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TEOs_in_massive_EEVs +©ESO 2023 +January 3, 2023 +Theoretical investigation of the occurrence of tidally excited +oscillations in massive eccentric binary systems +P. A. Kołaczek-Szyma´nski and T. Ró˙za´nski +Astronomical Institute, University of Wrocław, Kopernika 11, 51-622 Wrocław, Poland +e-mail: kolaczek@astro.uni.wroc.pl +Received 16.10.2022; Revised 01.12.2022; Re-revised 28.12.2022; Accepted 02.01.2023 +ABSTRACT +Context. Massive and intermediate-mass stars reside in binary systems much more frequently than low-mass stars. At the same +time, binaries containing massive main-sequence (MS) component(s) are often characterised by eccentric orbits, and can therefore be +observed as eccentric ellipsoidal variables (EEVs). The orbital phase-dependent tidal potential acting on the components of EEV can +induce tidally excited oscillations (TEOs), which can affect the evolution of the binary system. +Aims. We investigate how the history of resonances between the eigenmode spectra of the EEV components and the tidal forcing +frequencies depends on the initial parameters of the system, limiting our study to MS. Each resonance is a potential source of TEO. +We are particularly interested in the total number of resonances, their average rate of occurrence and their distribution in time. +Methods. We synthesised 20,000 evolutionary models of the EEVs across the MS using Modules for Experiments in Stellar Astro- +physics (MESA) software for stellar structure and evolution. We considered a range of masses of the primary component from 5 to +30 M⊙. Later, using the GYRE stellar non-adiabatic oscillations code, we calculated the eigenfrequencies for each model recorded by +MESA. We focused only on the l = 2, m = 0, +2 modes, which are suspected of being dominant TEOs. Knowing the temporal changes +in the orbital parameters of simulated EEVs and the changes of the eigenfrequency spectra for both components, we were able to +determine so-called ‘resonance curves’, which describe the overall chance of a resonance occurring and therefore of a TEO occurring. +We analysed the resonance curves by constructing basic statistics for them and analysing their morphology using machine learning +methods, including the Uniform Manifold Approximation and Projection (UMAP) tool. +Results. The EEV resonance curves from our sample are characterised by striking diversity, including the occurrence of exceptionally +long resonances or the absence of resonances for long evolutionary times. We found that the total number of resonances encountered +by components in the MS phase ranges from ∼102 to ∼103, mostly depending on the initial eccentricity. We also noticed that the +average rate of resonances is about an order of magnitude higher (∼102 Myr−1) for the most massive components in the assumed +range than for EEVs with intermediate-mass stars (∼101 Myr−1). The distribution of resonances over time is strongly inhomogeneous +and its shape depends mainly on whether the system is able to circularise its orbit before the primary component reaches the terminal- +age MS (TAMS). Both components may be subject to increased resonance rates as they approach the TAMS. Thanks to the low- +dimensional UMAP embeddings performed for the resonance curves, we argue that their morphology changes smoothly across the +resulting manifold for different initial EEV conditions. The structure of the embeddings allowed us to explore the whole space of +resonance curves in terms of their morphology and to isolate some extreme cases. +Conclusions. Resonances between tidal forcing frequencies and stellar eigenfrequencies cannot be considered rare events for EEVs +with massive and intermediate-mass MS stars. On average, we should observe TEOs more frequently in EEVs containing massive +components than intermediate-mass ones. TEOs will be particularly well-pronounced for EEVs with the component(s) close to the +TAMS, which begs for observational verification. Given the total number of resonances and their rates, TEOs may play an important +role in the transport of angular momentum within massive and intermediate-mass stars (mainly near TAMS). +Key words. binaries: close – stars: early-type – stars: massive – stars: oscillations – stars: evolution – methods: numerical +1. Introduction +For many reasons, massive stars (≳ 8 M⊙) are of particular in- +terest to modern astrophysics. Primarily, they are progenitors of +core-collapse supernovae (e.g., Janka et al. 2007; Smartt 2009) +and long γ-ray bursts (e.g., Fruchter et al. 2006; Yoon et al. +2006). For billions of years, they contributed to the chemical +evolution of the entire Universe and interacted mechanically +with the surrounding interstellar medium (e.g., Ouellette et al. +2007; Svirski et al. 2012), also through their intense line-driven +stellar winds (Vink 2021). Furthermore, most of their remnants +are compact objects, such as neutron stars (NSs, and among them +magnetars and pulsars) and black holes (BHs), which allow the +empirical study of effects of general relativity. Finally, massive +stars can be observed at cosmological distances due to their enor- +mous luminosities, hence they dominate in the spectra of distant +starburst galaxies (see Eldridge & Stanway 2022, for a recent re- +view). These features of massive stars (and many others) demon- +strate that understanding the structure and evolution of massive +stars is one of the key tasks of astronomy. +As is well known, massive and intermediate-mass (≳ 2 M⊙ +and ≲ 8 M⊙) stars reside in binary systems much more fre- +quently than their lower-mass counterparts (Duchêne & Kraus +2013; Sana et al. 2012). Moreover, as shown, for example, by +Sana et al. (2014) and Moe & Di Stefano (2017), O-type dwarfs +in particular are often found in multiple systems. This shows that +binarity is inherent in the evolution of massive stars and cannot +be ignored when studying these objects as well as their final out- +comes. Many interesting phenomena in the Universe are the re- +Article number, page 1 of 24 +arXiv:2301.00733v1 [astro-ph.SR] 2 Jan 2023 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +sult of binarity among massive and/or intermediate-mass stars. +These include Be stars (Kriz & Harmanec 1975; Bodensteiner +et al. 2020), so-called ‘stripped stars’ (Götberg et al. 2020; +Shenar et al. 2020; El-Badry & Burdge 2022), BH-BH/BH- +NS/NS-NS mergers (progenitors of gravitational-wave events, +Abbott et al. 2016, 2019), ‘early’ stellar mergers (Tokovinin & +Moe 2020; Sen et al. 2022; Li et al. 2022), Ib/c supernova pro- +genitors (Langer 2012; Yoon et al. 2012), ‘massive Algols’ (de +Mink et al. 2007; Skowron et al. 2017; Sen et al. 2022), and even +Wolf-Rayet stars (Shenar et al. 2019; Pauli et al. 2022). +At the same time, binary systems that contain massive main- +sequence (MS) component(s) are often characterised by eccen- +tric orbits due to their relatively young age and the presence of +radiative outer layers, which are less vulnerable to tidal dissi- +pation compared to convective envelopes (e.g., Van Eylen et al. +2016). Both observational studies of large samples of massive bi- +naries (e.g., Moe & Di Stefano 2017) and hydrodynamical sim- +ulations of their formation (see Oliva & Kuiper 2020, and ref- +erences therein) suggest significantly non-zero eccentricities at +their birth. +Assuming that the periastron distance between the compo- +nents is sufficiently small1, the combined proximity effects, such +as ellipsoidal distortion, irradiation/reflection effect and Doppler +beaming/boosting, make such a system an eccentric ellipsoidal +variable (hereafter EEV, e.g., Nicholls & Wood 2012). Due to +the characteristic shape of the light curve of EEV during the +periastron passage (which can resemble an electrocardiogram), +EEVs are sometimes referred to as ‘heartbeat stars’ (Welsh et al. +2011; Thompson et al. 2012; Beck et al. 2014; Kirk et al. 2016; +Kołaczek-Szyma´nski et al. 2021; Wrona et al. 2022b). +The orbital phase-dependent tidal potential acting on the +components of EEV can induce tidally excited oscillations +(TEOs) in their interiors (Zahn 1975; Kumar et al. 1995; Fuller +2017; Guo 2021), which in turn can affect the evolution of the +binary system. However, many details of TEOs in massive and +intermediate-mass stars are still poorly understood including the +total number of TEOs and their frequency of occurrence. In our +study, we aim to shed light on this issue based on theoretical +modelling combined with machine learning (ML) techniques. +The paper is organised as follows. Section 2 provides a con- +cise characterisation of TEOs and specifies the purpose of our +paper. In Sect. 3 we present a detailed description of the adopted +methodology, including the assumptions made and the software +used to generate the theoretical models. We then analyse the +obtained models and present our findings in Sect. 4. Finally, +we summarise the entire work and draw several conclusions in +Sect. 5. +2. Properties of TEOs and the purpose of the paper +TEOs are tidally forced eigenmodes of a star with frequencies, +σnlm (in the co-rotating frame of the star), coinciding with inte- +ger multiples, N, of the orbital frequency, forb2. The resonance +condition can be written as follows: +fNm ≡ N forb − m fs ≈ σnlm, +(1) +where fNm corresponds to the frequency of the tidal forcing in +the rotating frame, fs stands for the rotational frequency of the +star, while the subscripts n, l and m denote the radial order, de- +gree, and azimuthal order of the specific eigenmode, respec- +tively. This property of TEOs makes them relatively easy to dis- +1 That is, of the order of a few radii of the larger component. +2 We denote the corresponding orbital period as Porb = 1/forb. +tinguish from other types of pulsations (e.g., self-excited oscilla- +tions) in frequency spectra, provided the orbital period is known. +There are numerous examples of photometrically or spectro- +scopically detected TEOs (e.g., Handler et al. 2002; Welsh et al. +2011; Hambleton et al. 2013; Fuller et al. 2017; Guo et al. 2019, +2020; Wrona et al. 2022a), also in massive binary systems (e.g., +Willems & Aerts 2002; Pablo et al. 2017; Kołaczek-Szyma´nski +et al. 2021, 2022). Most TEOs are damped normal modes, mean- +ing that without constant tidal forcing they would not be ob- +served in the star. More importantly, because of their damped na- +ture, TEOs dissipate the total orbital energy making the system +tighter, more circular, and synchronised with time. On the other +hand, if the TEO is naturally an overstable mode it can transfer +thermal energy from the star to the orbit via so-called ‘inverse +tides’ (Fuller 2021). Regardless of the type of TEOs, they unde- +niably contribute to the evolution of the (massive) binary system, +and can therefore influence the characteristics of the phenomena +and objects mentioned above. The efficiency of energy transfer +between the stellar interior and the orbit due to TEOs strongly +depends on the temporal behaviour of the resonance condition +given by Eq. (1). It is to be expected that most TEOs are ‘chance +resonances’, i.e. resonances in which the aforementioned condi- +tion is satisfied for a relatively short time. Under such circum- +stances, TEOs do not have enough time to reach high ampli- +tudes, hence their ability to dissipate orbital energy is somewhat +limited. However, if, after reaching resonance, both frequencies +on the left and right sides of Eq. (1) evolve at the same rate and +direction, TEOs can ‘tidally lock’ for a longer time compared to +the chance resonance scenario (Fuller 2017). This unique vari- +ety of TEOs is suspected to be responsible for occasional peri- +ods of rapid evolution of the orbital parameters in binary systems +(Fuller et al. 2017). +TEOs are are not only restricted to MS stars, they can also +occur in binaries with pre-MS stars (Zanazzi & Wu 2021), some +compact objects (white dwarfs, Yu et al. 2021), planetary sys- +tems (Ma & Fuller 2021) and even planet-moon systems (e.g., in +the Saturn-Titan system, Lainey et al. 2020). +Although the literature on theoretical studies of TEOs is in- +deed extensive (see e.g., Fuller 2017; Guo 2021, for recent re- +views), the question of their rate of occurrence and the role they +play in the evolution of massive stars is still a matter of debate. +Unfortunately, only a small number of papers refer exclusively to +massive EEVs. Witte & Savonije (1999a,b) studied gravity- (g) +and Rossby-mode TEOs in an uniformly rotating 10 M⊙ MS star +using their own two-dimensional (2D) code for different stellar +rotation rates and several orbital configurations. They found that +dynamical tides can effectively circularise and tighten the orbits +of EEVs in just a few Myrs if resonance locking occurs. How- +ever, these and many other previous works on TEOs were done +under the assumption of a compact (point-like) secondary com- +panion that is not subject to tidal perturbations during each peri- +astron passage. This is obviously not the case in real binary sys- +tems, where both components are responsible for the tidal evo- +lution of the orbit. As theoretically shown by Witte & Savonije +(2001), for an eccentric binary system consisting of two 10 M⊙ +stars, tidal dissipation can be further enhanced due to the simul- +taneous excitation of tidally-locked TEOs in both components. +In spite of the advanced mathematical formalism, the aforemen- +tioned papers only dealt with a few assumed component masses +and sets of orbital parameters. Only Willems (2003) attempted +a qualitative analysis of the hyperspace of the orbital parame- +ters favouring excitation of TEOs in massive EEVs on MS. He +found that for a mass range of 2 – 20 M⊙, the favourable orbital +period interval lies between ∼5 and ∼12 d when both compo- +Article number, page 2 of 24 + +Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +nents are zero-age MS (ZAMS) stars. This interval shifts towards +longer orbital periods (up to ∼70 d) for components approaching +terminal-age MS (TAMS). +Although, as argued above, the role of TEOs in the life of +massive binary systems is still not well understood, we are not +aware of any published work that develops the qualitative analy- +sis carried out by Willems (2003) based on state-of-the-art stel- +lar evolution and oscillations codes. We would like to fill this +gap by combining the Modules for Experiments in Stellar As- +trophysics3 (MESA, Paxton et al. 2011, 2013, 2015, 2018, 2019) +stellar structure and evolution code with the GYRE4 (Townsend & +Teitler 2013; Townsend et al. 2018) non-adiabatic stellar oscil- +lation code. Our study aims to answer the following three ques- +tions: +1. How many resonances (given by Eq. (1)) can EEVs experi- +ence during their lifetime between ZAMS and TAMS? How +does this picture change with different initial parameters of +the binary system? +2. Can we distinguish several distinct types of EEV resonance +histories that are statistically related to the initial physical +and orbital parameters of binary systems? +3. Does the resonance history correlate in any way with the +properties of EEV near TAMS? For instance, are systems +that undergo mass transfer before reaching TAMS also sys- +tems with a higher total number of resonances encountered? +In order to answer the last two questions, we use of ML tech- +niques by performing a Uniform Manifold Approximation and +Projection5 (UMAP, McInnes et al. 2018) dimension reduction +analysis of the resonance histories obtained for simulated binary +systems. +3. Methods +Assuming that the dynamical tide excited in the component is +dominated by a single TEO close to resonance with the orbital +harmonic N, one can express its amplitude of luminosity change +as proportional to (after Fuller 2017, his eq. 2) +AN ∝ q +�R +a +�l+1 +|Qnlm|FNmLN, +(2) +where q is the ratio of the masses of the two components, R +stands for the radius of the component in which the TEO is +excited, and a is the semi-major axis of the relative orbit. The +quantity denoted Qnlm is known as the so-called overlap integral, +which describes the spatial coupling between a given oscillation +mode and the actual geometry of the tidal potential (Fuller 2017, +his eq. 4). In general, the larger the value of |n|6, the smaller the +Qnlm, hence eigenmodes with a large number of nodes in the ra- +dial direction have a much lower probability of tidal excitation7. +In addition to Qnlm, the Hansen coefficient FNm (Fuller 2017, +his eq. 5) is responsible for the temporal coupling of the forced +normal mode and the Nth component in the Fourier expansion +3 https://docs.mesastar.org/en/latest/ +4 https://gyre.readthedocs.io/en/stable/ +5 https://umap-learn.readthedocs.io/en/latest/ +6 We use |n| instead of n because GYRE assigns negative values of n to +g modes and positive ones to p modes. +7 More precisely, Qnlm does not vary strictly monotonic with n and +can change significantly between consecutive modes for given l and m. +However, the overall trend of Qnlm peaks for low values of |n| and falls +sharply for |n| ≫ 0. For a more detailed discussion on the behaviour of +Qnlm see, e.g., Burkart et al. (2012). +of the orbital motion. Quantitatively, it expresses the intuitive +principle that for more eccentric orbits, TEOs with larger orbital +harmonic numbers will be excited. This is because, as the ec- +centricity increases, the periastron passage takes less time for a +given orbital period, so eigenmodes with higher frequencies bet- +ter ‘match’ rapidly changing gravitational field, in terms of time +scale. Nevertheless, for very high N, the FNm drops rapidly (al- +most exponentially). This particular property of FNm is respon- +sible for the lack of excitation of the TEOs with extremely high +N. It is clear here that the frequency range of TEOs in massive +and intermediate-mass MS stars is limited on two sides inde- +pendently by Qnlm and FNm. On the low-frequency side, Qnlm +prevents the excitation of g modes with very high |n|, while on +the high-frequency side FNm decreases sharply, strongly limit- +ing the possible excitation of pressure (p) modes characterised +by high radial orders. The last term in Eq. (2), i.e. LN, denotes +the detuning factor given by the following formula, +LN = +fNm +� +(σnlm − fNm)2 + γ2 +nlm +, +(3) +where γnlm stands for damping/growth rate of the normal mode. +This Lorentzian-like factor reflects the mismatch between fNm +and σnlm. Given the typical values of |γnlm| for g and p modes +in massive and intermediate-mass MS stars (of the order of +∼ 10−7 – 10−3 d−1), LN is extremely sensitive to the difference +(σnlm− fNm). Hence, among many other factors, the LN undoubt- +edly plays a key role in the excitation of TEOs. +While the precise prediction of TEO amplitude is a difficult +task8, we are interested in analysing the changes in resonance +conditions dictated by the sum of all the contributing detuning +factors with passing time, t. Let us define the following dimen- +sionless quantity, +L(t) ≡ +� +nlm +Nmax +� +N=1 +LN(t). +(4) +In contrast to LN, associated with a single orbital harmonic, L +reflects the overall chance of TEOs being excited in the EEV +component. However, we must stress at this point that it does +not carry direct information on the amplitude of potential TEOs. +The first summation in Eq. (4) applies to all the normal modes +we consider in the modelling (Sect. 3.3). Obviously, the second +summation in Eq. (4) should run from N = 1 to +∞, but due +to time and physical constraints one has to truncate the series at +some reasonably chosen Nmax. For a detailed description of the +selection of Nmax values see Sect. 3.3. +In order to try to answer the questions raised in Sect. 2, we +have synthesised 20,000 binary evolution models and calculated +L(t) for both components in each of them. The whole procedure +is described extensively in the next four subsections. +3.1. General assumptions +From a practical point of view, a fully consistent calculation +of the evolution of binary systems taking TEOs into account is +8 In order to reliably predict the photometric amplitude of a TEO, one +needs to: (1) determine the exact value of LN, which is almost impossi- +ble given the uncertainties in both observations and stellar models, (2) +calculate the corresponding Qnlm and (3) know the eigenfunction of lu- +minosity fluctuations at the photospheric level, which is a challenge for +radiation pressure-dominated atmospheres of early-type stars (with in- +tense stellar winds). In addition, the equilibrium amplitude of a linearly +driven TEO is determined by various non-linear effects, for instance by +multi-mode coupling (Guo 2020; Guo et al. 2022). +Article number, page 3 of 24 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +very time-consuming, as it requires time steps shorter than the +times at which the resonances occur (several orders of magni- +tude shorter than the nuclear time scale, cf. Fig. 3). It would take +an enormous amount of time to perform such consistent calcula- +tions for 20,000 binaries with hundreds of resonances occurring +in each of them. Therefore, to make our project both feasible +and still scientifically useful, the models were synthesised un- +der the general assumption that each resonance encountered by +the EEV components is a chance resonance. By sacrificing the +ability to track resonantly-locked TEOs, we are able to decou- +ple evolutionary and seismic calculations and run them indepen- +dently, greatly simplifying the whole problem. We believe that +we can to do this for three reasons: (1) we are only interested +in obtaining some general statistical information about the reso- +nance conditions in a large number of simulated binary systems, +(2) the phenomenon of resonance locking is rare compared to the +rate of chance-resonance events, and (3) the impact of chance- +resonance TEOs on the orbit is limited due to their relatively +short time of existence (e.g., Witte & Savonije 1999b). In con- +clusion, we focus on finding candidate binaries that may or may +not experience numerous TEOs, rather than precisely predicting +their actual evolutionary histories, which is beyond the scope of +this paper. We believe that our results will serve as a starting +point for more detailed calculations performed for the most in- +teresting cases of massive EEVs. +3.2. Synthesis of binary evolution models +Since we assumed that we could separate stellar and orbital evo- +lution from seismic calculations, we first generated a set of bi- +nary evolutionary tracks and only then performed seismic anal- +ysis on them to find L(t). +3.2.1. Initialisation of models +We used the latest open-source 1D stellar evolution code MESA +(release 15140) compiled with the MESA Software Development +Kit (version 21.4.1, Townsend 2021) to compute a set of 20,000 +binary evolution models. The MESAbinary module (Paxton et al. +2015) allows the simultaneous evolution of binary system com- +ponents and their orbits. Throughout this paper, we use the sub- +scripts ‘1’ and ‘2’ to denote the primary (initially more massive) +and secondary components, respectively. +We assumed that both components have the same chemical +composition with metallicity Z = 0.02 and a solar-scaled mix- +ture of elements taken from Grevesse & Sauval (1998). Since +we were only interested in massive and intermediate-mass MS +EEVs that can exhibit TEOs during their lifetime, the initial sys- +tems consisted of two stars lying on the ZAMS and were charac- +terised by parameters randomly drawn from the following uni- +form distributions, U[α,β], on specific intervals [α, β]. +– Mass of primary component, log(M1/M⊙) ∼ U[log 5,log 30]. A +uniform distribution on a logarithmic scale was used instead +of a linear scale to cover the Hertzsprung-Russell diagram +(HRD) with more evenly distributed evolutionary tracks. +– The mass ratio, q ≡ M2/M1 ∼ U[0.2,0.95], where M2 corre- +sponds to the mass of the secondary component. The lower +limit for q was introduced due to the fact that the efficiency +in driving TEOs scales with q (cf. Eq. (2)), so it is less likely +to observe TEOs in a binary system at a small value of the +mass ratio. Moreover, if the generated q corresponded to +M2 < 2M⊙, a redraw was performed. +– Eccentricity, e ∼ U[0.2,0.8]. Range typical of the observed +EEVs. +– Periastron distance, rperi, normalised to the sum of compo- +nents’ radii, �rperi ≡ rperi/(R1 + R2) ∼ U[1,5.5]. However, +if the generated system was initially Roche-lobe overflow- +ing (RLOF) at the periastron, a redraw was performed. We +also assumed an upper value of �rperi because the overall +strength of tidal forces decays as r−3 +peri and simulating widely- +separated systems would contradict the aim of this paper. +– Tidally-enhanced wind factor, Bwind ∼ U[32,896]. Introduced +by Tout & Eggleton (1988) for red giants residing in binary +systems, it accounts for the tidal enhancement of the stellar +wind mass-loss rate due to the presence of a nearby com- +panion. The ad hoc chosen range of Bwind corresponds to a +maximum amplification of the ‘nominal’ wind mass-loss rate +by a factor of 1.5 – 10 (cf. Tout & Eggleton 1988, their eq. 2). +– The angular rotational velocity divided by its critical value9, +Ω/Ωcrit ∼ U[0.1,0.5]. The assumed range of initial Ω/Ωcrit +translates into the linear equatorial velocities between +∼50 km/s and ∼320 km/s in our simulations and reflects the +significant non-zero rotation velocities observed in massive +young MS stars (e.g., Dufton et al. 2006; Hunter et al. 2008). +– The overshoot mixing parameter, fov ∼ U[0.015,0.025]. In our +calculations, the overshooting of the material above the con- +vective, hydrogen-burning core was treated in the exponen- +tial diffusion formalism developed by Herwig (2000). An ad- +justable parameter, fov, describes the spatial extent of the +overshoot layer in terms of the local pressure-scale height, +but its value for massive stars is still under debate. We +adopted the range of fov after Ostrowski et al. (2017). +The parameters presented above were generated independently +for each EEV system. Moreover, the last two parameters, Ω/Ωcrit +and fov, were drawn independently for each component, so the +final hyperspace of parameters explored in our simulations in- +cluded {M1, q, e,�rperi, Ω/Ωcrit,1, Ω/Ωcrit,2, fov,1, fov,2, Bwind}. Fig- +ure 1 shows our initial sample of generated EEVs in the orbital +period versus eccentricity diagram. As expected, they occupy +the upper envelope of the aforementioned plane with the upper +boundary dictated by the onset of periastron RLOF on ZAMS. +The rest of the necessary parameters and ‘physics switches’ were +identical for each simulated binary. We will now briefly describe +them below. +3.2.2. Integration of the evolution +Nuclear reaction rates were calculated using ‘basic.net’ op- +tion in MESA. We used a convective premixing scheme (Paxton +et al. 2019, their Sect. 5) in combination with the Ledoux cri- +terion to define the boundaries of convective instability. This +specific approach of treating convection agrees with the results +of modern 3D hydrodynamic simulations (Anders et al. 2022). +Convective mixing was incorporated into the models via mix- +ing length theory (MLT) in the ‘Cox’ formalism (Cox & Giuli +1968, their chap. 14) with the value of the solar-calibrated mix- +ing length parameter αMLT = 1.8210 (Choi et al. 2016). As +9 By critical rotational velocity we mean the situation when the effec- +tive gravity at the stellar equator is zero, i.e. the centrifugal force and the +Eddington factor, Γ, balance the true gravity. MESA estimates this quan- +tity as Ωcrit = +� +(1 − Γ)GM/R3, where G is the gravitational constant +and Γ ≡ Lrad/LEdd. The Lrad and LEdd denote the radiative luminosity +and Eddington luminosity of the star, respectively. +10 There is some evidence that αMLT may depend on global stellar pa- +rameters such as mass (Yıldız et al. 2006) or metallicity (Viani et al. +Article number, page 4 of 24 + +Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 1. Distribution of initial orbital period and eccentricity for a sample +of 20,000 binaries evolved in our project. The initial normalised sepa- +ration at the periastron is colour-coded. The upper left-hand corner cor- +responds to the ZAMS EEVs, which experience RLOF at the periastron +and should therefore rapidly circularise their orbits. The lower right- +hand corner, on the other hand, is where relatively widely-separated +binaries can be found. +mentioned earlier, exponential overshoot mixing above the con- +vective core was also included11, but we neglected overshoot- +ing in the non-burning convection zones. For stars with masses +⩾ 15 M⊙, we activated the treatment of convection as ‘MLT++’ +(Paxton et al. 2013, their Sect. 7.2) to reduce superadiabaticity +in convective zones dominated by radiation pressure. Since we +used Ledoux criterion, semiconvection could appear in our stars +with its efficiency parameter, αsc = 0.01 (Langer et al. 1985). +In our case, semiconvection sometimes occurred in chemically- +modified layers left by the shrinking core. +Upon initialisation at the ZAMS, we relaxed both compo- +nents in ∼100 steps so that they rotated rigidly. Later, we al- +lowed our stars to rotate differentially during their evolution, +according to the so-called shellular approximation of rotation +(Meynet & Maeder 1997). Throughout the entire evolution, we +assumed that the rotation axes of the stars are perpendicular to +the orbital plane. MESAstar uses the mathematical formalism +of Heger et al. (2000) and Heger et al. (2005) to apply struc- +tural corrections, perform different types of rotationally induced +mixing and „diffusion” of angular momentum between adjacent +shells. The following rotational mixing mechanisms were taken +into account in MESA: dynamic shear instability, secular shear +instability, Eddington-Sweet circulation, Solberg-Høiland insta- +bility, and Goldreich-Schubert-Fricke instability (all described +in detail by Heger et al. 2000). Even the combined mixing coef- +ficients of the aforementioned rotational instabilities can be zero +in some parts of the star. However, this is clearly unrealistic due +to the presence of a nearby companion that induces additional +mixing throughout the star. To at least approximately account for +this process, we did not allow the total mixing coefficient, Dmix, +to fall below 105 cm2/s. This particular arbitrarily-selected value +is related to the mixing time scale, τmix ∼ (∆r)2/Dmix ≈ 15 Myr +at radial distance, ∆r = 0.1 R⊙. We cannot conceal here that ro- +2018). It is also very likely that αMLT is sensitive to the evolutionary +stage of the star and the type of convection zone (e.g., Wu et al. 2015). +Here, we have assumed a constant value of αMLT for simplicity. +11 Similarly to the αMLT, fov also can depend on different stellar param- +eters (e.g., Castro et al. 2014). +tation and mixing profiles in MS stars are still poorly understood +(except in the solar case). There are no definitive conclusions +as to what mixing mechanisms and whether they actually occur +in massive and intermediate-mass MS stars (see e.g., Pedersen +et al. 2021; Pedersen 2022, for a discussion of this problem and +its asteroseismic inference from B-type dwarfs). +Mass losses due to the radiation-driven stellar wind were +calculated according to the prescription given by Vink et al. +(2001). Their formulae take into account the presence of a +so-called bi-stability jump around the effective temperature of +Teff ≈ 26,000 K, caused by ionization and recombination of some +Fe ions. Nevertheless, the presence of a bi-stability jump is still +questionable and there is some evidence that the associated al- +most instantaneous change in the mass-loss rate may not be real +(cf. Krtiˇcka et al. 2021; Björklund et al. 2022). ‘Nominal’ wind +mass-loss rates in our simulations were modified in two ways: +(1) the rate was amplified by the aforementioned tidal mech- +anism, parametrized by Bwind (Tout & Eggleton 1988) and (2) +the effect of fast rotation at the surface, which can amplify the +mass-loss rate, was accounted for by the simplified power-law +description given by Heger et al. (2000) (their Sect. 2.6). We +assumed that the mass loss through the wind is completely non- +conservative, i.e. there is no mechanism that could transfer some +material back to the star or to a companion. +As we already mentioned above, MESAbinary allows the si- +multaneous integration of some stellar and orbital parameters +that are coupled to each other in a binary system. We have +switched on the MESA controls responsible for changes in the to- +tal orbital angular momentum caused by: (1) gravitational wave +radiation, (2) wind mass loss and (3) tidal spin-orbit coupling. +For the first process, the rate of orbital momentum loss was cal- +culated assuming point masses. The mass loss through the stellar +wind was completely non-conservative, so the angular momen- +tum lost via this channel was equal to the angular momentum +carried by the wind. The phenomenon (3), contributing to the +evolution of eccentricity, orbital and spin angular momenta, was +modelled using the theory of tidal interactions for radiative en- +velopes developed by Zahn (1977), Hut (1981) and Hurley et al. +(2002), after being adapted to the shellular approximation of ro- +tation. For stars with radiative envelopes, tidal dissipation pro- +cesses are dominated by tidally excited gravity modes that prop- +agate to the stellar surface, where they gain relatively large am- +plitudes and experience effective radiative damping (due to the +short local thermal time scale) and nonlinear damping. Conse- +quently, they deposit their energy and angular momentum in the +outer layers of the envelope. Following earlier calculations of +Zahn (1977), Hurley et al. (2002) delivered convenient formu- +lae to describe the tidal synchronisation and circularisation time +scales associated with the aforementioned phenomenon. Using +these time scales combined with the formalism presented by Hut +(1981), MESAbinary integrates the evolution of the eccentricity +and updates spin angular frequency of each shell in the stellar +model. Therefore, our calculations in MESAbinary took into ac- +count the approximate influence of the dynamical tide on the +orbit, at least up to the lowest possible order. Of course, the tidal +evolution formalism implemented in MESAbinary does not in- +clude the effects of resonance locking. For explicit formulae de- +scribing tidal processes in MESAbinary, we refer to Paxton et al. +(2015) (their Sect. 2). +We have completely ignored the effects of magnetic fields, +while bearing in mind that they may mainly affect the actual stel- +lar wind mass-loss rates, the efficiency of internal mixing pro- +cesses and synchronisation/circularisation time scales (e.g. via +the magnetic braking mechanism). The impact of fossil mag- +Article number, page 5 of 24 + +0.8 +RLOF on ZAMS +5.0 +0.7 - +4.5 +0.6- +4.0 +e +0.5 +3.5 +3.0 +0.4 - +wide +even +2.5 +0.3 +2.0 +0.2 +1.5 +100 +101 +102 +Porb (d)A&A proofs: manuscript no. TEOs_in_massive_EEVs +netic fields on the evolution of massive and intermediate-mass +stars was recently described by Keszthelyi et al. (2022). +All details on the parameters of our models in MESA can be +found in Appendix A, where we present the contents of our MESA +inlists. A concise description of the micro- and macrophysics +data sources used by MESA is provided in Appendix C. +3.2.3. Termination conditions +The evolution of the binary system was carried out until at least +one of the following termination conditions was met for any of +the components: +1. The component reached TAMS, i.e. the central mass abun- +dance of hydrogen fell below Xc ⩽ 10−4. +2. The eccentricity was reduced to e ⩽ 0.01. +3. The rotation velocity reached Ω/Ωcrit = 0.75 at the stellar +surface. +4. Episodic mass transfer between components due to the +RLOF in the periastron began. +The reasons behind providing the termination conditions out- +lined above are as follows. Our study is exclusively dedicated to +the MS phase of the evolution of EEVs, hence the first condi- +tion has to be enforced. The second condition is self-explanatory, +since we are interested in non-zero eccentricities that allow for +TEO excitation12. The third condition is related to the conver- +gence problems that can occur in MESAbinary when one of +the components nearly approaches the break-up velocity of rota- +tion. Numerous assumptions and descriptions of rotation-related +phenomena reach the limits of their applicability in MESA for +Ω/Ωcrit ≈ 1. Since for Ω/Ωcrit ≳ 0.75 the deviation from +spherical symmetry becomes significant, a 1D treatment of the +problem is no longer adequate. For instance, the way in which +such a star loses mass becomes fundamentally different from the +isotropic case. We have therefore decided to stop integrations un- +der such circumstances. The last condition is related to the diffi- +culty in correctly describing an episodic (near-periastron) RLOF, +when a ‘blob’ of material could be ejected from the RL-filling +component during each periastron passage. However, this kind +of orbital phase-dependent RLOF is not expected to be observed +in a binary for a long time due to strong tidal forces. They should +effectively suppress the eccentricity, making the system circular +(and so the second condition can be quickly met). +3.3. Asteroseismic calculations +A consequence of the assumption of the aligned vectors of the or- +bital and spin angular momenta is a rule for selecting the geome- +try of modes that can be tidally excited. Under such conditions, a +normal mode can be tidally excited only if +mod (|l+m|, 2) = 0, +e.g. the l = 2 TEOs will be characterised only by m = −2, 0, +2. +Here we restrict our study to only l = 2, m = 0, +2 modes be- +cause of two reasons. First, l = 2 modes correspond to the dom- +inant component in the series expansion of the variable tidal po- +tential. Modes with l > 2 undergo much weaker excitation due +to the dependence on (R/a)l+1, which enters Eq. (2). Second, +the values of FN,−2 are very small compared to their m = 0, +2 +counterparts. This can be easily seen in Fig. 2a, where we have +12 In theory, components of circular systems (e = 0) can also exhibit +TEOs, provided they do not rotate synchronously. However, the number +of modes observable as TEOs in these systems is much smaller than the +number of potential TEOs in EEVs. +plotted the maximum values of FNm for m = −2, 0, +2 and dif- +ferent eccentricities. FN,−2 is approximately at least 2 – 3 orders +of magnitude smaller than FN,0 or FN,2. +For each model of the stellar internal structure that was saved +during the synthesis of binaries in MESA, we calculated the oscil- +lation spectrum using the GYRE code in the non-adiabatic regime +and the second-order Gauss-Legendre Magnus integrator. The +frequencies σn,2,0 and σn,2,+2 corresponding to the non-adiabatic +calculations were searched by GYRE based on the preliminary +adiabatic calculations. Rotational effects (Coriolis force) were +taken into account using the so-called traditional approximation +of rotation (e.g., Aerts et al. 2010, their Sect. 3.8). We searched +for eigenvalues in the family of gravito-acoustic solutions. We +assumed the necessary (differential) rotation profile inside the +star from the MESA model. +As we argued in Sect. 3, it is necessary to choose a rea- +sonable range of frequencies to scan for eigenvalues based on +Qnlm and FNm. Therefore, we only searched for modes with +|n| ⩽ 30 and frequencies, σn,2,0 ∈ (forb, Nm=0 +max forb) or σn,2,+2 ∈ +(max{0, forb − 2fs,core}, Nm=+2 +max +forb − 2 fs,core). In the ranges shown, +Nm=0 +max and Nm=+2 +max +refer to the limits of N due to the decrease in +FN,0 and FN,2, respectively. The fs,core is the core rotation fre- +quency. We defined Nm=0 +max and Nm=+2 +max +as N for which FN,0 or FN,2 +is equal to 10−8, i.e. FNm starts to effectively prevent excitation +of TEOs. In practice, we numerically calculated the FNm func- +tions13 for different eccentricities and obtained the log Nm +max(e) +relations as a fit of a fourth-degree polynomial to a set of its dis- +crete points. A summary of this process is shown in Fig. 2b. For +low-e orbits, the typical range of N favourable for the excitation +of TEOs reaches N ∼ 101, in contrast to highly eccentric orbits, +which may exhibit as much as N ≈ 100 – 200 TEOs. Figure 2b +also shows another feature of m = −2 modes that makes them in- +ferior candidates for TEOs compared to axisymmetric and pro- +grade modes – as potential TEOs they always span a narrower +range of orbital harmonic numbers. +Defining the frequency range for σn,2,0 is quite straightfor- +ward, as these are axisymmetric modes and their frequencies +do not change when switching between inertial and co-rotating +frames. The situation is quite different when it comes to the +m = +2 modes. This time, due to the differential rotation inside +the star, σnlm = σnlm(r) = σnlm − mfs(r), where r is the radial co- +ordinate in the stellar interior and σnlm is oscillation frequency +in the inertial frame. For some eigenmodes, σnlm may change its +sign somewhere in the star, depending on the shape of the rota- +tional profile. This location is known as the critical layer, where +σnlm(r) → 0, and such a mode experiences severe damping due +to its very short radial wavelength (e.g., Alvan et al. 2013). To +exclude these modes from our experiment, the maximum fre- +quency of σn,2,+2 was set to (Nm=+2 +max +forb − 2 fs,core)14. This is be- +cause during evolution the core rotates almost rigidly and faster +than the envelope, hence the difference (Nm=+2 +max +forb − 2 fs,core) ⩽ +(Nm=+2 +max +forb−2fs,env), where fs,env stands for the rotation frequency +of the outermost part of the envelope. +More details of our calculations performed in GYRE can be +found in Appendix B, where we present the explicit contents of +our GYRE input file. +13 Using eq. 5 presented by Fuller (2017). +14 We note that this frequency is expressed in a rest frame co-rotating +with the stellar core. +Article number, page 6 of 24 + +Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 2. (a) Maximum values of the Hansen coefficients FNm versus +eccentricity for l = 2 modes and three different azimuthal orders, +m = −2, 0, +2 denoted by blue, green, and red points, respectively. We +note the marginal contribution of the m = −2 modes; (b) Dependence of +log Nm +max on eccentricity with the same colour-coding as in panel a. The +colour solid lines represent the best fits of the fourth-degree polynomi- +als, which we used to determine frequency ranges in the asteroseismic +calculations. +3.4. Derivation of L(t) +The introduction of differential rotation also has consequences +when it comes to interpreting the resonance condition from +Eq. (1). The quantity fs is no longer a constant value, so one has +to decide which fs to choose. Theoretical studies imply the in- +duction of g-mode TEOs (especially of high radial order) primar- +ily near the convective core boundary (e.g., Goldreich & Nichol- +son 1989) for stars with radiative envelopes. However, the res- +onances in our simulations are also due to p or g modes with +small radial order. Therefore, we decided to apply our resonance +condition to the envelope15 (not to the interface region near the +core boundary), rewriting Eq. (1) more accurately as +fNm ≡ N forb − m fs,env ≈ σnlm, +(5) +and use it in the subsequent modelling of L(t). It is essential to +note at this point that the resonance condition given by Eq. (5) +refers to fNm and σnlm expressed in a frame co-rotating with +the outer stellar envelope. Although in principle the morphol- +ogy of L(t) depends on the choice of the specific resonance con- +dition, we note that it does not affect at all resonances due to +15 In the exact approach, different resonance conditions would have to +be used for different modes, depending on the radial coordinate inside +the star where a given TEO is dominantly excited. Here we assume a +single form of resonance condition for all modes. +m = 0 modes and should not significantly affect resonances cor- +responding to p modes or low-|n|, m = +2 g modes. +Having a set of eigenfrequencies calculated by GYRE and +knowing the history of the binary evolution from MESA, we per- +formed the summation shown in Eq. (4). However, this was not +a direct summation running over the models saved by MESA and +GYRE, as their temporal resolution was still too coarse compared +to the duration of a typical resonance. To circumvent this prob- +lem, we interpolated the temporal variations of each oscillation +frequency and all necessary parameters of the binary system us- +ing Akima cubic spline functions (Akima 1970). Then, we were +able to calculate the values of L(t) on a uniformly-spaced time +grid with a constant time step of 2,000 years, which we assumed +to be identical for all EEVs in our simulations. From here on, +we will use the term ‘resonance curve’ as a proxy for the L(t) +time series. Figure 3 shows a compilation of example resonance +curves, although we postpone discussion of these to Sect. 4. To- +gether with the initial parameters of binary systems, resonance +curves are particularly important to us in this study. +3.5. ML analysis of the resonance curves +Although in Sects. 