with Mbg = 3.5×107 M⊙ and Mbg = 7.0×107 M⊙, and varying Msol/Mbg; the +results of this are shown in figure 4. In the interest of computational time, we run simulations +for only 7.6 Gyr; we do not assign a finite decay time when the vortex persists up to the +end of our simulations, as in cases where Msol/Mbg ≲ 1. The trend is as expected: vortices +persist for shorter times when the soliton mass is large relative to the central mass. There +is some scatter in the decay times, so the curves are not perfectly monotonic. This is due to +the nonlinear nature of this scenario. +In figure 5, we explore the relation between central mass and decay time, at fixed +Msol/Mbg = 8. We see a negative correlation between Mbg and tdecay. This appears to be +a power-law with tdecay ∼ (80 Gyr)(Mbg/107 M⊙)−2., but we caution against extrapolating +outside this small mass range. This can be explained by the increased density of solitons with +higher mass. As the mass of the rotating soliton increases, the radius of the torus shrinks. So +at the location of the soliton’s maximum density, the soliton’s self-gravity is a higher fraction +of the background potential, leading to a greater influence of nonlinearities in the system’s +evolution and a shorter decay time. +– 13 – + +Mbg [M ¯ £ 107] +10⁰⋅⁶ +10⁰⋅⁷ +10⁰⋅⁸ +10⁰⋅⁹ +tdecay [Gyr] +10⁰⋅⁰ +10⁰⋅² +10⁰⋅⁴ +10⁰⋅⁶ +Figure 5. Plot of decay time against the central mass Mbg. In each case, Msol/Mbg = 8. The bold +(faint) curve has winding number computed with an aperture of 10 (12) grid points. Increasing the +central mass also decreases the stability time at fixed mass ratio. +4 +Discussion +Our calculations show that a background gravitational potential suppresses the decay of +vortices in SDM. Our simulations show that this suppression is sufficient to give vortices +in soliton cores long lifetimes (compared e.g. to the dynamical/rotational time of the Milky +Way, ∼ 0.25 Gyr) when the central gravitational field is generated by a black hole with a +mass of order that of the soliton itself. The vortices are even longer-lived when the black-hole +mass is greater than the soliton mass. The lifetime of the vortices can exceed the Hubble +time [12, 72]. So, for all practical purposes, the vortices can be treated as stable, even though +they are not the true lowest-energy state that carries angular momentum. +However, in an ultralight dark matter (ULDM) scenario of a 10−22 eV axion, using +current estimates, supermassive black holes would typically be a few orders of magnitude too +light to stabilize vortex-solitons in most galaxies. In the Milky Way, of halo mass ∼ 1012M⊙, +Sgr A* has a mass of ∼ 106M⊙, while its soliton is estimated from simulations to be ∼ +109M⊙[17]. For other supermassive black holes, given fiducial scalings of Mcore ∝ M1/3 +halo [17] +and Mblack hole ∝ M1.55±0.05 +halo +[73], we would expect black holes to typically achieve equal mass +to soliton cores only for the most massive halos, with halo mass ≳ 1015M⊙, and black-hole +and soliton masses of ≳ 1010M⊙. There are few halos thought to be as massive as 1015M⊙; +perhaps the largest cluster known, El Gordo, is thought to have mass ∼ 2 × 1015M⊙ (each of +two merging parts with mass ∼ 1015M⊙). There are more than a handful of ‘ultramassive’ +(M ≥ 1010M⊙) black holes known currently (e.g. 7 in [74]). As candidates for the most- +massive, TON 618 has been measured at ∼ 4×1010M⊙ [75] and Phoenix A has mass perhaps +1011M⊙ [76], even though a theoretical upper limit of 5×1010 has been estimated for a black +hole accreting its mass through a disk [77]. Even the first-imaged black hole, M87*, has +– 14 – + +mass (6.5 ± 1) × 109M⊙ [78], within striking distance of 1010; recall that the stabilization +mechanism turns on gradually, still providing some stability when a black hole is lighter +than the vortex-soliton. The assembly histories in these extreme clusters may be particularly +complicated, but still, it seems worth considering whether these could have vortex-solitons. +Several uncertainties need to be kept in mind, though, e.g. the scaling of soliton-core mass +with halo mass has substantial uncertainty and scatter (e.g. [79]). The black-hole mass scaling +is uncertain as well; another estimate of the scaling exponent is 1.62 ([80], as used in Ref. +[81]). +We originally conceived this stabilization mechanism for central ULDM solitons in galax- +ies, since ordinary matter and black holes would typically inhabit their centers as well. But it +may be even more applicable to lower-mass solitons from higher-mass SDM particles. Due to +the scale-invariance of the Gross–Pitaevskii–Poisson equations, the simulations in section 3 +solve the dynamics of a family of systems of characterized by the particle mass m. The +QCD axion has a much heavier particle mass, m = 10−4 eV, and forms solitons of a typical +mass on the order of 10−14 M⊙ and radius on the order of 300 km [82–86]. Our simulations +suggest that in a gravitational well caused by ordinary hadronic matter of comparable mass, +the soliton can support vortices in its cores with lifetimes of many dynamical times, going +up to effectively infinite lifetime if the central mass is far-dominant. In an intermediate-mass +scenario, a Solar-System-scale soliton may exist around the Sun. An axion mass of ∼ 10−14 +eV would give an AU-scale vortex-less soliton, possibly detectable even if it is ∼ 12 orders +of magnitude less massive than the Sun [87]. A rotating version of this soliton seems quite +plausible and would have maximum density at ∼ 1 AU. +All of our analysis was with zero SDM self-interaction. Previous work found that vortices +are stable with repulsive self-interactions, but only when the mass of the soliton is larger that +some critical mass [29]. Adding a central mass should reduce this critical mass. On the other +hand, we found that vortices without self-interactions are long-lived when the soliton mass is +less than a different critical mass, roughly equal to the central mass. Adding self-interaction +to our scenario would change this critical central mass too. Attractive self-interactions would +destabilize the vortex, increasing the central mass threshold giving stability, and repulsive +self-interactions would tend to stabilize the vortex, decreasing this threshold. +A full answer to the question of whether our gravitational route to SDM vortex stability +actually enables stable spin-driven vortices in our Universe may require self-consistent cos- +mological simulations including SDM, hydrodynamics in the baryons, and black holes. It also +remains a question how a vortex-soliton around a black hole would form in the first place; it +is plausible but not clear quantitatively when dynamical friction from rotating baryons would +torque up SDM [88]. We think it possible that vortex-solitons naturally arise in a sufficiently +fast-rotating halo, but have not shown an explicit formation mechanism or investigated how +common the required level of rotation is in realistic halos. +In our study, we have treated the black hole as a nonrelativistic point mass poten- +tial, which is valid when the radius of the vortex-soliton is significantly longer than the +Schwarzschild radius. When this assumption does not hold, a richer phenomenology can en- +sue. In this case, superradiance can convert a rotating black hole’s spin to gravitational radi- +ation and may generate a rotationally synchronized bosonic dark matter halo, with stability +lifetimes possibly exceeding the age of the Universe [89, 90]. The gravitational stabilization +we show here may contribute to the stability of such systems of ‘black holes with synchro- +nized hair.’ On the other hand, if a vortex-soliton surrounding a black hole has radius much +larger than the black hole, the rotation would nearly evacuate the immediate surroundings +– 15 – + +of the black hole of dark matter, suppressing its interaction with the black hole. +Also, if black holes sometimes carry dark-matter vortex-solitons around with them, that +is relevant to black-hole mergers and the role that dark matter plays in them [62] including +closing the ‘final parsec’ of black-hole mergers [33]. +In conclusion, we have demonstrated an alternative mechanism for the long-term stabil- +ity of vortices in SDM: a background gravitational potential suppresses their dominant decay +mode. We have primarily concerned ourselves with stabilization of vortex-solitons at the +centers of dark matter halos comprised of particles with mass ∼ 10−22 eV by supermassive +black holes, but have highlighted other scenarios where this mechanism may be relevant. +This introduces a new mechanism by which black holes and other pointlike masses might +connect to their surroundings. +Author Contributions +All authors contributed substantial ideas across the various concepts in the paper, but here +we list particular contributions. NG contributed expertise with SDM simulations. AEM +and NM did the bulk of the analysis and writing; AEM concentrated on the analytic energy +arguments; NM wrote, designed, ran, and analysed simulations. MN initially conceived of the +project that evolved into the current paper, and contributed large-scale-structure expertise +and writing. +CPW organized the working group, contributed ideas about SDM and its +phenomenology from a particle-theory perspective, and guided the paper’s and project’s +coherence. +Acknowledgments +We thank Luna Zagorac and Dmitry Levkov for helpful discussions. We would also like to +thank the administrative and facilities staff at the University of New Hampshire including +Katie Makem-Boucher and Michelle Mancini. +Computations were performed on Marvin, a Cray CS500 supercomputer at UNH sup- +ported by the NSF MRI program under grant AGS-1919310. AEM’s contributions to this +project were supported by DOE Grant DE-SC0020220. NG’s participation was supported in +part by the National Science Foundation under Grant No. 1929080. This work was initiated +and performed in part at Aspen Center for Physics, which is supported by National Science +Foundation under Grant No. PHY-1607611. CPW thanks the late Karsten Pohl for actively +supporting the application for NSF grant No. 1929080. +This paper honors the memory of both Keenan Anderson and Tyre Nichols. +References +[1] P. J. E. Peebles, Origin of the Angular Momentum of Galaxies, ApJ 155 (Feb., 1969) 393. +[2] M. Neyrinck, M. A. Aragon-Calvo, B. Falck, A. S. Szalay and J. Wang, Halo Spin from +Primordial Inner Motions, The Open Journal of Astrophysics 3 (Mar., 2020) 3, [1904.03201]. +[3] D. H. Weinberg, J. S. Bullock, F. Governato, R. Kuzio de Naray and A. H. Peter, Cold dark +matter: Controversies on small scales, Proceedings of the National Academy of Sciences 112 +(2015) 12249–12255, [1306.0913]. +[4] B. 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Lett. 119 (Dec., 2017) 261101, [1706.06597]. +– 21 – + diff --git a/w9FPT4oBgHgl3EQf_zUO/content/tmp_files/load_file.txt b/w9FPT4oBgHgl3EQf_zUO/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8867e020751c55a3ecaee5cbf3f28db4c2853f3f --- /dev/null +++ b/w9FPT4oBgHgl3EQf_zUO/content/tmp_files/load_file.txt @@ -0,0 +1,1242 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf,len=1241 +page_content='Prepared for submission to JCAP Scalar dark matter vortex stabilization with black holes Noah Glennon,a Anthony E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Mirasola,b Nathan Musoke,a Mark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Neyrinck,c Chanda Prescod-Weinsteina aDepartment of Physics & Astronomy University of New Hampshire Durham, NH 03824, USA bDepartment of Physics University of Illinois at Urbana-Champaign Urbana, IL 6180, USA cIkerbasque, the Basque Foundation for Science 48009, Bilbao, Spain E-mail: nglennon@wildcats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='unh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='edu, aem8@illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='edu, nathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='musoke@unh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='edu, Mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='Neyrinck@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='com, chanda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='prescod-weinstein@unh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Galaxies and their dark-matter halos are commonly presupposed to spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' But it is an open question how this spin manifests in halos and soliton cores made of scalar dark matter (SDM, including fuzzy/wave/ultralight-axion dark matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' One way spin could manifest in a necessarily irrotational SDM velocity field is with a vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' But recent results have cast doubt on this scenario, finding that vortices are generally unstable except with substantial repulsive self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In this paper, we introduce an alternative route to stability: in both (non-relativistic) analytic calculations and simulations, a black hole or other central mass at least as massive as a soliton can stabilize a vortex within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This conclusion may also apply to stellar-scale Bose stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Keywords: scalar dark matter, fuzzy dark matter, vortex, black hole arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='13220v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='CO] 30 Jan 2023 Contents 1 Introduction 1 2 Analytic description of vortices in SDM 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='1 Rotating solutions to Gross–Pitaevskii–Poisson equations 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='2 Instability of vortex-solitons 5 3 Simulations of vortex solitons 8 4 Discussion 14 1 Introduction Astronomical objects such as galaxies and stars in the Universe generally have some nonzero angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' If an object’s angular momentum is drawn from some continuous dis- tribution about zero, it has formally zero chance of being exactly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' It is not trivial to explain cosmological-scale spin in galaxies, because the primordial velocity field is thought to have had negligible vorticity since it stretched away during and after inflation, leaving only a gravity-sourced potential flow until collapsed (multistream) ob- jects form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' These then have angular momenta distributed with some nonzero width around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The standard, widely accepted explanation for how a cosmological-scale object comes to rotate as it forms in a primordially irrotational velocity field is the tidal torque theory [1]: protogalaxies and protohalos, the regions of the primordial, homogeneous density field that collapse to form galaxies and the dark-matter halos around them, are never perfectly spher- ical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Their aspherical protuberances are generally torqued up by gravitational tidal fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This can also be understood as the protuberances carrying some nonzero primordial gravity- sourced velocities, whose contribution to the final collapsed object’s angular momentum does not cancel out [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Intergalactic-scale torques should spin up both baryons and dark matter similarly, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' with similar angular momentum per unit mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' One way to understand this is that the torquing is gravitational, which by the equivalence principle should affect all matter equiva- lently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' But this spin presents a question in a scalar dark matter (SDM) scenario, applicable to fuzzy, wave, or ultralight-axion dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This is because an SDM velocity field is irrotational, being defined as a gradient, with zero curl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' How can an object that is part of an irrotational flow be said to spin?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' SDM models have seen substantial recent attention as a promising alternative to the longstanding WIMP cold dark matter (WIMP CDM) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Simulations of SDM show that these alternative models behave similarly to WIMP CDM in large-scale structure formation (where WIMP CDM has been successful), but ultralight scalars radically differ on galactic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' WIMP CDM encounters well-known issues on these scales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' there is an apparent deficit of observed satellite galaxies compared to simple estimates from N-body simulations of WIMPs, and there is a disagreement between simulated and observed density profiles in galactic cores [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Some work has suggested that baryonic effects could possibly account for these discrepancies [6, 7], but this motivation for SDM or warm dark matter remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Some have even argued that a 10−25 eV particle comprising a small fraction of the dark – 1 – matter density would affect the H0 tension [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' There also are other motivations for SDM, beyond addressing observational problems such as the missing-satellite problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' They are well-motivated in string theory [9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Coming back to the question of spin in SDM, there are three mechanisms for a patch of an irrotational fluid such as SDM to carry angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' First, it can sport vor- tices, where the density goes to zero, and the vorticity to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This mechanism is unusual from an astrophysical standpoint, but it is seen in laboratory superfluid-torquing experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' A second mechanism to exhibit angular momentum in SDM is in an inhomoge- neous density field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' If the density is spherically symmetric, the angular momentum will be zero within a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' But if we increase the density in a moving lump near the edge of the sphere, that will generally unbalance the angular momentum integral and make the total nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' A third mechanism, possible even in a homogeneous density field, is if the patch has an aspherical boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In this case, regions outside a maximal sphere about the center generally will contribute angular momentum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' An example of such an object is a Riemann S-ellipsoid [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Torqued-up SDM subhalos in cosmological simulations have been observed to resemble these ellipsoids [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' While these latter two mechanisms exist, we imagine them not to be able to carry arbitrarily large spin except in contrived cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We are most interested in the first, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' ‘spin- driven vortices’ that would be produced when an object is sufficiently spun up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' They are particularly interesting because of the exotic phenomenology, that has the most plausible observational relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Spin-driven vortices contrast conceptually with ‘random vortices,’ meandering loci where the wave function happens to go precisely to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' these are straight- forward to study with random wave interference [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In an interfering jumble far from the center of a dark-matter halo, it is not completely clear how to distinguish random from spin-driven vortices apart from their different formation mechanism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=', from a single snap- shot of the wave-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' But if the vortex threads a soliton core, another piece of SDM phenomenology, it would be hard not to identify it with physical spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Solitons are stable structures thought to reside in the centers of SDM halos, from both simulations and theoretical arguments [17–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' They are supported against gravi- tational collapse by “quantum” pressure [23, 24], and in some models, by repulsive self- interactions [25] so long as certain soliton mass limits are not exceeded (in the case of at- tractive self-interactions) [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' These structures are long-lived, localized solutions which are stable against perturbations, and so are commonly referred to as solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' These solitons are expected to form through gravitational thermalization in the centers of SDM halos, and they have a variety of observational effects which could lead to their identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' It would be a major change in the standard SDM soliton picture if vortices commonly inhabited solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Some previous studies have argued that this arrangement is unstable (and therefore uncommon) unless the SDM has sufficiently strong repulsive self-interactions [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' That is, a soliton with one vortex inside the core transitions to a system with the same angular momentum and many vortices outside the soliton core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' While vortices still exist in such a system, when they are far away from the soliton core, they exist in regions of much lower density, or can become lost in the chaotic halo that surrounds the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Thus for all practical purposes the vortex has decayed out of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In this work, we show that there is another scenario besides strongly repulsive self- interactions that can stabilize vortices within soliton cores: a background gravitational well generated not by the SDM itself, but by other matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' While this does not change the energy analysis, and the rotating soliton still is not the energy minimum with fixed mass and – 2 – angular momentum, it does suppress the decay channel, causing the rotating soliton to have an extremely long lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' When the soliton lives in such a gravitational potential sourced by other matter (baryonic, black-hole, or ambient non-solitonic dark matter), the vortex can have a long lifetime before decaying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This stabilizing mechanism is relevant because all SDM solitons are expected to form in galactic cores where there is a concentration of other matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Particularly, supermassive black holes (mass 106 to 109 M⊙,) are thought to be present in the centers of almost all galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' [31]), plausibly coincident with soliton centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Simulations exist that include SDM, hydrodynamics, and star formation [22, 32], but none as far as we know that self-consistently include central black holes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Thus, even if the situation of a vortex-soliton supported by a supermassive black hole were common, we would not expect to find them in previous cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Interactions between the angular momentum of supermassive black holes and their en- vironments is important, for example to the merger of such black holes [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The literature contains other studies of the interaction between scalar and ultralight dark matter and black holes, including: dynamical friction due to black holes moving through scalar dark matter fields [34–36];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' the formation of black holes in scalar dark matter halos [37, 38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' the formation of vortex-less solitons through accretion of dark matter on black holes and interpretation as black-hole hair [39–50];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' black-hole superradiance and interactions between spinning black holes and scalar halos [46, 51–59], and the merger of black holes in wave dark matter en- vironments [60–62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The last two classes are particularly pertinent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' they are scenarios with significant angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Another, recently found example of cosmological-scale rotation is in filaments (com- prised of gas, dark matter, and small galaxies) that generally connect neighboring galax- ies [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Simulations in a SDM scenario that include hydrodynamics [22] have found that intergalactic filaments often contain soliton tubes in their centers, although these seem to fragment into cores on long timescales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' It would be quite interesting if spin-driven vortex tubes often resided in filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' But in a 2D version of the 3D discussion above about halo spin, an irrotational SDM velocity field could carry some amount of filamentary angular mo- mentum without vortex tubes, through velocity or density asymmetry in the filament’s cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The route to stability we find in this paper does not immediately pertain to vortex tubes within intergalactic filaments, because there is no known stringlike analog of a black hole that would gravitationally support a vortex tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Still, it would be interesting to further investigate in simulations what form filament spin takes in SDM, and if vortex tubes might often inhabit filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' It is possible that they could have long decay/fragmentation lifetimes, as the soliton tubes themselves have, or could be supported by repulsive self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In section 2 we analyse theoretical considerations for rotating solitons in the presence of a background potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='1 we present an approximate solution for vortices stabilised by the gravitational potential of a point mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='2 we discuss the energy and decay modes of a rotating soliton in an central gravi- tational potential, making analytic arguments that support the results found in simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In section 3, we use simulations from UltraDark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='jl to demonstrate that vortex solitons can be long-lived even in the presence of perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We end with a discussion of possible applications and future directions in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' – 3 – 2 Analytic description of vortices in SDM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='1 Rotating solutions to Gross–Pitaevskii–Poisson equations We are interested in scalar particles with mass ma and Lagrangian density L = 1 2gµν∂µφ∂νφ − 1 2m2 aφ2 − λ 4 φ4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='1) In the non-relativistic