4.3 and 4.4 we analyse the resonance curves +based on various statistics, due to their global nature we do +not distinguish many details that are ‘hidden’ in the resonance +curves. To characterise the morphology of all resonance curves +in more detail (without having to perform a visual classification, +which is almost impossible due to the number and complexity +of the data set), we applied dimensionality reduction methods. +With these, we were able to explore the topology spanned by +the morphological features of the resonance curves. We carried +out the entire analysis described here separately for the sets of +curves L1(t) and L2(t), corresponding to the primary and sec- +ondary components, respectively. +As a first step, we summarised each resonance curve with a +vector Q that described its morphological features. We focused +our attention on two particular features: (1) the distribution of +log(L) values and (2) the distribution of apparent maxima at a +normalised time, t/Tmax, where Tmax stands for the max{t}. In +practice, we calculated vectors Qx and Qy which contained sets +of 1,000 quantiles of normalised times corresponding to local +maxima of L(t) and 1,000 quantiles of log(L), respectively. The +levels of both calculated quantiles were spanned evenly from 0 +to 1. Qx describes the overall distribution of apparent maxima +in time, reporting changes in the rate of resonance occurrence. +We deliberately used normalisation by Tmax because we want +the results to be sensitive only to the relative distribution of the +resonance events over the lifetime of the EEV. Otherwise, its val- +ues would be strongly correlated with the length of the resonance +curve itself16, rather than with the distribution of resonances over +time. On the other hand, Qy encapsulated the combined informa- +tion about the mean level of log(L), any long-term trends in the +resonance curve and the distribution of the heights of the max- +ima. In contrast to the Qx, we did not apply any normalisation +to Qy as its absolute values carry valuable information about the +strength of the resonances and the average level of the entire +resonance curve. The final vector Q was constructed as the con- +catenation of Qx and Qy, which had previously been scaled using +the variance in the sets of all Qx and Qy. The resulting Q has a +total of 2,000 dimensions. +16 Which in turn is an almost a direct approximation for the mass of the +primary component. +Article number, page 7 of 24 + +0 +log (max[FNm )) +-6 +-8 +a) +2.5 +m=+2 +m=0 +2.0 +m=-2 +max +1.0 +0.5 +(b) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +eA&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 3. Sample of resonance curves obtained as described in Sect. 3. Each panel corresponds to a different arbitrarily-chosen binary system with +the rounded values of their initial parameters given on the right. The dark red and blue curves reflect the behaviour of L(t) for the primary and +secondary components, respectively. For clarity, L2(t) has been shifted vertically by three orders of magnitude downwards. Time t = 0 indicates +ZAMS. A sudden break in L2(t) on the bottom panel (after about 5.5 Myr) indicates L2 = 0, i.e. the absence of any resonances. The differences in +the height of the peaks are due to different values of γnlm and min{|σnlm − fNm|} for excited TEOs. +Article number, page 8 of 24 + +14.6Mo +M1 +M2 +3.0M +105. +0.32Qcrit +21 +22 +0.31Qcrit +fov,1 +0.022 +fov, 2 +0.019 +primary +secondary +103- +0.52 +e +Tperi +5.23 +Bwind +583.9 +101- +2 +0 +4 +6 +8 +10 +12 +M1 +5.8Mo +M2 +4.2Mo +105. +0.11 2crit +21 +22 +0.162crit +fov, 1 +0.016 +fov,2 +0.015 +103- +0.29 +e +uody +1.53 +Bwind +828.3 +101. +C2 / 103 +10 +20 +25 +5 +15 +30 +L +23.5 Mo +M1 +106. +M2 +13.1 Mo +L +0.43 2crit +21 +105. +0.44 2crit +22 +fov,1 +0.022 +104 +fov,2 +0.019 +0.62 +103. +Tperi +3.32 +Bwind +102. +806.0 +101. +2 +3 +4 +5 +6 +0 +M1 +19.8Mo +M2 +5.5Mo +105. +21 +0.24 Qcrit +22 +0.18Qcrit +104. +0.018 +fov, 1 +0.015 +fov,2 +103- +0.31 +e +Tperi +2.68 +102 +Bwind +893.8 +101- +5 +6 +0 +3 +4 +8 +t (Myr)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +In the next step, we performed a preliminary dimensional- +ity reduction of Q by means of the Principal Component Analy- +sis (PCA, Pearson 1901), obtaining pre-processed ‘morphology’ +vectors, θPCA. PCA is a method that orthogonally projects the +data into a coordinate system in which successive vector com- +ponents explain a smaller and smaller part of the data variance. +The target number of its dimensions returned by PCA for each +Q was set to 10. This value was chosen experimentally by ex- +amining the percentage of the total variance of the data set ex- +plained by successive PCA components. For L1(t), the first ten +PCA components explained a total of 99.8% of variance (first +component – 79% and second component – 19%). For L2(t), the +corresponding value was 98.5% (in this case, the first PCA com- +ponent explained 60% of the total variance, while the second +explained 22%). +We then performed the final 2D embedding by applying +UMAP on the collection of θPCA vectors. UMAP is a multi- +purpose non-linear dimensionality reduction technique that con- +structs a low-dimensional projection that preserves as accurately +as possible the topological structure of the input data. For in- +stance, in this case, a pair of embeddings of resonance curves +with similar properties (in the sense of their summary statistics +described above) are expected to lie in mutual vicinity on the +2D UMAP plane. Let us denote the UMAP results as θUMAP. +The manifold spanned by θUMAP (Sect. 4.5) allowed us to ef- +fectively examine the differences in the morphology of the res- +onance curves and their dependence on the initial parameters of +the simulated EEVs. +Unlike PCA, UMAP is a complex method, with many free +parameters that need to be adjusted with care, as the resulting +embedding may depend heavily on their choice. Appendix D +provides all the ‘technical’ details of this process, including the +values of the most important UMAP parameters adopted in this +study. +4. Results +4.1. General properties of synthesised EEVs +Before going into a detailed analysis of the resonance curves, we +briefly characterise the general properties of the models we have +synthesised using the MESAbinary and GYRE codes. +4.1.1. Evolutionary tracks in HRD +Figure 4 shows a pair of HRDs with a compilation of all 20,000 +evolutionary tracks that we obtained in our simulations for the +primary (Fig. 4a) and secondary (Fig. 4b) components. Although +it is impossible to clearly present thousands of evolutionary +tracks on a single HRD, we have highlighted and colour-coded a +small fraction of them in order to describe some of their features. +First of all, only a fraction of the primaries reached TAMS +when the central mass abundance of hydrogen dropped be- +low 10−4 (according to the first of our termination conditions, +Sect. 3.2.3). Many evolutionary tracks were interrupted at MS +due to the fulfilment of one of the other termination conditions. +Secondly, a number of tracks clearly change their character after +crossing the line corresponding to the bi-stability jump (around +Teff = 26,000 K). This is due to the associated sharp increase in +the wind mass-loss rate, as it tries to keep the stellar luminosity +constant. In some circumstances, the mass-loss rate is so high +that the star loses a significant part of its envelope17. This effect +17 We recall that these high mass-loss rates are not exclusively derived +from the description of Vink et al. (2001). Rotational and tidal amplifi- +‘pushes’ the star back to the high effective temperature region +and is particularly pronounced for the most massive stars in our +sample (cf. Fig. 4a, evolutionary tracks that ‘turn around’ and +cross the bi-stability jump for a second time). +4.1.2. EEV groups in terms of the termination condition +Only four of the seven18 termination conditions actually oc- +curred in our simulations. The majority of our EEVs (∼67.1 %) +ended up as MS RLOF systems in which the primary compo- +nent filled its Roche lobe during the periastron passage. The +next most numerous group (∼22.6 %) were systems in which the +primary component successfully reached TAMS (Xc ⩽ 10−4). +About 10.2 % of the binaries managed to circularise their orbits +before any other termination condition was met. The last group +contains only about 0.04 % of the total sample. This is the group +where the primary’s rotation velocity exceeded the maximum al- +lowed angular velocity (Ω / Ωcrit ⩾ 0.75). +Figure 5 presents these four groups of EEVs on the Porb-e +plane and allows a comparison of the initial (Fig. 5a) and final +(Fig. 5b) states of the aforementioned distribution. As expected, +the EEVs with the shortest orbital periods and high eccentricities +tended to circularise their orbits before leaving the MS. Their +trajectories in the Porb-e diagram (Fig. 5c) follow smooth, almost +vertical lines due to the strong tidal damping of eccentricity. On +the other hand, the integration of the evolution of systems with +large distances between components at periastron (�rperi ≳ 3.5) +has been terminated mainly due to the exhaustion of hydrogen +in the primary’s core. Although the majority of EEVs belonging +to this group do not significantly change their orbital parameters +during evolution, there is a subgroup of them that behaves quite +differently. It can be recognised as the distinct ‘cloud’ of green +dots in Fig. 5b, represented by the mainly horizontal green tracks +in Fig. 5c. These are systems that were characterised by very +strong stellar winds at the end of the MS phase and have lost +much of their envelopes, so that their orbital period has increased +significantly (Kepler’s third law). +The most numerous group of EEVs, in which the primary +component has filled its Roche lobe in the MS phase, forms a +kind of ‘bridge’ between the two previously mentioned groups +and merges with them. The shapes of the corresponding trajec- +tories on the Porb-e plane may vary from system to system, de- +pending on the interplay between tidal forces and the intensity +of stellar winds, so no single ‘type’ of track can be assigned +to them. However, many of them resemble the inverted Greek +letter ‘Γ’ – initially, the system drifts horizontally (towards the +longer orbital period) as a result of the mass loss and/or spin- +orbit coupling, and then undergoes more or less rapid circulari- +sation (moves vertically downwards) under the influence of the +intense tides, which come to the fore when the primary compo- +nent almost fills its Roche lobe. +Only 8 out of 20,000 EEVs underwent efficient spin up of +both components due to the pseudo-synchronisation (when the +rotation period of the star ‘matches’ the rate of orbital motion +at periastron, so that there is no net torque over an orbital cycle, +e.g., Hut 1981). These few systems are located in the upper right +cation mechanisms can significantly intensify stellar winds in our sim- +ulations. These phenomena are particularly well-pronounced when the +component is close to TAMS, i.e., its radius approaches the Roche-lobe +radius. +18 In Sect. 3.2.3 we give four types of termination conditions, but three +of them apply independently to both the primary and secondary compo- +nents. +Article number, page 9 of 24 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 4. (a) HRD with the evolutionary tracks of primary components. The grey area corresponds to the region occupied by the full set of 20,000 +tracks, while a subsample of 100 randomly-selected tracks is indicated with coloured points connected by black solid lines. Each point rep- +resents one saved MESA model. The colour coding reflects the central hydrogen abundance. The effective temperature of the bi-stability jump +(Teff ≈ 26,000 K) is marked with the vertical dashed line. The abrupt change in the behaviour of some evolutionary tracks after crossing the bi- +stability jump region is due to a significant change in the wind mass-loss rate; (b) The same as panel (a), but for a set of secondary components. +We note the difference in the ranges of the two axes in panels (a) and (b). More details are discussed in the main text. +corner of Figs. 5a and b. Their orbits were initially highly eccen- +tric yet relatively widely-separated at periastron (�rperi ≈ 4.5 – +5.0). Thus, in combination with the lower masses of the pri- +mary components (M1 ≈ 5 M⊙), there was no effective tidal +dissipation. However, the envelopes of these stars tended to ro- +tate pseudo-synchronously with the orbit (due to the relatively +long nuclear time scale of the evolution of a 5 M⊙, the primaries +had enough time to do so). Consequently, this led to a very fast +rotation of the primary component, exceeding the threshold of +Ω/Ωcrit = 0.75. +4.1.3. Internal structure and asteroseismic properties +The shape of the resonance curve depends not only on the global +properties of the components and the orbit, but also on the inter- +nal structure of the stars, which directly affects seismic proper- +ties (i.e. the spectrum of eigenmodes). Therefore, within the lim- +ited volume of this paper, we would like to show at least one rep- +resentative example of the evolution of the internal properties of +the primary component for an arbitrarily-chosen EEV. Figure 6 +shows the evolution of a primary with mass M1 ≈ 13.6 M⊙ in a +system with an initial eccentricity e ≈ 0.4 and an initial orbital +period Porb ≈ 4.0 d. In our simulations, this particular system +finished its evolution due to the circularisation of its orbit after +about 12 Myrs. The HRD in Fig. 6 reveals the ‘non-standard’ +evolutionary track of the primary due to the sharp change in the +mass-loss rate after crossing the bi-stability jump (right panel in +the top row of Fig. 6). The same panel also shows how the pri- +mary’s surface rotation rate varies over time – as the mass-loss +rate increases, it loses a lot of spin angular momentum and slows +down its rotation. The eccentricity and orbital period monotoni- +cally decrease with time (middle panel in the top row of Fig. 6), +except for a short episode of increase in Porb caused by the ir- +reversible loss of a large part of the envelope. We have selected +three epochs in the evolutionary history of this EEV (labelled A, +B, C on the HRD), for which we have presented the appearance +of the rotational profiles, mode propagation diagrams and oscil- +lation spectra of the primary component in the bottom part of +Fig. 6. Epoch A corresponds to the phase of evolution just af- +ter leaving the ZAMS, epoch B is characterised by Xc,1 ≈ 0.45, +and finally, epoch C marks the situation just before the complete +circularisation of the EEV. Let us briefly describe the changes +occurring in each of the three types of diagram below. +The internal rotation profile of the primary is almost constant +for epoch A, but by then a division between a faster-rotating core +and a slower-rotating envelope begins to emerge. The aforemen- +tioned division becomes particularly apparent in epoch B, when +the core has developed a rotation rate approximately 1.25 times +that of the surface layers. As can be seen, the contracting core ro- +tates as a rigid body throughout the MS lifetime due to efficient +angular momentum transport supported by convection. The outer +part of the envelope also rotates almost rigidly, but this time it is +due to large-scale Eddington-Sweet meridional flows. The angu- +lar velocity gradient in the primary starts to gradually decrease +as the star reaches epoch C. Various mixing processes in the +chemically-modified layer left by the core lead to the diffusion +of angular momentum from the core to the envelope. Moreover, +the rotational profile inside the star becomes a smooth function +of the radius (rather than a step-like function as for epoch B). +Article number, page 10 of 24 + +0.7 +5.5 +(a) +(b) +5 - +- 0.6 +5.0 - + 0.5 +4 - +4.5 - +-0.42 +X +1og ( +3.5 - +- 0.2 +2 - +3.0 - +- 0.1 + bi-stability jump +1- +0.0 +4.6 +4.5 +4.4 +4.3 +4.2 +4.1 +4.6 +4.5 +4.4 +4.3 +4.2 +4.1 +4.0 +3.9 +log (Teff, 1 / K) +log (Teff, 2 / K)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 5. Orbital period-eccentricity distributions of 20,000 modelled +EEVs; (a) Initial distribution of e as a function of Porb. Colour-coding +corresponds to the termination conditions described in Sect 3.2.3, i.e. +the RLOF of the primary component during periastron passages before +reaching TAMS (black), exhaustion of hydrogen in the primary’s core +(primary at TAMS, green), almost complete circularization of the or- +bit (e = 0.01, magenta), and the maximum allowed rotation rate of the +primary component (Ω / Ωcrit = 0.75, orange). A pair of dashed hori- +zontal lines mark the boundary values of the initial eccentricity, e = 0.8 +and e = 0.2; (b) Same as in panel (a), but for the final state of each +modelled binary system; (c) Random selection of 400 orbital evolution +tracks with the same colour-coding as in panels (a) and (b). +The majority of TEOs in our simulations belong to the g- +mode family of oscillations, so it is very important to control the +behaviour of the Brunt-Väisälä buoyancy frequency, NBV, in our +models. Together with Lamb frequency for l = 2 modes, S l=2, +they carry information about g and p mode cavities and their +evanescence regions (e.g., Aerts et al. 2010, their Sect. 3.4). The +evolution of NBV and S l=2 is presented in the middle column +of Fig. 6. The blue and grey shaded regions denote the posi- +tion of the l = 2 p-mode and g-mode propagation cavities, re- +spectively. The white areas that lie between the Brunt-Väisälä +and Lamb frequencies correspond to the evanescence regions. +During evolution, the receding convective core builds up a large +g-mode trapping cavity, which is very important for their fre- +quency spectrum. Additionally, the behaviour of the NBV just be- +low the stellar photosphere reveals a pair of thin subsurface con- +vection zones, expected for this type of star (e.g., Jermyn et al. +2022). Comparing the mode propagation diagrams for epochs A +and C, it can be seen that also the p modes can penetrate deeper +and deeper into the primary as it gradually depletes the hydrogen +in its core. +The right column in Fig. 6 contains most of the information +that is directly used to obtain the resonance curve. The horizontal +bars at the top of each panel correspond to the frequency range in +which GYRE looked for potential TEOs (according to the criteria +adopted in Sect. 3.3). We recall that that their width depends on +the Nm +max(e) functions, so as the system evolves towards lower ec- +centricities, these bars are shorter and shorter (i.e. fewer harmon- +ics of the orbital frequency can effectively drive TEOs). With the +thick, short vertical lines we mark the location of the tidal forcing +frequencies. As can be seen, the separation between successive +values of fNm becomes larger with passing time due to the in- +crease in forb. The eigenfrequencies found by GYRE are marked +with the long thin vertical lines, while the linear damping rates +of these modes are shown as black solid and dotted lines. The +presented set of three synthetic oscillation spectra reveals a typ- +ical structure for g modes with their asymptotic behaviour for +large radial orders (which correspond to lower frequencies). It +may appear that the dense ‘forests’ of eigenfrequencies end too +early relative to the left limits of horizontal bars. However, this +is not a mistake, but a direct consequence of the maximum |n| +we allowed in the calculations – modes with lower frequencies +would have larger radial orders than thirty. During the evolution +of the EEV, both the forcing frequencies and the oscillation spec- +trum shift, so the intersection of these two vertical line patterns is +virtually inevitable in most cases. Each of these intersections is +the source of a single resonance that can give rise to a noticeable +TEO at the level of the photosphere. +4.2. The ‘visual inspection’ of resonance curves +The resonance curves are characterised by a striking diversity in +terms of morphology, which is already partly evident in Fig. 3. +The four examples of L1(t) and L2(t) shown in this figure show +that the components of the EEVs can experience, firstly, a very +different number of resonances and, secondly, their distribution +in time can take various forms. The heights of the maxima of +the resonance curves are mainly dictated by the γnlm of the mode +to which the smallest difference corresponds, (σnlm − fNm). Sta- +tistically speaking, modes with larger |n| are more strongly non- +adiabatic (have larger damping rates), hence the maxima they +induce in the resonance curves are lower (cf. Eq. (3)). Another +factor determines the extent of the resonant maximum in time. It +is determined by the relative ‘velocity’ with which the eigenfre- +quency spectrum crosses the fNm spectrum. By ‘velocity’ here +we mean the rate of change of these two independent frequency +spectra. +It should be emphasised that there are also numerous cases in +which L(t) drops sharply to zero at some point (cf. L2(t) curve +in the bottom panel of Fig. 3) or resonances do not occur at all +(see Sect. 4.3 and Fig. 8). Such a situation can occur, for exam- +Article number, page 11 of 24 + +0.8 +initial +distribution +0.7 +0.6 +0.5 - +e 0.4- +0.3 +0.2 +0.1 - +RLOF of the primary +minimum eccentricity +(a) +primary at TAMS +maximum rate of rotation +0.0 +0.8 +final +0.7 +distribution +0.6 - +0.5 - +e0.4l +0.3 - +0.2 +0.1 - +(b) +0.0 +0.8 - +sample +tracks +0.7 - +0.6 - +0.5 - +e0.4l +0.3 - +0.2 - +0.1 - +(c) +0.0 +100 +101 +102 +Porb (d)A&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 6. Summary plot of the properties of the primary component of one of the arbitrarily selected binary systems from our simulations. The +approximate initial parameters of this particular system were as follows: M1 ≈ 13.6 M⊙, M2 ≈ 3.6 M⊙, e ≈ 0.4, and �rperi ≈ 2.3. The integration +of the system was terminated because of its circularisation. The top row of panels shows, from left to right, evolutionary track in the HRD, the +evolution of the orbital period and eccentricity, and the temporal changes of the wind mass-loss rate and surface rotation velocity. The vertical +dashed lines in the latter two diagrams correspond to epochs A, B, C in the HRD. The lower part of the figure shows the internal rotation profile +(left column), the mode-propagation diagram (middle column) and the synthetic oscillation spectrum (right column) for epochs A, B, C (shown +in consecutive rows labelled with these letters). The rotation frequency inside the primary is drawn as a function of fractional radius, r / R. The +range of rotation frequency is different in the three panels. The mode-propagation diagram shows the dependence of the Brunt-Väisälä frequency +(black line) and Lamb frequency for l = 2 modes (blue line) on the fractional radius. The grey and blue shaded areas correspond to the propagation +cavities of the g and l = 2 p modes, respectively. The synthetic oscillation spectrum diagrams contain several different pieces of information. The +light blue and light red horizontal bars delineate the range of frequencies allowed by the FNm values. In the background of each, the blue and red +short vertical lines indicate tidal-forcing frequencies lying within these ranges. The synthetic oscillation spectra calculated by GYRE are marked +with red (σn,2,0) and blue (σn,2,+2) long vertical lines. Their corresponding linear damping rates are plotted as solid (γn,2,0) and dashed (γn,2,2) black +lines. The frequency scale on the abscissa axis refers to the rest frame co-rotating with the primary’s core. +Article number, page 12 of 24 + +4.0 - + 0.40 +F 0.350 +-6.25 - +C +3.9 - + 0.35 +V! +B +4.35 - +6.50 - +F 0.325 +3.8 - +- 0.30 +-6.75 - +F 0.300 +B +yr- +3.7 - +- 0.25 +log (M / (Mo) +-7.00 - + 0.275 +C +e + 0.20 +-7.25 - +F 0.250 Ci +3.5 - +- 0.15 +-7.50 - +F 0.225 +3.4 + 0.10 +-7.75 - +4.20 - +F 0.200 +iA +IB +3.3 - + 0.05 +C! +-8.00 - +dA +F 0.175 +3.2 → + 0.00 +4.15 +4.46 4.44 4.42 4.40 + 4.38 +2 +4 +4.36 +0 +2 +6 +10 +6 +8 +10 +12 +4 +8 +t (Myr) +log (Teff /K) +t (Myr) +Rotational profile +Mode propagation +Oscillation spectra +140 - +10-8 - +0.487 +120 - +-10-7 , +0.486 - +）100- +9-01- +p +(d-l) +C +08 +Fs-0[- +S +60 +0.484 - +10-4 - +40 - +0.483 - +-10-3 - +20 - +-10-2 +0.482 +-0 +140 - +0.46 - +-10-8 - +120 - +-10-7 , +0.44 - +）100- +-10-6 _ + (d-1) +80 - +-10-5 +S +uul +C +-10-4 +0.40 - +40 - +ε-01- +B +20 - +0.38 - +-10-2, +-0 +0.310 : +140 - +F8-01 +NBV +l=2, m=0 modes +0.305 - +120 - +St=2 +l= 2, m= 2 modes +-10-7 +g-modes +0.300 - +100 +n,2,0 + propagation cavity +-10-6 - +→-. n,2,2 +p +I = 2 p-modes +）0.295 - +fn,0 +- 08 + propagation cavity +2 +p +10-5. +fn,2 +S +C 0.290 - + 09 +range between +-10-4 . +0.285 - +40 - +range between +fmi=? and fmax? +0.280 - +-10-3 _ +c +20 - +0.275 - +-10-2 - +-0 +0.2 +0.4 +0.8 +0.0 +0.2 +0.6 +80 +1.0 +0.0 +1.0 +0 +0.4 +2 +6 +r/R +r/R +Frequency (d-1)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +ple, when the oscillation spectrum lies completely outside the +frequency range allowed by the FNm coefficients or the nuclear +timescale of the secondary is much longer than the same time +scale for the primary. Under such circumstances, the secondary +component will remain close to the ZAMS until the termina- +tion condition is met. Thus, it will not significantly change its +internal structure and oscillation spectrum. This in turn means +that the oscillation spectrum will not move relative to the tidal +forcing frequencies, effectively reducing the number of possible +resonance events. +4.2.1. ‘Long’ resonances +Our sample of resonance curves includes a particular group of +L(t) curves that exhibit exceptionally long duration resonances +compared to typical ones (we will refer to them as ‘long res- +onances’). Figure 7 presents parts of three representative res- +onance curves belonging to this group. The shaded regions in +the figure mark the position of the long resonances. As can be +clearly seen, the typical resonance usually lasts for about 103 – +104 years, which is approximately 100 times shorter than the du- +ration of a long resonance (of the order of 105 – 106 years). They +originate from the intersection of one of the fNm frequencies with +the σnlm frequency at a very small angle, in terms of their tempo- +ral evolution. As a result, they remain for a relatively long time +in very close vicinity, leading to a broad resonance overlapping +with narrower ones (originating from other intersections of the +fNm and σnlm frequency spectra; cf. especially the middle panel +in Fig. 7). The long resonances are interesting for at least two +reasons. First of all, they are natural candidates for resonantly- +locked TEOs. However, based on our simulations, it is difficult +to say whether an extended resonance would persist when the +back-reaction of a TEO on the orbit is taken into account. Sec- +ondly, if the energy exchange between the eigenmode and the or- +bit that corresponds to a long resonance is not efficient (i.e. there +is a small chance that a long resonance will be lost), such a reso- +nance should lead to a high-amplitude TEO without the need for +resonance locking. This is simply because it has enough time to +reach its saturation level due to the non-linear effects. However, +we did not find any significant correlations between the occur- +rence of a long resonance in L(t) and the initial parameters of +our EEVs. +4.3. Total number of resonances and the average rate of +their occurrence +The first feature of the morphology of the resonance curves that +we investigated is the total number of resonances that occurred in +the primary and secondary components, Nres,1 and Nres,2. How- +ever, we did not calculate these statistics directly from L1(t) and +L2(t), because some of the apparent maxima may actually be a +blend of more than one resonance event. This is especially true +when the involved γnlm differ by orders of magnitude. Then one +of the resonances is characterised by a notably smaller maxi- +mum, which seems to ‘hide’ in the dominant one. Instead, we +used a different approach that did not underestimate the ac- +tual number of resonances. When post-processing the generated +models, we simply counted each intersection of the σn,2,0 and +σn,2,+2 frequency spectra with their counterparts fN,0 and fN,2, re- +spectively. The results of such an analysis are depicted in Fig. 8. +The most important thing about Fig. 8 is that it shows the +absolute number of resonances. EEVs can experience hundreds +or even thousands of resonances during their evolution on MS. +Fig. 7. Example resonance curves of the primary component for three +different EEVs from our simulations that exhibit long resonances (high- +lighted by the shaded areas in each panel). We note a substantial differ- +ence in the width of typical and long resonances. These long resonances +are good candidate for excitation of high-amplitude or resonantly- +locked TEOs. +It would therefore be wrong to claim that these phenomena are +rare in massive and intermediate-mass EEVs, although in gen- +eral, resonances are quite short-lived compared to the nuclear +time scale. The total number of resonances experienced by the +primary component (Fig. 8a) shows a correlation with both its +initial mass and the initial eccentricity of the system. The mildly +decreasing trend of Nres,1 towards higher M1 originates from the +fact that the mean lifetime of the star on MS shortens with in- +creasing mass. On the other hand, the wide range of Nres is +mainly due to differences in initial eccentricity. The closer the +system is to a circular geometry at the beginning of evolution, +the statistically lower the value of Nres, which is self-explanatory +and also applies to the secondary component (Fig. 8b). EEVs +that have managed to circularise their orbits in the MS phase +(magenta dots in Fig. 8c) have on average lower initial eccentric- +ities and thus fewer resonances. The opposite behaviour is exhib- +ited by EEVs in which the primary component has had a chance +to reach TAMS (green dots in Fig. 8c). The secondary compo- +nents experience a slightly fewer resonances compared to the +primaries (Fig. 8b) and there is no clear division of the Nres,2 dis- +tribution with respect to the termination criterion (Fig. 8d). The +noticeably smaller number of resonances for secondary compo- +nents with masses M2 < 5 M⊙ comes from the conditions of +our simulations, i.e. secondaries with these masses occur in sys- +Article number, page 13 of 24 + +107. +106. +L +105. +104- +22.0 +22.5 +23.0 +23.5 +24.0 +24.5 +25.0 +25.5 +106 - +L 105. +104→ +3 +4 +5 +6 +7 +8 +9 +105. +① +L +104- +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +t(Myr)A&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 8. Total number of resonances in the pri- +mary (a, c) and secondary (b, d) components +that we detected in our simulations as a function +of the initial masses of the components. The +initial eccentricity of the EEV is colour-coded +in panels (a) and (b), while the corresponding +scale is shown at the top of the figure. Pan- +els (c) and (d) are analogous to their counter- +parts in the top row, but the colours used reflect +the termination criterion (same as in Fig. 5). +The ordinate scale is logarithmically-scaled for +Nres > 10. Below this value, a linear scale was +applied in order to present components without +resonances, i.e. Nres,1 or Nres,2 equal to zero. +tems with decreasing mass ratios19. Hence, the large difference +in nuclear time scales between the components means that the +secondary component does not significantly change its eigenfre- +quency spectrum, resulting in a smaller number of resonances. +As we already mentioned in Sect. 4.2, some of the L1(t) and +L2(t) curves do not reveal any resonances, which is why they +lie in Fig. 8 on the horizontal line Nres = 0. This behaviour oc- +curred for only 0.07% of our primaries. They are all EEVs with +highly eccentric orbits that quickly filled their Roche lobes at pe- +riastron. There was much more such behaviour for secondaries, +about 7%, mainly for the intermediate-mass companions of the +much more massive primaries. +From an observational point of view, even more important +than the total number of resonances is the rate at which they +occur. Knowing this rate, for a given population of MS EEVs, +we can approximately say where we have a statistically higher +chance of observing TEOs. After all, our observations only cor- +respond to one particular moment in time, not the entire evolu- +tion. Knowing the values of Nres for each component and the age +of each system at termination, Tmax, we can calculate the average +rate of resonances as Rres ≡ Nres/Tmax. We show the distribution +of Rres,1 and Rres,2 in Fig. 9. It is very difficult to predict what the +dependence of Rres on the mass of the component will look like, +as it is the result of a complex interplay between many related +factors. On the one hand, it can be said that massive stars should +have a smaller Rres because their lifetimes are shorter and they +fill their Roche lobes relatively easily (in the considered range of +orbital parameters). On the other hand, however, massive stars +quickly change their internal structure (i.e. asteroseismic prop- +erties), so that their eigenfrequency spectra evolve rapidly, in- +creasing the likelihood of interaction with the structure of the +19 We recall that the minimum mass of the primary component consid- +ered in our study was equal to 5 M⊙. +tidal forcing frequencies. The question is, which of these pro- +cesses prevails? As can be seen in Fig. 9, it is the more massive +stars that are more likely to undergo resonances. Both primary +and secondary components with masses around 30 M⊙ have on +average an order of magnitude higher Rres (∼102 Myr−1) than +components with masses around 5 M⊙ (∼101 Myr−1, Fig. 9a and +b). Moreover, the dependence of the distributions shown in Fig. 9 +on the initial eccentricity and termination condition is inherited +from Fig. 8. +At this point, we can venture the conclusion that in the case +of MS EEVs, TEOs should be observed mostly in the upper part +of the MS (among early B- and O-type dwarfs), which still re- +quires observational verification on a large sample of massive +EEVs. Although we cannot extrapolate the obtained distributions +of Rres towards lower masses, these stars have an increasingly +extended convective envelope, which in turn should effectively +limit the photometric detection of g-mode TEOs. On the con- +trary, the envelopes of massive stars are radiative, which should +not prevent g-mode TEOs from propagating up to the vicinity +of the photosphere. Thus, they can be more easily detected by +analysing the light curves, especially in the era of high-quality +space-borne photometry. +4.4. Distribution of resonances over time +Since the average rate of resonances we have studied so far has +effectively obliterated any differences in the corresponding tem- +poral distribution, we can ask another important question: Are +there any distinctive moments in the evolution of the simulated +EEVs during which the systems experienced temporally higher +resonance rates? After visually inspecting hundreds of resonance +curves, we noticed that the aforementioned rate changes dramat- +ically in many cases (cf. the top panel of Fig. 3 as an example). In +Article number, page 14 of 24 + +e +0.3 +0.4 +0.6 +0.5 +0.7 +(a) +103 +102 +101 +0 +103 +102 +101 +RLOF of the primary +minimum eccentricity +primary at TAMS +maximum rate of rotation +0 +5 +15 +20 +25 +30 +5 +10 +10 +15 +20 +25 +30 +Mi/Mo +M2/MoKołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 9. Summary plots analogous to Fig. 8, but +showing the average rate of resonances occur- +ring in the simulated EEVs (average number +of resonances per Myr). Components that did +not exhibit any resonances during the simula- +tion have been omitted here as their Rres value +would simply be zero. The colour-coding is the +same as in Fig. 8. +order to compare the temporal distribution of resonance events +for the various EEVs we are dealing with, we performed this +type of analysis on subgroups of systems divided according to +the termination condition. We also normalised the time variable +by dividing it by Tmax of each resonance curve. This allowed +us to present the whole evolution of components on a convenient +and uniform interval, [0, 1]. Figure 10 shows the results obtained +for the primary components that have managed to deplete hydro- +gen in their cores. +Figure 10 also demonstrates that the distribution discussed +here is not uniform over time. Specific areas in this diagram are +clearly distinguishable. Nevertheless, this figure still contains in- +formation on the total number of resonances, which makes it +somewhat problematic to compare the shapes of these distribu- +tions for different masses of the components. We have, therefore, +prepared histograms of the times of resonances for five inter- +vals of the primary’s initial mass (every 5 M⊙). Separate sets of +histograms were generated for the primary and secondary com- +ponents and the three main termination conditions20. All his- +tograms are shown in Fig. 11. +The most diverse structure of the temporal distribution of res- +onances is shown by systems in which the primary component +has completed its evolution in our simulations at TAMS (Fig. 11a +and b). In fact, for all mass ranges, the distribution has two dis- +tinct maxima. The smaller of the two is located near the ZAMS, +while the other is just before reaching the TAMS. Their presence +can be explained by the rate of change in the stellar eigenspec- +trum, which is the highest (after averaging over all modes) at the +aforementioned moments of evolution. In particular, the rapid +changes in the radius of the star when it is close to complete de- +20 We did not prepare separate histograms for the EEVs, whose calcu- +lations were terminated due to the maximum allowed rotation rate. The +size of this group (only eight systems) was insufficient for this task. +Fig. 10. Time distribution of the resonances of the primary component +that reached TAMS. The abscissa axis corresponds to the normalised +time and the ordinate shows the initial mass of the primary compo- +nent. In addition, the ordinate is logarithmically scaled, so the set of +resonance curves is almost uniformly distributed in the vertical direc- +tion. The total number of resonances contained in one hexagonal bin is +colour-coded according to the scale on the right. +pletion of hydrogen in its core cause a very high ‘concentration’ +of resonances in the final MS phase. The height of this domi- +nant maximum decreases towards higher masses, but at the same +time it becomes wider and wider. The distribution for the sec- +Article number, page 15 of 24 + +3×101 +2×101 - +104 +Mi /Mo +101. +103 +6× 100 - +0.2 +0.4 +0.6 +0.8 +1.0 +t / Tmaxe +0.3 +0.4 +0.5 +0.6 +0.7 +102 +101 +0 +(a) +10-2 +102 +101 +109 +C +RLOF of the primary +minimum eccentricity +10-2 +primary at TAMS +maximum rate of rotation +5 +10 +15 +20 +25 +30 +5 +10 +15 +20 +25 +30 +Mi/Mo +M2/MA&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 11. Histograms of the normalised times of resonances occurring +in the primary (left column) and secondary (right column) components. +The consecutive rows (from top to bottom) correspond to EEVs satis- +fying different termination conditions, as labelled in panels (a), (c), and +(e). The colour of the histogram is related to the initial mass range of +the primary and is described in the legend in panel (b). We note that +the histograms on the right (corresponding to the secondaries) refer to +the different mass ranges of the primary component, not the secondary. +For example, the yellowish histogram in panel (b) summarises the be- +haviour of all secondaries of the systems with the primaries having mass +M1 > 25 M⊙, i.e. without distinguishing the mass ranges of M2. The +range of the ordinate axes is the same in each panel. +ondary components also reveals this kind of maximum near the +TAMS, which is particularly well pronounced for companions +of primaries with masses ≳ 25 M⊙. Given these facts, an inter- +esting conclusion can be drawn. Massive and intermediate-mass +EEVs with at least one component leaving the MS should expe- +rience an increased rate of encountered resonances. One might +therefore suspect that there is a statistically higher chance of ob- +serving TEOs in more evolved EEVs. The properties of the his- +tograms for EEVs in which RLOF eventually occurred at the +periastron (Fig. 11c and d) are very similar to the case described +above. The only evident difference between the two is the reduc- +tion in maximum of the distribution near TAMS. This is due to +the fact that the primary is likely to start the RLOF earlier than it +reaches the TAMS, preventing the occurrence of a large number +of resonances in a relatively short time, as mentioned earlier. +Eccentric systems that are subject to effective circularisation +(Fig. 11e and f) behave quite differently from the two previous +cases. They experience the vast majority of their resonance phe- +nomena at the beginning of evolution, and then reduce the num- +ber of resonances almost monotonically, as the orbital eccentric- +ity becomes smaller and smaller with time. Hence, the chance of +observing TEOs in initially relatively tight EEVs (cf. Fig. 5a) is +largest in the vicinity of ZAMS, which stays in contrast to the +systems described above. +4.5. Investigation of the morphology of resonance curves +using UMAP +All the analysis described above was based solely on the distri- +bution of resonances in time, i.e. neglecting the actual morphol- +ogy of the resonance curves, e.g. differences in the height and +width of resonance maxima, mean level of L(t), long-term trends +in L(t), etc. Using the dimensionality reduction techniques pre- +sented in Sect. 3.5, we constructed 2D UMAP embeddings of +the space of resonance curves in terms of their morphological +features. Figures 12 and 14 show the results obtained for the res- +onance curves of the primary and secondary component, respec- +tively. We recall that the idea of the low-dimensional embedding +performed here is to preserve the distances between two points in +the original space as accurately as possible, so that the distances +in the 2D plane reflect the distances in the full (original) space +of morphological features (the vector of 2,000 quantiles, Q). In +other words, a pair of distant points in Figs. 12 and 14 should cor- +respond to resonance curves with notably different morphologies +and ,vice versa, a pair of resonance curves with similar proper- +ties is expected to lie in mutual vicinity on the 2D UMAP plane. +Thanks to this key property of UMAP and many other dimen- +sionality reduction methods we can effectively explore the entire +space of resonance curve morphologies. +4.5.1. UMAP plane for primary components +We begin with a discussion of Fig. 12. Firstly, the presented 2D +embedding does not indicate the presence of any well-separated +groups among the resonance curves for the primary components. +This is an observation that is true over the entire range of differ- +ent values of the UMAP free parameters (Appendix D) as well as +for the different summary statistics of the resonance curves that +were considered during the preliminary experiments. The mor- +phology of the resonance curves changes smoothly depending +on to the initial parameters of the simulated EEVs. +Secondly, as can be seen in Figs. 12b and c, the initial +eccentricity and normalised periastron distance are parameters +strongly correlated with the overall morphology of the resonance +curves of the primary components. Moreover, their gradients in +the UMAP plane are approximately orthogonal. Therefore, the +pair of these parameters is the primary factor that determines +the shape of L1(t). The termination condition (Fig. 12e) gener- +ally follows the behaviour of �rperi except at small periastron dis- +tances, when the morphology remains similar but the simulations +were terminated due to hydrogen depletion in the primary’s core +or near-complete circularisation of the orbit. The initial mass of +the primary component (Fig. 12a) and its initial angular velocity +of rotation (Fig. 12d) are second-order factors shaping the mor- +phology of the L1(t) resonance curves. In the inner part of the +plane, M1 is distributed almost randomly. The clear exception is +Article number, page 16 of 24 + +(b) +Mi/Mo≤10 +(a) +5 +1025 +3 +2. +0 +(c) +(d) +5 +RLOF of the primary +0 +(e) +(f) +5 +minimum eccentricity +4 - +3 - +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +t/ TmaxKołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +the boundary of the plane which can be roughly divided into two +parts of mostly high or low initial mass of the primary. A simi- +lar conclusion can be drawn for the initial Ω1/Ωcrit,1. This time, +however, the upper right part of Fig. 12d reveals a well-defined +group of high initial Ω1 / Ωcrit,1 and�rperi. +It is difficult to include here a complete presentation of the +changes in the morphology of L1(t) as a function of their posi- +tion on the UMAP plane. Therefore, we only focus on some ex- +treme points to present some boundary cases. Figure 13 shows +examples of L1(t) from different areas of the morphological +plane. Primary components with lower masses and high initial +eccentricities are generally characterised by resonance curves +with high mean levels and a rich set of resonances, as shown +in Fig. 13a. The resonance curve depicted in Fig. 13b repre- +sents intermediate-mass fast-rotating primary with large initial +�rperi and low initial eccentricity. Here, the base level of L1(t) +increases by an order of magnitude and then the system expe- +riences a large number of resonances, during the evolution near +TAMS. The increase in the mean value of L1(t) is characteris- +tic of stars with a high initial rotation rates. Primary components +lying between points (a) and (b) generally do not manifest this +characteristic. Moving along a straight line on the plane from (a) +to (b), the increase in the number of resonances during the evo- +lution near TAMS becomes more and more apparent. Figure 13c +shows an example of a system with a small initial eccentricity +and a short periastron distance at ZAMS that is rapidly circu- +larising. As expected, the L1(t) resonance curves for such ob- +jects have a small number of resonance maxima and a low base +level. The case corresponding to the larger initial eccentricity is +shown in Fig. 13e, where the total number of resonances is much +greater. In this case, the evolution is mainly distinguished by a +decrease in the frequency of resonances with time, related to the +efficient circularisation of the orbit, and therefore a decrease in +the mean level of L1(t). Finally, the resonance curve in Fig. 13d +is representative of the most EEVs between points (c) and (d). +They are characterised by an approximately uniform distribution +of resonances over time and an almost constant base level of the +resonance curve. +4.5.2. UMAP plane for the secondary components +The situation for the secondary component (Fig. 14) is quite dif- +ferent from the previous case. The UMAP manifold obtained for +the set of L2(t) reveals slightly more complex structure than the +shape of embedding in Fig. 12. Since the time span of L2(t) is +largely determined by the mass of the primary component, the +resonance curves for the secondary components were terminated +at times not necessarily related to their actual evolutionary sta- +tus and are statistically shorter than they could be for primaries +of the same mass. For this reason, secondary components ex- +perience, on average, fewer resonances, but, at the same time, +their resonance curves can take more diverse forms compared to +L1(t). Undoubtedly, the main factor shaping the morphology of +the L2(t) is the initial eccentricity (Fig. 14c), which with the ex- +ception for two small areas, varies smoothly across the UMAP +plane. The other parameters (Fig. 14a, b and d) play a secondary +role, showing the complex and fine structures on the plane. As +can easily be seen in Fig. 14a, the extreme cases of L2(t) in terms +of their morphology belong almost exclusively to the EEVs with +intermediate-mass secondary components hat gather at the pe- +riphery of the plane. Some of these objects even form slightly +better separated groups, isolating from the central part of the +area. +Five limiting examples of resonance curves for secondary +components are shown in Fig. 15. Panels (a), (c) and (e) to- +gether with the area they approximately enclose contain res- +onance curves morphology very similar to that described for +primary components. The resonance curves belonging to the +‘clouds’ of points labelled as (b) and (d) in Fig. 15 are com- +pletely different. They are distinguished by the complete absence +of resonances during a certain period of the evolution of the +system. The differentiating feature of these cases is the disap- +pearance of resonances from some time to the end of evolution +(Fig. 15b) or the presence of resonances only around the middle +of the considered evolution time (Fig. 15d). +5. Summary and conclusions +In our paper, we aimed to investigate the temporal variation of +conditions that favour excitation of TEOs in EEVs with massive +and intermediate-mass MS components (Sect. 2) and see how +their picture changes with different initial parameters of the sys- +tem. In order to achieve this goal, we simulated the evolution of +20,000 EEVs using the MESA software in combination with the +GYRE stellar oscillations code (Sect. 3). Our calculations started +at ZAMS and were terminated if one of the conditions presented +in Sect. 3.2.3 was met. We considered only modes with l = 2, +m = 0, +2 because they are expected to be dominant TEOs. +We also assumed that all TEOs are due to chance resonances, +i.e. we neglected the effect of TEO on the orbit. Knowing the +evolution of the orbital parameters of simulated EEVs and the +temporal changes in the eigenmode spectra of the components, +we were able to derive resonance curves L1(t) and L2(t) defined +by Eqs. (4) and (3). The equations reflect the overall resonance +conditions, and thus indirectly also the chance of TEOs, sepa- +rately for the primary and secondary components of our simu- +lated EEVs. +After visually inspecting the obtained resonance curves, cal- +culating basic statistics for them and applying ML-based meth- +ods to the entire data set, our main results can be summarised as +follows. +1. Resonance curves are characterised by striking diversity in +terms of their morphology (Sect. 4.2). EEV components can +experience a very different number of resonances, and their +distribution over time can take various forms, including the +lack of resonances over a long periods of time. We also +distinguished a group of resonance curves that exhibit pro- +longed resonances, about two orders of magnitude longer +than typical (Sect. 4.2.1, Fig. 7). These long resonances +are the potential sources of high-amplitude and resonantly- +locked TEOs. +2. Resonances between tidal forcing frequencies and the spec- +trum of stellar normal modes are not rare events among mas- +sive and intermediate-mass MS EEVs (Sect. 4.3). Although +the total number of resonances depends mostly on the initial +orbital parameters, it is typically of the order of 102 – 103 for +a given system during the entire MS phase (Fig. 8). Let us +emphasise at this point that these numbers are rather lower +limits for the actual Nres in EEVs because we considered +only l = 2 TEOs. Taking higher degree modes into account +will certainly increase the reported values of Nres. +3. On average, the more massive a star is, the higher the rate of +resonances it experiences (Sect. 4.3). For the most massive +stars in our sample (≈ 30 M⊙), the average rate of resonances +can reach ∼ 102 Myr−1, which is approximately an order of +magnitude higher than for intermediate-mass stars (Fig. 9). +Article number, page 17 of 24 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +Fig. 12. 2D UMAP embedding of the manifold spanned by the morphological features of the resonance curves of the primary components. For +details on how to obtain the presented embedding, see Sect. 3.5. Panels (a) – (d) are colour-coded with respect to the initial parameters of the +simulated EEVs, as shown on the corresponding colour bars. The other initial parameters were omitted as they were not significantly related to +the location of the points on the presented map. The different colours of points in panel (e) correspond to the termination condition, as shown in +the legend on the right. The values on the abscissa and ordinate axes were omitted as they have no physical meaning. For clarity, the colour-coded +features have been averaged within the small hexagonal areas in each panel. A discussion of the figure can be found in Sect. 4.5. +Fig. 13. Variations in the morphology of the resonance curve for the primary component across the 2D UMAP plane from Fig. 12. The middle +panel in the bottom row shows the plane with colour-coding identical to that in Fig. 12a (without hexagonal binning). Panels (a) – (e), which +surround the area, show example resonance curves that correspond to the locations on the area masked with large red dots and labelled according +to the associated panel. The positions of points (a) – (d) have been chosen in such a way as to correspond to different extreme positions in the plain, +while point (e) refers to one of the intermediate cases. A discussion of the figure can be found in Sect. 4.5. +Article number, page 18 of 24 + +(a) +(b) +(c) + 5.0 +- 0.7 +25 + 4.5 + 0.6 + 4.0 +20 +1(M) +rperi + 3.5 +10.5 e +- +- 3.0 + 0.4 + 2.5 +10 +0.3 +2.0 +1.5 +(d) +(e) + 0.45 + RLOF of the primary +- 0.40 +0.35 +0.25 +- minimum eccentricity +- 0.20 +0.15 +: maximum rate of rotation(a) +(c) +105 +106 - +L +104 +105 +0 +20 +40 +60 +0 +8. +10 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +2 +6 +(b) 106 +(p) +LLF +(e) +105 +(a) +(d) +(e) +(C +103 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +t (Myr) +t (Myr)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +Fig. 14. Same as Fig. 12, but for a set of resonance curves of the secondary components. +Fig. 15. Same as Fig. 13, but for a set of resonance curves of the secondary components. +4. The distribution of resonances over time is not homogeneous +and depends primarily on whether the system circularises be- +fore the primary reaches the TAMS or RLOF occurs at the +periastron (Sect. 4.4, Fig. 11). We noticed a particular mo- +ment in the evolution of our EEVs near the TAMS, when the +components undergo an increased number of resonances in a +relatively short time (Fig. 11a and b). +5. The low-dimensional representation of the morphology of +the resonance curves, summarised by quantile-based statis- +tics and subsequently processed by UMAP, shows that its +manifold forms a rather smooth distribution without well de- +fined (separated) groups (Sect. 4.5, Figs. 12 and 14). Less +differentiated, at least in terms of the adopted method, are +the resonance curves of the primary components, for which +the initial eccentricity and the normalised periastron dis- +tance largely determine their morphological features. Al- +though secondary components experience far fewer reso- +nances, their shapes are generally more complex due to the +Article number, page 19 of 24 + +(a) +25 +(b) +(c) + 5.0 +- 0.7 + 4.5 + 20 + 0.6 +M2(Mo) + 4.0 +15 + 3.5 , +10.5 e +3.0 + 0.4 +10 + 2.5 +- 0.3 +2.0 +a +(e) + 0.45 + RLOF of the primary +- 0.40 +0.35 +crit, 2 + 0.25 +- minimum eccentricity +- 0.20 + 0.15 +- maximum rate of rotation106 +(a) +(b) +(c) +105 +105, +L2( +104 +104 - +20 +40 +60 +80 +100 +0 +2 +6 +8 +10 +(b) +106 +(p) +(e) +106 +2(t) +(a) +L +104 +105 = +(p) +0 +10 +20 +15 +0 +2 +4. +6 +8 +t (Myr) +t (Myr)A&A proofs: manuscript no. TEOs_in_massive_EEVs +predominant influence of the primary component on evolu- +tion time. +In light of the results obtained in our study, we can draw sev- +eral interesting conclusions. Firstly, statistically speaking, TEOs +are more likely to be discovered in more massive EEVs, as their +components have a higher average rate of resonances. This does +not necessarily mean that a higher absolute number of more mas- +sive EEVs exhibiting TEOs than less massive ones will be ob- +served21. However, when comparing two particular systems, one +with intermediate-mass components and the other with much +higher masses of the components, it is for the latter that we have +a statistically higher chance that some resonance is currently +underway there. Secondly, it seems that TEOs should be espe- +cially well visible in EEVs that contain a component approach- +ing TAMS. Given these facts, the ‘extreme-amplitude’ massive +EEV, MACHO 80.7443.1718 (Jayasinghe et al. 2021; Kołaczek- +Szyma´nski et al. 2022) fits this picture almost perfectly. Its pri- +mary component is a B0.5 Ib-II supergiant leaving the MS and, +more importantly, many high-amplitude TEOs have now been +detected in this extreme system. It is possible that what the pri- +mary component of MACHO 80.7443.1718 is currently under- +going corresponds to the resonance curve shown in Fig. 13b, i.e. +it is in the phase of a high resonance rate caused by relatively fast +changes in its radius and orbital parameters. Moreover, the am- +plitudes of TEOs observed in MACHO 80.7443.1718 vary over +time notably, suggesting that we may witness rapid changes in +the resonance conditions for the primary component of this par- +ticular EEV. It would therefore be very valuable to carry out +an observational study for a large sample of EEVs to verify +whether TEOs are common in massive and intermediate-mass +EEVs whose components have already depleted most of the hy- +drogen in their cores. With high-quality space-borne photometric +observations, both operational, such as the Transiting Exoplanet +Survey Satellite (Ricker et al. 2015) or BRITE-Constellation +(Weiss et al. 2014), and planned missions (e.g. Planetary Tran- +sits and Oscillations of Stars, Rauer et al. 2014), it is definitely a +feasible task. +The excitation of g-mode TEOs, which propagate deep in- +side the star, may be an underestimated mechanism for angular +momentum (AM) transport inside the components of EEVs. It +has long been suspected that self-excited oscillations and inter- +nal gravity waves22 efficiently redistribute AM in the radial di- +rection of the star (e.g., Rogers et al. 2013; Rogers & McElwaine +2017). Although the majority of our resonances have relatively +short durations (of the order of 103 – 104 years), they can be quite +frequent (especially near the TAMS), hence the question of their +contribution to AM transport and mixing processes becomes ur- +gent for the components of EEVs. Performing calculations that +would treat the evolution of the orbit, components and TEOs in a +fully self-consistent way seems particularly interesting for mas- +sive eccentric systems leaving the MS. +We have already entered the era of observational stud- +ies of distant star-bursting galaxies and stellar populations in +low-metallicity environments, that shaped the Universe in its +early epochs. Recalling that the metal-poor stars were much +more massive than their current metal-rich counterparts (e.g., +Hosokawa et al. 2013; Susa et al. 2014), we can ask what ef- +fect metallicity has on the occurrence of TEOs in massive EEVs +21 Due to the rapid decrease of the mass function towards larger stellar +masses (e.g., Chabrier 2003). +22 Gravity waves which are stochastically driven by the turbulent con- +vective motions near the interface of the convective core and the enve- +lope (see Bowman et al. 2020, for a recent review). +and how the results we presented depend on metals content. +Therefore, future studies of the importance of TEOs in massive, +metal-poor EEVs seems worthy further investigation, especially +because of the ongoing James Webb Space Telescope mission23 +(JWST, Gardner et al. 2006), which is certain to bring many dis- +coveries in the stellar astrophysics of early stellar populations, +including stars in eccentric binary systems. +Acknowledgements. PKS is indebted to his brother, Adam Karol Kołaczek- +Szyma´nski, who oversaw the purchase and assembly of a dedicated PC +workstation to enable the efficient calculation of models in MESA and GYRE. +Without his generous help, this project would literally never have been +completed. +The authors are thankful to Prof. Andrzej Pigulski for many important sugges- +tions and fruitful discussions that made this manuscript more comprehensible, +and to the anonymous referee for many inspiring comments that helped to +improve the manuscript. +PKS +was +supported +by +the +Polish +National +Science +Center +grant +no. 2019/35/N/ST9/03805. TR was partly founded from budgetary funds +for science in 2018-2022 in a research project under the program „Diamentowy +Grant”, no. DI2018 024648. Much of this work was developed and written +during IAU Symposium 361, „Massive Stars Near & Far”, held in Ballyconnell, +Ireland, 8 – 13 May, 2022. 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However, we have intentionally omitted any controls re- +lated to the names of the files or directories where the results +should be stored. All values that changed in the inlists depending +on the simulated binary system were enclosed in square brackets +– [ · ]. The parameters adopted below resulted in a typical num- +ber of about 2,400 zones in the radial direction of the star. The +number of models calculated per single EEV was typically of the +order of several hundred, mainly depending on the termination +condition. +A.1. MESAstar inlist +&star_job +!OUTPUT +history_columns_file = "my_history_columns.list" +profile_columns_file = "my_profile_columns.list" +show_log_description_at_start = .false. +save_photo_when_terminate=.false. +!MODIFICATIONS TO MODEL +new_rotation_flag=.true. +change_rotation_flag=.true. +new_omega_div_omega_crit=[Ω/Ωcrit] +num_steps_to_relax_rotation=100 +relax_omega_max_yrs_dt = 1d4 +relax_omega_div_omega_crit=.true. +set_initial_cumulative_energy_error = .true. +new_cumulative_energy_error = 0d0 +/ ! end of star_job namelist +&eos +/ ! end of eos namelist +&kap +use_Type2_opacities = .true. +Zbase = 0.02 +/ ! end of kap namelist +&controls +!SPECIFICATIONS FOR STARTING MODEL +initial_z=0.02d0 +!CONTROLS FOR OUTPUT +terminal_interval=100 +write_header_frequency=1 +photo_interval=100000 +history_interval=5 +star_history_dbl_format = "(1pes40.6e3, 1x)" +profile_interval=10 +max_num_profile_models=5000 +write_pulse_data_with_profile=.true. +24 Details of each keyword in MESAstar v. r15140 inlist can be +found at https://docs.mesastar.org/en/r15140/reference/ +star_job.html and https://docs.mesastar.org/en/r15140/ +reference/controls.html +25 Details of each keyword in MESAbinary v. r15140 inlist can be +found at https://docs.mesastar.org/en/r15140/reference/ +binary_job.html +and +https://docs.mesastar.org/en/ +r15140/reference/binary_controls.html +pulse_data_format="GYRE" +add_double_points_to_pulse_data=.true. +!WHEN TO STOP +max_model_number = 5000 +xa_central_lower_limit_species(1)="h1" +xa_central_lower_limit(1)=1d-4 +omega_div_omega_crit_limit=0.75 +!MIXING PARAMETERS +mixing_length_alpha=1.82d0 +use_Ledoux_criterion=.true. +num_cells_for_smooth_gradL_composition_term = 0 +alpha_semiconvection=0.01d0 +okay_to_reduce_gradT_excess=.true. +mlt_make_surface_no_mixing = .true. +overshoot_scheme(1)="exponential" +overshoot_zone_type(1) = "burn_H" +overshoot_zone_loc(1) = "core" +overshoot_bdy_loc(1) = "top" +overshoot_f(1) = [fov] +overshoot_f0(1) = 0.005 +do_conv_premix=.true. +set_min_D_mix=.true. +min_D_mix=1d5 +!ROTATION CONTROLS +am_D_mix_factor=0.0333333d0 +D_DSI_factor = 1 +D_SH_factor = 1 +D_SSI_factor = 1 +D_ES_factor = 1 +D_GSF_factor = 1 +!ATMOSPHERE BOUNDARY CONDITION +atm_option="table" +atm_table="photosphere" +!MASS GAIN OR LOSS +hot_wind_scheme="Vink" +hot_wind_full_on_T=1.2d4 +cool_wind_full_on_T=0.9d3 +Vink_scaling_factor=1d0 +no_wind_if_no_rotation=.true. +mdot_omega_power=0.43d0 +max_mdot_jump_for_rotation=5d0 +rotational_mdot_kh_fac = 1.0d3 +!MESH ADJUSTMENT +max_delta_x_for_merge = 0.01d0 +max_dq=1d-3 +min_dq=1d-16 +min_dq_for_split=1d-16 +!ASTEROSEISMOLOGY CONTROLS +num_cells_for_smooth_brunt_B = 0 +!STRUCTURE EQUATIONS +use_dedt_form_of_energy_eqn = .true. +!TIMESTEP CONTROLS +min_timestep_factor=0.5d0 +max_timestep_factor=2.0d0 +dH_div_H_limit=0.5d0 +delta_lgL_phot_limit = 0.05d0 +/ ! end of controls namelist +A.2. MESAbinary inlist +&binary_job +!OUTPUT/INPUT FILES +show_binary_log_description_at_start = .false. +binary_history_columns_file = +Article number, page 22 of 24 + +Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs +"my_binary_history_columns.list" +!STARTING MODEL +evolve_both_stars=.true. +change_ignore_rlof_flag = .true. +new_ignore_rlof_flag = .true. +/ ! end of binary_job namelist +&binary_controls +!SPECIFICATIONS FOR STARTING MODEL +m1=[M1] +m2=[M2] +initial_eccentricity=[e] +initial_period_in_days=-1 +initial_separation_in_Rsuns=[a] +!CONTROLS FOR OUTPUT +history_interval=5 +photo_interval=100000 +terminal_interval=100 +write_header_frequency=1 +!TIMESTEP CONTROLS +fa=0.02d0 +fa_hard=0.03d0 +fr=0.10d0 +fj=0.001d0 +fj_hard=0.005d0 +fe=0.02d0 +fr_dt_limit = 1.0d2 +fdm = 1d-3 +fdm_hard = 5d-3 +dt_softening_factor = 0.3d0 +varcontrol_ms=5d-4 +varcontrol_post_ms=5d-4 +dt_reduction_factor_for_j=5d-2 +!MASS TRANSFER CONTROLS +do_enhance_wind_1=.true. +do_enhance_wind_2=.true. +tout_B_wind_1 = [Bwind] +tout_B_wind_2 = [Bwind] +!ORBITAL JDOT CONTROLS +do_jdot_gr=.true. +do_jdot_ls=.true. +do_jdot_ml=.true. +do_jdot_mb=.false. +!ROTATION AND SYNC CONTROLS +do_tidal_sync=.true. +sync_type_1="Hut_rad" +sync_type_2="Hut_rad" +!ECCENTRICITY CONTROLS +do_tidal_circ=.true. +circ_type_1="Hut_rad" +circ_type_2="Hut_rad" +anomaly_steps=300 +/ ! end of binary_controls namelist +Appendix B: GYRE input file +The GYRE stellar oscillations code requires a single input file that +collects all the user-specified parameters of the asteroseismic +calculations being performed26. Below is our example file +gyre.in. As in Appendix A, we have omitted any keywords +related to specific file names and have highlighted variables by +26 Details of each keyword in GYRE v. 6.0.1 input file can be found +at +https://gyre.readthedocs.io/en/v6.0.1/ref-guide/ +input-files.html +enclosing them in square brackets. +&constants +/ +&model +model_type = "EVOL" +file_format = "MESA" +/ +&mode +l = 2 +m = 0 +n_pg_min = -30 +n_pg_max = 30 +tag = "m0" +/ +&mode +l = 2 +m = 2 +n_pg_min = -30 +n_pg_max = 30 +tag = "m2" +/ +&osc +inner_bound = "REGULAR" +outer_bound = "VACUUM" +adiabatic = .true.’ +nonadiabatic = .true.’ +/ +&rot +coriolis_method = "TAR" +Omega_rot_source = "MODEL" +/ +&num +ad_search = "BRACKET" +nad_search = "AD" +diff_scheme = "MAGNUS_GL2" +/ +&scan +grid_type = "LINEAR" +freq_min = [ f m=0 +min ] +freq_max = [ f m=0 +max ] +n_freq = [Nm=0 +freq ] +freq_units = "CYC_PER_DAY" +grid_frame = "INERTIAL" +freq_frame = "INERTIAL" +tag_list = "m0" +/ +&scan +grid_type = "LINEAR" +freq_min = [ f m=+2 +min +] +freq_max = [ f m=+2 +max ] +n_freq = [Nm=+2 +freq ] +freq_units = "CYC_PER_DAY" +grid_frame = "COROT_I" +freq_frame = "COROT_I" +tag_list = "m2" +/ +&grid +/ +&ad_output +/ +&nad_output +summary_file_format = "TXT" +Article number, page 23 of 24 + +A&A proofs: manuscript no. TEOs_in_massive_EEVs +summary_item_list = "freq,l,m,n_p,n_g,n_pg" +freq_units = "CYC_PER_DAY" +freq_frame = "INERTIAL" +The frequency scan limits, f m=0 +min , f m=0 +max , f m=+2 +min +, and f m=+2 +max , +were calculated as described in Sect. 3.3. The total numbers of +discrete frequency points, Nm=0 +freq , and Nm=+2 +freq , were obtained as +follows, +Nm=0,+2 +freq += ⌈( f m=0,+2 +max +− f m=0,+2 +min +)/(0.005 d−1)⌉, +(B.1) +where ⌈ · ⌉ denotes the ceiling function. +Appendix C: Data used by MESA +Our work uses the MESA stellar evolution code, which incorpo- +rates a vast compilation of knowledge, mainly from micro- and +macrophysics, collected by many authors. The MESAeos mod- +ule is a mixture of OPAL (Rogers & Nayfonov 2002), SCVH +(Saumon et al. 1995), FreeEOS (Irwin 2004), HELM (Timmes +& Swesty 2000), PC (Potekhin & Chabrier 2010) and Skye +(Jermyn et al. 2021) equation of states. Radiative opacities are +taken primarily from OPAL (Iglesias & Rogers 1993, 1996), +with low-temperature data from Ferguson et al. (2005) and +the high-temperature, Compton-scattering dominated regime by +Poutanen (2017). Electron conduction opacities are from Cas- +sisi et al. (2007) and Blouin et al. (2020). Nuclear reaction rates +are from JINA REACLIB (Cyburt et al. 2010), NACRE (An- +gulo et al. 1999) and additional tabulated weak reaction rates +from Fuller et al. (1985), Oda et al. (1994) and Langanke & +Martínez-Pinedo (2000). Screening is included via the prescrip- +tion of Chugunov et al. (2007). Thermal neutrino loss rates are +taken from Itoh et al. (1996). Roche lobe radii in binary systems +are computed using the fit of Eggleton (1983). +Appendix D: Adjustable parameters of the UMAP +UMAP, as a highly flexible method, is prone to returning mis- +leading results in the case of inappropriately set free parame- +ters. On the one hand, they can lead to the appearance of spuri- +ous groups and, on the other hand, to the loss of finer topologi- +cal structure. The vital UMAP parameters that need adjustment +are n_neighbors, min_dist, n_components and metric. The +n_neighbors parameter is the most important, as it controls +the balance between the local and global structure present in +the data that will be mapped to the embedding. We experi- +mented with different values of this parameter ranging from 5 to +1,000 (the default is 15) and concluded that the resonance curves +(summarised by the proposed statistics) always form a single +group, almost independently of the choice of n_neighbors. +Later, min_dist sets the minimum distance between two dif- +ferent points on the embedding. We tested its values from 0.0 to +0.5 and do not observe any significant effect on manifold. We +took the last two parameters, namely n_components that spec- +ifies the number of dimensions of the embedding, and metric +specifying the metric used for similarity calculation, as default +values. Finally, we used the following set of free parameters: +n_neighbors = 500, min_dist = 0.1, n_components = 2 +and metric = ’euclidean’. +Article number, page 24 of 24 + diff --git a/1NAyT4oBgHgl3EQf1PnY/content/tmp_files/load_file.txt b/1NAyT4oBgHgl3EQf1PnY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08ba6e1e8c83e7bf7c0e041edb4dd71d4c5e3a8b --- /dev/null +++ b/1NAyT4oBgHgl3EQf1PnY/content/tmp_files/load_file.txt @@ -0,0 +1,2369 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf,len=2368 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs ©ESO 2023 January 3, 2023 Theoretical investigation of the occurrence of tidally excited oscillations in massive eccentric binary systems P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Kołaczek-Szyma´nski and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Ró˙za´nski Astronomical Institute, University of Wrocław, Kopernika 11, 51-622 Wrocław, Poland e-mail: kolaczek@astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='wroc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='pl Received 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Revised 01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Re-revised 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Accepted 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Massive and intermediate-mass stars reside in binary systems much more frequently than low-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' At the same time, binaries containing massive main-sequence (MS) component(s) are often characterised by eccentric orbits, and can therefore be observed as eccentric ellipsoidal variables (EEVs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The orbital phase-dependent tidal potential acting on the components of EEV can induce tidally excited oscillations (TEOs), which can affect the evolution of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We investigate how the history of resonances between the eigenmode spectra of the EEV components and the tidal forcing frequencies depends on the initial parameters of the system, limiting our study to MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Each resonance is a potential source of TEO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We are particularly interested in the total number of resonances, their average rate of occurrence and their distribution in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We synthesised 20,000 evolutionary models of the EEVs across the MS using Modules for Experiments in Stellar Astro- physics (MESA) software for stellar structure and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We considered a range of masses of the primary component from 5 to 30 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Later, using the GYRE stellar non-adiabatic oscillations code, we calculated the eigenfrequencies for each model recorded by MESA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We focused only on the l = 2, m = 0, +2 modes, which are suspected of being dominant TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Knowing the temporal changes in the orbital parameters of simulated EEVs and the changes of the eigenfrequency spectra for both components, we were able to determine so-called ‘resonance curves’, which describe the overall chance of a resonance occurring and therefore of a TEO occurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We analysed the resonance curves by constructing basic statistics for them and analysing their morphology using machine learning methods, including the Uniform Manifold Approximation and Projection (UMAP) tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The EEV resonance curves from our sample are characterised by striking diversity, including the occurrence of exceptionally long resonances or the absence of resonances for long evolutionary times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We found that the total number of resonances encountered by components in the MS phase ranges from ∼102 to ∼103, mostly depending on the initial eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We also noticed that the average rate of resonances is about an order of magnitude higher (∼102 Myr−1) for the most massive components in the assumed range than for EEVs with intermediate-mass stars (∼101 Myr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The distribution of resonances over time is strongly inhomogeneous and its shape depends mainly on whether the system is able to circularise its orbit before the primary component reaches the terminal- age MS (TAMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Both components may be subject to increased resonance rates as they approach the TAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Thanks to the low- dimensional UMAP embeddings performed for the resonance curves, we argue that their morphology changes smoothly across the resulting manifold for different initial EEV conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The structure of the embeddings allowed us to explore the whole space of resonance curves in terms of their morphology and to isolate some extreme cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Resonances between tidal forcing frequencies and stellar eigenfrequencies cannot be considered rare events for EEVs with massive and intermediate-mass MS stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On average, we should observe TEOs more frequently in EEVs containing massive components than intermediate-mass ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs will be particularly well-pronounced for EEVs with the component(s) close to the TAMS, which begs for observational verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Given the total number of resonances and their rates, TEOs may play an important role in the transport of angular momentum within massive and intermediate-mass stars (mainly near TAMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' binaries: close – stars: early-type – stars: massive – stars: oscillations – stars: evolution – methods: numerical 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Introduction For many reasons, massive stars (≳ 8 M⊙) are of particular in- terest to modern astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Primarily, they are progenitors of core-collapse supernovae (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Janka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Smartt 2009) and long γ-ray bursts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Fruchter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For billions of years, they contributed to the chemical evolution of the entire Universe and interacted mechanically with the surrounding interstellar medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Ouellette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Svirski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2012), also through their intense line-driven stellar winds (Vink 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Furthermore, most of their remnants are compact objects, such as neutron stars (NSs, and among them magnetars and pulsars) and black holes (BHs), which allow the empirical study of effects of general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Finally, massive stars can be observed at cosmological distances due to their enor- mous luminosities, hence they dominate in the spectra of distant starburst galaxies (see Eldridge & Stanway 2022, for a recent re- view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' These features of massive stars (and many others) demon- strate that understanding the structure and evolution of massive stars is one of the key tasks of astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As is well known, massive and intermediate-mass (≳ 2 M⊙ and ≲ 8 M⊙) stars reside in binary systems much more fre- quently than their lower-mass counterparts (Duchêne & Kraus 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Moreover, as shown, for example, by Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2014) and Moe & Di Stefano (2017), O-type dwarfs in particular are often found in multiple systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This shows that binarity is inherent in the evolution of massive stars and cannot be ignored when studying these objects as well as their final out- comes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Many interesting phenomena in the Universe are the re- Article number, page 1 of 24 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='00733v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='SR] 2 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs sult of binarity among massive and/or intermediate-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' These include Be stars (Kriz & Harmanec 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Bodensteiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2020), so-called ‘stripped stars’ (Götberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' El-Badry & Burdge 2022), BH-BH/BH- NS/NS-NS mergers (progenitors of gravitational-wave events, Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2016, 2019), ‘early’ stellar mergers (Tokovinin & Moe 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022), Ib/c supernova pro- genitors (Langer 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2012), ‘massive Algols’ (de Mink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Skowron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022), and even Wolf-Rayet stars (Shenar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Pauli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' At the same time, binary systems that contain massive main- sequence (MS) component(s) are often characterised by eccen- tric orbits due to their relatively young age and the presence of radiative outer layers, which are less vulnerable to tidal dissi- pation compared to convective envelopes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Van Eylen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Both observational studies of large samples of massive bi- naries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Moe & Di Stefano 2017) and hydrodynamical sim- ulations of their formation (see Oliva & Kuiper 2020, and ref- erences therein) suggest significantly non-zero eccentricities at their birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Assuming that the periastron distance between the compo- nents is sufficiently small1, the combined proximity effects, such as ellipsoidal distortion, irradiation/reflection effect and Doppler beaming/boosting, make such a system an eccentric ellipsoidal variable (hereafter EEV, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Nicholls & Wood 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Due to the characteristic shape of the light curve of EEV during the periastron passage (which can resemble an electrocardiogram), EEVs are sometimes referred to as ‘heartbeat stars’ (Welsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Thompson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Beck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Kirk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Kołaczek-Szyma´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Wrona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The orbital phase-dependent tidal potential acting on the components of EEV can induce tidally excited oscillations (TEOs) in their interiors (Zahn 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Kumar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fuller 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Guo 2021), which in turn can affect the evolution of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, many details of TEOs in massive and intermediate-mass stars are still poorly understood including the total number of TEOs and their frequency of occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In our study, we aim to shed light on this issue based on theoretical modelling combined with machine learning (ML) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Section 2 provides a con- cise characterisation of TEOs and specifies the purpose of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3 we present a detailed description of the adopted methodology, including the assumptions made and the software used to generate the theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We