limit, the field φ can be rewritten as φ = 1 √ 2m � e−imtψ + e+imtψ∗� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='2) and transformed to a field ψ whose equations of motion are the Gross–Pitaevskii–Poisson equations i∂ψ ∂t = − 1 2m∇2ψ + mψ (Φ + Φbg) + λ 2m2 |ψ|2ψ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='3) ∇2Φ(r) = 4πGρ(r) = 4π|ψ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='4) Here Φ is the gravitational potential sourced by the ULDM density |ψ|2, Φbg is the gravi- tational potential due to a mass with density ρbg, and λ parametrizes the self-interactions arising from a quartic term in the SDM Lagrangian, with λ > 0 corresponding to repulsive self-interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Such a quartic self-interaction is common in axion models arising from string theory [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We do not consider backreaction on the background density, instead treating ρbg as a fixed background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The simplest solutions to the Gross–Pitaevskii equations that carry non-zero velocity circulation are axially symmetric solutions with a central vortex line, characterized by cir- culation l, which is an integer defining the winding number of the phase around the vortex line [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' These solutions have the form ψl(r, z)e−iωlt+ilφ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5) where a vortex line of circulation l is located at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The phase-independent amplitude ψl(r, z) can be explicitly written if self-interactions vanish (λ = 0) and the background po- tential in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='3) is due to a central point mass with Φbg ≫ |Φ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Then equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='3) is approximately i∂ψ ∂t ≈ − 1 2m∇2ψ − mψMbg r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='6) The gravitational potential from the point mass is analogous to the Coulomb potential, so in this limit, the rotating soliton is well-described by hydrogen atom wavefunctions [65–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The lowest energy solution exhibiting a vortex with circulation 1 is ψ2,1,±1 = 1 2 √ 6a−3/2 �r a � e−r/2a � − � 3 8π �1/2 sin θe±iφ � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='7) where a = 1/Mbg, r is the radial coordinate, θ is the polar angle, and φ is the azimuthal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Due to the nonlinearities in the Gross–Pitaevskii–Poisson equations, the exact solution exhibits deviations from the hydrogen wavefunctions in its radial profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Indeed, we observe – 4 – in simulations that starting from a radial profile that deviates from that of the ground state results in radial-profile oscillations, but not azimuthal perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In the absence of a background potential, there are significant deviations from these hydrogen-like solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' However, qualitative properties such as axial symmetry, and the presence of a central vortex in the middle of the densest region of the soliton remain [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We refer to all such systems as “vortex solitons” to indicate the essential point that the vortex is located in the center of the soliton, although we note that these objects may not satisfy all properties normally associated with solitons, in particular, their stability against collisions and perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' When the vortex line passes through the soliton core, as it does in these hydrogen-like wavefunctions, the angular momentum Ltot is quantized to integer multiples of the particle number N in the soliton, Ltot = ℏNl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' However, the system can carry a lower value of angular momentum if the vortex line departs from the center of the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In the limit that the vortex is far from the soliton center, the angular momentum asymptotically approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' When the vortex is far away from the soliton, there is hardly any mass near the vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The mass, and hence angular momentum density, is concentrated in the soliton core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' But the velocity field is highest near the vortex, because the velocity is proportional to the phase gradient, and the velocity falls off as one over the distance from the vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Therefore when the vortex is displaced from the soliton core, the total angular momentum can take any value L < Ltot = Nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='2 Instability of vortex-solitons Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' [29] demonstrated that in the absence of repulsive self-interactions, the vortices in ro- tating solitons of scalar field dark matter are unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' They showed that the vortex state is not the lowest energy state;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' it can attain lower energy by shedding angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' They did not consider the influence of a central background potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We show first that when an background potential Φbg caused by point mass Mbg is included, the vortex state is still not the lowest energy state, so it cannot truly be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We then show that the decay channel is suppressed, leading to the long lifetimes of the vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Begin with an initial state in the configuration ψ′ s, with circulation l > 0 and particle number Ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We will construct a configuration with the same particle number Ns and total angular momentum ℏlNs, but lower energy, and without a vortex in the soliton core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This shows that the vortex is not a global minimum energy state in the sector with angular momentum l, so it is not stable to sufficiently large perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' To construct the lower-energy configuration, we remove dNs particles from the initial state, and add dN0 particles to the l = 0 mode and dNl′ to the l′ ≫ l mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In order to conserve particle number, we require dNs = dN0 + dNl′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='8) In order to conserve angular momentum, we require ldNs = l′dNl′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='9) Thus the final wavefunction is ψ′ s → ψ = ψs + � dN0Ψ0 + � dNl′Ψl′, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='10) where the magnitude of the original soliton configuration has decreased, |ψs(x)| < |ψ′ s(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' – 5 – We will show that the energy of the initial rotating soliton is greater than the energy of this new configuration that has the same angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The energy of the final configuration is Ef = Es + ω0dN0 + ωl′dNl′, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='11) where ωl′ is the energy of a single particle in state l′, and Es is the energy of all particles remaining in the initial state of the rotating soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The initial energy is Ei = Es + ωsdNs + O(dN2 s ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='12) so the change in energy is ∆E = (ω0 − ωs)dN0 + (ωl′ − ωs)dNl′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='13) To show that this configuration has a lower energy than the vortex soliton, we must estimate the energy differences ωl′ − ωs, for l′ = 0, and for l′ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The modes Ψl′ and energies ωl′, are the solutions to the non-interacting Gross–Pitaevskii equation with background potential generated by the soliton and (in our case) the background central potential: ωl′Ψl′ = −∇2Ψl′ 2m + m(Φs + Φbg)Ψl′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='14) For any l′ < l, where l is the angular momentum of the soliton, we must have ωl′ < ωs, because the centrifugal barrier is weaker, while the other energy terms are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' So for l = 0, we have ω0 − ωs < − � d3xl2|Ψl|2 2mr2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='15) The background potential does not depend on l, so it does not affect this estimate of the energy difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In fact, it further lowers the energy of the l′ = 0 state, because that state has more mass closer to the center of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Now consider the l′ ≫ 1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The wavefunctions become more and more spatially spread out as l′ increases, so we can approximate both the wavefunctions and the energies as the eigenstates of the hydrogen atom with mass Ms + Mbg (since the soliton behaves like a point mass when viewed from long distances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The energies are therefore ωl′ ≈ −m3G2(Ms + Mbg)2 2(l′ + 1)2 ∼ O(l′−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='16) Note that the background potential significantly increases the energy gaps between states with increasing l′ (by adding a term proportional to M2 bg, but does not affect the scaling of the energies with l′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In particular, as as l′ → ∞, the energies asymptotically approach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Returning to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='13), we can re-write the energy difference as ∆E = (ω0 − ωs)dN0 + (O(l′−2) − ωs)dNl′ = (ω0 − ωs)dN0 + (O(l′−2) − ωs) l l′ dNs = (ω0 − ωs)dNs + O(l′−1)dNs < 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='17) where in the second line we used Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='9), which ensures conservation of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Therefore, for sufficiently large l′, the energy gap is dominated by the first term, which we – 6 – estimated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='15) is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This shows that the initial vortex state is not a global energy minimum over l, so it is not stable against perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' While the vortex state is not a global energy minimum, it might still be extraordinarily long lived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' To decay from the initial vortex state into the lower energy configuration con- structed above, we would require transfers from the initial l = 1 state into an l′ ≫ 1 state, but these are not strongly coupled together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In simulations of vortex solitons without back- ground potential (which do show an instability) [29], the decay has been demonstrated to occur due to pairwise transitions from l state to neighboring l′ = l ± 1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In particular, an initial l = 1 loses occupancy to l′ = 0, 2 states, whose occupation numbers grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' At later times, the occupation of the l = 2 state slows while l′ = 1, 3 begins to grow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The transitions from the initial state into a decay state can only take place when there are finite transition matrix elements between the coupled states, so that there is a decay channel between the coupled states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' These transition matrix elements can be estimated in a perturbation theory [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In the limit where the background mass is significantly greater than the soliton mass, Mbg/Ms ≫ 1, all transition matrix elements vanish, because the background potential dominates the self-potential of the soliton and the Gross–Pitaevskii– Poisson equations become linear equations with exact eigenstates corresponding to hydrogen atom wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Thus in this limit we expect the soliton to be long-lived and stable against all perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' However, the transition matrix elements are suppressed to the first order in perturbation theory even when the nonlinearities of the system are still significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In particular, the decay mode identified by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' [29] is suppressed by the background potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' At the first order in perturbation theory, the stationary states are the eigenstates of the background potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Since our background potential is that of a point mass, these are the hydrogen atom wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The self-gravity of the soliton creates a perturbative potential ∆V that introduces transition matrix elements between the unperturbed stationary states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In our initial state in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='7), the mass distribution is axially symmetric, ρ(r, θ, φ) = ρ(r, θ), and hence the gravitational potential is axially symmetric as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In this case, the mass distribution and gravitational potential admits a multipole expansion in terms of Legendre polynomials Pn(cos θ), ρ(r, θ) = � n ρn(r)Pn(cos θ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='18) Φ(r, θ) = � n Φn(r)Pn(cos θ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='19) where the components Φn of the potential are determined by Φn(r) = − 2πG n + 1 2 r−n−1 � r 0 r′n+2ρn(r′)dr′ − 2πG n + 1 2 rn � ∞ r r′n−1ρn(r′)dr′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='20) In our case, the initial state has a mass distribution with only P0 and P2 components nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Therefore these are the only nonvanishing components of the potential caused by the rotating soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Moreover, the background gravitational potential caused by the point mass is a central potential and so only has nonvanishing P0 component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' These monopole moments conserve angular momentum and do not lead to any transitions between states of different l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The component proportional to P2 does allow transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The transitional matrix element is ⟨l′m′|∆V |lm⟩ = � f′∗(r)Φ2(r)f(r)r2dr � Y m′ l′ ∗Y 0 2 Y m l dΩ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='21) – 7 – where f, f′ are the radial wavefunctions of the initial and final states and Y m l are spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The integral over angles is a Wigner 3-j symbol, ⟨l′m′|Φ|lm⟩ ∝ � l 2 l′ m 0 −m′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='22) The Wigner 3-j symbols are nonzero only when their selection rules are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' For our initial rotating soliton state |l = 1, m = 1⟩, we have permitted transitions to |l = 2, m = 1⟩, |l = 3, m = 1⟩, and no others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In particular, the pairwise transition which leads to the decay of the rotating soliton without background potential is not permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Thus the dominant instability mode observed by Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' [29] is not initially present in this system, and is suppressed by the background potential, at least at first order in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This suppression of the dominant instability mode of the soliton without background potential leads to the long lifetime of our vortex solitons compared to those in other studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' 3 Simulations of vortex solitons We use UltraDark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='jl to simulate these approximate solutions and understand their dynam- ics and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' UltraDark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='jl is a pseudospectral solver for the Gross–Pitaevskii–Poisson equations that allows for contributions from background gravitational fields, such as the potential Φbg in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='4) [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' UltraDark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='jl uses code units for time, length and mass, T = � 3 8πH2 0Ωm,0 �−1/2 ≈ 74 Gyr (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='1) L = � ℏ m �1/2� 3 8πΩm,0H2 0 �−1/4 ≈ 38 �10−22 eV m �1/2 kpc (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='2) M = � ℏ m �3/2 1 G � 3 8πΩm,0H2 0 �1/4 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='2 × 106 �10−22 eV m �3/2 M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='3) In these units, which we write as primed, equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='4) become i∂ψ′ ∂t′ = − 1 a2 ∇′2ψ′ + ψ′ � Φ′(r′) + Φ′ bg(r) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='4) ∇′2Φ′(r) = 1 a4π|ψ′|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5) We use H0 = 70 km/s/Mpc and Ωm,0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In these units, the critical density of the Universe today is 1 M/L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' These units are dependent on the mass m of SDM particle considered, and so the following simulations can be interpreted as corresponding to different scenarios for different values of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In the interest of clarity, units in figures assume m = 10−22, with the understanding that a different choice of m would scale the units as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We run simulations with a resolution of 2563 and a box length of lbox = 22/Mbg set by the length scale in the initial wavefunction equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' UltraDark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='jl has periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The density at the edge of the grid is ≲ 1% of the maximum density, and so only a small amount of matter passes through the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The periodic boundary conditions also cause the solutions to see gravitational fields due to the scalar field in neighbouring boxes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' this adds a violation of spherical symmetry – 8 – beyond that of the Cartesian grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The gravitational potential due to the central mass does not experience these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Our simulations assume Newtonian gravity, but are sufficient to model black holes on these scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The Schwarzschild radius of such a black hole would be ≲ 1 × 10−6 kpc, far smaller than the grid spacing of ∼ 10−1 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Furthermore, the vortex has vanishing density at the center so accretion would be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We consider a vortex to be stable up to a decay time tdecay if the winding number does not change in this time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We measure the winding number by integrating the phase difference between neighbouring grid points around a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The choice of loop around which to compute the winding number is important but somewhat subjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' When perturbations are introduced, the vortex orbits the central mass with a small but non-zero radius, even when has not decayed (see for example the second snapshot in figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' If the loop is too small, the vortex moves outside of it even though it has not decayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' If the loop is too large, it includes transient vortices in the underdense outskirts of the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' These can increase the winding number if the have the same chirality as the central vortex, or decrease it if they have opposite chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We used loops with radius similar to the radius of a ground state soliton of the same mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In practice, this corresponded to a radius of 10 to 12 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Figures 1 to 3 show a circle with a radius of 10 grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In figure 1 we show the results of evolving the initial conditions of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='7) forward with Mbg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5×107 M⊙ × � 10−22 eV/m �3/2 and Msol = √ 2Mbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' One can see that although there are radial oscillations, the vortex persists to the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' There are also perturbations aligned with the simulation grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' these break the axial symmetry of the ansatz, so should not increase the stability of the vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Verifying stability requires the addition of perturbations that break the cylindrical sym- metry of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We perturb the simulation with 10 small Gaussian overdensities with random position and velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In particular, δψ = A Npert � j=1 ρc exp � −(rj − r)2 σ � exp (i(rj − r) · vj) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='6) where σ = lbox/50, and for each perturbation the spherical radius rj is uniformly sampled from the interval [lbox/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5, lbox/2 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='9], the azimuthal angle from [0, 2π] and the polar an- gle from [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Each perturbation is assumed to have an angular velocity ωj, where each component of ωj is drawn from a normal distribution with mean 0 and standard deviation 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' vj = rj × ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The same positions and velocities of random perturbations are used for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The overall scaling A is set such that the ratio of the mass of the perturbations and soliton is |δψ|2/|ψ|2 = 1/100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' A realistic galaxy core might be subject to much larger perturbations, but preliminary simulations suggest that a central mass prolongs the lifetime of vortex-solitons even in the presence of much larger perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We have not yet fully investigated this area of parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Figure 2 shows the result of adding this perturbation to the approximate solution in equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='7) and evolving it forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The other initial conditions are the same as in figure 1: Mbg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5 × 107 M⊙ × � 10−22 eV/m �3/2 and Msol = √ 2Mbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' One may note that perturbations are not obvious in the initial phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' this is because the perturbation densities and velocities are small, and many of the perturbations lie outside the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The vortex persists until the end of the simulation, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='6 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' – 9 – t = 0:0 [Gyr] t = 3:7 [Gyr] x [kpc] 20 10 0 10 20 t = 7:6 [Gyr] ½=½crit 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁴ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁴ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁴ y [kpc] 20 10 0 10 20 y [kpc] 20 10 0 10 20 x [kpc] 20 10 0 10 20 y [kpc] 20 10 0 10 20 arg(Ã) ¡ ¼ ¡ ¼=2 0 ¼=2 ¼ Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (Stable vortex-soliton, with no initial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=') Snapshots of a simulation with initial conditions equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='7) with Mbg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5 × 107 M⊙ × � 10−22 eV/m �3/2 and Msol = √ 2Mbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Time increases from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The x- and y-grids are the same in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The left column shows the density projected into the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The right column is the phase in the x-y plane in a slice through z = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' the red curve is that used to compute the winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This initial condition is not an exact equilibrium because Msol > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' One can see that although there are radial fluctuations, the vortex persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='v=dEHL1Io0akY for an animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' – 10 – t = 0:0 [Gyr] t = 3:7 [Gyr] x [kpc] 20 10 0 10 20 t = 7:6 [Gyr] ½=½crit 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁴ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁴ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁴ y [kpc] 20 10 0 10 20 y [kpc] 20 10 0 10 20 x [kpc] 20 10 0 10 20 y [kpc] 20 10 0 10 20 arg(Ã) ¡ ¼ ¡ ¼=2 0 ¼=2 ¼ Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (Stable vortex-soliton, with initial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=') Snapshots of a simulation with initial conditions equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='7) with Mbg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5 × 107 M⊙ × � 10−22 eV/m �3/2, Msol = √ 2Mbg and pertur- bations as in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Time increases from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The left column is the projected density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The right column is the phase in the x-y plane in a slice through z = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' the red curve is that used to compute the winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This scenario does not have axial symmetry, but the central vortex persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='v=DYeL5UHQjdE for an animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' – 11 – .