then analyse the obtained models and present our findings in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Finally, we summarise the entire work and draw several conclusions in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Properties of TEOs and the purpose of the paper TEOs are tidally forced eigenmodes of a star with frequencies, σnlm (in the co-rotating frame of the star), coinciding with inte- ger multiples, N, of the orbital frequency, forb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The resonance condition can be written as follows: fNm ≡ N forb − m fs ≈ σnlm, (1) where fNm corresponds to the frequency of the tidal forcing in the rotating frame, fs stands for the rotational frequency of the star, while the subscripts n, l and m denote the radial order, de- gree, and azimuthal order of the specific eigenmode, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This property of TEOs makes them relatively easy to dis- 1 That is, of the order of a few radii of the larger component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2 We denote the corresponding orbital period as Porb = 1/forb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' tinguish from other types of pulsations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', self-excited oscilla- tions) in frequency spectra, provided the orbital period is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' There are numerous examples of photometrically or spectro- scopically detected TEOs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Handler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Welsh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Hambleton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Wrona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022a), also in massive binary systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Willems & Aerts 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Pablo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Kołaczek-Szyma´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Most TEOs are damped normal modes, mean- ing that without constant tidal forcing they would not be ob- served in the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' More importantly, because of their damped na- ture, TEOs dissipate the total orbital energy making the system tighter, more circular, and synchronised with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the other hand, if the TEO is naturally an overstable mode it can transfer thermal energy from the star to the orbit via so-called ‘inverse tides’ (Fuller 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Regardless of the type of TEOs, they unde- niably contribute to the evolution of the (massive) binary system, and can therefore influence the characteristics of the phenomena and objects mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The efficiency of energy transfer between the stellar interior and the orbit due to TEOs strongly depends on the temporal behaviour of the resonance condition given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It is to be expected that most TEOs are ‘chance resonances’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' resonances in which the aforementioned condi- tion is satisfied for a relatively short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Under such circum- stances, TEOs do not have enough time to reach high ampli- tudes, hence their ability to dissipate orbital energy is somewhat limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, if, after reaching resonance, both frequencies on the left and right sides of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (1) evolve at the same rate and direction, TEOs can ‘tidally lock’ for a longer time compared to the chance resonance scenario (Fuller 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This unique vari- ety of TEOs is suspected to be responsible for occasional peri- ods of rapid evolution of the orbital parameters in binary systems (Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs are are not only restricted to MS stars, they can also occur in binaries with pre-MS stars (Zanazzi & Wu 2021), some compact objects (white dwarfs, Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021), planetary sys- tems (Ma & Fuller 2021) and even planet-moon systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', in the Saturn-Titan system, Lainey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Although the literature on theoretical studies of TEOs is in- deed extensive (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Fuller 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Guo 2021, for recent re- views), the question of their rate of occurrence and the role they play in the evolution of massive stars is still a matter of debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Unfortunately, only a small number of papers refer exclusively to massive EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Witte & Savonije (1999a,b) studied gravity- (g) and Rossby-mode TEOs in an uniformly rotating 10 M⊙ MS star using their own two-dimensional (2D) code for different stellar rotation rates and several orbital configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' They found that dynamical tides can effectively circularise and tighten the orbits of EEVs in just a few Myrs if resonance locking occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' How- ever, these and many other previous works on TEOs were done under the assumption of a compact (point-like) secondary com- panion that is not subject to tidal perturbations during each peri- astron passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is obviously not the case in real binary sys- tems, where both components are responsible for the tidal evo- lution of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As theoretically shown by Witte & Savonije (2001), for an eccentric binary system consisting of two 10 M⊙ stars, tidal dissipation can be further enhanced due to the simul- taneous excitation of tidally-locked TEOs in both components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In spite of the advanced mathematical formalism, the aforemen- tioned papers only dealt with a few assumed component masses and sets of orbital parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Only Willems (2003) attempted a qualitative analysis of the hyperspace of the orbital parame- ters favouring excitation of TEOs in massive EEVs on MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' He found that for a mass range of 2 – 20 M⊙, the favourable orbital period interval lies between ∼5 and ∼12 d when both compo- Article number, page 2 of 24 Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs nents are zero-age MS (ZAMS) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This interval shifts towards longer orbital periods (up to ∼70 d) for components approaching terminal-age MS (TAMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Although, as argued above, the role of TEOs in the life of massive binary systems is still not well understood, we are not aware of any published work that develops the qualitative analy- sis carried out by Willems (2003) based on state-of-the-art stel- lar evolution and oscillations codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We would like to fill this gap by combining the Modules for Experiments in Stellar As- trophysics3 (MESA, Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2011, 2013, 2015, 2018, 2019) stellar structure and evolution code with the GYRE4 (Townsend & Teitler 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Townsend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2018) non-adiabatic stellar oscil- lation code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Our study aims to answer the following three ques- tions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' How many resonances (given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (1)) can EEVs experi- ence during their lifetime between ZAMS and TAMS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' How does this picture change with different initial parameters of the binary system?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Can we distinguish several distinct types of EEV resonance histories that are statistically related to the initial physical and orbital parameters of binary systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Does the resonance history correlate in any way with the properties of EEV near TAMS?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For instance, are systems that undergo mass transfer before reaching TAMS also sys- tems with a higher total number of resonances encountered?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In order to answer the last two questions, we use of ML tech- niques by performing a Uniform Manifold Approximation and Projection5 (UMAP, McInnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2018) dimension reduction analysis of the resonance histories obtained for simulated binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Methods Assuming that the dynamical tide excited in the component is dominated by a single TEO close to resonance with the orbital harmonic N, one can express its amplitude of luminosity change as proportional to (after Fuller 2017, his eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2) AN ∝ q �R a �l+1 |Qnlm|FNmLN, (2) where q is the ratio of the masses of the two components, R stands for the radius of the component in which the TEO is excited, and a is the semi-major axis of the relative orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The quantity denoted Qnlm is known as the so-called overlap integral, which describes the spatial coupling between a given oscillation mode and the actual geometry of the tidal potential (Fuller 2017, his eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In general, the larger the value of |n|6, the smaller the Qnlm, hence eigenmodes with a large number of nodes in the ra- dial direction have a much lower probability of tidal excitation7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In addition to Qnlm, the Hansen coefficient FNm (Fuller 2017, his eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5) is responsible for the temporal coupling of the forced normal mode and the Nth component in the Fourier expansion 3 https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='mesastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='org/en/latest/ 4 https://gyre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='io/en/stable/ 5 https://umap-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='io/en/latest/ 6 We use |n| instead of n because GYRE assigns negative values of n to g modes and positive ones to p modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 7 More precisely, Qnlm does not vary strictly monotonic with n and can change significantly between consecutive modes for given l and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, the overall trend of Qnlm peaks for low values of |n| and falls sharply for |n| ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For a more detailed discussion on the behaviour of Qnlm see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Burkart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' of the orbital motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Quantitatively, it expresses the intuitive principle that for more eccentric orbits, TEOs with larger orbital harmonic numbers will be excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is because, as the ec- centricity increases, the periastron passage takes less time for a given orbital period, so eigenmodes with higher frequencies bet- ter ‘match’ rapidly changing gravitational field, in terms of time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Nevertheless, for very high N, the FNm drops rapidly (al- most exponentially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This particular property of FNm is respon- sible for the lack of excitation of the TEOs with extremely high N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It is clear here that the frequency range of TEOs in massive and intermediate-mass MS stars is limited on two sides inde- pendently by Qnlm and FNm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the low-frequency side, Qnlm prevents the excitation of g modes with very high |n|, while on the high-frequency side FNm decreases sharply, strongly limit- ing the possible excitation of pressure (p) modes characterised by high radial orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' LN, denotes the detuning factor given by the following formula, LN = fNm � (σnlm − fNm)2 + γ2 nlm , (3) where γnlm stands for damping/growth rate of the normal mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This Lorentzian-like factor reflects the mismatch between fNm and σnlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Given the typical values of |γnlm| for g and p modes in massive and intermediate-mass MS stars (of the order of ∼ 10−7 – 10−3 d−1), LN is extremely sensitive to the difference (σnlm− fNm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Hence, among many other factors, the LN undoubt- edly plays a key role in the excitation of TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' While the precise prediction of TEO amplitude is a difficult task8, we are interested in analysing the changes in resonance conditions dictated by the sum of all the contributing detuning factors with passing time, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Let us define the following dimen- sionless quantity, L(t) ≡ � nlm Nmax � N=1 LN(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (4) In contrast to LN, associated with a single orbital harmonic, L reflects the overall chance of TEOs being excited in the EEV component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, we must stress at this point that it does not carry direct information on the amplitude of potential TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The first summation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (4) applies to all the normal modes we consider in the modelling (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Obviously, the second summation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (4) should run from N = 1 to +∞, but due to time and physical constraints one has to truncate the series at some reasonably chosen Nmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For a detailed description of the selection of Nmax values see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In order to try to answer the questions raised in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2, we have synthesised 20,000 binary evolution models and calculated L(t) for both components in each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The whole procedure is described extensively in the next four subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' General assumptions From a practical point of view,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' a fully consistent calculation of the evolution of binary systems taking TEOs into account is 8 In order to reliably predict the photometric amplitude of a TEO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' one needs to: (1) determine the exact value of LN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' which is almost impossi- ble given the uncertainties in both observations and stellar models,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2) calculate the corresponding Qnlm and (3) know the eigenfunction of lu- minosity fluctuations at the photospheric level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' which is a challenge for radiation pressure-dominated atmospheres of early-type stars (with in- tense stellar winds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In addition, the equilibrium amplitude of a linearly driven TEO is determined by various non-linear effects, for instance by multi-mode coupling (Guo 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 3 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs very time-consuming, as it requires time steps shorter than the times at which the resonances occur (several orders of magni- tude shorter than the nuclear time scale, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It would take an enormous amount of time to perform such consistent calcula- tions for 20,000 binaries with hundreds of resonances occurring in each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Therefore, to make our project both feasible and still scientifically useful, the models were synthesised un- der the general assumption that each resonance encountered by the EEV components is a chance resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' By sacrificing the ability to track resonantly-locked TEOs, we are able to decou- ple evolutionary and seismic calculations and run them indepen- dently, greatly simplifying the whole problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We believe that we can to do this for three reasons: (1) we are only interested in obtaining some general statistical information about the reso- nance conditions in a large number of simulated binary systems, (2) the phenomenon of resonance locking is rare compared to the rate of chance-resonance events, and (3) the impact of chance- resonance TEOs on the orbit is limited due to their relatively short time of existence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Witte & Savonije 1999b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In con- clusion, we focus on finding candidate binaries that may or may not experience numerous TEOs, rather than precisely predicting their actual evolutionary histories, which is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We believe that our results will serve as a starting point for more detailed calculations performed for the most in- teresting cases of massive EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Synthesis of binary evolution models Since we assumed that we could separate stellar and orbital evo- lution from seismic calculations, we first generated a set of bi- nary evolutionary tracks and only then performed seismic anal- ysis on them to find L(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Initialisation of models We used the latest open-source 1D stellar evolution code MESA (release 15140) compiled with the MESA Software Development Kit (version 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1, Townsend 2021) to compute a set of 20,000 binary evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The MESAbinary module (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2015) allows the simultaneous evolution of binary system com- ponents and their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Throughout this paper, we use the sub- scripts ‘1’ and ‘2’ to denote the primary (initially more massive) and secondary components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We assumed that both components have the same chemical composition with metallicity Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='02 and a solar-scaled mix- ture of elements taken from Grevesse & Sauval (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Since we were only interested in massive and intermediate-mass MS EEVs that can exhibit TEOs during their lifetime, the initial sys- tems consisted of two stars lying on the ZAMS and were charac- terised by parameters randomly drawn from the following uni- form distributions, U[α,β], on specific intervals [α, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' – Mass of primary component, log(M1/M⊙) ∼ U[log 5,log 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A uniform distribution on a logarithmic scale was used instead of a linear scale to cover the Hertzsprung-Russell diagram (HRD) with more evenly distributed evolutionary tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' – The mass ratio, q ≡ M2/M1 ∼ U[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='95], where M2 corre- sponds to the mass of the secondary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The lower limit for q was introduced due to the fact that the efficiency in driving TEOs scales with q (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2)), so it is less likely to observe TEOs in a binary system at a small value of the mass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Moreover, if the generated q corresponded to M2 < 2M⊙, a redraw was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' – Eccentricity, e ∼ U[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Range typical of the observed EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' – Periastron distance, rperi, normalised to the sum of compo- nents’ radii, �rperi ≡ rperi/(R1 + R2) ∼ U[1,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, if the generated system was initially Roche-lobe overflow- ing (RLOF) at the periastron, a redraw was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We also assumed an upper value of �rperi because the overall strength of tidal forces decays as r−3 peri and simulating widely- separated systems would contradict the aim of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' – Tidally-enhanced wind factor, Bwind ∼ U[32,896].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Introduced by Tout & Eggleton (1988) for red giants residing in binary systems, it accounts for the tidal enhancement of the stellar wind mass-loss rate due to the presence of a nearby com- panion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The ad hoc chosen range of Bwind corresponds to a maximum amplification of the ‘nominal’ wind mass-loss rate by a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 – 10 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Tout & Eggleton 1988, their eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' – The angular rotational velocity divided by its critical value9, Ω/Ωcrit ∼ U[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The assumed range of initial Ω/Ωcrit translates into the linear equatorial velocities between ∼50 km/s and ∼320 km/s in our simulations and reflects the significant non-zero rotation velocities observed in massive young MS stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Dufton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' – The overshoot mixing parameter, fov ∼ U[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='015,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='025].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In our calculations, the overshooting of the material above the con- vective, hydrogen-burning core was treated in the exponen- tial diffusion formalism developed by Herwig (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' An ad- justable parameter, fov, describes the spatial extent of the overshoot layer in terms of the local pressure-scale height, but its value for massive stars is still under debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We adopted the range of fov after Ostrowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The parameters presented above were generated independently for each EEV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Moreover, the last two parameters, Ω/Ωcrit and fov, were drawn independently for each component, so the final hyperspace of parameters explored in our simulations in- cluded {M1, q, e,�rperi, Ω/Ωcrit,1, Ω/Ωcrit,2, fov,1, fov,2, Bwind}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fig- ure 1 shows our initial sample of generated EEVs in the orbital period versus eccentricity diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As expected, they occupy the upper envelope of the aforementioned plane with the upper boundary dictated by the onset of periastron RLOF on ZAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The rest of the necessary parameters and ‘physics switches’ were identical for each simulated binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We will now briefly describe them below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Integration of the evolution Nuclear reaction rates were calculated using ‘basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='net’ op- tion in MESA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We used a convective premixing scheme (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2019, their Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5) in combination with the Ledoux cri- terion to define the boundaries of convective instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This specific approach of treating convection agrees with the results of modern 3D hydrodynamic simulations (Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Convective mixing was incorporated into the models via mix- ing length theory (MLT) in the ‘Cox’ formalism (Cox & Giuli 1968, their chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 14) with the value of the solar-calibrated mix- ing length parameter αMLT = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8210 (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As 9 By critical rotational velocity we mean the situation when the effec- tive gravity at the stellar equator is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' the centrifugal force and the Eddington factor, Γ, balance the true gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' MESA estimates this quan- tity as Ωcrit = � (1 − Γ)GM/R3, where G is the gravitational constant and Γ ≡ Lrad/LEdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The Lrad and LEdd denote the radiative luminosity and Eddington luminosity of the star, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 10 There is some evidence that αMLT may depend on global stellar pa- rameters such as mass (Yıldız et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2006) or metallicity (Viani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 4 of 24 Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Distribution of initial orbital period and eccentricity for a sample of 20,000 binaries evolved in our project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The initial normalised sepa- ration at the periastron is colour-coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The upper left-hand corner cor- responds to the ZAMS EEVs, which experience RLOF at the periastron and should therefore rapidly circularise their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The lower right- hand corner, on the other hand, is where relatively widely-separated binaries can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' mentioned earlier, exponential overshoot mixing above the con- vective core was also included11, but we neglected overshoot- ing in the non-burning convection zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For stars with masses ⩾ 15 M⊙, we activated the treatment of convection as ‘MLT++’ (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2013, their Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2) to reduce superadiabaticity in convective zones dominated by radiation pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Since we used Ledoux criterion, semiconvection could appear in our stars with its efficiency parameter, αsc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='01 (Langer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In our case, semiconvection sometimes occurred in chemically- modified layers left by the shrinking core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Upon initialisation at the ZAMS, we relaxed both compo- nents in ∼100 steps so that they rotated rigidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Later, we al- lowed our stars to rotate differentially during their evolution, according to the so-called shellular approximation of rotation (Meynet & Maeder 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Throughout the entire evolution, we assumed that the rotation axes of the stars are perpendicular to the orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' MESAstar uses the mathematical formalism of Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2000) and Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2005) to apply struc- tural corrections, perform different types of rotationally induced mixing and „diffusion” of angular momentum between adjacent shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The following rotational mixing mechanisms were taken into account in MESA: dynamic shear instability, secular shear instability, Eddington-Sweet circulation, Solberg-Høiland insta- bility, and Goldreich-Schubert-Fricke instability (all described in detail by Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Even the combined mixing coef- ficients of the aforementioned rotational instabilities can be zero in some parts of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, this is clearly unrealistic due to the presence of a nearby companion that induces additional mixing throughout the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' To at least approximately account for this process, we did not allow the total mixing coefficient, Dmix, to fall below 105 cm2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This particular arbitrarily-selected value is related to the mixing time scale, τmix ∼ (∆r)2/Dmix ≈ 15 Myr at radial distance, ∆r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 R⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We cannot conceal here that ro- 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It is also very likely that αMLT is sensitive to the evolutionary stage of the star and the type of convection zone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Here, we have assumed a constant value of αMLT for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 11 Similarly to the αMLT, fov also can depend on different stellar param- eters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' tation and mixing profiles in MS stars are still poorly understood (except in the solar case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' There are no definitive conclusions as to what mixing mechanisms and whether they actually occur in massive and intermediate-mass MS stars (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Pedersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Pedersen 2022, for a discussion of this problem and its asteroseismic inference from B-type dwarfs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Mass losses due to the radiation-driven stellar wind were calculated according to the prescription given by Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Their formulae take into account the presence of a so-called bi-stability jump around the effective temperature of Teff ≈ 26,000 K, caused by ionization and recombination of some Fe ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Nevertheless, the presence of a bi-stability jump is still questionable and there is some evidence that the associated al- most instantaneous change in the mass-loss rate may not be real (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Krtiˇcka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Björklund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' ‘Nominal’ wind mass-loss rates in our simulations were modified in two ways: (1) the rate was amplified by the aforementioned tidal mech- anism, parametrized by Bwind (Tout & Eggleton 1988) and (2) the effect of fast rotation at the surface, which can amplify the mass-loss rate, was accounted for by the simplified power-law description given by Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2000) (their Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We assumed that the mass loss through the wind is completely non- conservative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' there is no mechanism that could transfer some material back to the star or to a companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As we already mentioned above, MESAbinary allows the si- multaneous integration of some stellar and orbital parameters that are coupled to each other in a binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We have switched on the MESA controls responsible for changes in the to- tal orbital angular momentum caused by: (1) gravitational wave radiation, (2) wind mass loss and (3) tidal spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For the first process, the rate of orbital momentum loss was cal- culated assuming point masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The mass loss through the stellar wind was completely non-conservative, so the angular momen- tum lost via this channel was equal to the angular momentum carried by the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The phenomenon (3), contributing to the evolution of eccentricity, orbital and spin angular momenta, was modelled using the theory of tidal interactions for radiative en- velopes developed by Zahn (1977), Hut (1981) and Hurley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2002), after being adapted to the shellular approximation of ro- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For stars with radiative envelopes, tidal dissipation pro- cesses are dominated by tidally excited gravity modes that prop- agate to the stellar surface, where they gain relatively large am- plitudes and experience effective radiative damping (due to the short local thermal time scale) and nonlinear damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Conse- quently, they deposit their energy and angular momentum in the outer layers of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Following earlier calculations of Zahn (1977), Hurley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2002) delivered convenient formu- lae to describe the tidal synchronisation and circularisation time scales associated with the aforementioned phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Using these time scales combined with the formalism presented by Hut (1981), MESAbinary integrates the evolution of the eccentricity and updates spin angular frequency of each shell in the stellar model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Therefore, our calculations in MESAbinary took into ac- count the approximate influence of the dynamical tide on the orbit, at least up to the lowest possible order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Of course, the tidal evolution formalism implemented in MESAbinary does not in- clude the effects of resonance locking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For explicit formulae de- scribing tidal processes in MESAbinary, we refer to Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2015) (their Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We have completely ignored the effects of magnetic fields, while bearing in mind that they may mainly affect the actual stel- lar wind mass-loss rates, the efficiency of internal mixing pro- cesses and synchronisation/circularisation time scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' via the magnetic braking mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The impact of fossil mag- Article number, page 5 of 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 RLOF on ZAMS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6- 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 - wide even 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 100 101 102 Porb (d)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs netic fields on the evolution of massive and intermediate-mass stars was recently described by Keszthelyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' All details on the parameters of our models in MESA can be found in Appendix A, where we present the contents of our MESA inlists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A concise description of the micro- and macrophysics data sources used by MESA is provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Termination conditions The evolution of the binary system was carried out until at least one of the following termination conditions was met for any of the components: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The component reached TAMS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' the central mass abun- dance of hydrogen fell below Xc ⩽ 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The eccentricity was reduced to e ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The rotation velocity reached Ω/Ωcrit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='75 at the stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Episodic mass transfer between components due to the RLOF in the periastron began.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The reasons behind providing the termination conditions out- lined above are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Our study is exclusively dedicated to the MS phase of the evolution of EEVs, hence the first condi- tion has to be enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The second condition is self-explanatory, since we are interested in non-zero eccentricities that allow for TEO excitation12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The third condition is related to the conver- gence problems that can occur in MESAbinary when one of the components nearly approaches the break-up velocity of rota- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Numerous assumptions and descriptions of rotation-related phenomena reach the limits of their applicability in MESA for Ω/Ωcrit ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Since for Ω/Ωcrit ≳ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='75 the deviation from spherical symmetry becomes significant, a 1D treatment of the problem is no longer adequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For instance, the way in which such a star loses mass becomes fundamentally different from the isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We have therefore decided to stop integrations un- der such circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The last condition is related to the diffi- culty in correctly describing an episodic (near-periastron) RLOF, when a ‘blob’ of material could be ejected from the RL-filling component during each periastron passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, this kind of orbital phase-dependent RLOF is not expected to be observed in a binary for a long time due to strong tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' They should effectively suppress the eccentricity, making the system circular (and so the second condition can be quickly met).