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='t = 0:0 [Gyr] t = 3:7 [Gyr] x [kpc] 20 10 0 10 20 t = 7:6 [Gyr] ½=½crit 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁶ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁶ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×10⁶ y [kpc] 20 10 0 10 20 y [kpc] 20 10 0 10 20 x [kpc] 20 10 0 10 20 y [kpc] 20 10 0 10 20 arg(Ã) ¡ ¼ ¡ ¼=2 0 ¼=2 ¼ Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' (Unstable vortex-soliton, with initial perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=') Snapshots of a simulation with initial conditions equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='7) with Mbg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5 × 107 M⊙ × � 10−22 eV/m �3/2, Msol = 8Mbg and perturbations as in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Time increases from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The left column is the projected density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The right column is the phase in the x-y plane in a slice through z = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' the red curve is that used to compute the winding number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The ratio of the soliton mass and central mass is sufficiently large that the vortex is unstable even in the presence of the central mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='youtube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='com/watch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='v=g9NVU4LK2Lc for an animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' – 12 – Msol=Mbg 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0 tdecay [Gyr] 10⁰⋅⁰ 10⁰⋅⁵ 10¹⋅⁰ Mbg [M ¯ £ 107] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='52 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='04 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Plot of decay time against the mass ratio Msol/Mbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The solid blue curve with circles has a central mass of Mbg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5 × 107 M⊙ × � 10−22 eV/m �3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The dashed orange curve with triangles has a Mbg = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0 × 107 M⊙ × � 10−22 eV/m �3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In each case, the bold (faint) curve has winding number computed with an aperture of 10 (12) grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In both cases, increasing the mass of the soliton relative to the central mass decreases the time taken for the vortex to decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Figure 3 shows snapshots of a scenario which is unstable because the central mass is not dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' As in figures 1 and 2, Mbg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5 × 107 M⊙ × � 10−22 eV/m �3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' However, the vortex-soliton is much more massive, with Msol = 8Mbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' As the simulation proceeds, the initial dark matter distribution contracts to form a genus 0 soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This soliton falls to the bottom of the central potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Even before the initial central vortex dissipates, it orbits the central potential, instead of remaining centred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' To explore the relation between the mass ratio Msol/Mbg and decay time, we ran a set of simulations with Mbg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5×107 M⊙ and Mbg = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='0×107 M⊙, and varying Msol/Mbg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' the results of this are shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In the interest of computational time, we run simulations for only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='6 Gyr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' we do not assign a finite decay time when the vortex persists up to the end of our simulations, as in cases where Msol/Mbg ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The trend is as expected: vortices persist for shorter times when the soliton mass is large relative to the central mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' There is some scatter in the decay times, so the curves are not perfectly monotonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This is due to the nonlinear nature of this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In figure 5, we explore the relation between central mass and decay time, at fixed Msol/Mbg = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We see a negative correlation between Mbg and tdecay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This appears to be a power-law with tdecay ∼ (80 Gyr)(Mbg/107 M⊙)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=', but we caution against extrapolating outside this small mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This can be explained by the increased density of solitons with higher mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' As the mass of the rotating soliton increases, the radius of the torus shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' So at the location of the soliton’s maximum density, the soliton’s self-gravity is a higher fraction of the background potential, leading to a greater influence of nonlinearities in the system’s evolution and a shorter decay time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' – 13 – Mbg [M ¯ £ 107] 10⁰⋅⁶ 10⁰⋅⁷ 10⁰⋅⁸ 10⁰⋅⁹ tdecay [Gyr] 10⁰⋅⁰ 10⁰⋅² 10⁰⋅⁴ 10⁰⋅⁶ Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Plot of decay time against the central mass Mbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In each case, Msol/Mbg = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The bold (faint) curve has winding number computed with an aperture of 10 (12) grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Increasing the central mass also decreases the stability time at fixed mass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' 4 Discussion Our calculations show that a background gravitational potential suppresses the decay of vortices in SDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Our simulations show that this suppression is sufficient to give vortices in soliton cores long lifetimes (compared e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' to the dynamical/rotational time of the Milky Way, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='25 Gyr) when the central gravitational field is generated by a black hole with a mass of order that of the soliton itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The vortices are even longer-lived when the black-hole mass is greater than the soliton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The lifetime of the vortices can exceed the Hubble time [12, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' So, for all practical purposes, the vortices can be treated as stable, even though they are not the true lowest-energy state that carries angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' However, in an ultralight dark matter (ULDM) scenario of a 10−22 eV axion, using current estimates, supermassive black holes would typically be a few orders of magnitude too light to stabilize vortex-solitons in most galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In the Milky Way, of halo mass ∼ 1012M⊙, Sgr A* has a mass of ∼ 106M⊙, while its soliton is estimated from simulations to be ∼ 109M⊙[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' For other supermassive black holes, given fiducial scalings of Mcore ∝ M1/3 halo [17] and Mblack hole ∝ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='55±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='05 halo [73], we would expect black holes to typically achieve equal mass to soliton cores only for the most massive halos, with halo mass ≳ 1015M⊙, and black-hole and soliton masses of ≳ 1010M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' There are few halos thought to be as massive as 1015M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' perhaps the largest cluster known, El Gordo, is thought to have mass ∼ 2 × 1015M⊙ (each of two merging parts with mass ∼ 1015M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' There are more than a handful of ‘ultramassive’ (M ≥ 1010M⊙) black holes known currently (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' 7 in [74]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' As candidates for the most- massive, TON 618 has been measured at ∼ 4×1010M⊙ [75] and Phoenix A has mass perhaps 1011M⊙ [76], even though a theoretical upper limit of 5×1010 has been estimated for a black hole accreting its mass through a disk [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Even the first-imaged black hole, M87*, has – 14 – mass (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='5 ± 1) × 109M⊙ [78], within striking distance of 1010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' recall that the stabilization mechanism turns on gradually, still providing some stability when a black hole is lighter than the vortex-soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The assembly histories in these extreme clusters may be particularly complicated, but still, it seems worth considering whether these could have vortex-solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Several uncertainties need to be kept in mind, though, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' the scaling of soliton-core mass with halo mass has substantial uncertainty and scatter (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' [79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The black-hole mass scaling is uncertain as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' another estimate of the scaling exponent is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content='62 ([80], as used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' [81]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We originally conceived this stabilization mechanism for central ULDM solitons in galax- ies, since ordinary matter and black holes would typically inhabit their centers as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' But it may be even more applicable to lower-mass solitons from higher-mass SDM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Due to the scale-invariance of the Gross–Pitaevskii–Poisson equations, the simulations in section 3 solve the dynamics of a family of systems of characterized by the particle mass m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The QCD axion has a much heavier particle mass, m = 10−4 eV, and forms solitons of a typical mass on the order of 10−14 M⊙ and radius on the order of 300 km [82–86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Our simulations suggest that in a gravitational well caused by ordinary hadronic matter of comparable mass, the soliton can support vortices in its cores with lifetimes of many dynamical times, going up to effectively infinite lifetime if the central mass is far-dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In an intermediate-mass scenario, a Solar-System-scale soliton may exist around the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' An axion mass of ∼ 10−14 eV would give an AU-scale vortex-less soliton, possibly detectable even if it is ∼ 12 orders of magnitude less massive than the Sun [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' A rotating version of this soliton seems quite plausible and would have maximum density at ∼ 1 AU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' All of our analysis was with zero SDM self-interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Previous work found that vortices are stable with repulsive self-interactions, but only when the mass of the soliton is larger that some critical mass [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Adding a central mass should reduce this critical mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' On the other hand, we found that vortices without self-interactions are long-lived when the soliton mass is less than a different critical mass, roughly equal to the central mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Adding self-interaction to our scenario would change this critical central mass too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Attractive self-interactions would destabilize the vortex, increasing the central mass threshold giving stability, and repulsive self-interactions would tend to stabilize the vortex, decreasing this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' A full answer to the question of whether our gravitational route to SDM vortex stability actually enables stable spin-driven vortices in our Universe may require self-consistent cos- mological simulations including SDM, hydrodynamics in the baryons, and black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' It also remains a question how a vortex-soliton around a black hole would form in the first place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' it is plausible but not clear quantitatively when dynamical friction from rotating baryons would torque up SDM [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We think it possible that vortex-solitons naturally arise in a sufficiently fast-rotating halo, but have not shown an explicit formation mechanism or investigated how common the required level of rotation is in realistic halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In our study, we have treated the black hole as a nonrelativistic point mass poten- tial, which is valid when the radius of the vortex-soliton is significantly longer than the Schwarzschild radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' When this assumption does not hold, a richer phenomenology can en- sue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In this case, superradiance can convert a rotating black hole’s spin to gravitational radi- ation and may generate a rotationally synchronized bosonic dark matter halo, with stability lifetimes possibly exceeding the age of the Universe [89, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' The gravitational stabilization we show here may contribute to the stability of such systems of ‘black holes with synchro- nized hair.’ On the other hand, if a vortex-soliton surrounding a black hole has radius much larger than the black hole, the rotation would nearly evacuate the immediate surroundings – 15 – of the black hole of dark matter, suppressing its interaction with the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Also, if black holes sometimes carry dark-matter vortex-solitons around with them, that is relevant to black-hole mergers and the role that dark matter plays in them [62] including closing the ‘final parsec’ of black-hole mergers [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' In conclusion, we have demonstrated an alternative mechanism for the long-term stabil- ity of vortices in SDM: a background gravitational potential suppresses their dominant decay mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We have primarily concerned ourselves with stabilization of vortex-solitons at the centers of dark matter halos comprised of particles with mass ∼ 10−22 eV by supermassive black holes, but have highlighted other scenarios where this mechanism may be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This introduces a new mechanism by which black holes and other pointlike masses might connect to their surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Author Contributions All authors contributed substantial ideas across the various concepts in the paper, but here we list particular contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' NG contributed expertise with SDM simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' AEM and NM did the bulk of the analysis and writing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' AEM concentrated on the analytic energy arguments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' NM wrote, designed, ran, and analysed simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' MN initially conceived of the project that evolved into the current paper, and contributed large-scale-structure expertise and writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' CPW organized the working group, contributed ideas about SDM and its phenomenology from a particle-theory perspective, and guided the paper’s and project’s coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Acknowledgments We thank Luna Zagorac and Dmitry Levkov for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' We would also like to thank the administrative and facilities staff at the University of New Hampshire including Katie Makem-Boucher and Michelle Mancini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' Computations were performed on Marvin, a Cray CS500 supercomputer at UNH sup- ported by the NSF MRI program under grant AGS-1919310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' AEM’s contributions to this project were supported by DOE Grant DE-SC0020220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' NG’s participation was supported in part by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' 1929080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' This work was initiated and performed in part at Aspen Center for Physics, which is supported by National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' PHY-1607611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' CPW thanks the late Karsten Pohl for actively supporting the application for NSF grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/w9FPT4oBgHgl3EQf_zUO/content/2301.13220v1.pdf'} +page_content=' 1929080.' metadata={'source': 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0000000000000000000000000000000000000000..4ba2f9f239d7c6a3c2eacad94e1f7d3aa010c28b --- /dev/null +++ b/wtFST4oBgHgl3EQfRjgB/content/tmp_files/2301.13762v1.pdf.txt @@ -0,0 +1,799 @@ +arXiv:2301.13762v1 [math.GR] 31 Jan 2023 +ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF +FINITE SIMPLE EXCEPTIONAL GROUPS OF LIE TYPE +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +In memory of Irina Dmitrievna Suprunenko +Abstract. The Gruenberg–Kegel graph Γ(G) of a finite group G is the graph whose vertex +set is the set of prime divisors of |G| and in which two distinct vertices r and s are adjacent +if and only if there exists an element of order rs in G. +A finite group G is called almost recognizable (by Gruenberg–Kegel graph) if there is only +finite number of pairwise non-isomorphic finite groups having Gruenberg–Kegel graph as G. +If G is not almost recognizable, then it is called unrecognizable (by Gruenberg–Kegel graph). +Recently P. J. Cameron and the first author have proved that if a finite group is almost +recognizable, then the group is almost simple. Thus, the question of which almost simple +groups (in particular, finite simple groups) are almost recognizable is of prime interest. We +prove that every finite simple exceptional group of Lie type, which is isomorphic to neither +2B2(22n+1) with n ≥ 1 nor G2(3) and whose Gruenberg–Kegel graph has at least three +connected components, is almost recognizable. Moreover, groups 2B2(22n+1), where n ≥ 1, +and G2(3) are unrecognizable. +Throughout the paper we consider only finite groups and simple graphs, and henceforth +the term group means finite group and the term graph means simple graph, that is undirected +graph without loops and multiple edges. +Let G be a group. The spectrum ω(G) is the set of all element orders of G. The prime +spectrum π(G) is the set of all primes belonging to ω(G). A graph Γ(G) whose vertex set +is π(G) and in which two distinct vertices r and s are adjacent if and only if rs ∈ ω(G) is +called the Gruenberg–Kegel graph or the prime graph of G. +We say that the group G is +• recognizable (by Gruenberg–Kegel graph) if for every group H the equality Γ(H) = +Γ(G) implies that G ∼= H; +• k-recognizable (by Gruenberg–Kegel graph), where k is a positive integer, if there are +exactly k pairwise non-isomorphic groups having the same Gruenberg–Kegel graph +as G; +• almost recognizable (by Gruenberg–Kegel graph) if it is k-recognizable by Gruenberg– +Kegel graph for a positive integer k; +• unrecognizable (by Gruenberg–Kegel graph) if there are infinitely many pairwise non- +isomorphic groups having the same Gruenberg–Kegel graph as G. +Note that groups can be characterized by various numerical sets. For example, if we replace +Γ(G) in these definitions with ω(G), then we obtain the corresponding definitions for recog- +nizability by spectrum. Nevertheless, if the characterization set is not specified, we suppose +that it is the Grunberg-Kegel graph. It is easy to see that if ω(G) = ω(H) for groups G and +H, then Γ(G) = Γ(H), however, the converse implication is not true in general. Consider +alternating groups A5 and A6. Clearly, Γ(A5) = Γ(A6), where both graphs are empty graphs +on three vertices 2, 3, and 5, and, on the other hand, we see that 4 ∈ ω(A6) \ ω(A5). +1 + +2 +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +Recently P. J. Cameron and the first author have proved [7] that a group G is almost +recognizable if and only if each group H with Γ(G) = Γ(H) is almost simple. Thus, the +question of which almost simple groups are almost recognizable is of prime interest. At the +same paper [7], a survey of known results on recognition of simple groups has been presented. +Note that there are not many completed results at the moment. The situation is much better +in the case of the characterization problem by spectrum, where recognition is established for +many nonabelian simple groups [15]. +In [36], V. D. Mazurov conjectured that if a simple group G is not isomorphic to A6 and +Γ(G) has at least three connected components, then G is recognizable by its spectrum. This +conjecture was proved in a series of papers, the final result was obtained in [28], where the +author proved that groups E7(2) and E7(3) are recognizable by Gruenberg–Kegel graph, +therefore, these groups are recognizable by spectrum. +Connected components of Gruenberg–Kegel graphs of simple groups were described in [46, +27]. A complete result after corrections of mistakes can be found, for example, in [2, Tables 1– +3]. If G is a simple group, then Γ(G) has at least three connected components if and only if +one of the following statements holds: +(1) G ∼= G2(q), where q is a power of 3, or G ∼= 2G2(q), where q = 32n+1 > 3; +(2) G ∼= 2B2(q) ∼= Sz(q), where q = 22n+1 > 2; +(3) G ∼= F4(q), where q is even, or G ∼= 2F4(q) for q = 22n+1 > 2; +(4) G ∼= E8(q); +(5) G ∼= A1(q) ∼= PSL2(q), where q > 3; +(6) G ∼= 2Dn(3) ∼= PΩ− +2n(3), where n = 2m + 1 ≥ 3 is a prime; +(7) G is one of the following finite simple groups of Lie type: 2A5(2) ∼= PSU6(2), E7(2), +E7(3), A2(4) ∼= PSL3(4), 2E6(2); +(8) G is one of the following finite simple sporadic groups: M11, M23, M24, J3, HiS, Suz, +Co2, Fi23, F3, F2, M22, J1, O′N, LyS, Fi′ +24, F1, J4; +(9) G ∼= An, where n > 6 and both n and n − 2 are primes. +We consider the recognition problem of the simple exceptional groups of Lie type whose +Gruenberg–Kegel graphs have at least three connected components. The main result of this +paper is the following statement. +Main Theorem. Every finite simple exceptional group of Lie type, which is isomorphic to +neither 2B2(22n+1) with n ≥ 1 nor G2(3) and whose Gruenberg–Kegel graph has at least three +connected components, is almost recognizable by Grunberg–Kegel graph. Moreover, groups +2B2(22n+1), where n ≥ 1, and G2(3) are unrecognizable by Gruenberg–Kegel graph. +In fact, much has been known about the groups in the theorem before in the context of +the recognition problem. The groups E7(2), E7(3), and 2E6(2) are known to be recognizable +(see [28] and [29]). The recognizability of simple groups 2G2(q) with q > 3 was established +in [48]. +If G is a nonabelian simple group, then we say that G is quasirecognizable by its Gruenberg– +Kegel graph if every group H with Γ(G) = Γ(H) has a unique nonablelian composition factor +S and S ∼= G. In [50], it was proved that each group G2(q), where q is an odd power of 3, +is quasirecognizable; however, this result contains an error since Γ(G2(3)) = Γ(PSL2(13)). +Quasirecognizability of the simple groups F4(q) and 2F4(q), where q > 2 is even, was proved +in [24] and [1], respectively. We prove Main Theorem for groups G2(q), where q > 3, F4(q), +and 2F4(q) in Section 3. The quasirecognizability of groups 2B2(q) was proved in [50]. We +consider the groups 2B2(q) and G2(3) in Section 4. + +ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS +3 +In [47], A. V. Zavarnitsine proved that if G = E8(q), where q ≡ 0, ±1 (mod 5), and +H is a group such that Γ(H) = Γ(G), then H ∼= E8(u) for a prime power u ≡ 0, ±1 +(mod 5). In Section 5, we prove a similar result for groups E8(q), where q ≡ ±2 (mod 5). +Therefore, as a corollary, Main Theorem holds for the groups E8(q). However, the question +of quasirecognizability for the groups E8(q) is still open (see Section 5 for details). +Provide a mini-survey of known results on characterization by Gruenberg–Kegel graph of +the remaining simple groups whose Gruenberg–Kegel graphs have at least three connected +components. +First consider groups PSL2(q), where q = pk and p is a prime. The groups PSL2(4) ∼= +PSL2(5), PSL2(7), PSL2(8), and PSL2(9) ∼= A6 are unrecognizable while the groups +PSL2(11) and PSL2(25) are 2-recognizable (see, for example, [7, Theorem 1.7], [31, Theo- +rem 5] and [17]). The group PSL2(49) is 5-recognizable (see [34], this result is obtained as a +consequence of [31, Theorem 5] and direct calculations with using Lemma 1.15). Let k = 1 +and p > 11. In [25], it was proved that if p ̸≡ 1 (mod 12), then G is recognizable, and if +p ≡ 1 (mod 12), then G is quasirecognizable. However, this result has at least one mistake +since the group PSL2(13) is not quasirecognizable. If p is odd and k > 2 or k = 2 and p > 7, +then G is (k, 2)-recognizable (see [22] and [21]). If q > 8 is even, then G is quasirecognizable +(see [26, Theorem 3.3]). In general, we suggest that the results on recognition of the groups +PSL2(q) by Gruenberg–Kegel graph need a careful revision since the authors in their proofs +refer to the paper [18] which contains a number of rather serious inaccuracies. +If G ∼= PΩ− +2n(3), where n ≥ 3 is a prime, then G is recognizable (see [11]). The group +PSU6(2) is 2-recognizable (see [30, Theorem 2]), the group PSL3(4) is 5-recognizable since +Γ(PSL3(4)) = Γ(PSL2(49)). +The problem of recognition by Gruenberg–Kegel graph has been solved for all finite simple +sporadic groups. In particular, the groups J1, M22, M23, M24, Co2, J4, J3, Suz, O′N, LyS, +F3, Fi23, Fi′ +24, F2, and F1 are recognizable (see [17, Theorem 3], [48, Theorem B], [30, +Theorem 1], and [32]) while the groups M11 and HiS are 2-recognizable (see [17, Theorem 3] +and [30, Theorem 2], respectively). +Let G ∼= An, where n > 6 and both n and n − 2 are primes. If n = 7, then Γ(G) = +Γ(PSL2(49)) and, therefore, G is 5-recognizable. If n = 13, then G is recognizable (see [39, +Lemma 24]), and if n ≥ 19, then G is known to be quasirecognizable (see [39, Theorem 1]). +The question of recognizability by Gruenberg–Kegel graph of the groups An, where n ≥ 19 +is a prime and n − 2 is also a prime, is still open, and the conjecture [39, Conjecture 1] is +that these groups are recognizable. +1. Preliminaries +Let n be an integer. Denote by π(n) the set of all prime divisors of n. Let π be a set of +primes. The largest divisor m of n such that π(m) ⊆ π is called a π-part of n and is denoted +by nπ. By π′ we denote the set of primes which do not belong to π. If π consists of a unique +element p, then we will write np and np′ instead of n{p} and n{p}′, respectively. +If n is an integer and r is an odd prime with (r, n) = 1, then e(r, n) denotes the multi- +plicative order of n modulo r. Given an odd integer n, we put e(2, n) = 1 if n ≡ 1 (mod 4), +and e(2, n) = 2 otherwise. +The following lemma is proved in [3], and also in [51]. + +4 +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +Lemma 1.1 (Bang–Zsigmondy). Let q be an integer greater than 1. For every positive integer +m there exists a prime r with e(r, q) = m but for the cases q = 2 and m = 1, q = 3 and +m = 1, and q = 2 and m = 6. +Fix an integer a with |a| > 1. A prime r is said to be a primitive prime divisor of ai − 1 +if e(r, a) = i. We write ri(a) to denote some primitive prime divisor of ai − 1 if such a prime +exists, and Ri(a) to denote the set of all such divisors. +Lemma 1.2 ([14, Lemma 6]). Let q, m, and k be positive integers. Then Rmk(q) ⊆ Rm(qk). +If, in addition, (m, k) = 1, then Rm(q) ⊂ Rm(qk). +Lemma 1.3. If r ∈ Ri(q), then r = ik + 1, where k is a positive integer. +Proof. This is a consequence of Fermat’s little theorem. +□ +Given a positive integer i ̸= 2, denote by ki(a) the product of all primitive prime divisors +of ai − 1 with multiplicities counted. In case i = 2 put k2(a) = k1(−a). It is not difficult +to verify that if i is divisible by 4 then ki(a) = ki(−a) and if i is odd then ki(a) = k2i(−a). +The following general formula [38] expresses ki(a), where i > 2, in terms of cyclotomic +polynomials: +(1) +ki(a) = +Φi(a) +(r, Φ(i)r′(a)), +where r is the greatest prime divisor of i. +The following assertion is well-known and its proof is elementary. +Lemma 1.4. Suppose that q > 1 is an integer. For a positive integer i, an odd prime r +divides qi − 1 if and only if e(r, q) divides i. +The following lemma is a particular case of the well-known Nagell–Ljunggren equation. +Lemma 1.5 ([37]). Suppose that x, y, and k are positive integers. If x2 + x + 1 = yk, then +either k = 1 or k = 3, x = 18, and y = 7. +The following technical lemma follows from Lemma 1.5, but we give an independent proof. +This statement will be needed in the proof of Main Theorem for the group G2(q). +Lemma 1.6. Suppose that q is an integer greater than 2 and π(q2 + εq + 1) = {r}, where +ε ∈ {+, −} and r is a prime. Then r ≡ 1 (mod 6). Moreover, if q ≡ 1 (mod 8) then either +r ≡ 1 (mod 8) or r ≡ 3 (mod 8). +Proof. By assumption, there exists a positive integer n such that q2 + εq + 1 = rn. Clearly, +r is odd. Note that q2 + εq + 1 is not divisible by 9, so if r = 3, then n = 1 and q ∈ {1, 2}. +If r ̸= 3, then r divides k3(εq) = q2+εq+1 +(q−ε1,3) and hence r ∈ R3(εq). It follows from Lemma 1.3 +that r ≡ 1 (mod 6). +Suppose that q ≡ 1 (mod 8). Then either q2 + εq + 1 ≡ 1 (mod 8) or q2 + εq + 1 ≡ 3 +(mod 8). Therefore, if r ≡ 5 (mod 8) or r ≡ 7 (mod 8), then n is even. On the other hand, +(q − 1) < q2 − q + 1 < q2 and q2 < q2 + q + 1 < (q + 1)2, so n cannot be even. This implies +that either r ≡ 1 (mod 8) or r ≡ 3 (mod 8). +□ +Let G be a finite group. Denote the number of connected components of Γ(G) by s(G), +and the set of connected components of Γ(G) by {πi(G) | 1 ≤ i ≤ s(G)}; for a group G of +even order, we assume that 2 ∈ π1(G). Denote by t(G) the independence number of Γ(G), + +ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS +5 +that is the greatest size of a coclique (i. e. induced subgraph with no edges) in Γ(G). If +r ∈ π(G), then denote by t(r, G) the greatest size of a coclique in Γ(G) containing r. +Lemma 1.7. Let A and B be normal subgroups of a group G such that A ≤ B. If r, s ∈ +π(B/A) \ (π(A) ∪ π(G/B)), then r and s are adjacent in Γ(G) if and only if r and s are +adjacent in Γ(B/A). +Proof. The proof of this lemma is elementary. +□ +Lemma 1.8 ([42]). Let G be a finite group with t(G) ≥ 3 and t(2, G) ≥ 2. Then the following +statements hold. +(1) There exists a nonabelian simple group S such that S ⊴ G = G/K ≤ Aut(S), where K +is the solvable radical of G (i. e., the largest solvable normal subgroup of G). +(2) For every coclique ρ of Γ(G) of size at least three, at most one prime in ρ divides the +product |K| · |G/S|. In particular, t(S) ≥ t(G) − 1. +(3) One of the following two conditions holds: +(3.1) S ∼= A7 or L2(q) for some odd q, and t(S) = t(2, S) = 3. +(3.2) Every prime p ∈ π(G) nonadjacent to 2 in Γ(G) does not divide the product +|K| · |G/S|. In particular, t(2, S) ≥ t(2, G). +Lemma 1.9 ([39, Lemma 1]). Let N ⊴ G be an elementary abelian subgroup and H = G/N. +Define a homomorphism φ : H → Aut(N) as follows nφ(gN) = ng. Then Γ(G) = Γ(N ⋊φ H). +Lemma 1.10 ([7, Proposition 3.1]). Let π be a finite set of primes. The number of pairwise +nonisomorphic nonabelian simple groups S with π(S) ⊆ π is finite, and is at most O(|π|3). +Let S be a finite simple group of Lie type in characteristic p. Let A be any abelian p-group +with an S-action. Any element s ∈ S is said to be unisingular on A if s has a (nonzero) fixed +point on A. The group S is said to be unisingular if every element s ∈ S acts unisingularly +on every finite abelian p-group A with an S-action. Denote by PSLε +n(q), where ε ∈ {+, −}, +the group PSLn(q) if ε = 1 and PSUn(q) if ε = −1. Similarly, Eε +6(q) denotes the simple +group E6(q) if ε = 1 and 2E6(q) if ε = −1. +Lemma 1.11 ([16, Theorem 1.3]). A finite simple group S of Lie type of characteristic p is +unisingular if and only if S is one of the following: +(i) PSLε +n(p) with ε ∈ {+, −} and n divides p − ε1; +(ii) PΩ2n+1(p), PSp2n(p) with p odd; +(iii) PΩε +2n(p) with ε ∈ {+, −}, p odd, and ε = (−1)n(p−1)/2; +(iv) 2G2(q), F4(q), 2F4(q), E8(q) with q arbitrary; +(v) G2(q) with q odd; +(vi) Eε +6(p) with ε ∈ {+, −} and 3 divides p − ε1; +(vii) E7(p) with p odd. +Lemma 1.12 ([47, Proposition 2]). Let G = 3D4(q) act on a nonzero vector space V over +a field of characteristic not dividing q (possibly, zero). Then each element of G of order +q4 − q2 + 1 fixes on V a nonzero vector. +The following lemma is also well-known, but we provide its proof for completeness. +Lemma 1.13. Let G be a group, g ∈ G an element of order r, and φ a non-trivial irreducible +representation of G on a nonzero vector space V . If the minimum polynomial degree of φ(g) +equals to r, then g fixes on V a nonzero vector. + +6 +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +Proof. Let A = φ(g). Since gr = 1, we have Ar = 1, and, therefore, the minimal polynomial +for A divides the polynomial xr − 1. Since the minimum polynomial degree of A equals to r, +we have that the minimum polynomial for A is xr − 1. +By the Cayley–Hamilton theorem, A is a root of its characteristic polynomial. In particular, +1 is an eigenvalue of A, and, therefore, each eigenvector of A which corresponds to the +eigenvalue 1, is fixed by A = φ(g). +□ +Lemma 1.14 ([41, Theorem 1.1]). Let G be one of the groups 2B2(q), where q > 2, 2G2(q), +where q > 3, 2F4(q), G2(q), 3D4(q). Let g ∈ G an element of prime power order coprime +to q. Let φ be a non-trivial irreducible representation of G over a field F of characteristic l +coprime to q. Then the minimum polynomial degree of φ(g) equals |g|, unless possibly when +G = 2F4(8), l = 3, p = 109 and φ(1) < 64692. +Lemma 1.15 ([10, Lemma 4]). Let G be a finite simple group, F a field of characteristic +p > 0, V an absolute irreducible GF-module, and β a Brauer character of V . If g ∈ G is an +element of prime order distinct from p, then +dimCV (g) = (β⟨g⟩, 1⟨g⟩) = 1 +|g| +� +x∈⟨g⟩ +β(x). +Lemma 1.16. Let G be a group with a non-trivial nilpotent normal subgroup K such that +G/K has a subgroup H isomorphic to E8(q), where q is a prime power. Then R24(q) ⊂ π1(G). +Proof. Note that R24(q) ⊂ π(H). +Let q = pl, where p is a prime. Suppose that K is a p-group. Factoring G and K by K′, we +can assume that K is abelian. According to [2, Table 3], p lies in π1(G). By Lemma 1.11, H +is unisingular and hence p is adjacent to each element of π(H). Therefore, R24(q) ⊂ π1(G). +Take any r ∈ π(K). Since Sylow 2-subgroups of H are non-cyclic, we infer that either +r = 2 or 2 and r are adjacent in Γ(G) (see, for example, [12, Theorem 10.3.1]). Therefore, +r ∈ π1(G). +Suppose that there exists r ∈ π(K)∩R24(q). Since π(K) is a clique in Γ(G) and r ∈ π1(G), +we get that R24(q) ⊂ π1(G). +If K is not a p-group and π(K)∩R24(q) = ∅, then we can choose r ∈ π(K)\({p}∪R24(q)). +By Lemma 1.12, r is adjacent in Γ(G) to any prime from R24(q). This implies that R24(q) ⊂ +π1(G). +□ +2. The Grunberg–Kegel graphs of some exceptional groups of Lie type +A criterion for the adjacency of vertices in the Gruenberg–Kegel graph for all finite non- +abelian simple groups was obtained in [44]. Based on this paper and [45], in this section, we +collect the necessary information for exceptional groups of Lie type from Main Theorem. +By the compact form of the Gruenberg–Kegel graph Γ(G) for a group G we mean a graph +whose vertices are labeled with sets of primes. A vertex labeled by a set τ represents the +clique of Γ(G) such that every vertex in this clique labeled by a prime from τ. An edge +connecting two sets represents the set of edges of Γ(G) that connect each vertex in the first +set with each vertex in the second. +Lemma 2.1 ([45, Proposition 2.7] and [44, Proposition 3.2]). Let G ∼= F4(q), where q = 2n, +r, s ∈ π(G) and r ̸= s. Then r and s are nonadjacent if and only if one of the following +conditions holds (up to permutation): +(1) 2 = r, and e(s, q) ∈ {8, 12}. + +ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS +7 +(2) s, r ̸= 2, k = e(r, q), l = e(s, q), 1 ≤ k < l and either l ∈ {8, 12}, or l = 6 and +k ∈ {3, 4}, or l = 4 and k = 3. +In particular, the compact form for Γ(F4(q)) is the following. +R1 +R2 +R6 +R3 +R4 +2 +R12 +R8 +Remark. Note that R6(2) and R1(2) are empty sets. +Lemma 2.2 ([44, Proposition 3.3] and [45, Proposition 2.9]; see also [9, Lemma 3]). Let +G ∼= 2F4(q), where q = 22n+1, r, s ∈ π(G) and r ̸= s. Put m1(n) = q − 1, m2(n) = q + 1, +m3(n) = q2 + 1, m4(n) = q2 − q + 1, m5(n) = q2 − +� +2q3 + q − √2q + 1, m6(n) = q2 + +� +2q3 + q + √2q + 1. Then r and s are nonadjacent in Γ(G) if an only if one of the following +conditions holds: +(1) 2 = r, s divides mk(n), s ̸= 3, and k > 3. +(2) 2 ̸= s, r; either 3 ̸= r ∈ π(mk(n)), 3 ̸= s ∈ π(ml(n)) for k ̸= l, and {k, l} ̸= +{1, 2}, {1, 3}; or r = 3 and s ∈ π(ml(n)), where l ∈ {3, 5, 6}. +In particular, the compact form for Γ(2F4(q)) is the following. +2 +π(q + 1)\{3} +3 +π(q − 1) +π(q2 − q + 1)\{3} +π(q2 + 1) +π(q2 + +� +2q3 + q + √2q + 1) +π(q2 − +� +2q3 + q − √2q + 1) +Lemma 2.3 ([45, Proposition 2.7] and [44, Propositions 3.2, 4.5]). Let G ∼= G2(q), where +q = 3k, r, s ∈ π(G) and r ̸= s. Then r and s are nonadjacent in Γ(G) if and only if e(r, q) +or e(s, q) ∈ {3, 6}. +In particular, the compact form for Γ(G2(q)) is the following. +R1 +R2 +{3} +R3 +R6 +Remark. Note that the set R1(3) is empty. + +8 +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +Lemma 2.4 ([45, Proposition 2.7] and [44, Propositions 3.2, 4.5]). Let G ∼= E8(q), where q +is a power of a prime p. Suppose that r, s ∈ π(G) with r ̸= s. Then r and s are nonadjacent +in Γ(G) if and only if one of the following conditions holds: +(1) r ∈ {2, p}, s ̸= p and e(s, q) ∈ {15, 20, 24, 30}. +(2) s, r ̸∈ {2, p}, k = e(r, q), l = e(s, q), 1 ≤ k < l, and either l = 6 and k = 5, or +l ∈ {7, 14} and k ≥ 3, or l = 9 and k ≥ 4, or l ∈ {8, 12} and k ≥ 5, k ̸= 6, or +l = 10 and k ≥ 3, k ̸∈ {4, 6}, or l = 18 and k ̸∈ {1, 2, 6}, or l = 20 and r · k ̸= 20, or +l ∈ {15, 24, 30}. +In particular, the compact form for Γ(E8(q)) is the following. Here, R(q) = R1(q)∪R2(q)∪ +{p} and the vector from 5 to R4(q) and the dotted edge {5, R20(q)} indicate that R4(q) and +R20(q) are not adjacent, but if 5 ∈ R4(q) (i.e., q2 ≡ −1 (mod 5)), then there exist edges +between 5 and the primes from R20(q). +R +R18 +R5 +R3 +R8 +R12 +R6 +R10 +R9 +R14 +R7 +R4 +5 +R20 +R15 +R24 +R30 +3. Almost recognizability of groups G2(q), F4(q), and 2F4(q) by +Gruenberg–Kegel graph +In this section, we prove Main Theorem for groups L = F4(q), where q ≥ 2 is a power of +2, L = 2F4(q), where q = 22m+1 > 2, and L = G2(q), where q > 3 is a power of 3. +Theorem 1. If G is a group with Γ(G) = Γ(L), then L ∼= Inn(L) ⊴ G ≤ Aut(L). +In +particular, L is almost recognizable by Gruenberg–Kegel graph. +By Lemmas 2.1, 2.2, and 2.3, we see that t(L) ≥ 3 and t(2, L) ≥ 2. +It follows from +Lemma 1.8 that there exists a nonabelian simple group S such that S ∼= Inn(S) ≤ G/K ≤ +Aut(S), where K is the solvable radical of G. Moreover, by the Thompson theorem on finite +groups with fixed-point-free automorphisms of prime order [40, Theorem 1], K is nilpotent. +To prove Theorem 1, it suffices to show that S ∼= L and K = 1. +These two facts are +established in the following four lemmas. +Lemma 3.1. S ∼= L. +Proof. If q > 2 is even and L = 2F4(q) or L = F4(q), then Lemma follows from [1, Theo- +rem 3.4] and [24, Theorem 3.3], respectively. +Suppose that L = F4(2). Then π(S) ⊆ π(L) = {2, 3, 5, 7, 13, 17}. Note that 13 ∈ R12(2) +and 17 ∈ R8(2). By Lemma 2.1, we find that 13 and 17 are nonadjacent to all vertices +in Γ(G). Therefore, {13, 17} ⊂ π(S) by Lemma 1.8. Inspecting [49, Table 1], we see that +S ∈ {PSU4(4), PSU3(17), PSL2(132), PSp4(13), PSL3(16), PSp6(4), PΩ+ +8 (4), F4(2)}. + +ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS +9 +Using [4, Corollary 3], we find that 5 and 17 are adjacent in Γ(PSL2(132)) and Γ(PSL3(16)), +while 3 and 17 are adjacent in Γ(PSU3(17)) and Γ(PSU4(4)). According to [5, Corollaries 2– +4], 5 and 17 are adjacent in Γ(PSp4(13)), Γ(PSp6(4)), and Γ(PΩ+ +8 (4)). This implies that +S ∼= F4(2), as claimed. +It remains to consider the case L ∼= G2(q), where q > 3 is a power of 3. If q > 3 is an +odd power of 3, then S ∼= L by [50, Theorem 1.1]. Suppose that L ∼= G2(q), where q = 32k +for a positive integer k. By Lemma 2.3, sets R3(q) and R6(q) are connected components +in Γ(L). It follows from Lemma 1.8 that sets R3(q) and R6(q) are connected components +in Γ(S) and hence Γ(S) has at least three connected components. Therefore, S isomorphic +to a group listed in Introduction before Main Theorem. We show that S ∼= L considering +each case for S separately. We will extensively use that k3(q) = q2+q+1 +(q−1,3) = q2 + q + 1 and +k6(q) = q2−q+1 +(q+1,3) = q2 − q + 1 (see equation (1)). +Case S ∼= Ap, where p > 6 and both p and p − 2 are primes. The connected components of +Γ(S) are π1(S), {p}, and {p − 2}. We know that R3(q) and R6(q) are connected components +in Γ(S), so either p − 2 ∈ R3(q) or p − 2 ∈ R6(q). It follows from Lemma 1.3 that p − 3 is +divisible by 3; a contradiction since p ̸= 3. +Case S ∼= E8(u), where u is a prime power. Since 32k ≡ (−1)k (mod 5), we infer that +q ≡ ±1 (mod 5). This implies that there exists ε ∈ {+, −} such that q2+εq+1 ≡ 3 (mod 5). +Take any prime divisor r of k3(εq) = q2 + εq + 1 such that r ̸≡ 1 (mod 5). Since r ∈ R3(εq), +Lemma 2.4 implies that r ∈ Rj(u), where j ∈ {15, 20, 24, 30}. Then r−1 is divisible by j and +hence j = 24. Since R24(u) is a connected component in Γ(S), we find that R3(εq) = R24(u). +If K ̸= 1, then Lemma 1.16 implies that R24(u) ⊆ π1(G); a contradiction since r ̸∈ π1(G). +Assume that there exists s ∈ π(G/S) such that s > 3. It follows from Lemma 1.8 that +s ∈ π1(G). By [13, Theorem 2.5.12, Definition 2.5.13], there exists an element g ∈ G\S such +that |g| = s and g acts on S as a field automorphism. It is known that CS(g) ∼= E8(u1/s) +(see, e.g., [13, Proposition 4.9.1]). By Lemma 1.2, we see that R24(u1/s) ⊆ R24(u) and hence +s is adjacent to r in Γ(G); we arrive at a contradiction since s ∈ π1(G) and r ̸∈ π1(G). +Now consider a coclique ρ = {s1, s2, . . . , s12} of maximal size in Γ(S). By Lemma 2.4, we +have (si, 6) = 1 for every i ∈ {1, . . . , 12}. Applying Lemma 1.7, we find that t(G) ≥ 12; a +contradiction with Lemma 2.3. +Cases S ∈ {PSU6(2), PSL3(4), M11, M23, M24, J3, HiS, Suz, Co2, Fi23, F2, M22, Fi′ +24, F1}. +By Lemma 1.8, we infer that {2, r3(q), r6(q)} is a coclique in Γ(S). It follows from Lemma 1.3 +that r3(q) ≡ 1 (mod 6) and r6(q) ≡ 1 (mod 6). Using [8], we see that there is no such a +coclique in Γ(S); contradiction. +Case S ∼= F4(u), where u is even. Since q2 + q + 1 ≡ 3 (mod 8), there exists r ∈ R3(q) +such that r ̸≡ 1 (mod 8). By Lemmas 1.8, 2.1, and 1.3, we infer that r ∈ R12(u). Then +π(q2 +q +1) = R12(u). On the other hand, we have q2 +q +1 ≡ 7 (mod 12) and hence there +exists s ∈ π(q2 + q + 1) such that s ̸≡ 1 (mod 12); a contradiction with Lemma 1.3. +Case S ∼= 2F4(u), where u = 22m+1 > 2. By Lemma 2.2, it is true that π2(S) ∪ π3(S) = +π(u4 − u2 + 1) ⊆ π(u6 + 1). Consider any prime r ∈ R3(q). Then r ∈ π2(S) ∪ π3(S) and r +divides (u3)2 + 1. Since (u3)2 ≡ −1 (mod r), we find that −1 is a quadratic residue modulo +r. This implies that r ≡ 1 (mod 4). Since r is an arbitrary element of R3(q), we infer that +q2 + q + 1 ≡ 1 (mod 4). On the other hand, q2 + q + 1 ≡ 34k + 32k + 1 ≡ 3 (mod 4); a +contradiction. +Case S ∼= 2B2(u), where u = 22m+1 > 2. According to [2, Table 3], we can assume that +π2(S) = π(u − 1), π3(S) = π(u − +√ +2u + 1), and π4(S) = π(u + +√ +2u + 1). +Note that + +10 +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +π(u − +√ +2u + 1) ∪ π(u + +√ +2u + 1) = π(u2 + 1). If R3(q) ̸= π2(S), then R3(q) ⊆ π(u2 + 1) +and we get a contradiction arguing as in the case S ∼= 2F4(u). Therefore, we can assume +that R3(q) = π(u − 1). +Suppose that r | u − 1 = 22m+1 − 1. +Then r | 22n+2 − 2 and +hence 2 is a quadratic residue modulo r. This implies that r ≡ ±1 (mod 8). It follows that +q2 + q + 1 ≡ ±1 (mod 8); a contradiction since q2 + q + 1 ≡ 34k + 32k + 1 ≡ 3 (mod 8). +Case S ∼= 2G2(u), where u = 32m+1 > 3. According to [2, Table 2], we can assume that +π2(S) = π(u − +√ +3u + 1) and π3(S) = π(u + +√ +3u + 1). Take any prime r ∈ R12k(3). By +Lemma 1.2, we infer that r ∈ R6(q). We know that R6(q) ⊆ π2(S) ∪ π3(S) = π(u2 − u + 1) = +R6(u). By Lemma 1.4, we conclude that 12k | 6 · (2m + 1); a contradiction. +Case S ∼= 2Dp(3), where p = 2m + 1 ≥ 3 is prime. According to [2, Table 2], we can +assume that π2(S) = π((3p−1 + 1)/2) and π3(S) = π((3p + 1)/4). Take any r ∈ R12k(3). By +Lemma 1.2, we infer that r ∈ R6(q). Since r ∈ π2(S)∪π3(S), we find that r divides 32(p−1)−1 +or 32p − 1. It follows from Lemma 1.4 that 12k | 2(p − 1) or 12k | 2p; a contradiction. +Case S ∼= A1(u) ∼= PSL2(q), where u = 2m > 2. According to [2, Table 2], we can assume +that π2(S) = π(u − 1) and π3(S) = π(u + 1). Note that 3 divides u2 − 1. On the other hand, +π(u2 − 1) = π2(S) ∪ π3(S) = π2(G) ∪ π3(G); a contradiction since 3 ∈ π1(G). +Case S ∼= A1(u) ∼= PSL2(u), where 3 < u = vn and u is odd. Consider ε ∈ {+, −} such +that u ≡ ε1 (mod 4). According to [2, Table 2], we can assume that π1(S) = π(u − ε1), +π2(S) = {v}, and π3(S) = π( u+ε1 +2 ). Therefore, there exists τ ∈ {+, −} such that π(q2 + τq + +1) = {v} and π(q2 − τq + 1) = π( u+ε1 +2 ). Moreover, we know that π(u − ε1) ⊆ π(3(q2 − 1)). +Since q2 − q + 1 = (q − 1)2 + (q − 1) + 1, Lemma 1.5 implies that either q2 + τq + 1 = v or +q = 19 and v = 7. By assumption, q = 32k and hence q2 + τq + 1 = v. Then v − 1 is divisible +by 9. Therefore, v ≥ 19 and 3 ∈ π( u−1 +2 ) \ π(q2 − τq + 1). This implies that ε = + since, by +Lemma 2.3, 3 ∈ π1(G). +Suppose that n is even. Then v2 − 1 divides u − 1, so π(v + 1) ⊆ π(q2 − 1). Take any +r ∈ π(v+1). Since q ≡ ±1 (mod r) and r divides q2 +τq+2, we find that r divides 3±1 and +hence r = 2. Therefore, q2 +τq +2 is a power of 2. On the other hand, q2 +τq +2 ≡ 1±1+2 +(mod 8). This implies that q2 + τq + 2 ≤ 4; a contradiction. +We can assume that n is odd. +Then +v+1 +2 +divides +u+1 +2 . +Take any r ∈ π( v+1 +2 ). +Then +r divides both q2 + τq + 2 and q2 − τq + 1. +Therefore, 2τq ≡ −1 (mod r) and hence +0 ≡ 4q2 + 4τq + 8 ≡ 4τq + 9 ≡ 7 (mod r). This implies that r = 7 and v + 1 = 2 · 7m for a +positive integer m. Since q ≡ 1 (mod 8) and q2+τq+2 = 2·7m, we infer that 3+τ1 ≡ 2·(−1)m +(mod 8). This implies that τ = −1 and m is even. A straightforward calculation shows that +92k − 9k + 2 is not divisible by 49 for all positive integer k; a contradiction. +Cases S ∈ {F3, O′N, J1, LyS, J4, 2E6(2), E7(2), E7(3)}. According to [2, Tables 2, 3], we +see that there exist primes r and s such that π(q2 + q + 1) = {r} and π(q2 − q + 1) = {s}. +Applying Lemma 1.6, we find that r ≡ s ≡ 1 (mod 6) and the remainders of r and s divided +by 8 belong to the set {1, 3}. Inspecting [2, Tables 2, 3], we see that in each case there are +no two primes satisfying these restrictions; a contradiction. +Thus, we conclude that S ∼= G2(u), where u is a power of 3. Now Lemma 1.1 implies +immediately that u = q and, therefore, S ∼= L. This completes the proof of the lemma. +□ +Remark. For q = 3, the paper [50] contains a mistake, and S ∈ {G2(3), PSL2(13)} by [20, +Table 1]. +Show that K = 1. Consider a minimal (by order) counterexample G to this claim. + +ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS +11 +Lemma 3.2. K is an elementary abelian r-group for some r ∈ π1(L). +Proof. Let r be a prime divisor of |K|. +Since K is nilpotent, we have a decomposition +K = P ×U, where P is a Sylow r-subgroup of K and U is a normal subgroup of K such that +r ̸∈ π(U). Since U and Φ(P) are characteristic subgroups of K, the subgroup N = U ×Φ(P) +is a normal subgroup of G. Then Γ(G/N) is a subgraph of Γ(G). On the other hand, Γ(L) +is a subgraph of Γ(G/N) and hence Γ(G/N) = Γ(L). By the minimality of G, we infer that +N = 1 and, therefore, K is an elementary abelian r-group. +□ +By Lemmas 1.9, 3.1 and 3.2, we can assume that G = K ⋊ X, where K is an elementary +abelian r-group for r ∈ π1(L) and Soc(X) = S ∼= L. +Lemma 3.3. K = 1 if L = F4(q). +Proof. According to [33, Table 5.1], S has a subgroup H isomorphic to 3D4(q). +Assume that r ̸∈ {2} ∪ R12(q). By Lemma 1.12, r is adjacent to each element from R12(q) +in Γ(KH). It follows from Lemma 2.1 that Γ(G) ̸= Γ(L); a contradiction. +Assume that r = 2. By Lemma 1.11, S is unisingular and, therefore, 2 is adjacent to all +other vertices in Γ(KS). We arrive at a contradiction with Lemma 2.1. +Assume that r ∈ R12(q). Since q is even, by [33, Table 5.1], S has a subgroup H1 ∼= PΩ+ +8 (q). +Now by [6, Table 8.50], H1 has a subgroup H2 ∼= PΩ− +4 (q)×PΩ− +4 (q) ∼= PSL2(q2)×PSL2(q2). +Therefore, for each s ∈ R4(q), a Sylow s-subgroup of S is non-cyclic. This implies that r and +s are adjacent in Γ(KH2) (see, for example, [12, Theorem 10.3.1]). Therefore, r and s are +adjacent in Γ(G); a contradiction with Lemma 2.1. Thus, K = 1. +□ +Lemma 3.4. K = 1 if L ∼= G2(q) for q > 3 or L ∼= 2F4(q) for q > 2. +Proof. Let p = 2 if L ∼= 2F4(q) and p = 3 if L ∼= G2(q). By Lemma 1.11, L is unisingular +and, therefore, if r = p, then Γ(G) is connected; a contradiction. Therefore, r ∈ π1(L) \ {p}. +By Lemmas 1.13 and 1.14, r is adjacent to a prime from π2(L); a contradiction. +Thus, +K = 1. +□ +This completes the proof of Theorem 1. +Since the group L is known to be almost recognizable, the following natural question arises. +Problem 1. Suppose that L is a group from the statement of Theorem 1. Find a positive +integer k such that L is k-recognizable by Gruenberg-Kegel graph. +4. Unrecognizability of groups 2B2(q) and G2(3) by Gruenberg–Kegel graphs +In this short section, we show that groups 2B2(q) with q > 2 and G2(3) are unrecognizable. +Proposition 4.1. Let G = 2B2(q), where q > 2 is an odd power of 2. Then G is unrecog- +nizable by Gruenberg–Kegel graph. +Proof. By [16, Lemma 3.6], there exists a 4-dimensional module V over the field of order q +such that each nontrivial element from each maximal torus of G acts fixed-point freely on V . +Thus, Γ(V ⋊ G) = Γ(G) and, therefore, G is unrecognizable by [7, Theorem 1.2]. +□ +Proposition 4.2. Let G ∼= G2(3). Then G is unrecognizable by Gruenberg–Kegel graph. +Proof. Using [8], we find that Γ(G2(3)) is the following: +2 +3 +7 +13 + +12 +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +Moreover, Γ(G) = Γ(PSL2(13)). By [19, P. 9] and Lemma 1.15, there is a 6-dimensional +irreducible PSL2(13)-module V over a field of characteristic two such that all elements in +PSL2(13) of orders 7 and 13 act fixed-point freely on V . Therefore, +Γ(V ⋊ PSL2(13)) = Γ(PSL2(13)) = Γ(G). +Thus, by [7, Theorem 1.2], G is unrecognizable by Gruenberg–Kegel graph. +□ +5. Almost recognizability of groups E8(q) by Gruenberg–Kegel graph +To complete the proof of Main Theorem, it remains to consider the case of groups E8(q). +In [47], A. V. Zavarnitsine proved that if G is a finite group such that Γ(G) = Γ(E8(q)), +where q ≡ 0, ±1 (mod 5), then G ∼= E8(u) for some u ≡ 0, ±1 (mod 5). The aim of this +section is to prove the following similar result for the remaining cases for q. +Theorem 2. Let L = E8(q), where q ≡ ±2 (mod 5) is a prime power, and G be a group +such that Γ(G) = Γ(L). Then G ∼= E8(u) for some prime power u with u ≡ ±2 (mod 5). +Proof. Since Γ(G) is disconnected, Lemma 1.8 implies that there exists a nonabelian simple +group S such that S ≤ G/K ≤ Aut(S), where K is the solvable radical of G. +By the +Thompson theorem on finite groups with fixed-point-free automorphisms of prime order [40, +Theorem 1], K is nilpotent. +By Lemmas 1.8 and 2.4, we find that s(S) ≥ s(L) = 4, +t(2, S) ≥ t(2, L) = 5 and t(S) ≥ t(L)−1 = 11. According to [2, Table 1] and [44, Tables 2, 4, +and 5], we find that either S ∼= F1 or S ∼= E8(u), where u is a prime power. +Suppose that S ∼= F1. According to [2, Table 3], we see that s(S) = 4, for each i ≥ 2, +|πi(S)| = 1, and π(S) \ π1(S) = {41, 59, 71}. At the same time, π2(G) = R15(q), π3(G) = +R24(q), and π4(G) = R30(q). By Lemma 1.3, the numbers 15, 24, and 30 divide ri − 1 for +pairwise distinct primes ri from π(S) \ π1(S); a contradiction. +Suppose that S ∼= E8(u), where u is a power of a prime v and u ≡ 0, ±1 (mod 5). Since +q ≡ ±2 (mod 5), we find that 5 ∈ R4(q). Denote +θ = {r9(q), r14(q), r7(q), r18(q), r15(q), r24(q), r30(q)}. +By Lemma 2.4, we see that θ ∪{5} is a coclique of size 8 in Γ(G). It follows from Lemma 1.8 +that at least six elements of θ belong to π(S). Take any r ∈ π(S) ∩ θ. Since r and 5 are +nonadjacent in Γ(G), they are nonadjacent in Γ(S). We know that 5 ∈ {v} ∪ R1(u) ∪ R2(u) +and hence r ∈ R20(u) ∪ R15(u) ∪ R24(u) ∪ R30(u) according to Lemma 2.4. This implies that +at least two elements from θ ∩ π(S) are adjacent in Γ(S); a contradiction. +Suppose that S ∼= E8(u), where vl = u ≡ ±2 (mod 5). According to [2, Table 3], we can +assume that π2(G) = R15(q), π3(G) = R24(q), and π4(G) = R30(q), while π2(S) = R15(u), +π2(S) = R24(u), and π4(S) = R30(u). If K ̸= 1, then Lemma 1.16 implies that R24(u) ⊂ +π1(G); a contradiction since R24(u) must coincide with a connected component of Γ(G) not +containing 2. Therefore, we can assume that S ≤ G ≤ Aut(S). +Now prove that G/S = 1. If a prime r divides |G : S|, then by [13, Theorem 2.5.12, +Definition 2.5.13, Proposition 4.9.1], G \ S contains a field automorphism x of S of order +r with CS(x) ≥ E8(u1/r). +This implies that r is adjacent in Γ(G) to each prime from +π(E8(u1/r)). +Suppose that r ̸∈ {2, 3, 5}. +By Lemma 1.2, Γ(G) is connected and hence +Γ(G) ̸= Γ(S). If r = 2, then by Lemma 1.2, R15(u) ⊂ π1(G); a contradiction. If r = 5, then +by Lemma 1.2, r is adjacent in Γ(G) to some primes from R24(u) and hence R24(u) ⊂ π1(G); +a contradiction. Therefore, we can assume that G/S is a 3-group. By Lemma 2.4, we see + +ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS +13 +that ri(q) is nonadjacent to 2 in Γ(G) if and only if i ∈ {15, 20, 24, 30}. Similarly, a prime +ri(u) is nonadjacent to 2 in Γ(S) if and only if i ∈ {15, 20, 24, 30}. Since R20(q) ⊂ π1(G) and +R20(u) ⊂ π1(G), we infer that R20(q) = R20(u). By Lemma 1.2, we infer that 3 is adjacent +in Γ(G) to a prime from R20(u), therefore, Γ(G) ̸= Γ(S). +Thus, G = S. The proof of Theorem 2 is complete. +□ +Corollary 1. For each value of q, the group E8(q) is almost recognizable by Gruenberg–Kegel +graph. +Proof. If G is a group such that Γ(G) = Γ(E8(q)), then by [47, Theorem 1] and Theorem 2, +we have G ∼= E8(u) for a prime power u. By Lemma 1.10, for a given q, the number of +possibilities for u is finite. This implies that there is only the finite number of possibilities +for G (up to isomorphism), in particular, E8(q) is almost recognizable by Gruenberg–Kegel +graph. +□ +Problem 2. Do there exist prime powers q and q1 with q ̸= q1 and Γ(E8(q)) = Γ(E8(q1))? +Corollary 2. If G is a finite group such that Γ(G) = Γ(E8(q)) and |G| = |E8(q)| for some +prime power q, then G ∼= E8(q). +Proof. By Theorem 2, if Γ(G) = Γ(E8(q)), then G ∼= E8(q1) for some prime power q1. It is +clear that a function +f(x) = x120(x2 − 1)(x8 − 1)(x12 − 1)(x14 − 1)(x18 − 1)(x20 − 1)(x24 − 1)(x30 − 1) +strictly monotonically increases if x ≥ 1. Thus, if +f(q1) = |G| = |E8(q)| = f(q), +then q1 = q, and, therefore, G ∼= E8(q). +□ +In [43], it was proved that if G is a simple group and H is a group such that ω(H) = ω(G) +and |H| = |G|, then H ∼= G. Thus, each simple group is uniquely determined by its order +and spectrum. It is known that if q is odd and n ≥ 3, then Γ(PΩ2n+1(q)) = Γ(PSp2n(q)) +and |PΩ2n+1(q)| = |PSp2n(q)| but these groups are not isomorphic. Therefore, it is natural +to consider the following problem. +Problem 3. For which simple groups G is the following true: if H is a group with Γ(H) = Γ(G) +and |H| = |G|, then H is isomorphic to G? +Problem 3 was formulated by B. Khosravi in his survey paper [23, Question 4.2], by +A.S. Kondrat’ev in frame of the open problems session of the 13th School-Conference on +Group Theory Dedicated to V. A. Belonogov’s 85th Birthday (see [35, Question 4]), and +was independently formulated by W. Shi in a personal communication with the first author. +Also Problem 3 was formulated in the paper by P. J. Cameron and the first author (see [7, +Problem 2]). It is clear that if a simple group is quasirecognizable by Gruenberg–Kegel graph, +then Problem 3 solves in the positive for this group. At the same time, Corollary 2 gives a +solution of Problem 3 for finite simple groups E8(q) which are not necessary quasirecognizable +by Gruenberg–Kegel graph. + +14 +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +6. Acknowledgements +The first author is supported by the Ministry of Science and Higher Education of the Rus- +sian Federation, project 075-02-2022-877 for the development of the regional scientific and +educational mathematical center ”Ural Mathematical Center” (for example, Section 5). The +second author is supported by the Mathematical Center in Akademgorodok under the agree- +ment No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian +Federation (for example, Section 3). The third author is supported by RAS Fundamental +Research Program, project FWNF-2022-0002 (for example, Section 4). +References +1. Z. Akhlaghi, M. Khatami, B. Khosravi, Quasirecognition by prime graph of the simple group 2F4(q), Acta +Math. Hungar., 122:4 (2009), 387–397. +2. O. A. Alekseeva and A. S. Kondrat’ev, On recognizability of some finite simple orthogonal groups by +spectrum, Proc. Steklov Inst. Math., 266 (2009), 10–23. +3. A.S. 