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Asteroseismic calculations A consequence of the assumption of the aligned vectors of the or- bital and spin angular momenta is a rule for selecting the geome- try of modes that can be tidally excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Under such conditions, a normal mode can be tidally excited only if mod (|l+m|, 2) = 0, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' the l = 2 TEOs will be characterised only by m = −2, 0, +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Here we restrict our study to only l = 2, m = 0, +2 modes be- cause of two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' First, l = 2 modes correspond to the dom- inant component in the series expansion of the variable tidal po- tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Modes with l > 2 undergo much weaker excitation due to the dependence on (R/a)l+1, which enters Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Second, the values of FN,−2 are very small compared to their m = 0, +2 counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This can be easily seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2a, where we have 12 In theory, components of circular systems (e = 0) can also exhibit TEOs, provided they do not rotate synchronously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, the number of modes observable as TEOs in these systems is much smaller than the number of potential TEOs in EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' plotted the maximum values of FNm for m = −2, 0, +2 and dif- ferent eccentricities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' FN,−2 is approximately at least 2 – 3 orders of magnitude smaller than FN,0 or FN,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For each model of the stellar internal structure that was saved during the synthesis of binaries in MESA, we calculated the oscil- lation spectrum using the GYRE code in the non-adiabatic regime and the second-order Gauss-Legendre Magnus integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The frequencies σn,2,0 and σn,2,+2 corresponding to the non-adiabatic calculations were searched by GYRE based on the preliminary adiabatic calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Rotational effects (Coriolis force) were taken into account using the so-called traditional approximation of rotation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2010, their Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We searched for eigenvalues in the family of gravito-acoustic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We assumed the necessary (differential) rotation profile inside the star from the MESA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As we argued in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3, it is necessary to choose a rea- sonable range of frequencies to scan for eigenvalues based on Qnlm and FNm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Therefore, we only searched for modes with |n| ⩽ 30 and frequencies, σn,2,0 ∈ (forb, Nm=0 max forb) or σn,2,+2 ∈ (max{0, forb − 2fs,core}, Nm=+2 max forb − 2 fs,core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In the ranges shown, Nm=0 max and Nm=+2 max refer to the limits of N due to the decrease in FN,0 and FN,2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The fs,core is the core rotation fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We defined Nm=0 max and Nm=+2 max as N for which FN,0 or FN,2 is equal to 10−8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' FNm starts to effectively prevent excitation of TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In practice, we numerically calculated the FNm func- tions13 for different eccentricities and obtained the log Nm max(e) relations as a fit of a fourth-degree polynomial to a set of its dis- crete points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A summary of this process is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For low-e orbits, the typical range of N favourable for the excitation of TEOs reaches N ∼ 101, in contrast to highly eccentric orbits, which may exhibit as much as N ≈ 100 – 200 TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 2b also shows another feature of m = −2 modes that makes them in- ferior candidates for TEOs compared to axisymmetric and pro- grade modes – as potential TEOs they always span a narrower range of orbital harmonic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Defining the frequency range for σn,2,0 is quite straightfor- ward, as these are axisymmetric modes and their frequencies do not change when switching between inertial and co-rotating frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The situation is quite different when it comes to the m = +2 modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This time, due to the differential rotation inside the star, σnlm = σnlm(r) = σnlm − mfs(r), where r is the radial co- ordinate in the stellar interior and σnlm is oscillation frequency in the inertial frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For some eigenmodes, σnlm may change its sign somewhere in the star, depending on the shape of the rota- tional profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This location is known as the critical layer, where σnlm(r) → 0, and such a mode experiences severe damping due to its very short radial wavelength (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Alvan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' To exclude these modes from our experiment, the maximum fre- quency of σn,2,+2 was set to (Nm=+2 max forb − 2 fs,core)14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is be- cause during evolution the core rotates almost rigidly and faster than the envelope, hence the difference (Nm=+2 max forb − 2 fs,core) ⩽ (Nm=+2 max forb−2fs,env), where fs,env stands for the rotation frequency of the outermost part of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' More details of our calculations performed in GYRE can be found in Appendix B, where we present the explicit contents of our GYRE input file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 13 Using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5 presented by Fuller (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 14 We note that this frequency is expressed in a rest frame co-rotating with the stellar core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 6 of 24 Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (a) Maximum values of the Hansen coefficients FNm versus eccentricity for l = 2 modes and three different azimuthal orders, m = −2, 0, +2 denoted by blue, green, and red points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We note the marginal contribution of the m = −2 modes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (b) Dependence of log Nm max on eccentricity with the same colour-coding as in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The colour solid lines represent the best fits of the fourth-degree polynomi- als, which we used to determine frequency ranges in the asteroseismic calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Derivation of L(t) The introduction of differential rotation also has consequences when it comes to interpreting the resonance condition from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The quantity fs is no longer a constant value, so one has to decide which fs to choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Theoretical studies imply the in- duction of g-mode TEOs (especially of high radial order) primar- ily near the convective core boundary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Goldreich & Nichol- son 1989) for stars with radiative envelopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, the res- onances in our simulations are also due to p or g modes with small radial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Therefore, we decided to apply our resonance condition to the envelope15 (not to the interface region near the core boundary), rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (1) more accurately as fNm ≡ N forb − m fs,env ≈ σnlm, (5) and use it in the subsequent modelling of L(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It is essential to note at this point that the resonance condition given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (5) refers to fNm and σnlm expressed in a frame co-rotating with the outer stellar envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Although in principle the morphol- ogy of L(t) depends on the choice of the specific resonance con- dition, we note that it does not affect at all resonances due to 15 In the exact approach, different resonance conditions would have to be used for different modes, depending on the radial coordinate inside the star where a given TEO is dominantly excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Here we assume a single form of resonance condition for all modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' m = 0 modes and should not significantly affect resonances cor- responding to p modes or low-|n|, m = +2 g modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Having a set of eigenfrequencies calculated by GYRE and knowing the history of the binary evolution from MESA, we per- formed the summation shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, this was not a direct summation running over the models saved by MESA and GYRE, as their temporal resolution was still too coarse compared to the duration of a typical resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' To circumvent this prob- lem, we interpolated the temporal variations of each oscillation frequency and all necessary parameters of the binary system us- ing Akima cubic spline functions (Akima 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Then, we were able to calculate the values of L(t) on a uniformly-spaced time grid with a constant time step of 2,000 years, which we assumed to be identical for all EEVs in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' From here on, we will use the term ‘resonance curve’ as a proxy for the L(t) time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 3 shows a compilation of example resonance curves, although we postpone discussion of these to Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' To- gether with the initial parameters of binary systems, resonance curves are particularly important to us in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' ML analysis of the resonance curves Although in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 we analyse the resonance curves based on various statistics, due to their global nature we do not distinguish many details that are ‘hidden’ in the resonance curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' To characterise the morphology of all resonance curves in more detail (without having to perform a visual classification, which is almost impossible due to the number and complexity of the data set), we applied dimensionality reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' With these, we were able to explore the topology spanned by the morphological features of the resonance curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We carried out the entire analysis described here separately for the sets of curves L1(t) and L2(t), corresponding to the primary and sec- ondary components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As a first step, we summarised each resonance curve with a vector Q that described its morphological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We focused our attention on two particular features: (1) the distribution of log(L) values and (2) the distribution of apparent maxima at a normalised time, t/Tmax, where Tmax stands for the max{t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In practice, we calculated vectors Qx and Qy which contained sets of 1,000 quantiles of normalised times corresponding to local maxima of L(t) and 1,000 quantiles of log(L), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The levels of both calculated quantiles were spanned evenly from 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Qx describes the overall distribution of apparent maxima in time, reporting changes in the rate of resonance occurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We deliberately used normalisation by Tmax because we want the results to be sensitive only to the relative distribution of the resonance events over the lifetime of the EEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Otherwise, its val- ues would be strongly correlated with the length of the resonance curve itself16, rather than with the distribution of resonances over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the other hand, Qy encapsulated the combined informa- tion about the mean level of log(L), any long-term trends in the resonance curve and the distribution of the heights of the max- ima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In contrast to the Qx, we did not apply any normalisation to Qy as its absolute values carry valuable information about the strength of the resonances and the average level of the entire resonance curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The final vector Q was constructed as the con- catenation of Qx and Qy, which had previously been scaled using the variance in the sets of all Qx and Qy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The resulting Q has a total of 2,000 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 16 Which in turn is an almost a direct approximation for the mass of the primary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 7 of 24 0 log (max[FNm )) 6 8 a) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 m=+2 m=0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 m=-2 max 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 eA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Sample of resonance curves obtained as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Each panel corresponds to a different arbitrarily-chosen binary system with the rounded values of their initial parameters given on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The dark red and blue curves reflect the behaviour of L(t) for the primary and secondary components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For clarity, L2(t) has been shifted vertically by three orders of magnitude downwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Time t = 0 indicates ZAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A sudden break in L2(t) on the bottom panel (after about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 Myr) indicates L2 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' the absence of any resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The differences in the height of the peaks are due to different values of γnlm and min{|σnlm − fNm|} for excited TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 8 of 24 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6Mo M1 M2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0M 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='32Qcrit 21 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='31Qcrit fov,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='022 fov, 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='019 primary secondary 103- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='52 e Tperi 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='23 Bwind 583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='9 101- 2 0 4 6 8 10 12 M1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8Mo M2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2Mo 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='11 2crit 21 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='162crit fov, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='016 fov,2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='015 103- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='29 e uody 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='53 Bwind 828.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' C2 / 103 10 20 25 5 15 30 L 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 Mo M1 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' M2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 Mo L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='43 2crit 21 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='44 2crit 22 fov,1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='022 104 fov,2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='62 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Tperi 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='32 Bwind 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2 3 4 5 6 0 M1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8Mo M2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5Mo 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='24 Qcrit 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='18Qcrit 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='018 fov, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='015 fov,2 103- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='31 e Tperi 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='68 102 Bwind 893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 101- 5 6 0 3 4 8 t (Myr)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs In the next step, we performed a preliminary dimensional- ity reduction of Q by means of the Principal Component Analy- sis (PCA, Pearson 1901), obtaining pre-processed ‘morphology’ vectors, θPCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' PCA is a method that orthogonally projects the data into a coordinate system in which successive vector com- ponents explain a smaller and smaller part of the data variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The target number of its dimensions returned by PCA for each Q was set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This value was chosen experimentally by ex- amining the percentage of the total variance of the data set ex- plained by successive PCA components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For L1(t), the first ten PCA components explained a total of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8% of variance (first component – 79% and second component – 19%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For L2(t), the corresponding value was 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5% (in this case, the first PCA com- ponent explained 60% of the total variance, while the second explained 22%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We then performed the final 2D embedding by applying UMAP on the collection of θPCA vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' UMAP is a multi- purpose non-linear dimensionality reduction technique that con- structs a low-dimensional projection that preserves as accurately as possible the topological structure of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For in- stance, in this case, a pair of embeddings of resonance curves with similar properties (in the sense of their summary statistics described above) are expected to lie in mutual vicinity on the 2D UMAP plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Let us denote the UMAP results as θUMAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The manifold spanned by θUMAP (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5) allowed us to ef- fectively examine the differences in the morphology of the res- onance curves and their dependence on the initial parameters of the simulated EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Unlike PCA, UMAP is a complex method, with many free parameters that need to be adjusted with care, as the resulting embedding may depend heavily on their choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Appendix D provides all the ‘technical’ details of this process, including the values of the most important UMAP parameters adopted in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' General properties of synthesised EEVs Before going into a detailed analysis of the resonance curves, we briefly characterise the general properties of the models we have synthesised using the MESAbinary and GYRE codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Evolutionary tracks in HRD Figure 4 shows a pair of HRDs with a compilation of all 20,000 evolutionary tracks that we obtained in our simulations for the primary (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4a) and secondary (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4b) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Although it is impossible to clearly present thousands of evolutionary tracks on a single HRD, we have highlighted and colour-coded a small fraction of them in order to describe some of their features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' First of all, only a fraction of the primaries reached TAMS when the central mass abundance of hydrogen dropped be- low 10−4 (according to the first of our termination conditions, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Many evolutionary tracks were interrupted at MS due to the fulfilment of one of the other termination conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Secondly, a number of tracks clearly change their character after crossing the line corresponding to the bi-stability jump (around Teff = 26,000 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is due to the associated sharp increase in the wind mass-loss rate, as it tries to keep the stellar luminosity constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In some circumstances, the mass-loss rate is so high that the star loses a significant part of its envelope17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This effect 17 We recall that these high mass-loss rates are not exclusively derived from the description of Vink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Rotational and tidal amplifi- ‘pushes’ the star back to the high effective temperature region and is particularly pronounced for the most massive stars in our sample (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4a, evolutionary tracks that ‘turn around’ and cross the bi-stability jump for a second time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' EEV groups in terms of the termination condition Only four of the seven18 termination conditions actually oc- curred in our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The majority of our EEVs (∼67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 %) ended up as MS RLOF systems in which the primary compo- nent filled its Roche lobe during the periastron passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The next most numerous group (∼22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 %) were systems in which the primary component successfully reached TAMS (Xc ⩽ 10−4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' About 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 % of the binaries managed to circularise their orbits before any other termination condition was met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The last group contains only about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='04 % of the total sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is the group where the primary’s rotation velocity exceeded the maximum al- lowed angular velocity (Ω / Ωcrit ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 5 presents these four groups of EEVs on the Porb-e plane and allows a comparison of the initial (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5a) and final (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5b) states of the aforementioned distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As expected, the EEVs with the shortest orbital periods and high eccentricities tended to circularise their orbits before leaving the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Their trajectories in the Porb-e diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5c) follow smooth, almost vertical lines due to the strong tidal damping of eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the other hand, the integration of the evolution of systems with large distances between components at periastron (�rperi ≳ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5) has been terminated mainly due to the exhaustion of hydrogen in the primary’s core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Although the majority of EEVs belonging to this group do not significantly change their orbital parameters during evolution, there is a subgroup of them that behaves quite differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It can be recognised as the distinct ‘cloud’ of green dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5b, represented by the mainly horizontal green tracks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' These are systems that were characterised by very strong stellar winds at the end of the MS phase and have lost much of their envelopes, so that their orbital period has increased significantly (Kepler’s third law).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The most numerous group of EEVs, in which the primary component has filled its Roche lobe in the MS phase, forms a kind of ‘bridge’ between the two previously mentioned groups and merges with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The shapes of the corresponding trajec- tories on the Porb-e plane may vary from system to system, de- pending on the interplay between tidal forces and the intensity of stellar winds, so no single ‘type’ of track can be assigned to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, many of them resemble the inverted Greek letter ‘Γ’ – initially, the system drifts horizontally (towards the longer orbital period) as a result of the mass loss and/or spin- orbit coupling, and then undergoes more or less rapid circulari- sation (moves vertically downwards) under the influence of the intense tides, which come to the fore when the primary compo- nent almost fills its Roche lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Only 8 out of 20,000 EEVs underwent efficient spin up of both components due to the pseudo-synchronisation (when the rotation period of the star ‘matches’ the rate of orbital motion at periastron, so that there is no net torque over an orbital cycle, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Hut 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' These few systems are located in the upper right cation mechanisms can significantly intensify stellar winds in our sim- ulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' These phenomena are particularly well-pronounced when the component is close to TAMS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', its radius approaches the Roche-lobe radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 18 In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 we give four types of termination conditions, but three of them apply independently to both the primary and secondary compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 9 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (a) HRD with the evolutionary tracks of primary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The grey area corresponds to the region occupied by the full set of 20,000 tracks, while a subsample of 100 randomly-selected tracks is indicated with coloured points connected by black solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Each point rep- resents one saved MESA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The colour coding reflects the central hydrogen abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The effective temperature of the bi-stability jump (Teff ≈ 26,000 K) is marked with the vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The abrupt change in the behaviour of some evolutionary tracks after crossing the bi- stability jump region is due to a significant change in the wind mass-loss rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (b) The same as panel (a), but for a set of secondary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We note the difference in the ranges of the two axes in panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' More details are discussed in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' corner of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Their orbits were initially highly eccen- tric yet relatively widely-separated at periastron (�rperi ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 – 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Thus, in combination with the lower masses of the pri- mary components (M1 ≈ 5 M⊙), there was no effective tidal dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, the envelopes of these stars tended to ro- tate pseudo-synchronously with the orbit (due to the relatively long nuclear time scale of the evolution of a 5 M⊙, the primaries had enough time to do so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Consequently, this led to a very fast rotation of the primary component, exceeding the threshold of Ω/Ωcrit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Internal structure and asteroseismic properties The shape of the resonance curve depends not only on the global properties of the components and the orbit, but also on the inter- nal structure of the stars, which directly affects seismic proper- ties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' the spectrum of eigenmodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Therefore, within the lim- ited volume of this paper, we would like to show at least one rep- resentative example of the evolution of the internal properties of the primary component for an arbitrarily-chosen EEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 6 shows the evolution of a primary with mass M1 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 M⊙ in a system with an initial eccentricity e ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 and an initial orbital period Porb ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In our simulations, this particular system finished its evolution due to the circularisation of its orbit after about 12 Myrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The HRD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6 reveals the ‘non-standard’ evolutionary track of the primary due to the sharp change in the mass-loss rate after crossing the bi-stability jump (right panel in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The same panel also shows how the pri- mary’s surface rotation rate varies over time – as the mass-loss rate increases, it loses a lot of spin angular momentum and slows down its rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The eccentricity and orbital period monotoni- cally decrease with time (middle panel in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6), except for a short episode of increase in Porb caused by the ir- reversible loss of a large part of the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We have selected three epochs in the evolutionary history of this EEV (labelled A, B, C on the HRD), for which we have presented the appearance of the rotational profiles, mode propagation diagrams and oscil- lation spectra of the primary component in the bottom part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Epoch A corresponds to the phase of evolution just af- ter leaving the ZAMS, epoch B is characterised by Xc,1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='45, and finally, epoch C marks the situation just before the complete circularisation of the EEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Let us briefly describe the changes occurring in each of the three types of diagram below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The internal rotation profile of the primary is almost constant for epoch A, but by then a division between a faster-rotating core and a slower-rotating envelope begins to emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The aforemen- tioned division becomes particularly apparent in epoch B, when the core has developed a rotation rate approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='25 times that of the surface layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As can be seen, the contracting core ro- tates as a rigid body throughout the MS lifetime due to efficient angular momentum transport supported by convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The outer part of the envelope also rotates almost rigidly, but this time it is due to large-scale Eddington-Sweet meridional flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The angu- lar velocity gradient in the primary starts to gradually decrease as the star reaches epoch C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Various mixing processes in the chemically-modified layer left by the core lead to the diffusion of angular momentum from the core to the envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Moreover, the rotational profile inside the star becomes a smooth function of the radius (rather than a step-like function as for epoch B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 10 of 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 (a) (b) 5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 4 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='42 X 1og ( 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 2 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 bi-stability jump 1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='9 log (Teff, 1 / K) log (Teff, 2 / K)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Orbital period-eccentricity distributions of 20,000 modelled EEVs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (a) Initial distribution of e as a function of Porb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Colour-coding corresponds to the termination conditions described in Sect 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' the RLOF of the primary component during periastron passages before reaching TAMS (black), exhaustion of hydrogen in the primary’s core (primary at TAMS, green), almost complete circularization of the or- bit (e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='01, magenta), and the maximum allowed rotation rate of the primary component (Ω / Ωcrit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='75, orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A pair of dashed hori- zontal lines mark the boundary values of the initial eccentricity, e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 and e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (b) Same as in panel (a), but for the final state of each modelled binary system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (c) Random selection of 400 orbital evolution tracks with the same colour-coding as in panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The majority of TEOs in our simulations belong to the g- mode family of oscillations, so it is very important to control the behaviour of the Brunt-Väisälä buoyancy frequency, NBV, in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Together with Lamb frequency for l = 2 modes, S l=2, they carry information about g and p mode cavities and their evanescence regions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Aerts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2010, their Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The evolution of NBV and S l=2 is presented in the middle column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The blue and grey shaded regions denote the posi- tion of the l = 2 p-mode and g-mode propagation cavities, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The white areas that lie between the Brunt-Väisälä and Lamb frequencies correspond to the evanescence regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' During evolution, the receding convective core builds up a large g-mode trapping cavity, which is very important for their fre- quency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Additionally, the behaviour of the NBV just be- low the stellar photosphere reveals a pair of thin subsurface con- vection zones, expected for this type of star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Jermyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Comparing the mode propagation diagrams for epochs A and C, it can be seen that also the p modes can penetrate deeper and deeper into the primary as it gradually depletes the hydrogen in its core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The right column in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6 contains most of the information that is directly used to obtain the resonance curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The horizontal bars at the top of each panel correspond to the frequency range in which GYRE looked for potential TEOs (according to the criteria adopted in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We recall that that their width depends on the Nm max(e) functions, so as the system evolves towards lower ec- centricities, these bars are shorter and shorter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' fewer harmon- ics of the orbital frequency can effectively drive TEOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' With the thick, short vertical lines we mark the location of the tidal forcing frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As can be seen, the separation between successive values of fNm becomes larger with passing time due to the in- crease in forb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The eigenfrequencies found by GYRE are marked with the long thin vertical lines, while the linear damping rates of these modes are shown as black solid and dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The presented set of three synthetic oscillation spectra reveals a typ- ical structure for g modes with their asymptotic behaviour for large radial orders (which correspond to lower frequencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It may appear that the dense ‘forests’ of eigenfrequencies end too early relative to the left limits of horizontal bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, this is not a mistake, but a direct consequence of the maximum |n| we allowed in the calculations – modes with lower frequencies would have larger radial orders than thirty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' During the evolution of the EEV, both the forcing frequencies and the oscillation spec- trum shift, so the intersection of these two vertical line patterns is virtually inevitable in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Each of these intersections is the source of a single resonance that can give rise to a noticeable TEO at the level of the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The ‘visual inspection’ of resonance curves The resonance curves are characterised by a striking diversity in terms of morphology, which is already partly evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The four examples of L1(t) and L2(t) shown in this figure show that the components of the EEVs can experience, firstly, a very different number of resonances and, secondly, their distribution in time can take various forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The heights of the maxima of the resonance curves are mainly dictated by the γnlm of the mode to which the smallest difference corresponds, (σnlm − fNm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Sta- tistically speaking, modes with larger |n| are more strongly non- adiabatic (have larger damping rates), hence the maxima they induce in the resonance curves are lower (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Another factor determines the extent of the resonant maximum in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It is determined by the relative ‘velocity’ with which the eigenfre- quency spectrum crosses the fNm spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' By ‘velocity’ here we mean the rate of change of these two independent frequency spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It should be emphasised that there are also numerous cases in which L(t) drops sharply to zero at some point (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' L2(t) curve in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3) or resonances do not occur at all (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Such a situation can occur, for exam- Article number, page 11 of 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 initial distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 - e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 - RLOF of the primary minimum eccentricity (a) primary at TAMS maximum rate of rotation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 final 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 distribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 - e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 - (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 - sample tracks 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 - e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 - (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 100 101 102 Porb (d)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Summary plot of the properties of the primary component of one of the arbitrarily selected binary systems from our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The approximate initial parameters of this particular system were as follows: M1 ≈ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 M⊙, M2 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 M⊙, e ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4, and �rperi ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The integration of the system was terminated because of its circularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The top row of panels shows, from left to right, evolutionary track in the HRD, the evolution of the orbital period and eccentricity, and the temporal changes of the wind mass-loss rate and surface rotation velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The vertical dashed lines in the latter two diagrams correspond to epochs A, B, C in the HRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The lower part of the figure shows the internal rotation profile (left column), the mode-propagation diagram (middle column) and the synthetic oscillation spectrum (right column) for epochs A, B, C (shown in consecutive rows labelled with these letters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The rotation frequency inside the primary is drawn as a function of fractional radius, r / R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The range of rotation frequency is different in the three panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The mode-propagation diagram shows the dependence of the Brunt-Väisälä frequency (black line) and Lamb frequency for l = 2 modes (blue line) on the fractional radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The grey and blue shaded areas correspond to the propagation cavities of the g and l = 2 p modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The synthetic oscillation spectrum diagrams contain several different pieces of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The light blue and light red horizontal bars delineate the range of frequencies allowed by the FNm values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In the background of each, the blue and red short vertical lines indicate tidal-forcing frequencies lying within these ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The synthetic oscillation spectra calculated by GYRE are marked with red (σn,2,0) and blue (σn,2,+2) long vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Their corresponding linear damping rates are plotted as solid (γn,2,0) and dashed (γn,2,2) black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The frequency scale on the abscissa axis refers to the rest frame co-rotating with the primary’s core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 12 of 24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='40 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='350 6.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='275 C e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='25 - F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='250 Ci 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='50 - F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='225 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='75 - 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='20 - F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='200 iA IB 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='05 C!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='00 - dA F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='175 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='15 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='487 120 - 10-7 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='486 - ）100- 9-01- p (d-l) C 08 Fs-0[- S 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='484 - 10-4 - 40 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='483 - 10-3 - 20 - 10-2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='482 0 140 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='46 - 10-8 - 120 - 10-7 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='44 - ）100- 10-6 _ (d-1) 80 - 10-5 S uul C 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='40 - 40 - ε-01- B 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='38 - 10-2, 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='310 : 140 - F8-01 NBV l=2, m=0 modes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='305 - 120 - St=2 l= 2, m= 2 modes 10-7 g-modes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='300 - 100 n,2,0 propagation cavity 10-6 - →-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' n,2,2 p I = 2 p-modes ）0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='295 - fn,0 08 propagation cavity 2 p 10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' fn,2 S C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='290 - 09 range between 10-4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='285 - 40 - range between fmi=?