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Vasil′ev, E.P. Vdovin, Cocliques of maximal size in the prime graph of a finite simple group, Algebra +and Logic, 50:4 (2011), 291–322. +46. J. S. Williams, Prime graph components of finite groups, J. Algebra, 69 (1981), 487–513. +47. A.V. Zavarnitsine, Finite groups with a five-component prime graph, Sib. Math. J., 54:1 (2013), 40–46. +48. A.V. Zavarnitsine, Recognition of finite groups by the prime graph, Algebra and Logic, 45:4 (2006), +220–231. +49. A.V. Zavarnitsine, Finite simple groups with narrow prime spectrum, Sib. Electron Mat. Izv., 6 (2009), +1–12. +50. Q. Zhang, W. Shi, R. Shen, Quasirecognition by prime graph of the simple groups G2(q) and 2B2(q), J. +Algebra Appl., 10:2 (2011), 309–317. +51. K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. f´’ur Math. und Phys., 3 (1892), 265–284. + +16 +NATALIA V. MASLOVA, VIKTOR V. PANSHIN, AND ALEXEY M. STAROLETOV +Natalia Vladimirovna Maslova +Krasovskii Institute of Mathematics and Mechanics UB RAS, +16, S. Kovalevskaja str., Yekaterinburg, 620108, Russia +Ural Federal University, +19, Mira str., Yekaterinburg, 620002, Russia +Email address: butterson@mail.ru +Viktor Vladimirovich Panshin +Novosibirsk State University, +1, Pirogova str., Novosibirsk, 630090, Russia +Sobolev Institute of Mathematics, +4, Acad. Koptyug ave., Novosibirsk, 630090, Russia +Email address: v.pansh1n@yandex.ru +Alexey Mikhailovich Staroletov +Sobolev Institute of Mathematics, +4, Acad. Koptyug ave., Novosibirsk, 630090, Russia +Email address: staroletov@math.nsc.ru + diff --git a/wtFST4oBgHgl3EQfRjgB/content/tmp_files/load_file.txt b/wtFST4oBgHgl3EQfRjgB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec71b8dd539df8aecffd1f244a517ef1cad9f149 --- /dev/null +++ b/wtFST4oBgHgl3EQfRjgB/content/tmp_files/load_file.txt @@ -0,0 +1,1082 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf,len=1081 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='13762v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='GR] 31 Jan 2023 ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE EXCEPTIONAL GROUPS OF LIE TYPE NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV In memory of Irina Dmitrievna Suprunenko Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The Gruenberg–Kegel graph Γ(G) of a finite group G is the graph whose vertex set is the set of prime divisors of |G| and in which two distinct vertices r and s are adjacent if and only if there exists an element of order rs in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' A finite group G is called almost recognizable (by Gruenberg–Kegel graph) if there is only finite number of pairwise non-isomorphic finite groups having Gruenberg–Kegel graph as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If G is not almost recognizable, then it is called unrecognizable (by Gruenberg–Kegel graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Recently P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Cameron and the first author have proved that if a finite group is almost recognizable, then the group is almost simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, the question of which almost simple groups (in particular, finite simple groups) are almost recognizable is of prime interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We prove that every finite simple exceptional group of Lie type, which is isomorphic to neither 2B2(22n+1) with n ≥ 1 nor G2(3) and whose Gruenberg–Kegel graph has at least three connected components, is almost recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Moreover, groups 2B2(22n+1), where n ≥ 1, and G2(3) are unrecognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Throughout the paper we consider only finite groups and simple graphs, and henceforth the term group means finite group and the term graph means simple graph, that is undirected graph without loops and multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G be a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The spectrum ω(G) is the set of all element orders of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The prime spectrum π(G) is the set of all primes belonging to ω(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' A graph Γ(G) whose vertex set is π(G) and in which two distinct vertices r and s are adjacent if and only if rs ∈ ω(G) is called the Gruenberg–Kegel graph or the prime graph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We say that the group G is recognizable (by Gruenberg–Kegel graph) if for every group H the equality Γ(H) = Γ(G) implies that G ∼= H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' k-recognizable (by Gruenberg–Kegel graph), where k is a positive integer, if there are exactly k pairwise non-isomorphic groups having the same Gruenberg–Kegel graph as G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' almost recognizable (by Gruenberg–Kegel graph) if it is k-recognizable by Gruenberg– Kegel graph for a positive integer k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' unrecognizable (by Gruenberg–Kegel graph) if there are infinitely many pairwise non- isomorphic groups having the same Gruenberg–Kegel graph as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that groups can be characterized by various numerical sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' For example, if we replace Γ(G) in these definitions with ω(G), then we obtain the corresponding definitions for recog- nizability by spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Nevertheless, if the characterization set is not specified, we suppose that it is the Grunberg-Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It is easy to see that if ω(G) = ω(H) for groups G and H, then Γ(G) = Γ(H), however, the converse implication is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Consider alternating groups A5 and A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Clearly, Γ(A5) = Γ(A6), where both graphs are empty graphs on three vertices 2, 3, and 5, and, on the other hand, we see that 4 ∈ ω(A6) \\ ω(A5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 1 2 NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV Recently P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Cameron and the first author have proved [7] that a group G is almost recognizable if and only if each group H with Γ(G) = Γ(H) is almost simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, the question of which almost simple groups are almost recognizable is of prime interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' At the same paper [7], a survey of known results on recognition of simple groups has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that there are not many completed results at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The situation is much better in the case of the characterization problem by spectrum, where recognition is established for many nonabelian simple groups [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In [36], V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Mazurov conjectured that if a simple group G is not isomorphic to A6 and Γ(G) has at least three connected components, then G is recognizable by its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This conjecture was proved in a series of papers, the final result was obtained in [28], where the author proved that groups E7(2) and E7(3) are recognizable by Gruenberg–Kegel graph, therefore, these groups are recognizable by spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Connected components of Gruenberg–Kegel graphs of simple groups were described in [46, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' A complete result after corrections of mistakes can be found, for example, in [2, Tables 1– 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If G is a simple group, then Γ(G) has at least three connected components if and only if one of the following statements holds: (1) G ∼= G2(q), where q is a power of 3, or G ∼= 2G2(q), where q = 32n+1 > 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (2) G ∼= 2B2(q) ∼= Sz(q), where q = 22n+1 > 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (3) G ∼= F4(q), where q is even, or G ∼= 2F4(q) for q = 22n+1 > 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (4) G ∼= E8(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (5) G ∼= A1(q) ∼= PSL2(q), where q > 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (6) G ∼= 2Dn(3) ∼= PΩ− 2n(3), where n = 2m + 1 ≥ 3 is a prime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (7) G is one of the following finite simple groups of Lie type: 2A5(2) ∼= PSU6(2), E7(2), E7(3), A2(4) ∼= PSL3(4), 2E6(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (8) G is one of the following finite simple sporadic groups: M11, M23, M24, J3, HiS, Suz, Co2, Fi23, F3, F2, M22, J1, O′N, LyS, Fi′ 24, F1, J4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (9) G ∼= An, where n > 6 and both n and n − 2 are primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We consider the recognition problem of the simple exceptional groups of Lie type whose Gruenberg–Kegel graphs have at least three connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The main result of this paper is the following statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Every finite simple exceptional group of Lie type, which is isomorphic to neither 2B2(22n+1) with n ≥ 1 nor G2(3) and whose Gruenberg–Kegel graph has at least three connected components, is almost recognizable by Grunberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Moreover, groups 2B2(22n+1), where n ≥ 1, and G2(3) are unrecognizable by Gruenberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In fact, much has been known about the groups in the theorem before in the context of the recognition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The groups E7(2), E7(3), and 2E6(2) are known to be recognizable (see [28] and [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The recognizability of simple groups 2G2(q) with q > 3 was established in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If G is a nonabelian simple group, then we say that G is quasirecognizable by its Gruenberg– Kegel graph if every group H with Γ(G) = Γ(H) has a unique nonablelian composition factor S and S ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In [50], it was proved that each group G2(q), where q is an odd power of 3, is quasirecognizable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' however, this result contains an error since Γ(G2(3)) = Γ(PSL2(13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Quasirecognizability of the simple groups F4(q) and 2F4(q), where q > 2 is even, was proved in [24] and [1], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We prove Main Theorem for groups G2(q), where q > 3, F4(q), and 2F4(q) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The quasirecognizability of groups 2B2(q) was proved in [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We consider the groups 2B2(q) and G2(3) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS 3 In [47], A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Zavarnitsine proved that if G = E8(q), where q ≡ 0, ±1 (mod 5), and H is a group such that Γ(H) = Γ(G), then H ∼= E8(u) for a prime power u ≡ 0, ±1 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In Section 5, we prove a similar result for groups E8(q), where q ≡ ±2 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, as a corollary, Main Theorem holds for the groups E8(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' However, the question of quasirecognizability for the groups E8(q) is still open (see Section 5 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Provide a mini-survey of known results on characterization by Gruenberg–Kegel graph of the remaining simple groups whose Gruenberg–Kegel graphs have at least three connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' First consider groups PSL2(q), where q = pk and p is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The groups PSL2(4) ∼= PSL2(5), PSL2(7), PSL2(8), and PSL2(9) ∼= A6 are unrecognizable while the groups PSL2(11) and PSL2(25) are 2-recognizable (see, for example, [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='7], [31, Theo- rem 5] and [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The group PSL2(49) is 5-recognizable (see [34], this result is obtained as a consequence of [31, Theorem 5] and direct calculations with using Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let k = 1 and p > 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In [25], it was proved that if p ̸≡ 1 (mod 12), then G is recognizable, and if p ≡ 1 (mod 12), then G is quasirecognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' However, this result has at least one mistake since the group PSL2(13) is not quasirecognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If p is odd and k > 2 or k = 2 and p > 7, then G is (k, 2)-recognizable (see [22] and [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If q > 8 is even, then G is quasirecognizable (see [26, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In general, we suggest that the results on recognition of the groups PSL2(q) by Gruenberg–Kegel graph need a careful revision since the authors in their proofs refer to the paper [18] which contains a number of rather serious inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If G ∼= PΩ− 2n(3), where n ≥ 3 is a prime, then G is recognizable (see [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The group PSU6(2) is 2-recognizable (see [30, Theorem 2]), the group PSL3(4) is 5-recognizable since Γ(PSL3(4)) = Γ(PSL2(49)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The problem of recognition by Gruenberg–Kegel graph has been solved for all finite simple sporadic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, the groups J1, M22, M23, M24, Co2, J4, J3, Suz, O′N, LyS, F3, Fi23, Fi′ 24, F2, and F1 are recognizable (see [17, Theorem 3], [48, Theorem B], [30, Theorem 1], and [32]) while the groups M11 and HiS are 2-recognizable (see [17, Theorem 3] and [30, Theorem 2], respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G ∼= An, where n > 6 and both n and n − 2 are primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If n = 7, then Γ(G) = Γ(PSL2(49)) and, therefore, G is 5-recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If n = 13, then G is recognizable (see [39, Lemma 24]), and if n ≥ 19, then G is known to be quasirecognizable (see [39, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The question of recognizability by Gruenberg–Kegel graph of the groups An, where n ≥ 19 is a prime and n − 2 is also a prime, is still open, and the conjecture [39, Conjecture 1] is that these groups are recognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Preliminaries Let n be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Denote by π(n) the set of all prime divisors of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let π be a set of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The largest divisor m of n such that π(m) ⊆ π is called a π-part of n and is denoted by nπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By π′ we denote the set of primes which do not belong to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If π consists of a unique element p, then we will write np and np′ instead of n{p} and n{p}′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If n is an integer and r is an odd prime with (r, n) = 1, then e(r, n) denotes the multi- plicative order of n modulo r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Given an odd integer n, we put e(2, n) = 1 if n ≡ 1 (mod 4), and e(2, n) = 2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The following lemma is proved in [3], and also in [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 4 NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1 (Bang–Zsigmondy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let q be an integer greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' For every positive integer m there exists a prime r with e(r, q) = m but for the cases q = 2 and m = 1, q = 3 and m = 1, and q = 2 and m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Fix an integer a with |a| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' A prime r is said to be a primitive prime divisor of ai − 1 if e(r, a) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We write ri(a) to denote some primitive prime divisor of ai − 1 if such a prime exists, and Ri(a) to denote the set of all such divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2 ([14, Lemma 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let q, m, and k be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then Rmk(q) ⊆ Rm(qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If, in addition, (m, k) = 1, then Rm(q) ⊂ Rm(qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If r ∈ Ri(q), then r = ik + 1, where k is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This is a consequence of Fermat’s little theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Given a positive integer i ̸= 2, denote by ki(a) the product of all primitive prime divisors of ai − 1 with multiplicities counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In case i = 2 put k2(a) = k1(−a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It is not difficult to verify that if i is divisible by 4 then ki(a) = ki(−a) and if i is odd then ki(a) = k2i(−a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The following general formula [38] expresses ki(a), where i > 2, in terms of cyclotomic polynomials: (1) ki(a) = Φi(a) (r, Φ(i)r′(a)), where r is the greatest prime divisor of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The following assertion is well-known and its proof is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that q > 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' For a positive integer i, an odd prime r divides qi − 1 if and only if e(r, q) divides i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The following lemma is a particular case of the well-known Nagell–Ljunggren equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5 ([37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that x, y, and k are positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If x2 + x + 1 = yk, then either k = 1 or k = 3, x = 18, and y = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The following technical lemma follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5, but we give an independent proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This statement will be needed in the proof of Main Theorem for the group G2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that q is an integer greater than 2 and π(q2 + εq + 1) = {r}, where ε ∈ {+, −} and r is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r ≡ 1 (mod 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Moreover, if q ≡ 1 (mod 8) then either r ≡ 1 (mod 8) or r ≡ 3 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By assumption, there exists a positive integer n such that q2 + εq + 1 = rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Clearly, r is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that q2 + εq + 1 is not divisible by 9, so if r = 3, then n = 1 and q ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If r ̸= 3, then r divides k3(εq) = q2+εq+1 (q−ε1,3) and hence r ∈ R3(εq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3 that r ≡ 1 (mod 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that q ≡ 1 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then either q2 + εq + 1 ≡ 1 (mod 8) or q2 + εq + 1 ≡ 3 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, if r ≡ 5 (mod 8) or r ≡ 7 (mod 8), then n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' On the other hand, (q − 1) < q2 − q + 1 < q2 and q2 < q2 + q + 1 < (q + 1)2, so n cannot be even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that either r ≡ 1 (mod 8) or r ≡ 3 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Denote the number of connected components of Γ(G) by s(G), and the set of connected components of Γ(G) by {πi(G) | 1 ≤ i ≤ s(G)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' for a group G of even order, we assume that 2 ∈ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Denote by t(G) the independence number of Γ(G), ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS 5 that is the greatest size of a coclique (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' induced subgraph with no edges) in Γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If r ∈ π(G), then denote by t(r, G) the greatest size of a coclique in Γ(G) containing r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let A and B be normal subgroups of a group G such that A ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If r, s ∈ π(B/A) \\ (π(A) ∪ π(G/B)), then r and s are adjacent in Γ(G) if and only if r and s are adjacent in Γ(B/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The proof of this lemma is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8 ([42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G be a finite group with t(G) ≥ 3 and t(2, G) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (1) There exists a nonabelian simple group S such that S ⊴ G = G/K ≤ Aut(S), where K is the solvable radical of G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', the largest solvable normal subgroup of G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (2) For every coclique ρ of Γ(G) of size at least three, at most one prime in ρ divides the product |K| · |G/S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, t(S) ≥ t(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (3) One of the following two conditions holds: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1) S ∼= A7 or L2(q) for some odd q, and t(S) = t(2, S) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2) Every prime p ∈ π(G) nonadjacent to 2 in Γ(G) does not divide the product |K| · |G/S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, t(2, S) ≥ t(2, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='9 ([39, Lemma 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let N ⊴ G be an elementary abelian subgroup and H = G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Define a homomorphism φ : H → Aut(N) as follows nφ(gN) = ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then Γ(G) = Γ(N ⋊φ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='10 ([7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let π be a finite set of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The number of pairwise nonisomorphic nonabelian simple groups S with π(S) ⊆ π is finite, and is at most O(|π|3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let S be a finite simple group of Lie type in characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let A be any abelian p-group with an S-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Any element s ∈ S is said to be unisingular on A if s has a (nonzero) fixed point on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The group S is said to be unisingular if every element s ∈ S acts unisingularly on every finite abelian p-group A with an S-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Denote by PSLε n(q), where ε ∈ {+, −}, the group PSLn(q) if ε = 1 and PSUn(q) if ε = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Similarly, Eε 6(q) denotes the simple group E6(q) if ε = 1 and 2E6(q) if ε = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='11 ([16, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' A finite simple group S of Lie type of characteristic p is unisingular if and only if S is one of the following: (i) PSLε n(p) with ε ∈ {+, −} and n divides p − ε1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (ii) PΩ2n+1(p), PSp2n(p) with p odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (iii) PΩε 2n(p) with ε ∈ {+, −}, p odd, and ε = (−1)n(p−1)/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (iv) 2G2(q), F4(q), 2F4(q), E8(q) with q arbitrary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (v) G2(q) with q odd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (vi) Eε 6(p) with ε ∈ {+, −} and 3 divides p − ε1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (vii) E7(p) with p odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='12 ([47, Proposition 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G = 3D4(q) act on a nonzero vector space V over a field of characteristic not dividing q (possibly, zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then each element of G of order q4 − q2 + 1 fixes on V a nonzero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The following lemma is also well-known, but we provide its proof for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G be a group, g ∈ G an element of order r, and φ a non-trivial irreducible representation of G on a nonzero vector space V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If the minimum polynomial degree of φ(g) equals to r, then g fixes on V a nonzero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 6 NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let A = φ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since gr = 1, we have Ar = 1, and, therefore, the minimal polynomial for A divides the polynomial xr − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since the minimum polynomial degree of A equals to r, we have that the minimum polynomial for A is xr − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By the Cayley–Hamilton theorem, A is a root of its characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, 1 is an eigenvalue of A, and, therefore, each eigenvector of A which corresponds to the eigenvalue 1, is fixed by A = φ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='14 ([41, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G be one of the groups 2B2(q), where q > 2, 2G2(q), where q > 3, 2F4(q), G2(q), 3D4(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let g ∈ G an element of prime power order coprime to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let φ be a non-trivial irreducible representation of G over a field F of characteristic l coprime to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then the minimum polynomial degree of φ(g) equals |g|, unless possibly when G = 2F4(8), l = 3, p = 109 and φ(1) < 64692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='15 ([10, Lemma 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G be a finite simple group, F a field of characteristic p > 0, V an absolute irreducible GF-module, and β a Brauer character of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If g ∈ G is an element of prime order distinct from p, then dimCV (g) = (β⟨g⟩, 1⟨g⟩) = 1 |g| � x∈⟨g⟩ β(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G be a group with a non-trivial nilpotent normal subgroup K such that G/K has a subgroup H isomorphic to E8(q), where q is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then R24(q) ⊂ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that R24(q) ⊂ π(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let q = pl, where p is a prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that K is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Factoring G and K by K′, we can assume that K is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 3], p lies in π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='11, H is unisingular and hence p is adjacent to each element of π(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, R24(q) ⊂ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Take any r ∈ π(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since Sylow 2-subgroups of H are non-cyclic, we infer that either r = 2 or 2 and r are adjacent in Γ(G) (see, for example, [12, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, r ∈ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that there exists r ∈ π(K)∩R24(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since π(K) is a clique in Γ(G) and r ∈ π1(G), we get that R24(q) ⊂ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If K is not a p-group and π(K)∩R24(q) = ∅, then we can choose r ∈ π(K)\\({p}∪R24(q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='12, r is adjacent in Γ(G) to any prime from R24(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that R24(q) ⊂ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The Grunberg–Kegel graphs of some exceptional groups of Lie type A criterion for the adjacency of vertices in the Gruenberg–Kegel graph for all finite non- abelian simple groups was obtained in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Based on this paper and [45], in this section, we collect the necessary information for exceptional groups of Lie type from Main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By the compact form of the Gruenberg–Kegel graph Γ(G) for a group G we mean a graph whose vertices are labeled with sets of primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' A vertex labeled by a set τ represents the clique of Γ(G) such that every vertex in this clique labeled by a prime from τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' An edge connecting two sets represents the set of edges of Γ(G) that connect each vertex in the first set with each vertex in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1 ([45, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='7] and [44, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G ∼= F4(q), where q = 2n, r, s ∈ π(G) and r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r and s are nonadjacent if and only if one of the following conditions holds (up to permutation): (1) 2 = r, and e(s, q) ∈ {8, 12}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS 7 (2) s, r ̸= 2, k = e(r, q), l = e(s, q), 1 ≤ k < l and either l ∈ {8, 12}, or l = 6 and k ∈ {3, 4}, or l = 4 and k = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, the compact form for Γ(F4(q)) is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' R1 R2 R6 R3 R4 2 R12 R8 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that R6(2) and R1(2) are empty sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2 ([44, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3] and [45, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='9];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' see also [9, Lemma 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G ∼= 2F4(q), where q = 22n+1, r, s ∈ π(G) and r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Put m1(n) = q − 1, m2(n) = q + 1, m3(n) = q2 + 1, m4(n) = q2 − q + 1, m5(n) = q2 − � 2q3 + q − √2q + 1, m6(n) = q2 + � 2q3 + q + √2q + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r and s are nonadjacent in Γ(G) if an only if one of the following conditions holds: (1) 2 = r, s divides mk(n), s ̸= 3, and k > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (2) 2 ̸= s, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' either 3 ̸= r ∈ π(mk(n)), 3 ̸= s ∈ π(ml(n)) for k ̸= l, and {k, l} ̸= {1, 2}, {1, 3};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' or r = 3 and s ∈ π(ml(n)), where l ∈ {3, 5, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, the compact form for Γ(2F4(q)) is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 2 π(q + 1)\\{3} 3 π(q − 1) π(q2 − q + 1)\\{3} π(q2 + 1) π(q2 + � 2q3 + q + √2q + 1) π(q2 − � 2q3 + q − √2q + 1) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3 ([45, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='7] and [44, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G ∼= G2(q), where q = 3k, r, s ∈ π(G) and r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r and s are nonadjacent in Γ(G) if and only if e(r, q) or e(s, q) ∈ {3, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, the compact form for Γ(G2(q)) is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' R1 R2 {3} R3 R6 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that the set R1(3) is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 8 NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4 ([45, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='7] and [44, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G ∼= E8(q), where q is a power of a prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that r, s ∈ π(G) with r ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r and s are nonadjacent in Γ(G) if and only if one of the following conditions holds: (1) r ∈ {2, p}, s ̸= p and e(s, q) ∈ {15, 20, 24, 30}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' (2) s, r ̸∈ {2, p}, k = e(r, q), l = e(s, q), 1 ≤ k < l, and either l = 6 and k = 5, or l ∈ {7, 14} and k ≥ 3, or l = 9 and k ≥ 4, or l ∈ {8, 12} and k ≥ 5, k ̸= 6, or l = 10 and k ≥ 3, k ̸∈ {4, 6}, or l = 18 and k ̸∈ {1, 2, 6}, or l = 20 and r · k ̸= 20, or l ∈ {15, 24, 30}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, the compact form for Γ(E8(q)) is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Here, R(q) = R1(q)∪R2(q)∪ {p} and the vector from 5 to R4(q) and the dotted edge {5, R20(q)} indicate that R4(q) and R20(q) are not adjacent, but if 5 ∈ R4(q) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', q2 ≡ −1 (mod 5)), then there exist edges between 5 and the primes from R20(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' R R18 R5 R3 R8 R12 R6 R10 R9 R14 R7 R4 5 R20 R15 R24 R30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Almost recognizability of groups G2(q), F4(q), and 2F4(q) by Gruenberg–Kegel graph In this section, we prove Main Theorem for groups L = F4(q), where q ≥ 2 is a power of 2, L = 2F4(q), where q = 22m+1 > 2, and L = G2(q), where q > 3 is a power of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If G is a group with Γ(G) = Γ(L), then L ∼= Inn(L) ⊴ G ≤ Aut(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In particular, L is almost recognizable by Gruenberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3, we see that t(L) ≥ 3 and t(2, L) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8 that there exists a nonabelian simple group S such that S ∼= Inn(S) ≤ G/K ≤ Aut(S), where K is the solvable radical of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Moreover, by the Thompson theorem on finite groups with fixed-point-free automorphisms of prime order [40, Theorem 1], K is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' To prove Theorem 1, it suffices to show that S ∼= L and K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' These two facts are established in the following four lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' S ∼= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If q > 2 is even and L = 2F4(q) or L = F4(q), then Lemma follows from [1, Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4] and [24, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that L = F4(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then π(S) ⊆ π(L) = {2, 3, 5, 7, 13, 17}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that 13 ∈ R12(2) and 17 ∈ R8(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1, we find that 13 and 17 are nonadjacent to all vertices in Γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, {13, 17} ⊂ π(S) by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Inspecting [49, Table 1], we see that S ∈ {PSU4(4), PSU3(17), PSL2(132), PSp4(13), PSL3(16), PSp6(4), PΩ+ 8 (4), F4(2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS 9 Using [4, Corollary 3], we find that 5 and 17 are adjacent in Γ(PSL2(132)) and Γ(PSL3(16)), while 3 and 17 are adjacent in Γ(PSU3(17)) and Γ(PSU4(4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [5, Corollaries 2– 4], 5 and 17 are adjacent in Γ(PSp4(13)), Γ(PSp6(4)), and Γ(PΩ+ 8 (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that S ∼= F4(2), as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It remains to consider the case L ∼= G2(q), where q > 3 is a power of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If q > 3 is an odd power of 3, then S ∼= L by [50, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that L ∼= G2(q), where q = 32k for a positive integer k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3, sets R3(q) and R6(q) are connected components in Γ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8 that sets R3(q) and R6(q) are connected components in Γ(S) and hence Γ(S) has at least three connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, S isomorphic to a group listed in Introduction before Main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We show that S ∼= L considering each case for S separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We will extensively use that k3(q) = q2+q+1 (q−1,3) = q2 + q + 1 and k6(q) = q2−q+1 (q+1,3) = q2 − q + 1 (see equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= Ap, where p > 6 and both p and p − 2 are primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The connected components of Γ(S) are π1(S), {p}, and {p − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We know that R3(q) and R6(q) are connected components in Γ(S), so either p − 2 ∈ R3(q) or p − 2 ∈ R6(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3 that p − 3 is divisible by 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction since p ̸= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= E8(u), where u is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since 32k ≡ (−1)k (mod 5), we infer that q ≡ ±1 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that there exists ε ∈ {+, −} such that q2+εq+1 ≡ 3 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Take any prime divisor r of k3(εq) = q2 + εq + 1 such that r ̸≡ 1 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since r ∈ R3(εq), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4 implies that r ∈ Rj(u), where j ∈ {15, 20, 24, 30}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r−1 is divisible by j and hence j = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since R24(u) is a connected component in Γ(S), we find that R3(εq) = R24(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If K ̸= 1, then Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='16 implies that R24(u) ⊆ π1(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction since r ̸∈ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Assume that there exists s ∈ π(G/S) such that s > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8 that s ∈ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='12, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='13], there exists an element g ∈ G\\S such that |g| = s and g acts on S as a field automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It is known that CS(g) ∼= E8(u1/s) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', [13, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, we see that R24(u1/s) ⊆ R24(u) and hence s is adjacent to r in Γ(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' we arrive at a contradiction since s ∈ π1(G) and r ̸∈ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Now consider a coclique ρ = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' , s12} of maximal size in Γ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4, we have (si, 6) = 1 for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' , 12}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Applying Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='7, we find that t(G) ≥ 12;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Cases S ∈ {PSU6(2), PSL3(4), M11, M23, M24, J3, HiS, Suz, Co2, Fi23, F2, M22, Fi′ 24, F1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8, we infer that {2, r3(q), r6(q)} is a coclique in Γ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3 that r3(q) ≡ 1 (mod 6) and r6(q) ≡ 1 (mod 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Using [8], we see that there is no such a coclique in Γ(S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= F4(u), where u is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since q2 + q + 1 ≡ 3 (mod 8), there exists r ∈ R3(q) such that r ̸≡ 1 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3, we infer that r ∈ R12(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then π(q2 +q +1) = R12(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' On the other hand, we have q2 +q +1 ≡ 7 (mod 12) and hence there exists s ∈ π(q2 + q + 1) such that s ̸≡ 1 (mod 12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction with Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= 2F4(u), where u = 22m+1 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, it is true that π2(S) ∪ π3(S) = π(u4 − u2 + 1) ⊆ π(u6 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Consider any prime r ∈ R3(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r ∈ π2(S) ∪ π3(S) and r divides (u3)2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since (u3)2 ≡ −1 (mod r), we find that −1 is a quadratic residue modulo r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that r ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since r is an arbitrary element of R3(q), we infer that q2 + q + 1 ≡ 1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' On the other hand, q2 + q + 1 ≡ 34k + 32k + 1 ≡ 3 (mod 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= 2B2(u), where u = 22m+1 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 3], we can assume that π2(S) = π(u − 1), π3(S) = π(u − √ 2u + 1), and π4(S) = π(u + √ 2u + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that 10 NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV π(u − √ 2u + 1) ∪ π(u + √ 2u + 1) = π(u2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If R3(q) ̸= π2(S), then R3(q) ⊆ π(u2 + 1) and we get a contradiction arguing as in the case S ∼= 2F4(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, we can assume that R3(q) = π(u − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that r | u − 1 = 22m+1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r | 22n+2 − 2 and hence 2 is a quadratic residue modulo r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that r ≡ ±1 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows that q2 + q + 1 ≡ ±1 (mod 8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction since q2 + q + 1 ≡ 34k + 32k + 1 ≡ 3 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= 2G2(u), where u = 32m+1 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 2], we can assume that π2(S) = π(u − √ 3u + 1) and π3(S) = π(u + √ 3u + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Take any prime r ∈ R12k(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, we infer that r ∈ R6(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We know that R6(q) ⊆ π2(S) ∪ π3(S) = π(u2 − u + 1) = R6(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4, we conclude that 12k | 6 · (2m + 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= 2Dp(3), where p = 2m + 1 ≥ 3 is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 2], we can assume that π2(S) = π((3p−1 + 1)/2) and π3(S) = π((3p + 1)/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Take any r ∈ R12k(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, we infer that r ∈ R6(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since r ∈ π2(S)∪π3(S), we find that r divides 32(p−1)−1 or 32p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4 that 12k | 2(p − 1) or 12k | 2p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= A1(u) ∼= PSL2(q), where u = 2m > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 2], we can assume that π2(S) = π(u − 1) and π3(S) = π(u + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Note that 3 divides u2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' On the other hand, π(u2 − 1) = π2(S) ∪ π3(S) = π2(G) ∪ π3(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction since 3 ∈ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Case S ∼= A1(u) ∼= PSL2(u), where 3 < u = vn and u is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Consider ε ∈ {+, −} such that u ≡ ε1 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 2], we can assume that π1(S) = π(u − ε1), π2(S) = {v}, and π3(S) = π( u+ε1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, there exists τ ∈ {+, −} such that π(q2 + τq + 1) = {v} and π(q2 − τq + 1) = π( u+ε1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Moreover, we know that π(u − ε1) ⊆ π(3(q2 − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since q2 − q + 1 = (q − 1)2 + (q − 1) + 1, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5 implies that either q2 + τq + 1 = v or q = 19 and v = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By assumption, q = 32k and hence q2 + τq + 1 = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then v − 1 is divisible by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, v ≥ 19 and 3 ∈ π( u−1 2 ) \\ π(q2 − τq + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that ε = + since, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3, 3 ∈ π1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then v2 − 1 divides u − 1, so π(v + 1) ⊆ π(q2 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Take any r ∈ π(v+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since q ≡ ±1 (mod r) and r divides q2 +τq+2, we find that r divides 3±1 and hence r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, q2 +τq +2 is a power of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' On the other hand, q2 +τq +2 ≡ 1±1+2 (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that q2 + τq + 2 ≤ 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We can assume that n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then v+1 2 divides u+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Take any r ∈ π( v+1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then r divides both q2 + τq + 2 and q2 − τq + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, 2τq ≡ −1 (mod r) and hence 0 ≡ 4q2 + 4τq + 8 ≡ 4τq + 9 ≡ 7 (mod r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that r = 7 and v + 1 = 2 · 7m for a positive integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since q ≡ 1 (mod 8) and q2+τq+2 = 2·7m, we infer that 3+τ1 ≡ 2·(−1)m (mod 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that τ = −1 and m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' A straightforward calculation shows that 92k − 9k + 2 is not divisible by 49 for all positive integer k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Cases S ∈ {F3, O′N, J1, LyS, J4, 2E6(2), E7(2), E7(3)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Tables 2, 3], we see that there exist primes r and s such that π(q2 + q + 1) = {r} and π(q2 − q + 1) = {s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Applying Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='6, we find that r ≡ s ≡ 1 (mod 6) and the remainders of r and s divided by 8 belong to the set {1, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Inspecting [2, Tables 2, 3], we see that in each case there are no two primes satisfying these restrictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, we conclude that S ∼= G2(u), where u is a power of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Now Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1 implies immediately that u = q and, therefore, S ∼= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' For q = 3, the paper [50] contains a mistake, and S ∈ {G2(3), PSL2(13)} by [20, Table 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Show that K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Consider a minimal (by order) counterexample G to this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS 11 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' K is an elementary abelian r-group for some r ∈ π1(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let r be a prime divisor of |K|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since K is nilpotent, we have a decomposition K = P ×U, where P is a Sylow r-subgroup of K and U is a normal subgroup of K such that r ̸∈ π(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since U and Φ(P) are characteristic subgroups of K, the subgroup N = U ×Φ(P) is a normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then Γ(G/N) is a subgraph of Γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' On the other hand, Γ(L) is a subgraph of Γ(G/N) and hence Γ(G/N) = Γ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By the minimality of G, we infer that N = 1 and, therefore, K is an elementary abelian r-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ By Lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='9, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, we can assume that G = K ⋊ X, where K is an elementary abelian r-group for r ∈ π1(L) and Soc(X) = S ∼= L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' K = 1 if L = F4(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [33, Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1], S has a subgroup H isomorphic to 3D4(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Assume that r ̸∈ {2} ∪ R12(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='12, r is adjacent to each element from R12(q) in Γ(KH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1 that Γ(G) ̸= Γ(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Assume that r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='11, S is unisingular and, therefore, 2 is adjacent to all other vertices in Γ(KS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We arrive at a contradiction with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Assume that r ∈ R12(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since q is even, by [33, Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1], S has a subgroup H1 ∼= PΩ+ 8 (q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Now by [6, Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='50], H1 has a subgroup H2 ∼= PΩ− 4 (q)×PΩ− 4 (q) ∼= PSL2(q2)×PSL2(q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, for each s ∈ R4(q), a Sylow s-subgroup of S is non-cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that r and s are adjacent in Γ(KH2) (see, for example, [12, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, r and s are adjacent in Γ(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' K = 1 if L ∼= G2(q) for q > 3 or L ∼= 2F4(q) for q > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let p = 2 if L ∼= 2F4(q) and p = 3 if L ∼= G2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='11, L is unisingular and, therefore, if r = p, then Γ(G) is connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, r ∈ π1(L) \\ {p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='13 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='14, r is adjacent to a prime from π2(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since the group L is known to be almost recognizable, the following natural question arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that L is a group from the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Find a positive integer k such that L is k-recognizable by Gruenberg-Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Unrecognizability of groups 2B2(q) and G2(3) by Gruenberg–Kegel graphs In this short section, we show that groups 2B2(q) with q > 2 and G2(3) are unrecognizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G = 2B2(q), where q > 2 is an odd power of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then G is unrecog- nizable by Gruenberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By [16, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='6], there exists a 4-dimensional module V over the field of order q such that each nontrivial element from each maximal torus of G acts fixed-point freely on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, Γ(V ⋊ G) = Γ(G) and, therefore, G is unrecognizable by [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let G ∼= G2(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then G is unrecognizable by Gruenberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Using [8], we find that Γ(G2(3)) is the following: 2 3 7 13 12 NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV Moreover, Γ(G) = Γ(PSL2(13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By [19, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 9] and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='15, there is a 6-dimensional irreducible PSL2(13)-module V over a field of characteristic two such that all elements in PSL2(13) of orders 7 and 13 act fixed-point freely on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, Γ(V ⋊ PSL2(13)) = Γ(PSL2(13)) = Γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, by [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2], G is unrecognizable by Gruenberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Almost recognizability of groups E8(q) by Gruenberg–Kegel graph To complete the proof of Main Theorem, it remains to consider the case of groups E8(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' In [47], A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Zavarnitsine proved that if G is a finite group such that Γ(G) = Γ(E8(q)), where q ≡ 0, ±1 (mod 5), then G ∼= E8(u) for some u ≡ 0, ±1 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The aim of this section is to prove the following similar result for the remaining cases for q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Let L = E8(q), where q ≡ ±2 (mod 5) is a prime power, and G be a group such that Γ(G) = Γ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Then G ∼= E8(u) for some prime power u with u ≡ ±2 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since Γ(G) is disconnected, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8 implies that there exists a nonabelian simple group S such that S ≤ G/K ≤ Aut(S), where K is the solvable radical of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By the Thompson theorem on finite groups with fixed-point-free automorphisms of prime order [40, Theorem 1], K is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemmas 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4, we find that s(S) ≥ s(L) = 4, t(2, S) ≥ t(2, L) = 5 and t(S) ≥ t(L)−1 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 1] and [44, Tables 2, 4, and 5], we find that either S ∼= F1 or S ∼= E8(u), where u is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that S ∼= F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 3], we see that s(S) = 4, for each i ≥ 2, |πi(S)| = 1, and π(S) \\ π1(S) = {41, 59, 71}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' At the same time, π2(G) = R15(q), π3(G) = R24(q), and π4(G) = R30(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='3, the numbers 