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' and fmax?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='280 - 10-3 _ c 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='275 - 10-2 - 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 2 6 r/R r/R Frequency (d-1)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs ple, when the oscillation spectrum lies completely outside the frequency range allowed by the FNm coefficients or the nuclear timescale of the secondary is much longer than the same time scale for the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Under such circumstances, the secondary component will remain close to the ZAMS until the termina- tion condition is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Thus, it will not significantly change its internal structure and oscillation spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This in turn means that the oscillation spectrum will not move relative to the tidal forcing frequencies, effectively reducing the number of possible resonance events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' ‘Long’ resonances Our sample of resonance curves includes a particular group of L(t) curves that exhibit exceptionally long duration resonances compared to typical ones (we will refer to them as ‘long res- onances’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 7 presents parts of three representative res- onance curves belonging to this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The shaded regions in the figure mark the position of the long resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As can be clearly seen, the typical resonance usually lasts for about 103 – 104 years, which is approximately 100 times shorter than the du- ration of a long resonance (of the order of 105 – 106 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' They originate from the intersection of one of the fNm frequencies with the σnlm frequency at a very small angle, in terms of their tempo- ral evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As a result, they remain for a relatively long time in very close vicinity, leading to a broad resonance overlapping with narrower ones (originating from other intersections of the fNm and σnlm frequency spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' especially the middle panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The long resonances are interesting for at least two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' First of all, they are natural candidates for resonantly- locked TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, based on our simulations, it is difficult to say whether an extended resonance would persist when the back-reaction of a TEO on the orbit is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Sec- ondly, if the energy exchange between the eigenmode and the or- bit that corresponds to a long resonance is not efficient (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' there is a small chance that a long resonance will be lost), such a reso- nance should lead to a high-amplitude TEO without the need for resonance locking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is simply because it has enough time to reach its saturation level due to the non-linear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, we did not find any significant correlations between the occur- rence of a long resonance in L(t) and the initial parameters of our EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Total number of resonances and the average rate of their occurrence The first feature of the morphology of the resonance curves that we investigated is the total number of resonances that occurred in the primary and secondary components, Nres,1 and Nres,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' How- ever, we did not calculate these statistics directly from L1(t) and L2(t), because some of the apparent maxima may actually be a blend of more than one resonance event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is especially true when the involved γnlm differ by orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Then one of the resonances is characterised by a notably smaller maxi- mum, which seems to ‘hide’ in the dominant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Instead, we used a different approach that did not underestimate the ac- tual number of resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' When post-processing the generated models, we simply counted each intersection of the σn,2,0 and σn,2,+2 frequency spectra with their counterparts fN,0 and fN,2, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The results of such an analysis are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The most important thing about Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8 is that it shows the absolute number of resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' EEVs can experience hundreds or even thousands of resonances during their evolution on MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Example resonance curves of the primary component for three different EEVs from our simulations that exhibit long resonances (high- lighted by the shaded areas in each panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We note a substantial differ- ence in the width of typical and long resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' These long resonances are good candidate for excitation of high-amplitude or resonantly- locked TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It would therefore be wrong to claim that these phenomena are rare in massive and intermediate-mass EEVs, although in gen- eral, resonances are quite short-lived compared to the nuclear time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The total number of resonances experienced by the primary component (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8a) shows a correlation with both its initial mass and the initial eccentricity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The mildly decreasing trend of Nres,1 towards higher M1 originates from the fact that the mean lifetime of the star on MS shortens with in- creasing mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the other hand, the wide range of Nres is mainly due to differences in initial eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The closer the system is to a circular geometry at the beginning of evolution, the statistically lower the value of Nres, which is self-explanatory and also applies to the secondary component (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' EEVs that have managed to circularise their orbits in the MS phase (magenta dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8c) have on average lower initial eccentric- ities and thus fewer resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The opposite behaviour is exhib- ited by EEVs in which the primary component has had a chance to reach TAMS (green dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The secondary compo- nents experience a slightly fewer resonances compared to the primaries (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8b) and there is no clear division of the Nres,2 dis- tribution with respect to the termination criterion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The noticeably smaller number of resonances for secondary compo- nents with masses M2 < 5 M⊙ comes from the conditions of our simulations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' secondaries with these masses occur in sys- Article number, page 13 of 24 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' L 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 104- 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 106 - L 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 104→ 3 4 5 6 7 8 9 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' ① L 104- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 t(Myr)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Total number of resonances in the pri- mary (a, c) and secondary (b, d) components that we detected in our simulations as a function of the initial masses of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The initial eccentricity of the EEV is colour-coded in panels (a) and (b), while the corresponding scale is shown at the top of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Pan- els (c) and (d) are analogous to their counter- parts in the top row, but the colours used reflect the termination criterion (same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The ordinate scale is logarithmically-scaled for Nres > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Below this value, a linear scale was applied in order to present components without resonances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Nres,1 or Nres,2 equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' tems with decreasing mass ratios19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Hence, the large difference in nuclear time scales between the components means that the secondary component does not significantly change its eigenfre- quency spectrum, resulting in a smaller number of resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As we already mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2, some of the L1(t) and L2(t) curves do not reveal any resonances, which is why they lie in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8 on the horizontal line Nres = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This behaviour oc- curred for only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='07% of our primaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' They are all EEVs with highly eccentric orbits that quickly filled their Roche lobes at pe- riastron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' There was much more such behaviour for secondaries, about 7%, mainly for the intermediate-mass companions of the much more massive primaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' From an observational point of view, even more important than the total number of resonances is the rate at which they occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Knowing this rate, for a given population of MS EEVs, we can approximately say where we have a statistically higher chance of observing TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' After all, our observations only cor- respond to one particular moment in time, not the entire evolu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Knowing the values of Nres for each component and the age of each system at termination, Tmax, we can calculate the average rate of resonances as Rres ≡ Nres/Tmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We show the distribution of Rres,1 and Rres,2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It is very difficult to predict what the dependence of Rres on the mass of the component will look like, as it is the result of a complex interplay between many related factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the one hand, it can be said that massive stars should have a smaller Rres because their lifetimes are shorter and they fill their Roche lobes relatively easily (in the considered range of orbital parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the other hand, however, massive stars quickly change their internal structure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' asteroseismic prop- erties), so that their eigenfrequency spectra evolve rapidly, in- creasing the likelihood of interaction with the structure of the 19 We recall that the minimum mass of the primary component consid- ered in our study was equal to 5 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' tidal forcing frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The question is, which of these pro- cesses prevails?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 9, it is the more massive stars that are more likely to undergo resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Both primary and secondary components with masses around 30 M⊙ have on average an order of magnitude higher Rres (∼102 Myr−1) than components with masses around 5 M⊙ (∼101 Myr−1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 9a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Moreover, the dependence of the distributions shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 9 on the initial eccentricity and termination condition is inherited from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' At this point, we can venture the conclusion that in the case of MS EEVs, TEOs should be observed mostly in the upper part of the MS (among early B- and O-type dwarfs), which still re- quires observational verification on a large sample of massive EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Although we cannot extrapolate the obtained distributions of Rres towards lower masses, these stars have an increasingly extended convective envelope, which in turn should effectively limit the photometric detection of g-mode TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the con- trary, the envelopes of massive stars are radiative, which should not prevent g-mode TEOs from propagating up to the vicinity of the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Thus, they can be more easily detected by analysing the light curves, especially in the era of high-quality space-borne photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Distribution of resonances over time Since the average rate of resonances we have studied so far has effectively obliterated any differences in the corresponding tem- poral distribution, we can ask another important question: Are there any distinctive moments in the evolution of the simulated EEVs during which the systems experienced temporally higher resonance rates?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' After visually inspecting hundreds of resonance curves, we noticed that the aforementioned rate changes dramat- ically in many cases (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' the top panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3 as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In Article number, page 14 of 24 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 (a) 103 102 101 0 103 102 101 RLOF of the primary minimum eccentricity primary at TAMS maximum rate of rotation 0 5 15 20 25 30 5 10 10 15 20 25 30 Mi/Mo M2/MoKołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Summary plots analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8, but showing the average rate of resonances occur- ring in the simulated EEVs (average number of resonances per Myr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Components that did not exhibit any resonances during the simula- tion have been omitted here as their Rres value would simply be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The colour-coding is the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' order to compare the temporal distribution of resonance events for the various EEVs we are dealing with, we performed this type of analysis on subgroups of systems divided according to the termination condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We also normalised the time variable by dividing it by Tmax of each resonance curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This allowed us to present the whole evolution of components on a convenient and uniform interval, [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 10 shows the results obtained for the primary components that have managed to deplete hydro- gen in their cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 10 also demonstrates that the distribution discussed here is not uniform over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Specific areas in this diagram are clearly distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Nevertheless, this figure still contains in- formation on the total number of resonances, which makes it somewhat problematic to compare the shapes of these distribu- tions for different masses of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We have, therefore, prepared histograms of the times of resonances for five inter- vals of the primary’s initial mass (every 5 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Separate sets of histograms were generated for the primary and secondary com- ponents and the three main termination conditions20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' All his- tograms are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The most diverse structure of the temporal distribution of res- onances is shown by systems in which the primary component has completed its evolution in our simulations at TAMS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 11a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In fact, for all mass ranges, the distribution has two dis- tinct maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The smaller of the two is located near the ZAMS, while the other is just before reaching the TAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Their presence can be explained by the rate of change in the stellar eigenspec- trum, which is the highest (after averaging over all modes) at the aforementioned moments of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In particular, the rapid changes in the radius of the star when it is close to complete de- 20 We did not prepare separate histograms for the EEVs, whose calcu- lations were terminated due to the maximum allowed rotation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The size of this group (only eight systems) was insufficient for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Time distribution of the resonances of the primary component that reached TAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The abscissa axis corresponds to the normalised time and the ordinate shows the initial mass of the primary compo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In addition, the ordinate is logarithmically scaled, so the set of resonance curves is almost uniformly distributed in the vertical direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The total number of resonances contained in one hexagonal bin is colour-coded according to the scale on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' pletion of hydrogen in its core cause a very high ‘concentration’ of resonances in the final MS phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The height of this domi- nant maximum decreases towards higher masses, but at the same time it becomes wider and wider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The distribution for the sec- Article number, page 15 of 24 3×101 2×101 - 104 Mi /Mo 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 103 6× 100 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 t / Tmaxe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 102 101 0 (a) 10-2 102 101 109 C RLOF of the primary minimum eccentricity 10-2 primary at TAMS maximum rate of rotation 5 10 15 20 25 30 5 10 15 20 25 30 Mi/Mo M2/MA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Histograms of the normalised times of resonances occurring in the primary (left column) and secondary (right column) components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The consecutive rows (from top to bottom) correspond to EEVs satis- fying different termination conditions, as labelled in panels (a), (c), and (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The colour of the histogram is related to the initial mass range of the primary and is described in the legend in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We note that the histograms on the right (corresponding to the secondaries) refer to the different mass ranges of the primary component, not the secondary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For example, the yellowish histogram in panel (b) summarises the be- haviour of all secondaries of the systems with the primaries having mass M1 > 25 M⊙, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' without distinguishing the mass ranges of M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The range of the ordinate axes is the same in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' ondary components also reveals this kind of maximum near the TAMS, which is particularly well pronounced for companions of primaries with masses ≳ 25 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Given these facts, an inter- esting conclusion can be drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Massive and intermediate-mass EEVs with at least one component leaving the MS should expe- rience an increased rate of encountered resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' One might therefore suspect that there is a statistically higher chance of ob- serving TEOs in more evolved EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The properties of the his- tograms for EEVs in which RLOF eventually occurred at the periastron (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 11c and d) are very similar to the case described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The only evident difference between the two is the reduc- tion in maximum of the distribution near TAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is due to the fact that the primary is likely to start the RLOF earlier than it reaches the TAMS, preventing the occurrence of a large number of resonances in a relatively short time, as mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Eccentric systems that are subject to effective circularisation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 11e and f) behave quite differently from the two previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' They experience the vast majority of their resonance phe- nomena at the beginning of evolution, and then reduce the num- ber of resonances almost monotonically, as the orbital eccentric- ity becomes smaller and smaller with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Hence, the chance of observing TEOs in initially relatively tight EEVs (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5a) is largest in the vicinity of ZAMS, which stays in contrast to the systems described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Investigation of the morphology of resonance curves using UMAP All the analysis described above was based solely on the distri- bution of resonances in time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' neglecting the actual morphol- ogy of the resonance curves, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' differences in the height and width of resonance maxima, mean level of L(t), long-term trends in L(t), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Using the dimensionality reduction techniques pre- sented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5, we constructed 2D UMAP embeddings of the space of resonance curves in terms of their morphological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figures 12 and 14 show the results obtained for the res- onance curves of the primary and secondary component, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We recall that the idea of the low-dimensional embedding performed here is to preserve the distances between two points in the original space as accurately as possible, so that the distances in the 2D plane reflect the distances in the full (original) space of morphological features (the vector of 2,000 quantiles, Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In other words, a pair of distant points in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12 and 14 should cor- respond to resonance curves with notably different morphologies and ,vice versa, a pair of resonance curves with similar proper- ties is expected to lie in mutual vicinity on the 2D UMAP plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Thanks to this key property of UMAP and many other dimen- sionality reduction methods we can effectively explore the entire space of resonance curve morphologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' UMAP plane for primary components We begin with a discussion of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Firstly, the presented 2D embedding does not indicate the presence of any well-separated groups among the resonance curves for the primary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This is an observation that is true over the entire range of differ- ent values of the UMAP free parameters (Appendix D) as well as for the different summary statistics of the resonance curves that were considered during the preliminary experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The mor- phology of the resonance curves changes smoothly depending on to the initial parameters of the simulated EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Secondly, as can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12b and c, the initial eccentricity and normalised periastron distance are parameters strongly correlated with the overall morphology of the resonance curves of the primary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Moreover, their gradients in the UMAP plane are approximately orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Therefore, the pair of these parameters is the primary factor that determines the shape of L1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The termination condition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12e) gener- ally follows the behaviour of �rperi except at small periastron dis- tances, when the morphology remains similar but the simulations were terminated due to hydrogen depletion in the primary’s core or near-complete circularisation of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The initial mass of the primary component (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12a) and its initial angular velocity of rotation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12d) are second-order factors shaping the mor- phology of the L1(t) resonance curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In the inner part of the plane, M1 is distributed almost randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The clear exception is Article number, page 16 of 24 (b) Mi/Mo≤10 (a) 5 1025 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 0 (c) (d) 5 RLOF of the primary 0 (e) (f) 5 minimum eccentricity 4 - 3 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 t/ TmaxKołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs the boundary of the plane which can be roughly divided into two parts of mostly high or low initial mass of the primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A simi- lar conclusion can be drawn for the initial Ω1/Ωcrit,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This time, however, the upper right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12d reveals a well-defined group of high initial Ω1 / Ωcrit,1 and�rperi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It is difficult to include here a complete presentation of the changes in the morphology of L1(t) as a function of their posi- tion on the UMAP plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Therefore, we only focus on some ex- treme points to present some boundary cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 13 shows examples of L1(t) from different areas of the morphological plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Primary components with lower masses and high initial eccentricities are generally characterised by resonance curves with high mean levels and a rich set of resonances, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 13a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The resonance curve depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 13b repre- sents intermediate-mass fast-rotating primary with large initial �rperi and low initial eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Here, the base level of L1(t) increases by an order of magnitude and then the system expe- riences a large number of resonances, during the evolution near TAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The increase in the mean value of L1(t) is characteris- tic of stars with a high initial rotation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Primary components lying between points (a) and (b) generally do not manifest this characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Moving along a straight line on the plane from (a) to (b), the increase in the number of resonances during the evo- lution near TAMS becomes more and more apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Figure 13c shows an example of a system with a small initial eccentricity and a short periastron distance at ZAMS that is rapidly circu- larising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As expected, the L1(t) resonance curves for such ob- jects have a small number of resonance maxima and a low base level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The case corresponding to the larger initial eccentricity is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 13e, where the total number of resonances is much greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In this case, the evolution is mainly distinguished by a decrease in the frequency of resonances with time, related to the efficient circularisation of the orbit, and therefore a decrease in the mean level of L1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Finally, the resonance curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 13d is representative of the most EEVs between points (c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' They are characterised by an approximately uniform distribution of resonances over time and an almost constant base level of the resonance curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' UMAP plane for the secondary components The situation for the secondary component (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 14) is quite dif- ferent from the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The UMAP manifold obtained for the set of L2(t) reveals slightly more complex structure than the shape of embedding in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Since the time span of L2(t) is largely determined by the mass of the primary component, the resonance curves for the secondary components were terminated at times not necessarily related to their actual evolutionary sta- tus and are statistically shorter than they could be for primaries of the same mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For this reason, secondary components ex- perience, on average, fewer resonances, but, at the same time, their resonance curves can take more diverse forms compared to L1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Undoubtedly, the main factor shaping the morphology of the L2(t) is the initial eccentricity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 14c), which with the ex- ception for two small areas, varies smoothly across the UMAP plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The other parameters (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 14a, b and d) play a secondary role, showing the complex and fine structures on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As can easily be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 14a, the extreme cases of L2(t) in terms of their morphology belong almost exclusively to the EEVs with intermediate-mass secondary components hat gather at the pe- riphery of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Some of these objects even form slightly better separated groups, isolating from the central part of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Five limiting examples of resonance curves for secondary components are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Panels (a), (c) and (e) to- gether with the area they approximately enclose contain res- onance curves morphology very similar to that described for primary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The resonance curves belonging to the ‘clouds’ of points labelled as (b) and (d) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 15 are com- pletely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' They are distinguished by the complete absence of resonances during a certain period of the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The differentiating feature of these cases is the disap- pearance of resonances from some time to the end of evolution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 15b) or the presence of resonances only around the middle of the considered evolution time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 15d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Summary and conclusions In our paper, we aimed to investigate the temporal variation of conditions that favour excitation of TEOs in EEVs with massive and intermediate-mass MS components (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2) and see how their picture changes with different initial parameters of the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In order to achieve this goal, we simulated the evolution of 20,000 EEVs using the MESA software in combination with the GYRE stellar oscillations code (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Our calculations started at ZAMS and were terminated if one of the conditions presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 was met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We considered only modes with l = 2, m = 0, +2 because they are expected to be dominant TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We also assumed that all TEOs are due to chance resonances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' we neglected the effect of TEO on the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Knowing the evolution of the orbital parameters of simulated EEVs and the temporal changes in the eigenmode spectra of the components, we were able to derive resonance curves L1(t) and L2(t) defined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (4) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The equations reflect the overall resonance conditions, and thus indirectly also the chance of TEOs, sepa- rately for the primary and secondary components of our simu- lated EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' After visually inspecting the obtained resonance curves, cal- culating basic statistics for them and applying ML-based meth- ods to the entire data set, our main results can be summarised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Resonance curves are characterised by striking diversity in terms of their morphology (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' EEV components can experience a very different number of resonances, and their distribution over time can take various forms, including the lack of resonances over a long periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We also distinguished a group of resonance curves that exhibit pro- longed resonances, about two orders of magnitude longer than typical (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' These long resonances are the potential sources of high-amplitude and resonantly- locked TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Resonances between tidal forcing frequencies and the spec- trum of stellar normal modes are not rare events among mas- sive and intermediate-mass MS EEVs (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Although the total number of resonances depends mostly on the initial orbital parameters, it is typically of the order of 102 – 103 for a given system during the entire MS phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Let us emphasise at this point that these numbers are rather lower limits for the actual Nres in EEVs because we considered only l = 2 TEOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Taking higher degree modes into account will certainly increase the reported values of Nres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On average, the more massive a star is, the higher the rate of resonances it experiences (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For the most massive stars in our sample (≈ 30 M⊙), the average rate of resonances can reach ∼ 102 Myr−1, which is approximately an order of magnitude higher than for intermediate-mass stars (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 17 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2D UMAP embedding of the manifold spanned by the morphological features of the resonance curves of the primary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For details on how to obtain the presented embedding, see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Panels (a) – (d) are colour-coded with respect to the initial parameters of the simulated EEVs, as shown on the corresponding colour bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The other initial parameters were omitted as they were not significantly related to the location of the points on the presented map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The different colours of points in panel (e) correspond to the termination condition, as shown in the legend on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The values on the abscissa and ordinate axes were omitted as they have no physical meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' For clarity, the colour-coded features have been averaged within the small hexagonal areas in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A discussion of the figure can be found in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Variations in the morphology of the resonance curve for the primary component across the 2D UMAP plane from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The middle panel in the bottom row shows the plane with colour-coding identical to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12a (without hexagonal binning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Panels (a) – (e), which surround the area, show example resonance curves that correspond to the locations on the area masked with large red dots and labelled according to the associated panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The positions of points (a) – (d) have been chosen in such a way as to correspond to different extreme positions in the plain, while point (e) refers to one of the intermediate cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A discussion of the figure can be found in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 18 of 24 (a) (b) (c) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 20 1(M) rperi 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 e 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 (d) (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='45 RLOF of the primary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='25 minimum eccentricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='15 : maximum rate of rotation(a) (c) 105 106 - L 104 105 0 20 40 60 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='50 2 6 (b) 106 (p) LLF (e) 105 (a) (d) (e) (C 103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 t (Myr) t (Myr)Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12, but for a set of resonance curves of the secondary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 13, but for a set of resonance curves of the secondary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The distribution of resonances over time is not homogeneous and depends primarily on whether the system circularises be- fore the primary reaches the TAMS or RLOF occurs at the periastron (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We noticed a particular mo- ment in the evolution of our EEVs near the TAMS, when the components undergo an increased number of resonances in a relatively short time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 11a and b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The low-dimensional representation of the morphology of the resonance curves, summarised by quantile-based statis- tics and subsequently processed by UMAP, shows that its manifold forms a rather smooth distribution without well de- fined (separated) groups (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 12 and 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Less differentiated, at least in terms of the adopted method, are the resonance curves of the primary components, for which the initial eccentricity and the normalised periastron dis- tance largely determine their morphological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Al- though secondary components experience far fewer reso- nances, their shapes are generally more complex due to the Article number, page 19 of 24 (a) 25 (b) (c) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6 M2(Mo) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 , 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 e 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='4 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 a (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='45 RLOF of the primary 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='35 crit, 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='25 minimum eccentricity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='15 maximum rate of rotation106 (a) (b) (c) 105 105, L2( 104 104 - 20 40 60 80 100 0 2 6 8 10 (b) 106 (p) (e) 106 2(t) (a) L 104 105 = (p) 0 10 20 15 0 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6 8 t (Myr) t (Myr)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs predominant influence of the primary component on evolu- tion time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In light of the results obtained in our study, we can draw sev- eral interesting conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Firstly, statistically speaking, TEOs are more likely to be discovered in more massive EEVs, as their components have a higher average rate of resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' This does not necessarily mean that a higher absolute number of more mas- sive