15, 24, and 30 divide ri − 1 for pairwise distinct primes ri from π(S) \\ π1(S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that S ∼= E8(u), where u is a power of a prime v and u ≡ 0, ±1 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since q ≡ ±2 (mod 5), we find that 5 ∈ R4(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Denote θ = {r9(q), r14(q), r7(q), r18(q), r15(q), r24(q), r30(q)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4, we see that θ ∪{5} is a coclique of size 8 in Γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='8 that at least six elements of θ belong to π(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Take any r ∈ π(S) ∩ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since r and 5 are nonadjacent in Γ(G), they are nonadjacent in Γ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' We know that 5 ∈ {v} ∪ R1(u) ∪ R2(u) and hence r ∈ R20(u) ∪ R15(u) ∪ R24(u) ∪ R30(u) according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that at least two elements from θ ∩ π(S) are adjacent in Γ(S);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that S ∼= E8(u), where vl = u ≡ ±2 (mod 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' According to [2, Table 3], we can assume that π2(G) = R15(q), π3(G) = R24(q), and π4(G) = R30(q), while π2(S) = R15(u), π2(S) = R24(u), and π4(S) = R30(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If K ̸= 1, then Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='16 implies that R24(u) ⊂ π1(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction since R24(u) must coincide with a connected component of Γ(G) not containing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, we can assume that S ≤ G ≤ Aut(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Now prove that G/S = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If a prime r divides |G : S|, then by [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='12, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='13, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='1], G \\ S contains a field automorphism x of S of order r with CS(x) ≥ E8(u1/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that r is adjacent in Γ(G) to each prime from π(E8(u1/r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Suppose that r ̸∈ {2, 3, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, Γ(G) is connected and hence Γ(G) ̸= Γ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If r = 2, then by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, R15(u) ⊂ π1(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If r = 5, then by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, r is adjacent in Γ(G) to some primes from R24(u) and hence R24(u) ⊂ π1(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, we can assume that G/S is a 3-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='4, we see ON CHARACTERIZATION BY GRUENBERG–KEGEL GRAPH OF FINITE SIMPLE GROUPS 13 that ri(q) is nonadjacent to 2 in Γ(G) if and only if i ∈ {15, 20, 24, 30}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Similarly, a prime ri(u) is nonadjacent to 2 in Γ(S) if and only if i ∈ {15, 20, 24, 30}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Since R20(q) ⊂ π1(G) and R20(u) ⊂ π1(G), we infer that R20(q) = R20(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2, we infer that 3 is adjacent in Γ(G) to a prime from R20(u), therefore, Γ(G) ̸= Γ(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, G = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The proof of Theorem 2 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' For each value of q, the group E8(q) is almost recognizable by Gruenberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If G is a group such that Γ(G) = Γ(E8(q)), then by [47, Theorem 1] and Theorem 2, we have G ∼= E8(u) for a prime power u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='10, for a given q, the number of possibilities for u is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' This implies that there is only the finite number of possibilities for G (up to isomorphism), in particular, E8(q) is almost recognizable by Gruenberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Do there exist prime powers q and q1 with q ̸= q1 and Γ(E8(q)) = Γ(E8(q1))?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' If G is a finite group such that Γ(G) = Γ(E8(q)) and |G| = |E8(q)| for some prime power q, then G ∼= E8(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' By Theorem 2, if Γ(G) = Γ(E8(q)), then G ∼= E8(q1) for some prime power q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It is clear that a function f(x) = x120(x2 − 1)(x8 − 1)(x12 − 1)(x14 − 1)(x18 − 1)(x20 − 1)(x24 − 1)(x30 − 1) strictly monotonically increases if x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, if f(q1) = |G| = |E8(q)| = f(q), then q1 = q, and, therefore, G ∼= E8(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' □ In [43], it was proved that if G is a simple group and H is a group such that ω(H) = ω(G) and |H| = |G|, then H ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Thus, each simple group is uniquely determined by its order and spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It is known that if q is odd and n ≥ 3, then Γ(PΩ2n+1(q)) = Γ(PSp2n(q)) and |PΩ2n+1(q)| = |PSp2n(q)| but these groups are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Therefore, it is natural to consider the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' For which simple groups G is the following true: if H is a group with Γ(H) = Γ(G) and |H| = |G|, then H is isomorphic to G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Problem 3 was formulated by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Khosravi in his survey paper [23, Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='2], by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Kondrat’ev in frame of the open problems session of the 13th School-Conference on Group Theory Dedicated to V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Belonogov’s 85th Birthday (see [35, Question 4]), and was independently formulated by W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Shi in a personal communication with the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Also Problem 3 was formulated in the paper by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Cameron and the first author (see [7, Problem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' It is clear that if a simple group is quasirecognizable by Gruenberg–Kegel graph, then Problem 3 solves in the positive for this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' At the same time, Corollary 2 gives a solution of Problem 3 for finite simple groups E8(q) which are not necessary quasirecognizable by Gruenberg–Kegel graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 14 NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Acknowledgements The first author is supported by the Ministry of Science and Higher Education of the Rus- sian Federation, project 075-02-2022-877 for the development of the regional scientific and educational mathematical center ”Ural Mathematical Center” (for example, Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The second author is supported by the Mathematical Center in Akademgorodok under the agree- ment No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation (for example, Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' The third author is supported by RAS Fundamental Research Program, project FWNF-2022-0002 (for example, Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Akhlaghi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Khatami, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Khosravi, Quasirecognition by prime graph of the simple 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' und Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', 3 (1892), 265–284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' 16 NATALIA V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' MASLOVA, VIKTOR V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' PANSHIN, AND ALEXEY M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' STAROLETOV Natalia Vladimirovna Maslova Krasovskii Institute of Mathematics and Mechanics UB RAS, 16, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Kovalevskaja str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', Yekaterinburg, 620108, Russia Ural Federal University, 19, Mira str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', Yekaterinburg, 620002, Russia Email address: butterson@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='ru Viktor Vladimirovich Panshin Novosibirsk State University, 1, Pirogova str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', Novosibirsk, 630090, Russia Sobolev Institute of Mathematics, 4, Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Koptyug ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', Novosibirsk, 630090, Russia Email address: v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='pansh1n@yandex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='ru Alexey Mikhailovich Staroletov Sobolev Institute of Mathematics, 4, Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=' Koptyug ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content=', Novosibirsk, 630090, Russia Email address: staroletov@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='nsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} +page_content='ru' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wtFST4oBgHgl3EQfRjgB/content/2301.13762v1.pdf'} diff --git a/xNFST4oBgHgl3EQfSDhf/vector_store/index.faiss b/xNFST4oBgHgl3EQfSDhf/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..33ef9695e0277cffd15b2fee89ed6f6e0c0f38a9 --- /dev/null +++ b/xNFST4oBgHgl3EQfSDhf/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0312758440c45e048b12eb9af7d4c1c14530749146ed7fc118ec499f811966e9 +size 4849709 diff --git a/ydFRT4oBgHgl3EQfizeW/content/tmp_files/2301.13588v1.pdf.txt b/ydFRT4oBgHgl3EQfizeW/content/tmp_files/2301.13588v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..17a9e8dfa7c1aade725da8225fb4e09d84536a2a --- /dev/null +++ b/ydFRT4oBgHgl3EQfizeW/content/tmp_files/2301.13588v1.pdf.txt @@ -0,0 +1,551 @@ +On the miscibility gap in tungsten-based alloys +Andrzej Piotr Kądzielawa1, 2, ∗ and Dominik Legut3, † +1IT4Innovations, VŠB - Technical University, Ostrava-Poruba, Czechia +2Institute of Theoretical Physics, Jagiellonian University, Kraków, Poland +3IT4Innovations $ Nanotechnology Centre, CEET, VŠB - Technical University, +17.listopadu 2172/15, Ostrava, CZ 708 00 Czech Republic +(Dated: February 1, 2023) +In this work we establish an approach to model miscibility gaps of alloys using statistical physics, +lattice dynamics from first-principles calculations. We carefully calculate the entropy to include +all processes introducing disorder to the system, i.e., combining the electronic, phononic, and con- +figuration entropies. Furthermore we present our algorithm for generating Special Quasirandom +Structures (SQS). We model the miscibility gap in tungsten - chromium and tungsten - molybde- +num systems, obtaining the agreement with the experimental data. Furthermore, we propose an +enhancement for the tungsten-chromium W70Cr30 alloy with tantalum and hafnium, leading to the +modified stabilization temperatures TS, where the solid solution is miscible. +I. +INTRODUCTION +With the rapid search of carbon-neutral power pro- +ducing facilities, the need for working fusion reactors +is higher than ever before. +Successful Ignition Events +[1, 2], followed by net-positive cycle [3], increased the +priority of research on the crucial elements of a fusion +reactor, e.g., the so-called first walls. One family of com- +pounds, namely tungsten-based alloys represent prospec- +tive candidates to replace pure tungsten, as the alloy’s +features (e.g., selfpassivation) make them resistant to the +Loss of Coolant Accident (LOCA) with concomitant air +ingress into the reactor vessel. Since tungsten undergoes +volatile oxidation when in a temperature above 500◦ C +(the nuclear afterheat causes a surge of temperature up +to 1200◦C), threatening structural damage. In case of +selfpassivating tungsten alloys, the oxidation of the ad- +ditional element forms a robust oxide scale [4, 5]. The +problem being that the alloy operates as purposed while +in extreme conditions (high temperature). On the other +hand in normal conditions it is not thermodynamically +stable, diffusing into W-rich and Cr-rich phases. +Finding this so-called miscibility gap, the research com- +munity of 20th century [6–15], did not propose any large- +scale applications for W-Cr binaries. In contrast, now, +the W-Cr alloy is a candidate for the first wall of a fu- +sion reactor vessel [16] (e.g., 620 m2 for the Interna- +tional Thermonuclear Experimental Reactor). The ther- +mal stability of W-Cr alloys was already studied exper- +imentally [17–20], showing fast decomposition kinetics +close to 1000◦ C, i.e., near the LOCA conditions. In such +a case, changes in the phase content and degradation of +oxidation performance are inevitable. +The density-functional-theory-based ab initio compu- +tational modelling [21–27] is now the state-of-the-art ap- +proach when high-throughput (hundreds of millions core- +∗ andrzej.piotr.kadzielawa@vsb.cz +† dominik.legut@vsb.cz +hours) calculations are to be considered. +DFT-based +methods already have their input into designing novel +materials [28–31]. +Treatment of alloys in real-space requires averaging +over different randomly-generated structures. +The so- +called special quasi-random structures (SQS) [32–34] re- +duce the computational-complexity by providing us with +a set of most "alloy-like" supercells. +In effort to decrease the closing temperature for the +miscibility gap (henceforth referred to as a stabilization +temperature TS), we consider the enhanced W-Cr-X al- +loy. For the purpose of this paper we chose two refrac- +tory metals, i.e., hafnium (Hf) and tantalum (Ta). In +Section II we discuss the thermodynamics behind our +calculations (Sec. II A), then the problem of geometry +of the thermodynamical potential in composition dia- +gram (Sec. II A 1). +Next we briefly focus on the gen- +eration of the SQSs (Sec. II B) and other computational +details (Sec. II C). In Section III, we discuss the pres- +ence of the miscibility gap, it closure in high temperature +(Sec. III A), and the impact on the TS of a small (∼ 2 +atomic %) enhancement of an W70Cr30 alloy with Hf and +Ta(Ssec. III B). +II. +MODEL +We tackle the problem of thermodynamics of a solid +solution, by starting from the choice of the Canonic En- +semble, with the Gibbs Free energy as our potential. +A. +Gibbs energy +We assume that (i) we can model the electronic struc- +ture of an alloy in T ̸= 0 K by artificially expand the +volume; (ii) we obtain the temperature dependence of +volume by minimizing the Helmholtz free energy; (iii) +the entropy is a sum of configuration, electronic, and +arXiv:2301.13588v1 [cond-mat.mtrl-sci] 31 Jan 2023 + +2 +FIG. 1. A general scheme for the computational approach. +Note that the input is not required to be specific. +phononic entropies, i.e., +S +� +T +� += Sconfiguration + Selectronic(T) + Sphononic(T), +(1) +note that the volume is a function of temperature (not +an independent variable), as our thermodynamic func- +tion is the Gibbs energy G(T, p) (here in p = 0), not +the Helmholtz Free energy F(T, V ). Cf. the graphical +scheme in Fig. 1, where the steps required by configura- +tion entropy are in orange, electronic entropy in green, +and phononic entropy in blue. +An exemplary alloy Ax1Bx2Cx3 . . . (� +i xi = 1) has +the configuration entropy +Sconfiguration ≡ −kB +� +i +xi log(xi). +(2) +For more details on various entropy calculations see Ap- +pendix A. +1. +Gibbs energy of formation +To analyze the instability - the likelihood to diffuse into +two (or more) subsystems, we start from the Gibbs En- +ergy of Formation (for an exemplary compound AxByCz) +∆GAxByCz ≡ GAxByCz − xGA − yGB − zGC, +(3) +where x + y + z ≡ 1, +(4) +while GA ≡ GA(T, p), GB ≡ GB(T, p), and GC ≡ +GC(T, p) are the Gibbs free energies of pure A, B, and C +systems respectively. If we consider a vector space: the +composition (x, y, since z ≡ z(x, y))supplemented with +the energy of formation (∆G), the point in that space +represents a single real-space system. We identify a sta- +ble point when its ∆G is negative and lower than a that +of a linear combination of its neighbors energies of forma- +tions +� +∆G +� +N (such as the linear composition is equated +to the point composition). Thus, the problem of stabil- +ity of a set of compounds on the composition-energy of +formation diagram reduces to a problem of finding the +convex hull of the set of points representing the physical +systems. +B. +Special Quasirandom Structures +Proposed in the early ’90 [32], the special quasirandom +structures (SQS) are now the established technique to +model dynamics of an alloy. They stem from the princi- +ple that the best crystalic structure to represent an alloy +AxByCz. . . will have the probability of finding an atom A +on any coordination sphere to be exactly x (etc. for B - +y,. . . ). Normally, generation of an SQS starts from some +random selection and then via the Metropolis-Hastings +algorithm [35, 36] optimizes the error function. +This +approach, while time and memory conserving, produces +structures from 1.P1 symmetry group, since all the other +structures consists a null set in the space of all possible +configurations. +Our approach to determine a fast SQS is to (i) cre- +ate a starting set from the single-element lattice with +N nodes (body-centered cubic - bcc - in this case with +tungsten), and (ii) randomly change n ≪ N atoms on +the starting element lattice, (iii) analyze symmetry of +the resultant set to remove redundant structures, as well +as the ones with 1.P1 symmetry, and finally (iv) calculate +error function for each surviving structure. The remain- +ing set constitutes a population. Each random structure +is associated with a number (the error function). If the +concentration is the one we are looking for, we have a +ranked list of structures and we quit this algorithm. If +not, we employ (v) extinction, i.e., ith structure has a +probability to survive +Pi ≡ E0 +Ei +, +(5) +where E0 is the lowest value of error function (i.e., +min +�� +Ei +�� +). Next, we (vi) create a new set of M best +surviving structures and use it as an input in point (ii), +until expected concentration is achieved. +This approach provides comparable results to the es- +tablished tool, namely sqsgenerator [37], correlation- +wise (see Fig. 2). +E.g., +our predictions of high- +temperature magnitudes of elastic moduli are close to +the experimental results (for both see [38]). +C. +Computational details +Using the Vienna ab initio simulation package (VASP) +[39] we performed the electronic calculations to equilib- +riate volume in T = 0K. For that we used PAW [40] +pseudopotentials supplied with VASP and the Perdew- +Burke-Ernzerhof (PBE) generalized gradient approxima- +tion (GGA) exchange-correlation functional [41]. +To + +neighborhood +approach +Our +relaxation +V = 95-110% Vo +phonon +spectrum3 +FIG. 2. Correlation function for different Special Quasirandom W70Cr30 Structures. We used two sqsgenerator [37] generated +1.P1-symmetry cells with 109 and 102 iterations. We included our own cells from 5.C2 and 38.Amm2 symmetry groups, while +the result for purely random 1.P1 cell is given as a reference point. Inset contains calculated values of the error function E. +model the temperature dependence on the crystals rep- +resenting W-Cr alloy, we employ the quasiharmonic ap- +proximation [42], i.e., we compute the phononic spectra +for volumes 90-110% of the T = 0K volume V0. We use +PHONOPY [43] to find the minimal number of atomic +displacements and analyze the output files. +For VASP calculations we used pseudopotentials with +6 valence electrons for W, Mo, and Cr, 5 for Ta, and 4 for +Hf. We chose 3×3×3 BCC supercell (i.e., with 54 lattice +nodes). The k-point mesh was set to 4×4×4 Γ-centered, +where preliminary calculations showed energy saturation. +The kinetic energy cut-off for the electrons was set to +320eV . We carried out the frozen-ion calculations with a +0.1 Å displacement, and we used a 69×69×69 reciprocal +cell mesh for estimation of the thermal properties up to +3000 K with a 1 K step. +III. +RESULTS +As aforementioned, in our approach there is no ab- +initio adjustable parameters. Below, we discuss the re- +sults obtained using our systematical approach. +A. +Miscibility gap +To assess the prediction capabilities of our method, we +reproduce the high-temperature miscibility gap of W-Cr +binary alloys (vide Fig. 3) and a low-temperature mis- +cibility gap of W-Mo binary alloys (vide Fig. 4). +Our +results (blue crosses in Fig. 3) lay closer to the experi- +mental points when compared to the previous parameter- +adjustable results [15, 44] (note that the variance of the +FIG. 3. Tungsten - chromium miscibility, doping with Tanta- +lum (Green) and Hafnium (Lime) modifies stabilization tem- +perature. +experimental points is of the same magnitude as our re- +sults error). +To be sure that a large miscibility gap in tungsten- +chromium system is not a coincidence we also performed +the same calculations for the tungsten-molybdenum sys- +tem. The W-Mo system behave in the same qualitative +manner as W-Cr - there is a dome-shaped misciblity gap +with a maximum stabilization temperature (TS) for an +atomic concentration close to 50-50 (atomic %). There +is yet a quantitative alteration - the temperature below +which the solid solution becomes immiscible is expected +to be two orders of magnitude smaller - a result we un- +equivocally obtain (cf., Fig. 4). + +[0;1 +Error function: +sqsgen.109 it. 1.P1 +5.76 10-4 +0.65 +sqsgen.102 it.1.P1 +1.0310-3 +function, +thiswork5.C2 +2.9210-4 +this work 38.Amm2 +3.9210-3 +andom1.P1 +1.11 10-3 +0.60 +ideal correlation +C = 0.58252 +correlation +0.55 +0.50 +this work 5.C2 +random 1.P1 +sqsgen. 109 it. 1.P1 +sqsgen. 102 it. 1.P1 +this work 38.Amm2 +2 +4 +6 +8 +10 +12Miscibility of Tungsten-Chromium alloys +W38Cr15Tai +Greenaway, J: Inst. Met. 80, 589 (1951-1952) +McQuillan, J. Inst. Met.;80, 694 (1951-1952) +W38Cr15Hf +Den Broeder, Acta Metall. 20, 319 (1972) +Hawkins et ali, Phys. Rev. B 33, 4782 (1986) +2200 +-- +WxCr1-x (at. %) +Margaria et al., High Temp.-High Pressures 8, 451 (1976) + Naidu et al., Bull. Alloy Phase Diagr. 5, 289 (1984) +miscible +2000 +Ts +f stabilization, +1800 +of +1600 +Temperature +1400 +imrmiscible +1200 + +0 +10 +15 +20 +25 +30 +35 +50 +55 +65 +75 +80 +85 +40 +60 +70 +90 +95 +45 +100 +Tungsten concentration, x (at. %)4 +FIG. 4. Tungsten - Molybdenum miscibility. Solid line marks +the absolute zero (0 K ≡ −273.15 ◦C. +B. +Enhancing W-Cr alloy +As previously stated, we are interested in decreasing of +the stabilization temperature TS for a W70Cr30 (atomic +%) alloy, by the means of enriching the system with small +amount (∼2 atomic %) of other (refractory) metals. Here +we consider hafnium and tantalum as such candidates. +We use the same approach as described in Sec. II A to +reliably study a convexity of a point (corresponding to +the composition in question) in the composition – en- +ergy of formation diagram, we generate (as we did in +Sec. II B) a set of compositions neighboring our point - +(W70Cr30)98X2 - up to the third nearest neighbor. The +analysis of the geometry in three-dimensional composi- +tion – energy of formation space, shows decisively dif- +ferent behavior - increase of TS : 1550 ◦C → 1800 ◦C +for 2 % Hf and decrease of TS : 1550 ◦C → 1150 ◦C for +2 % Ta, a result consistent with previous experimental +measurements [45]. +IV. +CONCLUSION +In this paper we have presented our approach to de- +crease the computational load for complicated problem +like temperature dependence of alloy stability. +Since +we have incorporated our method of generating special +quasirandom structures with some residual symmetry +left1 we are able to calculate whole phase diagram, close +to experiment (vide Fig. 3) or even modify parts of this +diagram by enhancing alloy with Hf (increasing the misci- +bility gap) and Ta (decreasing the miscibility gap). These +1 A note is here to make. +No real-space supercell structure is +perfectly disordered - translational symmetry always prevails. +So from that point of view, considering systems with additional - +though still low - symmetry does not introduce new qualitative +error. +results are in agreement with previous experimental work +[45]. Our approach is parameterless and general and can +be applied to other problems as well. +V. +ACKNOWLEDGEMENTS +Financial support by the Czech Science Foundation +through grant No. 20-18392S is acknowledged as well as +the project e-INFRA CZ (ID:90140) by Czech MŠMT. + +Miscibility of Tungsten-Molybdenum alloys +-160.0 +WxMo1-x (at. %) +180.0 +miscible +-200.0 +220.0 +immiscible +240.0 +-260.0 +-273.1 +-280.0 +25 +30 +35 +40 +45 +50 +55 +60 +65 +70 +75 +80 +85 +90 +95 +100 +Tungsten concentration, x (at. %)5 +[1] H. Abu-Shawareb et al., Phys. Rev. Lett. 129, 075001 +(2022). +[2] A. B. Zylstra et al., Phys. Rev. E 106, 025202 (2022). +[3] A. B. 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Hard Mater. 73, 29 (2018). +[47] A. A. Markov, Proc. Phys. Mat. Soc. Uni. Kazan, 2nd +series 15, 135 (1906). + +6 +Appendix A: Gibbs energy +S = Sconfiguration + Selectronic + Sphononic. +(A1) +With β ≡ +1 +kBT , an exemplary alloy Ax1Bx2Cx3 . . . (� +i xi = 1) has the configuration entropy +Sconfiguration ≡ −kB +� +i +xi log(xi), +(A2) +while the electronic entropy +Selectronic ≡ −kB +� +σ +� +dϵDOSσ(ϵ) +� +fFD(ϵ) log fFD(ϵ) + [1 − fFD(ϵ)] log[1 − fFD(ϵ)] +� +, +(A3) +where σ corresponds to spin, and fFD(E) ≡ (exp[(E − EFermi)β] + 1)−1 is the Fermi-Dirac distribution. Next, the +phononic term +Sphononic ≡ − kB +� +q,ν +� +log(fBE(ℏωq,ν)) − ℏωq,νβfBE(ℏωq,ν) exp(−ℏωq,νβ) +� +(A4) +where ωq,ν is the angular frequency of the ν-phonon at q direction in the reciprocal space. fBE(E) ≡ (exp(Eβ) − 1)−1 +is the Bose-Einstein distribution. + diff --git a/ydFRT4oBgHgl3EQfizeW/content/tmp_files/load_file.txt b/ydFRT4oBgHgl3EQfizeW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5e07eb008c7972fbb70e435306badc400d8234a5 --- /dev/null +++ b/ydFRT4oBgHgl3EQfizeW/content/tmp_files/load_file.txt @@ -0,0 +1,590 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf,len=589 +page_content='On the miscibility gap in tungsten-based alloys Andrzej Piotr Kądzielawa1, 2, ∗ and Dominik Legut3, † 1IT4Innovations, VŠB - Technical University, Ostrava-Poruba, Czechia 2Institute of Theoretical Physics, Jagiellonian University, Kraków, Poland 3IT4Innovations $ Nanotechnology Centre, CEET, VŠB - Technical University, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='listopadu 2172/15, Ostrava, CZ 708 00 Czech Republic (Dated: February 1, 2023) In this work we establish an approach to model miscibility gaps of alloys using statistical physics, lattice dynamics from first-principles calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We carefully calculate the entropy to include all processes introducing disorder to the system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', combining the electronic, phononic, and con- figuration entropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Furthermore we present our algorithm for generating Special Quasirandom Structures (SQS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We model the miscibility gap in tungsten - chromium and tungsten - molybde- num systems, obtaining the agreement with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Furthermore, we propose an enhancement for the tungsten-chromium W70Cr30 alloy with tantalum and hafnium, leading to the modified stabilization temperatures TS, where the solid solution is miscible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' INTRODUCTION With the rapid search of carbon-neutral power pro- ducing facilities, the need for working fusion reactors is higher than ever before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Successful Ignition Events [1, 2], followed by net-positive cycle [3], increased the priority of research on the crucial elements of a fusion reactor, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', the so-called first walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' One family of com- pounds, namely tungsten-based alloys represent prospec- tive candidates to replace pure tungsten, as the alloy’s features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', selfpassivation) make them resistant to the Loss of Coolant Accident (LOCA) with concomitant air ingress into the reactor vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Since tungsten undergoes volatile oxidation when in a temperature above 500◦ C (the nuclear afterheat causes a surge of temperature up to 1200◦C), threatening structural damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' In case of selfpassivating tungsten alloys, the oxidation of the ad- ditional element forms a robust oxide scale [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The problem being that the alloy operates as purposed while in extreme conditions (high temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' On the other hand in normal conditions it is not thermodynamically stable, diffusing into W-rich and Cr-rich phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Finding this so-called miscibility gap, the research com- munity of 20th century [6–15], did not propose any large- scale applications for W-Cr binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' In contrast, now, the W-Cr alloy is a candidate for the first wall of a fu- sion reactor vessel [16] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', 620 m2 for the Interna- tional Thermonuclear Experimental Reactor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The ther- mal stability of W-Cr alloys was already studied exper- imentally [17–20], showing fast decomposition kinetics close to 1000◦ C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', near the LOCA conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' In such a case, changes in the phase content and degradation of oxidation performance are inevitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The density-functional-theory-based ab initio compu- tational modelling [21–27] is now the state-of-the-art ap- proach when high-throughput (hundreds of millions core- ∗ andrzej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='piotr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='kadzielawa@vsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='cz † dominik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='legut@vsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='cz hours) calculations are to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' DFT-based methods already have their input into designing novel materials [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Treatment of alloys in real-space requires averaging over different randomly-generated structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The so- called special quasi-random structures (SQS) [32–34] re- duce the computational-complexity by providing us with a set of most "alloy-like" supercells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' In effort to decrease the closing temperature for the miscibility gap (henceforth referred to as a stabilization temperature TS), we consider the enhanced W-Cr-X al- loy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' For the purpose of this paper we chose two refrac- tory metals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', hafnium (Hf) and tantalum (Ta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' In Section II we discuss the thermodynamics behind our calculations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' II A), then the problem of geometry of the thermodynamical potential in composition dia- gram (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' II A 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Next we briefly focus on the gen- eration of the SQSs (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' II B) and other computational details (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' II C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' In Section III, we discuss the pres- ence of the miscibility gap, it closure in high temperature (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' III A), and the impact on the TS of a small (∼ 2 atomic %) enhancement of an W70Cr30 alloy with Hf and Ta(Ssec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' III B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' MODEL We tackle the problem of thermodynamics of a solid solution, by starting from the choice of the Canonic En- semble, with the Gibbs Free energy as our potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Gibbs energy We assume that (i) we can model the electronic struc- ture of an alloy in T ̸= 0 K by artificially expand the volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' (ii) we obtain the temperature dependence of volume by minimizing the Helmholtz free energy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' (iii) the entropy is a sum of configuration, electronic, and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='13588v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='mtrl-sci] 31 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' A general scheme for the computational approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Note that the input is not required to be specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' phononic entropies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', S � T � = Sconfiguration + Selectronic(T) + Sphononic(T), (1) note that the volume is a function of temperature (not an independent variable), as our thermodynamic func- tion is the Gibbs energy G(T, p) (here in p = 0), not the Helmholtz Free energy F(T, V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' the graphical scheme in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 1, where the steps required by configura- tion entropy are in orange, electronic entropy in green, and phononic entropy in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' An exemplary alloy Ax1Bx2Cx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' (� i xi = 1) has the configuration entropy Sconfiguration ≡ −kB � i xi log(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' (2) For more details on various entropy calculations see Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Gibbs energy of formation To analyze the instability - the likelihood to diffuse into two (or more) subsystems, we start from the Gibbs En- ergy of Formation (for an exemplary compound AxByCz) ∆GAxByCz ≡ GAxByCz − xGA − yGB − zGC, (3) where x + y + z ≡ 1, (4) while GA ≡ GA(T, p), GB ≡ GB(T, p), and GC ≡ GC(T, p) are the Gibbs free energies of pure A, B, and C systems respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' If we consider a vector space: the composition (x, y, since z ≡ z(x, y))supplemented with the energy of formation (∆G), the point in that space represents a single real-space system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We identify a sta- ble point when its ∆G is negative and lower than a that of a linear combination of its neighbors energies of forma- tions � ∆G � N (such as the linear composition is equated to the point composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Thus, the problem of stabil- ity of a set of compounds on the composition-energy of formation diagram reduces to a problem of finding the convex hull of the set of points representing the physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Special Quasirandom Structures Proposed in the early ’90 [32], the special quasirandom structures (SQS) are now the established technique to model dynamics of an alloy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' They stem from the princi- ple that the best crystalic structure to represent an alloy AxByCz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' will have the probability of finding an atom A on any coordination sphere to be exactly x (etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' for B - y,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Normally, generation of an SQS starts from some random selection and then via the Metropolis-Hastings algorithm [35, 36] optimizes the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' This approach, while time and memory conserving, produces structures from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 symmetry group, since all the other structures consists a null set in the space of all possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Our approach to determine a fast SQS is to (i) cre- ate a starting set from the single-element lattice with N nodes (body-centered cubic - bcc - in this case with tungsten), and (ii) randomly change n ≪ N atoms on the starting element lattice, (iii) analyze symmetry of the resultant set to remove redundant structures, as well as the ones with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 symmetry, and finally (iv) calculate error function for each surviving structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The remain- ing set constitutes a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Each random structure is associated with a number (the error function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' If the concentration is the one we are looking for, we have a ranked list of structures and we quit this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' If not, we employ (v) extinction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', ith structure has a probability to survive Pi ≡ E0 Ei , (5) where E0 is the lowest value of error function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', min �� Ei �� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Next, we (vi) create a new set of M best surviving structures and use it as an input in point (ii), until expected concentration is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' This approach provides comparable results to the es- tablished tool, namely sqsgenerator [37], correlation- wise (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', our predictions of high- temperature magnitudes of elastic moduli are close to the experimental results (for both see [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Computational details Using the Vienna ab initio simulation package (VASP) [39] we performed the electronic calculations to equilib- riate volume in T = 0K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' For that we used PAW [40] pseudopotentials supplied with VASP and the Perdew- Burke-Ernzerhof (PBE) generalized gradient approxima- tion (GGA) exchange-correlation functional [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' To neighborhood approach Our relaxation V = 95-110% Vo phonon spectrum3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Correlation function for different Special Quasirandom W70Cr30 Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We used two sqsgenerator [37] generated 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1-symmetry cells with 109 and 102 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We included our own cells from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='C2 and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='Amm2 symmetry groups, while the result for purely random 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 cell is given as a reference point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Inset contains calculated values of the error function E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' model the temperature dependence on the crystals rep- resenting W-Cr alloy, we employ the quasiharmonic ap- proximation [42], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', we compute the phononic spectra for volumes 90-110% of the T = 0K volume V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We use PHONOPY [43] to find the minimal number of atomic displacements and analyze the output files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' For VASP calculations we used pseudopotentials with 6 valence electrons for W, Mo, and Cr, 5 for Ta, and 4 for Hf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We chose 3×3×3 BCC supercell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', with 54 lattice nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The k-point mesh was set to 4×4×4 Γ-centered, where preliminary calculations showed energy saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The kinetic energy cut-off for the electrons was set to 320eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We carried out the frozen-ion calculations with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='1 Å displacement, and we used a 69×69×69 reciprocal cell mesh for estimation of the thermal properties up to 3000 K with a 1 K step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' RESULTS As aforementioned, in our approach there is no ab- initio adjustable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Below, we discuss the re- sults obtained using our systematical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Miscibility gap To assess the prediction capabilities of our method, we reproduce the high-temperature miscibility gap of W-Cr binary alloys (vide Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 3) and a low-temperature mis- cibility gap of W-Mo binary alloys (vide Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Our results (blue crosses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 3) lay closer to the experi- mental points when compared to the previous parameter- adjustable results [15, 44] (note that the variance of the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Tungsten - chromium miscibility, doping with Tanta- lum (Green) and Hafnium (Lime) modifies stabilization tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' experimental points is of the same magnitude as our re- sults error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' To be sure that a large miscibility gap in tungsten- chromium system is not a coincidence we also performed the same calculations for the tungsten-molybdenum sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The W-Mo system behave in the same qualitative manner as W-Cr - there is a dome-shaped misciblity gap with a maximum stabilization temperature (TS) for an atomic concentration close to 50-50 (atomic %).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' There is yet a quantitative alteration - the temperature below which the solid solution becomes immiscible is expected to be two orders of magnitude smaller - a result we un- equivocally obtain (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='1 Error function: sqsgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='109 it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='76 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='65 sqsgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='102 it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='0310-3 function, thiswork5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='C2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='9210-4 this work 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='Amm2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='9210-3 andom1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='11 10-3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='60 ideal correlation C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='58252 correlation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='50 this work 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='C2 random 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 sqsgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 109 it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 sqsgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 102 it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='P1 this work 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='Amm2 2 4 6 8 10 12Miscibility of Tungsten-Chromium alloys W38Cr15Tai Greenaway, J: Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 80, 589 (1951-1952) McQuillan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='80, 694 (1951-1952) W38Cr15Hf Den Broeder, Acta Metall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 20, 319 (1972) Hawkins et ali, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' B 33, 4782 (1986) 2200 +-- WxCr1-x (at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' %) Margaria et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', High Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='-High Pressures 8, 451 (1976) Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Alloy Phase Diagr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 5, 289 (1984) miscible 2000 Ts f stabilization, 1800 of 1600 Temperature 1400 imrmiscible 1200 + 0 10 15 20 25 30 35 50 55 65 75 80 85 40 60 70 90 95 45 100 Tungsten concentration, x (at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' %)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Tungsten - Molybdenum miscibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Solid line marks the absolute zero (0 K ≡ −273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='15 ◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Enhancing W-Cr alloy As previously stated, we are interested in decreasing of the stabilization temperature TS for a W70Cr30 (atomic %) alloy, by the means of enriching the system with small amount (∼2 atomic %) of other (refractory) metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Here we consider hafnium and tantalum as such candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' We use the same approach as described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' II A to reliably study a convexity of a point (corresponding to the composition in question) in the composition – en- ergy of formation diagram, we generate (as we did in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' II B) a set of compositions neighboring our point - (W70Cr30)98X2 - up to the third nearest neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' The analysis of the geometry in three-dimensional composi- tion – energy of formation space, shows decisively dif- ferent behavior - increase of TS : 1550 ◦C → 1800 ◦C for 2 % Hf and decrease of TS : 1550 ◦C → 1150 ◦C for 2 % Ta, a result consistent with previous experimental measurements [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' CONCLUSION In this paper we have presented our approach to de- crease the computational load for complicated problem like temperature dependence of alloy stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Since we have incorporated our method of generating special quasirandom structures with some residual symmetry left1 we are able to calculate whole phase diagram, close to experiment (vide Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 3) or even modify parts of this diagram by enhancing alloy with Hf (increasing the misci- bility gap) and Ta (decreasing the miscibility gap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' These 1 A note is here to make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' No real-space supercell structure is perfectly disordered - translational symmetry always prevails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' So from that point of view, considering systems with additional - though still low - symmetry does not introduce new qualitative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' results are in agreement with previous experimental work [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Our approach is parameterless and general and can be applied to other problems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' ACKNOWLEDGEMENTS Financial support by the Czech Science Foundation through grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 20-18392S is acknowledged as well as the project e-INFRA CZ (ID:90140) by Czech MŠMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Miscibility of Tungsten-Molybdenum alloys 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='0 WxMo1-x (at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' %) 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='0 miscible 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='0 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='0 immiscible 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='0 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='0 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='1 280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='0 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Tungsten concentration, x (at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' %)5 [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Abu-Shawareb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 129, 075001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' B.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Linsmeier, Rasinski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Kreter, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Unterberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Coenen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Du, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Mayer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' García-Rosales, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Calvo, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Ordas, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Fusion 57, 066020 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' García-Rosales, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' López-Ruiz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Alvarez-Martín, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Calvo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Ordás, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Koch, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Brinkmann, Fusion Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 89, 1611 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Den Broeder and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Burgers, Acta Metall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 16, 265 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Den Broeder, Acta Metall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 20, 319 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Porter, Acta Metall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 15, 721 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Rietveld, Acta Crystallogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 22, 151 (1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Margaria, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Allibert, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Ansara, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Driole, High Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' High Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 8, 451 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [11] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Greenaway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 80, 589 (1952).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Greenaway, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Johnstone, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' McQuillan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 80, 109 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' McQuillan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 79, 379 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Venkatraman and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Neumann, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Alloy Phase Diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 8, 216 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Naidu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Sriramamurthy, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Rao, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Alloy Phase Diag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 5, 289 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Tanure, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content='Bakaeva, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Lapeire, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Terentyev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Vilé- mová, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Matějíček, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Verbeken, Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Coat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Tech- nol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 355, 252 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Vilémová, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Lukáč, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Veverka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Illková, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Matějíček, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Refract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Hard Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 79, 217 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Vilémová, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Illková, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Lukáč, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Matějíček, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Klečka, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=', Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Refract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Hard Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 73, 29 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [20] E.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Des.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 146, 1596 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Hohenberg and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Kohn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Tejado, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Pintsuk, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Ordás, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Iturriza, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Neu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Pastor, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' García-Rosales, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Refract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Hard Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 73, 29 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Markov, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Kazan, 2nd series 15, 135 (1906).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' 6 Appendix A: Gibbs energy S = Sconfiguration + Selectronic + Sphononic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' (A1) With β ≡ 1 kBT , an exemplary alloy Ax1Bx2Cx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' (� i xi = 1) has the configuration entropy Sconfiguration ≡ −kB � i xi log(xi), (A2) while the electronic entropy Selectronic ≡ −kB � σ � dϵDOSσ(ϵ) � fFD(ϵ) log fFD(ϵ) + [1 − fFD(ϵ)] log[1 − fFD(ϵ)] � , (A3) where σ corresponds to spin, and fFD(E) ≡ (exp[(E − EFermi)β] + 1)−1 is the Fermi-Dirac distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' Next, the phononic term Sphononic ≡ − kB � q,ν � log(fBE(ℏωq,ν)) − ℏωq,νβfBE(ℏωq,ν) exp(−ℏωq,νβ) � (A4) where ωq,ν is the angular frequency of the ν-phonon at q direction in the reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} +page_content=' fBE(E) ≡ (exp(Eβ) − 1)−1 is the Bose-Einstein distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFRT4oBgHgl3EQfizeW/content/2301.13588v1.pdf'} diff --git a/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf b/zdFQT4oBgHgl3EQfzDZw/content/2301.13411v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1cc89ff73b788e209159e8d0e1f51bf5ea1b2548 --- /dev/null +++ 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