EEVs exhibiting TEOs than less massive ones will be ob- served21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, when comparing two particular systems, one with intermediate-mass components and the other with much higher masses of the components, it is for the latter that we have a statistically higher chance that some resonance is currently underway there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Secondly, it seems that TEOs should be espe- cially well visible in EEVs that contain a component approach- ing TAMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Given these facts, the ‘extreme-amplitude’ massive EEV, MACHO 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1718 (Jayasinghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Kołaczek- Szyma´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2022) fits this picture almost perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Its pri- mary component is a B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 Ib-II supergiant leaving the MS and, more importantly, many high-amplitude TEOs have now been detected in this extreme system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It is possible that what the pri- mary component of MACHO 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1718 is currently under- going corresponds to the resonance curve shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 13b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' it is in the phase of a high resonance rate caused by relatively fast changes in its radius and orbital parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Moreover, the am- plitudes of TEOs observed in MACHO 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='7443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1718 vary over time notably, suggesting that we may witness rapid changes in the resonance conditions for the primary component of this par- ticular EEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It would therefore be very valuable to carry out an observational study for a large sample of EEVs to verify whether TEOs are common in massive and intermediate-mass EEVs whose components have already depleted most of the hy- drogen in their cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' With high-quality space-borne photometric observations, both operational, such as the Transiting Exoplanet Survey Satellite (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2015) or BRITE-Constellation (Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2014), and planned missions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Planetary Tran- sits and Oscillations of Stars, Rauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2014), it is definitely a feasible task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The excitation of g-mode TEOs, which propagate deep in- side the star, may be an underestimated mechanism for angular momentum (AM) transport inside the components of EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' It has long been suspected that self-excited oscillations and inter- nal gravity waves22 efficiently redistribute AM in the radial di- rection of the star (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Rogers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Rogers & McElwaine 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Although the majority of our resonances have relatively short durations (of the order of 103 – 104 years), they can be quite frequent (especially near the TAMS), hence the question of their contribution to AM transport and mixing processes becomes ur- gent for the components of EEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Performing calculations that would treat the evolution of the orbit, components and TEOs in a fully self-consistent way seems particularly interesting for mas- sive eccentric systems leaving the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We have already entered the era of observational stud- ies of distant star-bursting galaxies and stellar populations in low-metallicity environments, that shaped the Universe in its early epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Recalling that the metal-poor stars were much more massive than their current metal-rich counterparts (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Hosokawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Susa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2014), we can ask what ef- fect metallicity has on the occurrence of TEOs in massive EEVs 21 Due to the rapid decrease of the mass function towards larger stellar masses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Chabrier 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 22 Gravity waves which are stochastically driven by the turbulent con- vective motions near the interface of the convective core and the enve- lope (see Bowman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2020, for a recent review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' and how the results we presented depend on metals content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Therefore, future studies of the importance of TEOs in massive, metal-poor EEVs seems worthy further investigation, especially because of the ongoing James Webb Space Telescope mission23 (JWST, Gardner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2006), which is certain to bring many dis- coveries in the stellar astrophysics of early stellar populations, including stars in eccentric binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' PKS is indebted to his brother, Adam Karol Kołaczek- Szyma´nski, who oversaw the purchase and assembly of a dedicated PC workstation to enable the efficient calculation of models in MESA and GYRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Without his generous help, this project would literally never have been completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The authors are thankful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Andrzej Pigulski for many important sugges- tions and fruitful discussions that made this manuscript more comprehensible, and to the anonymous referee for many inspiring comments that helped to improve the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' PKS was supported by the Polish National Science Center grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2019/35/N/ST9/03805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TR was partly founded from budgetary funds for science in 2018-2022 in a research project under the program „Diamentowy Grant”, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' DI2018 024648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Much of this work was developed and written during IAU Symposium 361, „Massive Stars Near & Far”, held in Ballyconnell, Ireland, 8 – 13 May, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' PKS is very grateful to the organisers for the opportunity to participate in this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' References Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Aerts, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Pápics, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021, Nature Astronomy, 5, 715 Potekhin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Y.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2009, ARA&A, 47, 63 Susa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Hasegawa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', & Tominaga, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2014, ApJ, 792, 32 Svirski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Nakar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', & Sari, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Kozyreva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', & Izzard, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2012, A&A, 544, L11 Yoon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Langer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', & Norman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2006, A&A, 460, 199 Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', Fuller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=', & Burdge, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021, MNRAS, 501, 1836 Zahn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 1975, A&A, 41, 329 Zahn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 1977, A&A, 57, 383 Zanazzi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' & Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021, AJ, 161, 263 Article number, page 21 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs Appendix A: MESA input files MESAbinary needs at least three input files (hereafter ‘inlists’) to start the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Two of them provide all the necessary parameters to perform the evolution of each component sepa- rately24, and the last one describes the evolution of the orbit as well as other processes that depend on binarity25 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' spin- orbit coupling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' In the following appendix, we present example MESAstar and MESAbinary inlists, which we used to generate a set of the evolutionary tracks of the components of the binary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' However, we have intentionally omitted any controls re- lated to the names of the files or directories where the results should be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' All values that changed in the inlists depending on the simulated binary system were enclosed in square brackets – [ · ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The parameters adopted below resulted in a typical num- ber of about 2,400 zones in the radial direction of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The number of models calculated per single EEV was typically of the order of several hundred, mainly depending on the termination condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' MESAstar inlist &star_job !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='OUTPUT history_columns_file = "my_history_columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='list" profile_columns_file = "my_profile_columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='list" show_log_description_at_start = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' save_photo_when_terminate=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='MODIFICATIONS TO MODEL new_rotation_flag=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' change_rotation_flag=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' new_omega_div_omega_crit=[Ω/Ωcrit] num_steps_to_relax_rotation=100 relax_omega_max_yrs_dt = 1d4 relax_omega_div_omega_crit=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' set_initial_cumulative_energy_error = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' new_cumulative_energy_error = 0d0 / !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' end of star_job namelist &eos / !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' end of eos namelist &kap use_Type2_opacities = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Zbase = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='02 / !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' end of kap namelist &controls !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='SPECIFICATIONS FOR STARTING MODEL initial_z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='02d0 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='CONTROLS FOR OUTPUT terminal_interval=100 write_header_frequency=1 photo_interval=100000 history_interval=5 star_history_dbl_format = "(1pes40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='6e3, 1x)" profile_interval=10 max_num_profile_models=5000 write_pulse_data_with_profile=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 24 Details of each keyword in MESAstar v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' r15140 inlist can be found at https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='mesastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='org/en/r15140/reference/ star_job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='html and https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='mesastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='org/en/r15140/ reference/controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='html 25 Details of each keyword in MESAbinary v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' r15140 inlist can be found at https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='mesastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='org/en/r15140/reference/ binary_job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='html and https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='mesastar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='org/en/ r15140/reference/binary_controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='html pulse_data_format="GYRE" add_double_points_to_pulse_data=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='WHEN TO STOP max_model_number = 5000 xa_central_lower_limit_species(1)="h1" xa_central_lower_limit(1)=1d-4 omega_div_omega_crit_limit=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='75 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='MIXING PARAMETERS mixing_length_alpha=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='82d0 use_Ledoux_criterion=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' num_cells_for_smooth_gradL_composition_term = 0 alpha_semiconvection=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='01d0 okay_to_reduce_gradT_excess=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' mlt_make_surface_no_mixing = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' overshoot_scheme(1)="exponential" overshoot_zone_type(1) = "burn_H" overshoot_zone_loc(1) = "core" overshoot_bdy_loc(1) = "top" overshoot_f(1) = [fov] overshoot_f0(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='005 do_conv_premix=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' set_min_D_mix=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' min_D_mix=1d5 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='ROTATION CONTROLS am_D_mix_factor=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0333333d0 D_DSI_factor = 1 D_SH_factor = 1 D_SSI_factor = 1 D_ES_factor = 1 D_GSF_factor = 1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='ATMOSPHERE BOUNDARY CONDITION atm_option="table" atm_table="photosphere" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='MASS GAIN OR LOSS hot_wind_scheme="Vink" hot_wind_full_on_T=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2d4 cool_wind_full_on_T=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='9d3 Vink_scaling_factor=1d0 no_wind_if_no_rotation=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' mdot_omega_power=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='43d0 max_mdot_jump_for_rotation=5d0 rotational_mdot_kh_fac = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0d3 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='MESH ADJUSTMENT max_delta_x_for_merge = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='01d0 max_dq=1d-3 min_dq=1d-16 min_dq_for_split=1d-16 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='ASTEROSEISMOLOGY CONTROLS num_cells_for_smooth_brunt_B = 0 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='STRUCTURE EQUATIONS use_dedt_form_of_energy_eqn = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='TIMESTEP CONTROLS min_timestep_factor=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5d0 max_timestep_factor=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0d0 dH_div_H_limit=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5d0 delta_lgL_phot_limit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='05d0 / !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' end of controls namelist A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' MESAbinary inlist &binary_job !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='OUTPUT/INPUT FILES show_binary_log_description_at_start = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' binary_history_columns_file = Article number, page 22 of 24 Kołaczek-Szyma´nski & Ró˙za´nski: Tidally excited oscillations in massive and intermediate-mass EEVs "my_binary_history_columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='list" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='STARTING MODEL evolve_both_stars=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' change_ignore_rlof_flag = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' new_ignore_rlof_flag = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' / !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' end of binary_job namelist &binary_controls !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='SPECIFICATIONS FOR STARTING MODEL m1=[M1] m2=[M2] initial_eccentricity=[e] initial_period_in_days=-1 initial_separation_in_Rsuns=[a] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='CONTROLS FOR OUTPUT history_interval=5 photo_interval=100000 terminal_interval=100 write_header_frequency=1 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='TIMESTEP CONTROLS fa=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='02d0 fa_hard=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='03d0 fr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='10d0 fj=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='001d0 fj_hard=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='005d0 fe=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='02d0 fr_dt_limit = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0d2 fdm = 1d-3 fdm_hard = 5d-3 dt_softening_factor = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3d0 varcontrol_ms=5d-4 varcontrol_post_ms=5d-4 dt_reduction_factor_for_j=5d-2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='MASS TRANSFER CONTROLS do_enhance_wind_1=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' do_enhance_wind_2=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' tout_B_wind_1 = [Bwind] tout_B_wind_2 = [Bwind] !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='ORBITAL JDOT CONTROLS do_jdot_gr=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' do_jdot_ls=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' do_jdot_ml=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' do_jdot_mb=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='ROTATION AND SYNC CONTROLS do_tidal_sync=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' sync_type_1="Hut_rad" sync_type_2="Hut_rad" !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='ECCENTRICITY CONTROLS do_tidal_circ=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' circ_type_1="Hut_rad" circ_type_2="Hut_rad" anomaly_steps=300 / !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' end of binary_controls namelist Appendix B: GYRE input file The GYRE stellar oscillations code requires a single input file that collects all the user-specified parameters of the asteroseismic calculations being performed26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Below is our example file gyre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' As in Appendix A, we have omitted any keywords related to specific file names and have highlighted variables by 26 Details of each keyword in GYRE v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1 input file can be found at https://gyre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='io/en/v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1/ref-guide/ input-files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='html enclosing them in square brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' &constants / &model model_type = "EVOL" file_format = "MESA" / &mode l = 2 m = 0 n_pg_min = -30 n_pg_max = 30 tag = "m0" / &mode l = 2 m = 2 n_pg_min = -30 n_pg_max = 30 tag = "m2" / &osc inner_bound = "REGULAR" outer_bound = "VACUUM" adiabatic = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.’ nonadiabatic = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='true.’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='/ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='&rot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='summary_file_format = "TXT" ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='Article number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' page 23 of 24 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' TEOs_in_massive_EEVs summary_item_list = "freq,l,m,n_p,n_g,n_pg" freq_units = "CYC_PER_DAY" freq_frame = "INERTIAL" The frequency scan limits, f m=0 min , f m=0 max , f m=+2 min , and f m=+2 max , were calculated as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The total numbers of discrete frequency points, Nm=0 freq , and Nm=+2 freq , were obtained as follows, Nm=0,+2 freq = ⌈( f m=0,+2 max − f m=0,+2 min )/(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='005 d−1)⌉, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1) where ⌈ · ⌉ denotes the ceiling function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Appendix C: Data used by MESA Our work uses the MESA stellar evolution code, which incorpo- rates a vast compilation of knowledge, mainly from micro- and macrophysics, collected by many authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The MESAeos mod- ule is a mixture of OPAL (Rogers & Nayfonov 2002), SCVH (Saumon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 1995), FreeEOS (Irwin 2004), HELM (Timmes & Swesty 2000), PC (Potekhin & Chabrier 2010) and Skye (Jermyn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2021) equation of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Radiative opacities are taken primarily from OPAL (Iglesias & Rogers 1993, 1996), with low-temperature data from Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2005) and the high-temperature, Compton-scattering dominated regime by Poutanen (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Electron conduction opacities are from Cas- sisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2007) and Blouin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Nuclear reaction rates are from JINA REACLIB (Cyburt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 2010), NACRE (An- gulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' 1999) and additional tabulated weak reaction rates from Fuller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (1985), Oda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (1994) and Langanke & Martínez-Pinedo (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Screening is included via the prescrip- tion of Chugunov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Thermal neutrino loss rates are taken from Itoh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Roche lobe radii in binary systems are computed using the fit of Eggleton (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Appendix D: Adjustable parameters of the UMAP UMAP, as a highly flexible method, is prone to returning mis- leading results in the case of inappropriately set free parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' On the one hand, they can lead to the appearance of spuri- ous groups and, on the other hand, to the loss of finer topologi- cal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The vital UMAP parameters that need adjustment are n_neighbors, min_dist, n_components and metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' The n_neighbors parameter is the most important, as it controls the balance between the local and global structure present in the data that will be mapped to the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We experi- mented with different values of this parameter ranging from 5 to 1,000 (the default is 15) and concluded that the resonance curves (summarised by the proposed statistics) always form a single group, almost independently of the choice of n_neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Later, min_dist sets the minimum distance between two dif- ferent points on the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We tested its values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='5 and do not observe any significant effect on manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' We took the last two parameters, namely n_components that spec- ifies the number of dimensions of the embedding, and metric specifying the metric used for similarity calculation, as default values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Finally, we used the following set of free parameters: n_neighbors = 500, min_dist = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content='1, n_components = 2 and metric = ’euclidean’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} +page_content=' Article number, page 24 of 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NAyT4oBgHgl3EQf1PnY/content/2301.00733v1.pdf'} diff --git a/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf b/2NFLT4oBgHgl3EQfqi-E/content/2301.12140v1.pdf new file mode 100644 index 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Composing Full Length Musical Piece +Abhinav Kaushal Keshari (Purdue University) +Abstract +In the task of generating music, the art factor plays +a big role and is a great challenge for AI. Previ- +ous work involving adversarial training (Dong +et al., 2018) to produce new music pieces and +modeling the compatibility (Huang et al., 2021) +of variety in music (beats, tempo, musical stems) +demonstrated great examples of learning this task. +Though this was limited to generating mashups +or learning features from tempo and key distri- +butions to produce similar patterns. Compound +Word Transformer (Hsiao et al., 2021) was able +to represent music generation task as a sequence +generation challenge involving musical events de- +fined by compound words. These musical events +give a more accurate description of notes progres- +sion, chord change, harmony and the art factor. +The objective of the project is to implement a +Multi-Genre Transformer which learns to produce +music pieces through more adaptive learning pro- +cess involving more challenging task where gen- +res or form of the composition is also considered. +We built a multi-genre compound word dataset, +implemented a linear transformer (Katharopoulos +et al., 2020) which was trained on this dataset. +We call this Multi-Genre Transformer, which was +able to generate full length new musical pieces +which is diverse and comparable to original tracks. +The model trains 2-5 times faster than other mod- +els discussed. +1. Related Work +Despite achieving great success in generation challenges +using Artificial Intelligence in Natural Language Genera- +tion (NLG) there is a factor of art that still makes them +different from human like performance. In terms of NLG +we can relate it to something like the difference between +computer generated article and a piece of art like novels, +biography, etc. For music art factor always come into ac- +count and despite able to produce musical compositions +through Adversarial networks or mixing stems using super- +vised learning the solution still is very different from an +original piece of music which we discuss below. +1.1. Music Generation using GANs +Generative adversarial networks (GANs) have provided sig- +nificant progress in producing text, videos and images. Sim- +ilar efforts have been made to bring neural networks to +artistic domain of music. MuseGAN(Dong et al., 2018) +brought a novel model for generating multi-track music. +Until 2018, the progress in using AI to compose music had +been able to produce +• Single-track (monophonic) music +• Multi-track (polyphonic) music by combining several +monophonic melodies in chronological order +Music usually being an art involving multiple instruments +played together requires music to be multi-track and because +music notes are made up of chords, arpeggios or melodies +the idea of using a chronological order setting prevents it +from being generalized. +The paper(Dong et al., 2018) address this challenge in gen- +eralising real music by discussing current technical lacks in +neural network models and how it relates to the real world +music. +1. Music is an art of time and has characteristics of coher- +ence, rhythm, tension and emotion flow. This requires +it to have a Temporal Model. +2. Music compositions usually involves different instru- +ments interacting with one another making the compo- +sitions to be harmonic. To solve this issue a Composer +Model is required. +3. Musical notes are built of chords, arpeggios or +melodies and how they unfold over time; thus introduc- +ing chronological generation of notes is not suitable. +To address this the paper introduces using bars (seg- +ment of time) instead of notes as the basic unit for +composition. And then generate music bar by bar us- +ing transposed convolutional neural networks to learn +translation-invariant patterns. +The paper(Dong et al., 2018) makes contributions in terms +of both ability to artificially compose realistic music and +use of generative adversarial framework with temporal and +composition models. In short the contributions are: +arXiv:2301.02385v1 [cs.SD] 6 Jan 2023 + +Multi-Genre Music Transformer - Composing Full Length Musical Piece +• First GAN based model for generating multi-track se- +quence. +• First model which can generate multi-track polyphonic +music. +• Same model can be used as a music accompaniment. +• Creates a new Lakh Pianoroll Dataset (LPD) for multi- +track piano-rolls +• For future work metrics in the domain of artificial mu- +sic a new set of objective metrics are proposed. +MuseGAN model proposed considers two sub-network gen- +erator Gtemp (temporal structure generator) and Gbar (bar +generator) making the overall generator: +G(z) = +� +Gbar(Gtemp(z)(t)) +�T +t=1 +where z is the input noise vector. The strength of the model +is the ability to generate samples having chord like inter- +vals (learning features from temporal model) and melodies +involving pitch overlap among guitar, piano and strings +(learning features from composer model). +The model introduces multi-track by modeling interdepen- +dency of tracks by proposing 3 different generator model +(Jamming, Composer and Hybrid), but the author brings up +these based on the understanding of pop music composition. +This possibly restricts the generator to explore on a broad +spectrum of music and prevents it from being generalised. +Also worth mentioning is that the work relies on multi-track +interdependency, but misses to study about the compatibility +of these tracks which can significantly increase the quality +of music being generated. We will see this issue being +addressed in the next paper. +1.2. Modeling the Compatibility of Stem Tracks to +Generate Music Mashups(Huang et al., 2021) +Source separation(Jansson et al., 2017; D´efossez et al., +2019) makes it possible to generate a music mashup with iso- +lated stems like vocals, drums, piano, etc. The challenge lies +in producing music which has compatibility between these +stems. This paper creates a mashup generation pipeline and +trains a model to predict the compatibility by automatically +learning to adjust key and tempo (characteristics of quality +mashups in real world). +General models trained for harmonic compatibility +(Bernardes et al., 2017; Macas et al., 2018) fails to con- +sider subtle features or surprise mixes of disparate samples +which is quite common in this art domain. Other issue that +arises is audio compatibility models like Neural Loop Com- +biner (Chen et al., 2020) having lack of vocal source and +variety of genres. +The authors designed a self supervised learning model +by recombining the original combination of stems before +source separation to serve as examples of ground truth. To +avoid highly polarized model, semi-supervised learning +was introduced which included producing several random +mashups by mixing different stems and treated them as +unlabeled instances. Label smoothing regularization for +outliers (Zheng et al., 2017) was used to assign uniform +distribution to the unlabeled data for loss computation. This +helps in regularization. +The final architecture consists of 3 modules: +1. Music Source Separation: +Uses MSS algorithm +(Jansson et al., 2017) to get different stems vocals, +drums, bass and other. +2. Mashup Database (MashupDB): Using Madmom +(B¨ock et al., 2016) different features from the music +clips are extracted like key, tempo and downbeat in- +formation. Using these features and separate stem +combinations a mashup database is created which will +act as either harmonic or percussion stem candidates +for mashup generation process. +3. Mashup Generation: It uses candidate stems from +MashupDB and adjusts key and tempo to produce +mashups within 3 conditions - original, matched and +unmatched. +The model (Huang et al., 2021) is defined by p(y|V, H, P) +where V , H, and P are input signals for respective stems +vocal, harmonic, and percussion. The output probability p +is used as the mashup compatibility and y ∈ {0, 1} stating +good or bad. +The model (Huang et al., 2021) implementation tries to +mimic learning compatibility for producing new mashups +and provides objective and subjective evaluation by cross +validation among multiple different datasets. This technique +becomes easier because of the ability of the model to ex- +tract different stems and features and build its own mashup +candidates. This also makes the model training process not +dependent on human labeled data. The model is also ro- +bust as negative data is added along with positive data for +supervised learning. The range of music coverage is also +extensive and the source separation step makes it easier for +the model to be extended to different genres for training. +But the current model design lacks the effective embedding +of different stems while producing a mashup and makes +it dependent on tuning of key and tempo. Currently the +implementation comes up with fixed range of key and tempo +difference for compatibility and does not explain in detail +how they came up with these numbers. Although defining +a range prevents large pitch shifting and time stretching. +Additionally the results of the model ranks positive labeled +data (original) over unlabeled data which might lead to + +Multi-Genre Music Transformer - Composing Full Length Musical Piece +concerns of flexibility. Another major challenge of the +model is the large training time which is around 3 days using +an NVIDIA Tesla-V100 GPU whereas using transformer +model significantly reduces the training time. +1.3. Music Transformers +With state-of-the art neural network we managed to learn +features in music by defining certain rules on matching +tempo, beats or compatibility. In the previous paper we +also tried to learn compatibility with the help of supervised +learning. The model though suffered with bias as compati- +bility was favoured for matched key or tempo and also lacks +generalization. Compound Word Transformer (Hsiao et al., +2021) considers music as sequence of events and uses a +Transformer (neural sequence model) (Vaswani et al., 2017) +to generate a new musical sequence. +A musical note can be described by note’s pitch, chord, bar, +duration, velocity (dynamics), placement (onset time). If +we consider these as tokens we can then define music as +sequence of tokens and these tokens are a part of pre-defined +vocabulary. As music is multi-faceted a particular type of +token can capture only a certain feature like melody, rhythm, +harmony. All the neural networks until now treated these +tokens as equal and thus lacked heterogeneity. +Compound Word Transformer (Hsiao et al., 2021) generates +music in a conceptually different way as it allows tokens +to be of specific types and let them have their own proper- +ties. Tokens can be of note type (pitch, duration) or metric +type (beginning of new beat, bar). We then defines a mu- +sical event by combination of such tokens which allows to +capture co-occurrence relationship among the tokens. This +combination of tokens are termed as compound words. So, +now we can represent a music piece (X) as a sequence (S) +of compound words (cp) or S = g(X) = {cpt}T +t=1 where +g(.) is the conversion function to convert music into time- +ordered sequence of musical events and T is the length of +the music sequence. +Theoretically, the model learns over discrete-time dynamic +directed hypergraphs. Consider a graph G = (V, E) (Figure +1) the vertices (V ) are tokens and edges (E) are sequence of +token. Collection of vertices can be defined as a compound +word and hyperedge in this graph represents sequence of +compound words. In figure 1 v1, v2, v5 are the tokens and +the edge E1 defines a sequence of tokens whereas e1, e2 +defines a hyperedge (connecting more than 2 nodes). And +transitioning from one hyperedge to another defines the +sequence of composition words which we are trying to learn. +Using a transformer we are trying to learn the next musi- +cal event or compound word (combination of tokens). The +self attention part of the transformer learns the dependency +among the elements in musical sequence and different feed- +Figure 1. Graphical Representation of Music Space +forward head is used for tokens of different type. In short +the implementation groups tokens to form compound words +and then perform sequence modeling in this sequence of +compound words, the major contributions are: +• Compose pop-piano music of full song length. +• Compound word sequencing with linear transformer +providing state-of-the-art results in terms of quality +with 5-10x faster training and inference time. +• Music defined as Dynamic Directed Hypergraph. +Generating a new musical event or a group of tokens to +be combined as a compound word at each time step is the +backbone of this model, but it relies on assuming that no +two musical events can occur together. The new hyperedge +generated by the Transformer decoder marks other tokens as +[ignore] once an event of a particular token type is detected. +Can this limit the music generation task? Additionally the +model is trained using only pop music which limits the +expressing power of the transformer. +2. Implementation +Compound Word Transformer (Hsiao et al., 2021) was able +to represent music generation task as a sequence generation +challenge involving musical events defined by compound +words. Leveraging this representation we implement a neu- +ral model which learns to produce music pieces through +more adaptive learning process involving more challenging +task where genres or form of the composition is also con- +sidered. This adds the richness of music art in the learning +process of attention driven sequential learning. We will call +this model Multi-Genre Music Transformer and following +are the steps involved for implementing this: +• Building Dataset: This involves generating compound +word dictionary for songs of different genres. + +Pitch +Duration +v1 +v2 +Velocity +e1 +EA +Chord +Beat +e2 +v5Multi-Genre Music Transformer - Composing Full Length Musical Piece +• Implementing Transformer Model: We implement +our Transformer class, the training steps and the gener- +ation logic for inference. +• Adaptive Learning: We allow our tuned model to +be adaptable by training on a smaller and multi-genre +dataset. +2.1. Building Dataset +To be able to provide a more generalised learning process +for our transformer it needs to be trained with a piano roll +dataset involving musical pieces of variety of genres/style. +The dataset should be based on compound words (Hsiao +et al., 2021) to represent different musical tokens as a com- +bined unit for sequence modeling which is different from +traditional musical dataset (MIDI, REMI). +Figure 2. Dataset Building Pipeline +This required us to build a dataset by selecting music clip- +pings and converting them to piano roll using Onsets and +Frames (Hawthorne et al., 2017). Extracting downbeat and +beat information from these songs using madmom, a mu- +sic signal processing library (B¨ock et al., 2016). Finally +representing these metadata into a compound word repre- +sentation using the dataset generation scripts provided in the +compound word transformer repository1. This also adds on +to the AILabs.tw Pop1K7 dataset (Hsiao et al., 2021) which +currently only includes pop music. Figure 2 demonstrates +the pipeline for creating a new dataset. +Following the pipeline above we managed to create a Com- +pound Word (Hsiao et al., 2021) dataset which involved +1https://github.com/YatingMusic/compound-word- +transformer/blob/main/dataset/Dataset.md +piano roll for 150 musical pieces from 3 different genres +including Electronic Dance Music (EDM), Indie and Hip- +Hop. +2.2. Implementing Transformer Model +We implement a linear transformer(Katharopoulos et al., +2020) to address long sequence dependency which is a very +relevant factor in music generation due to the presence of +a context or a rhythm in the entire musical piece. Hav- +ing an independent feed-forward head in the Transformer +Decoder allows to improve the loss of independent tokens. +This allows the model to scale for additional perspective +(like genre, form or involving a particular chord progres- +sion) in the music by adding an additional token type. We +implement our transformer model in a generic way which +allows user to define its own token sampling model, token +embedding model and these can be scalable for any number +of token types. The loss observed at each feed-forward head +is shown in Figure 6. This shows adding a new token (for +genre/style/form) for model to learn can be simply achieved +by adding an independent feed-forward head for the same. +2.2.1. TOKEN EMBEDDING +Figure 3. Demonstrates how each token undergoes independent +embedding before combining with Positional Encoding. Here +T1, T2...Tk are K different tokens for our Transformer each having +its own embedding function and dimension. We are assuming the +Transformer supports K type of tokens. +The input to a transformer requires positional encoding +added to the embedding vector of our input sequence el- +ements. As each element in our sequence is a compound +word (Hsiao et al., 2021) which is combined of different +tokens, we embed each token separately (allowing to have +adaptive size) and then concatenate them. Having an adap- + +Youtube +WAV Audio +MP3 Audio +Files +Files +Onsets and Frames +Madmom +Piano Transcription +Beat Tracking +Compound Word Transformer +Scripts +Training DataPositional +Transformer Input +Emedding +Feed-Forward Layer +Concatenate +T1 +T2 +Embedding +Embedding +Embedding +TK +Compound WordMulti-Genre Music Transformer - Composing Full Length Musical Piece +tive token size allows to use smaller embedding dimension +for a token type with smaller vocabulary and when we con- +catenate all of these we get an embedding dimension of 512 +for our model. Refer to Figure 3 for detailed steps of token +embedding. +2.2.2. TOKEN SAMPLING +For inference, sampling plays a crucial role to avoid degen- +eration and improve diversity. To avoid degeneration we +follow Nucleus Sampling (Holtzman et al., 2019), which is +a stochastic temperature controlled process. This method +samples from the smallest subset of tokens whose cumu- +lative probability mass exceeds a threshold. We also had +each token to have a separate sampling policy by defining +different threshold p and different temperature parameter +τ (Ackley et al., 1985) for reshaping the probability be- +fore sampling. We reused the inference implementation +from Compound Word Transformer (Hsiao et al., 2021) and +tweaked τ to have higher values for chord to allow more +diverse chord progressions. Figure 4 shows the sampling +process and individual feed-forward layer for each token in +the transformer. +Figure 4. Transformer with N self-attention layers and independent +feed-forward head for each token. We first predict the Token Type +for the particular time-step and then perform a nucleus sampling +before predicting the remaining tokens. +2.3. Adaptive Learning +After defining the model, the next important step is to imple- +ment the training steps. To support scalable token definition +in our generalised transformer we make the training steps +modular and general to variable number of token types. This +allows easy addition of a new token and independently mon- +itor gradient descent optimization for the respective loss. +We trained our model in parallel for 2 different conditions. +The first set of training was performed on the original AIL- +abs.tw Pop1K7 dataset (Hsiao et al., 2021). The second set +of training took into consideration to provide multi-genre +learning environment for the transformer as it involved train- +ing on a dictionary that was generated from 3 different +genres (EDM, Indie, Hip-Hop). +3. Evaluation and Results +To train a multi-genre transformer the primary objective +was to provide it with a dataset which is richer in variety +than the original pop only dataset. With the help of dataset +building pipeline we managed to create a token set which +has a higher variance allowing the model to have a broader +expressive power. Figure 5 shows the comparison of tokens +between the 2 datasets used. +Figure 5. Left image shows token distributions for the songs in the +generated multi-genre dataset and the right image shows similar +distribution for AILabs.tw Pop1K7 dataset (Hsiao et al., 2021). +After training the model for both the datasets we also ob- +serve (refer to Figure 6) the individual token loss and total +average loss is similar and indicates the model converging. +Additionally, the gradient descent is more gradual using the +multi-genre dataset displaying a more settled progression. +We trained the model with 12 self-attentions layers, 8 feed- +forward heads with model dimension of 512 and batch size +of 4 for 180 epochs which took around 17hrs. Then using +the trained model we generated 20 new full length musical +pieces with an average inference time of 12.56sec/song +which is faster than the compound-word transformer though +having slightly less number of average tokens per song. +Table 1 shows a more detailed comparison. + +T(k-1) +Feed-Forward Layer +Feed-Forward Layer +Feed-Forward Layer +Nucleus Sampling +Type +Token +Feed-Forward Layer +T +h +Layer 1 +Layer 2 +Layer N +Self-Attention Layersmean:2342.926std:1194.481 +mean:2138.370_std:775.472 +250 +10 +200 +8 +Number of +150 +songs 6 +Number of +songs +100 +4 +2 +50 ++0 +0 +0 +1000 +2000 +3000 +4000 +5000 +6000 +7000 +2000 +4000 +6000 +8000 +Number of Tokens +Number of TokensMulti-Genre Music Transformer - Composing Full Length Musical Piece +Figure 6. Loss vs Epoch for different token types. The last plot corresponds to the average loss for all different token types. +For a qualitative evaluation of the musical pieces that were +produced we compare (Figure 7) the piano rolls of these +with the piano rolls of original tracks that were used to train +the model. +Original Songs +Generated Songs +Figure 7. Piano roll of original and generated songs. We can see a +rich and complete content for the generated songs similar to some +original tracks. +4. Conclusion +In this project we produce music as a sequence of musical +events produced by a trained Transformer. We leverage the +definition of Compound Word (Hsiao et al., 2021) to define +musical event by grouping multiple tokens. This grouping +greatly reduces the size of our sequence and boosts long- +range learning. This also reduces the training and inference +time for our model remarkably. We also exploit the feature +of each token having its independent feed-forward head for +prediction to make the model scalable for new token types +that can be introduced in our dictionary. This allows to add +any new token for this transformer very easily which can be +used for musical form, chord progression, etc. Additionally, +we created an entire new dataset consisting of multi-genre +compound word dictionary and trained our model with this +to provide it a more adaptive learning environment. The +compositions that were generated were highly rich in musi- +cal events and were of good quality. +Table 1. Quantitative evaluation results for Multi-Genre Transformer and Compound Word Transformer. Results for Compound Word +Transformer comes from the implementation in the paper (Hsiao et al., 2021). +MODEL +TRAINING TIME +GPU +INFERENCE TIME (/SONG) +AVG TOKENS (/SONG) +MULTI-GENRE TRANSFORMER +17 HRS +9.8GB +12.56 SEC +9190 +COMPOUND TRANSFORMER +1.3 DAYS +9.5GB +19.8 SEC +9546 + +tempo loss vs epoch +chord loss vs epoch +bar-beat loss vs epoch +type loss vs epoch +Pop Dataset +14 +Pop Dataset +14 +Pop Dataset +Pop Dataset +14 + Multi-Genre Dataset + Multi-Genre Dataset + Multi-Genre Dataset +1.2 +0.6 +Multi-Genre Dataset +1.2 +1.2 +1.0 +1.0 +0.5 +10 +0.8 +0.8 +0.4 +0.8 +0.6 +0.6 +0.6 +0.4 +E'O +0.4 +0.4 +0.2 +0.2 +0.2 +0.2 +0.0 +75100125150175 +75100125150175 +0.1 +0 +25 +50 +75100125 +150175 +25 +50 +0 +25 +50 +0 +25 +50 +75100125150175 +pitch loss vs epoch +duration loss vs epoch +velocity loss vs epoch +average loss vs epoch +OE +Pop Dataset +18 + Pop Dataset +1.8 + Pop Dataset +16 +PopDataset +Multi-Genre Dataset + Multi-Genre Dataset +Multi-Genre Dataset +Multi-Genre Dataset +2.5 +16 +16 +14 +2.0 +14 +1.2 +14 +1.2 +1.0 +1.5 +1.2 +10 +0.8 +1.0 +0.8 +10 +0.6 +0.5 +0.8 +0.4 +0.6 +0 +25 +50 +100 +150 +175 +25 +50 +125 +150 +175 +0 +50 +75 +100125 150 +175 +0 +0 +50 +125150175.: +84 +(ow) +". +(ow) +. +72 +72 +. +60 +60 +48 +36 +LLLL +36 +: +24 +0 +20 +40 +60 +80 +100 +120 +140 +160 +0 +50 +100 +150 +200 +time (sec) +time (sec)96 +(aw) +72 +72 +itch +60 +60 +48 +8 +96 +36 +0 +50 +100 +150 +0 +50 +100 +150 +200 +time (sec) +time (sec)Multi-Genre Music Transformer - Composing Full Length Musical Piece +References +Ackley, D. 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N., Kaiser, Ł., and Polo- +sukhin, I. +Attention is all you need. +In Advances +in neural information processing systems, pp. 5998– +6008, +2017. +URL +https://proceedings. +neurips.cc/paper/2017/file/ +3f5ee243547dee91fbd053c1c4a845aa-Paper. +pdf. +Zheng, +Z., +Zheng, +L., +and Yang, +Y. +Unlabeled +samples generated by gan improve the person re- +identification baseline in vitro. +In Proceedings +of +the +IEEE +international +conference +on +com- +puter vision, pp. 3754–3762, 2017. +URL https: +//openaccess.thecvf.com/content_iccv_ +2017/html/Zheng_Unlabeled_Samples_ +Generated_ICCV_2017_paper.html. + diff --git a/3dE0T4oBgHgl3EQfeACo/content/tmp_files/load_file.txt b/3dE0T4oBgHgl3EQfeACo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..62f2b68b69fe863e57b964c32edd6ada00369f0e --- /dev/null +++ b/3dE0T4oBgHgl3EQfeACo/content/tmp_files/load_file.txt @@ -0,0 +1,507 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf,len=506 +page_content='Multi-Genre Music Transformer - Composing Full Length Musical Piece Abhinav Kaushal Keshari (Purdue University) Abstract In the task of generating music, the art factor plays a big role and is a great challenge for AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Previ- ous work involving adversarial training (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2018) to produce new music pieces and modeling the compatibility (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) of variety in music (beats, tempo, musical stems) demonstrated great examples of learning this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Though this was limited to generating mashups or learning features from tempo and key distri- butions to produce similar patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) was able to represent music generation task as a sequence generation challenge involving musical events de- fined by compound words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' These musical events give a more accurate description of notes progres- sion, chord change, harmony and the art factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The objective of the project is to implement a Multi-Genre Transformer which learns to produce music pieces through more adaptive learning pro- cess involving more challenging task where gen- res or form of the composition is also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We built a multi-genre compound word dataset, implemented a linear transformer (Katharopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2020) which was trained on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We call this Multi-Genre Transformer, which was able to generate full length new musical pieces which is diverse and comparable to original tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The model trains 2-5 times faster than other mod- els discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Related Work Despite achieving great success in generation challenges using Artificial Intelligence in Natural Language Genera- tion (NLG) there is a factor of art that still makes them different from human like performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' In terms of NLG we can relate it to something like the difference between computer generated article and a piece of art like novels, biography, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' For music art factor always come into ac- count and despite able to produce musical compositions through Adversarial networks or mixing stems using super- vised learning the solution still is very different from an original piece of music which we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Music Generation using GANs Generative adversarial networks (GANs) have provided sig- nificant progress in producing text, videos and images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Sim- ilar efforts have been made to bring neural networks to artistic domain of music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' MuseGAN(Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2018) brought a novel model for generating multi-track music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Until 2018, the progress in using AI to compose music had been able to produce Single-track (monophonic) music Multi-track (polyphonic) music by combining several monophonic melodies in chronological order Music usually being an art involving multiple instruments played together requires music to be multi-track and because music notes are made up of chords, arpeggios or melodies the idea of using a chronological order setting prevents it from being generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The paper(Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2018) address this challenge in gen- eralising real music by discussing current technical lacks in neural network models and how it relates to the real world music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Music is an art of time and has characteristics of coher- ence, rhythm, tension and emotion flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This requires it to have a Temporal Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Music compositions usually involves different instru- ments interacting with one another making the compo- sitions to be harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' To solve this issue a Composer Model is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Musical notes are built of chords, arpeggios or melodies and how they unfold over time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' thus introduc- ing chronological generation of notes is not suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' To address this the paper introduces using bars (seg- ment of time) instead of notes as the basic unit for composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' And then generate music bar by bar us- ing transposed convolutional neural networks to learn translation-invariant patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The paper(Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2018) makes contributions in terms of both ability to artificially compose realistic music and use of generative adversarial framework with temporal and composition models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' In short the contributions are: arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='02385v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='SD] 6 Jan 2023 Multi-Genre Music Transformer - Composing Full Length Musical Piece First GAN based model for generating multi-track se- quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' First model which can generate multi-track polyphonic music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Same model can be used as a music accompaniment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Creates a new Lakh Pianoroll Dataset (LPD) for multi- track piano-rolls For future work metrics in the domain of artificial mu- sic a new set of objective metrics are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' MuseGAN model proposed considers two sub-network gen- erator Gtemp (temporal structure generator) and Gbar (bar generator) making the overall generator: G(z) = � Gbar(Gtemp(z)(t)) �T t=1 where z is the input noise vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The strength of the model is the ability to generate samples having chord like inter- vals (learning features from temporal model) and melodies involving pitch overlap among guitar, piano and strings (learning features from composer model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The model introduces multi-track by modeling interdepen- dency of tracks by proposing 3 different generator model (Jamming, Composer and Hybrid), but the author brings up these based on the understanding of pop music composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This possibly restricts the generator to explore on a broad spectrum of music and prevents it from being generalised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Also worth mentioning is that the work relies on multi-track interdependency, but misses to study about the compatibility of these tracks which can significantly increase the quality of music being generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We will see this issue being addressed in the next paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Modeling the Compatibility of Stem Tracks to Generate Music Mashups(Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) Source separation(Jansson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' D´efossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2019) makes it possible to generate a music mashup with iso- lated stems like vocals, drums, piano, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The challenge lies in producing music which has compatibility between these stems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This paper creates a mashup generation pipeline and trains a model to predict the compatibility by automatically learning to adjust key and tempo (characteristics of quality mashups in real world).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' General models trained for harmonic compatibility (Bernardes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Macas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2018) fails to con- sider subtle features or surprise mixes of disparate samples which is quite common in this art domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Other issue that arises is audio compatibility models like Neural Loop Com- biner (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2020) having lack of vocal source and variety of genres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The authors designed a self supervised learning model by recombining the original combination of stems before source separation to serve as examples of ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' To avoid highly polarized model, semi-supervised learning was introduced which included producing several random mashups by mixing different stems and treated them as unlabeled instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Label smoothing regularization for outliers (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2017) was used to assign uniform distribution to the unlabeled data for loss computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This helps in regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The final architecture consists of 3 modules: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Music Source Separation: Uses MSS algorithm (Jansson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2017) to get different stems vocals, drums, bass and other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Mashup Database (MashupDB): Using Madmom (B¨ock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2016) different features from the music clips are extracted like key, tempo and downbeat in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Using these features and separate stem combinations a mashup database is created which will act as either harmonic or percussion stem candidates for mashup generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Mashup Generation: It uses candidate stems from MashupDB and adjusts key and tempo to produce mashups within 3 conditions - original, matched and unmatched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The model (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) is defined by p(y|V, H, P) where V , H, and P are input signals for respective stems vocal, harmonic, and percussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The output probability p is used as the mashup compatibility and y ∈ {0, 1} stating good or bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The model (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) implementation tries to mimic learning compatibility for producing new mashups and provides objective and subjective evaluation by cross validation among multiple different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This technique becomes easier because of the ability of the model to ex- tract different stems and features and build its own mashup candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This also makes the model training process not dependent on human labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The model is also ro- bust as negative data is added along with positive data for supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The range of music coverage is also extensive and the source separation step makes it easier for the model to be extended to different genres for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' But the current model design lacks the effective embedding of different stems while producing a mashup and makes it dependent on tuning of key and tempo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Currently the implementation comes up with fixed range of key and tempo difference for compatibility and does not explain in detail how they came up with these numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Although defining a range prevents large pitch shifting and time stretching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Additionally the results of the model ranks positive labeled data (original) over unlabeled data which might lead to Multi-Genre Music Transformer - Composing Full Length Musical Piece concerns of flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Another major challenge of the model is the large training time which is around 3 days using an NVIDIA Tesla-V100 GPU whereas using transformer model significantly reduces the training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Music Transformers With state-of-the art neural network we managed to learn features in music by defining certain rules on matching tempo, beats or compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' In the previous paper we also tried to learn compatibility with the help of supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The model though suffered with bias as compati- bility was favoured for matched key or tempo and also lacks generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) considers music as sequence of events and uses a Transformer (neural sequence model) (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2017) to generate a new musical sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' A musical note can be described by note’s pitch, chord, bar, duration, velocity (dynamics), placement (onset time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' If we consider these as tokens we can then define music as sequence of tokens and these tokens are a part of pre-defined vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' As music is multi-faceted a particular type of token can capture only a certain feature like melody, rhythm, harmony.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' All the neural networks until now treated these tokens as equal and thus lacked heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) generates music in a conceptually different way as it allows tokens to be of specific types and let them have their own proper- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Tokens can be of note type (pitch, duration) or metric type (beginning of new beat, bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We then defines a mu- sical event by combination of such tokens which allows to capture co-occurrence relationship among the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This combination of tokens are termed as compound words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' So, now we can represent a music piece (X) as a sequence (S) of compound words (cp) or S = g(X) = {cpt}T t=1 where g(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=') is the conversion function to convert music into time- ordered sequence of musical events and T is the length of the music sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Theoretically, the model learns over discrete-time dynamic directed hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Consider a graph G = (V, E) (Figure 1) the vertices (V ) are tokens and edges (E) are sequence of token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Collection of vertices can be defined as a compound word and hyperedge in this graph represents sequence of compound words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' In figure 1 v1, v2, v5 are the tokens and the edge E1 defines a sequence of tokens whereas e1, e2 defines a hyperedge (connecting more than 2 nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' And transitioning from one hyperedge to another defines the sequence of composition words which we are trying to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Using a transformer we are trying to learn the next musi- cal event or compound word (combination of tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The self attention part of the transformer learns the dependency among the elements in musical sequence and different feed- Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Graphical Representation of Music Space forward head is used for tokens of different type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' In short the implementation groups tokens to form compound words and then perform sequence modeling in this sequence of compound words, the major contributions are: Compose pop-piano music of full song length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Compound word sequencing with linear transformer providing state-of-the-art results in terms of quality with 5-10x faster training and inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Music defined as Dynamic Directed Hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Generating a new musical event or a group of tokens to be combined as a compound word at each time step is the backbone of this model, but it relies on assuming that no two musical events can occur together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The new hyperedge generated by the Transformer decoder marks other tokens as [ignore] once an event of a particular token type is detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Can this limit the music generation task?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Additionally the model is trained using only pop music which limits the expressing power of the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Implementation Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) was able to represent music generation task as a sequence generation challenge involving musical events defined by compound words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Leveraging this representation we implement a neu- ral model which learns to produce music pieces through more adaptive learning process involving more challenging task where genres or form of the composition is also con- sidered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This adds the richness of music art in the learning process of attention driven sequential learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We will call this model Multi-Genre Music Transformer and following are the steps involved for implementing this: Building Dataset: This involves generating compound word dictionary for songs of different genres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Pitch Duration v1 v2 Velocity e1 EA Chord Beat e2 v5Multi-Genre Music Transformer - Composing Full Length Musical Piece Implementing Transformer Model: We implement our Transformer class, the training steps and the gener- ation logic for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Adaptive Learning: We allow our tuned model to be adaptable by training on a smaller and multi-genre dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Building Dataset To be able to provide a more generalised learning process for our transformer it needs to be trained with a piano roll dataset involving musical pieces of variety of genres/style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The dataset should be based on compound words (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) to represent different musical tokens as a com- bined unit for sequence modeling which is different from traditional musical dataset (MIDI, REMI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Dataset Building Pipeline This required us to build a dataset by selecting music clip- pings and converting them to piano roll using Onsets and Frames (Hawthorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Extracting downbeat and beat information from these songs using madmom, a mu- sic signal processing library (B¨ock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Finally representing these metadata into a compound word repre- sentation using the dataset generation scripts provided in the compound word transformer repository1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This also adds on to the AILabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='tw Pop1K7 dataset (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) which currently only includes pop music.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Figure 2 demonstrates the pipeline for creating a new dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Following the pipeline above we managed to create a Com- pound Word (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) dataset which involved 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='com/YatingMusic/compound-word- transformer/blob/main/dataset/Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='md piano roll for 150 musical pieces from 3 different genres including Electronic Dance Music (EDM), Indie and Hip- Hop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Implementing Transformer Model We implement a linear transformer(Katharopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2020) to address long sequence dependency which is a very relevant factor in music generation due to the presence of a context or a rhythm in the entire musical piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Hav- ing an independent feed-forward head in the Transformer Decoder allows to improve the loss of independent tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This allows the model to scale for additional perspective (like genre, form or involving a particular chord progres- sion) in the music by adding an additional token type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We implement our transformer model in a generic way which allows user to define its own token sampling model, token embedding model and these can be scalable for any number of token types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The loss observed at each feed-forward head is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This shows adding a new token (for genre/style/form) for model to learn can be simply achieved by adding an independent feed-forward head for the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' TOKEN EMBEDDING Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Demonstrates how each token undergoes independent embedding before combining with Positional Encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Here T1, T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Tk are K different tokens for our Transformer each having its own embedding function and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We are assuming the Transformer supports K type of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The input to a transformer requires positional encoding added to the embedding vector of our input sequence el- ements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' As each element in our sequence is a compound word (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) which is combined of different tokens, we embed each token separately (allowing to have adaptive size) and then concatenate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Having an adap- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Youtube ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='WAV Audio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='MP3 Audio ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Files ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Files ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Onsets and Frames ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Madmom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Piano Transcription ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Beat Tracking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Compound Word Transformer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Scripts ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Training DataPositional ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Transformer Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Emedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Feed-Forward Layer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Concatenate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='T1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='T2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='TK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='Compound WordMulti-Genre Music Transformer - Composing Full Length Musical Piece ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='tive token size allows to use smaller embedding dimension ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='for a token type with smaller vocabulary and when we con- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='catenate all of these we get an embedding dimension of 512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Refer to Figure 3 for detailed steps of token embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' TOKEN SAMPLING For inference, sampling plays a crucial role to avoid degen- eration and improve diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' To avoid degeneration we follow Nucleus Sampling (Holtzman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2019), which is a stochastic temperature controlled process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This method samples from the smallest subset of tokens whose cumu- lative probability mass exceeds a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We also had each token to have a separate sampling policy by defining different threshold p and different temperature parameter τ (Ackley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 1985) for reshaping the probability be- fore sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We reused the inference implementation from Compound Word Transformer (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) and tweaked τ to have higher values for chord to allow more diverse chord progressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Figure 4 shows the sampling process and individual feed-forward layer for each token in the transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Transformer with N self-attention layers and independent feed-forward head for each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We first predict the Token Type for the particular time-step and then perform a nucleus sampling before predicting the remaining tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Adaptive Learning After defining the model, the next important step is to imple- ment the training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' To support scalable token definition in our generalised transformer we make the training steps modular and general to variable number of token types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This allows easy addition of a new token and independently mon- itor gradient descent optimization for the respective loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We trained our model in parallel for 2 different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The first set of training was performed on the original AIL- abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='tw Pop1K7 dataset (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The second set of training took into consideration to provide multi-genre learning environment for the transformer as it involved train- ing on a dictionary that was generated from 3 different genres (EDM, Indie, Hip-Hop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Evaluation and Results To train a multi-genre transformer the primary objective was to provide it with a dataset which is richer in variety than the original pop only dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' With the help of dataset building pipeline we managed to create a token set which has a higher variance allowing the model to have a broader expressive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Figure 5 shows the comparison of tokens between the 2 datasets used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Left image shows token distributions for the songs in the generated multi-genre dataset and the right image shows similar distribution for AILabs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='tw Pop1K7 dataset (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' After training the model for both the datasets we also ob- serve (refer to Figure 6) the individual token loss and total average loss is similar and indicates the model converging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Additionally, the gradient descent is more gradual using the multi-genre dataset displaying a more settled progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We trained the model with 12 self-attentions layers, 8 feed- forward heads with model dimension of 512 and batch size of 4 for 180 epochs which took around 17hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Then using the trained model we generated 20 new full length musical pieces with an average inference time of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='56sec/song which is faster than the compound-word transformer though having slightly less number of average tokens per song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Table 1 shows a more detailed comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' T(k-1) Feed-Forward Layer Feed-Forward Layer Feed-Forward Layer Nucleus Sampling Type Token Feed-Forward Layer T h Layer 1 Layer 2 Layer N Self-Attention Layersmean:2342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='926std:1194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='481 mean:2138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='370_std:775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='472 250 10 200 8 Number of 150 songs 6 Number of songs 100 4 2 50 +0 0 0 1000 2000 3000 4000 5000 6000 7000 2000 4000 6000 8000 Number of Tokens Number of TokensMulti-Genre Music Transformer - Composing Full Length Musical Piece Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Loss vs Epoch for different token types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The last plot corresponds to the average loss for all different token types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' For a qualitative evaluation of the musical pieces that were produced we compare (Figure 7) the piano rolls of these with the piano rolls of original tracks that were used to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Original Songs Generated Songs Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Piano roll of original and generated songs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We can see a rich and complete content for the generated songs similar to some original tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Conclusion In this project we produce music as a sequence of musical events produced by a trained Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We leverage the definition of Compound Word (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021) to define musical event by grouping multiple tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This grouping greatly reduces the size of our sequence and boosts long- range learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This also reduces the training and inference time for our model remarkably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' We also exploit the feature of each token having its independent feed-forward head for prediction to make the model scalable for new token types that can be introduced in our dictionary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' This allows to add any new token for this transformer very easily which can be used for musical form, chord progression, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Additionally, we created an entire new dataset consisting of multi-genre compound word dictionary and trained our model with this to provide it a more adaptive learning environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' The compositions that were generated were highly rich in musi- cal events and were of good quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Quantitative evaluation results for Multi-Genre Transformer and Compound Word Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' Results for Compound Word Transformer comes from the implementation in the paper (Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content=' MODEL TRAINING TIME GPU INFERENCE TIME (/SONG) AVG TOKENS (/SONG) MULTI-GENRE TRANSFORMER 17 HRS 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='8GB 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='56 SEC 9190 COMPOUND TRANSFORMER 1.' metadata={'source': 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50 0 25 50 0 25 50 75100125150175 pitch loss vs epoch duration loss vs epoch velocity loss vs epoch average loss vs epoch OE Pop Dataset 18 Pop Dataset 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='8 Pop Dataset 16 PopDataset Multi-Genre Dataset Multi-Genre Dataset Multi-Genre Dataset Multi-Genre Dataset 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='5 16 16 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='0 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='2 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE0T4oBgHgl3EQfeACo/content/2301.02385v1.pdf'} +page_content='2 1.' metadata={'source': 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0000000000000000000000000000000000000000..66c8f5ef9c31fb8aaba1b9b2e132995291b02a5d --- /dev/null +++ b/3dE3T4oBgHgl3EQfoQqU/content/tmp_files/2301.04632v1.pdf.txt @@ -0,0 +1,1845 @@ +Federated Learning under Heterogeneous and +Correlated Client Availability +Angelo Rodio∗, Francescomaria Faticanti∗, Othmane Marfoq∗†, Giovanni Neglia∗, Emilio Leonardi‡ +∗Inria, Universit´e Cˆote d’Azur, France. Email: {firstname.lastname}@inria.fr, +†Accenture Labs, Sophia-Antipolis, France. Email: {firstname.lastname}@accenture.com, +‡Politecnico di Torino, Turin, Italy. Email: {firstname.lastname}@polito.it +Abstract—The enormous amount of data produced by mobile and +IoT devices has motivated the development of federated learning +(FL), a framework allowing such devices (or clients) to collabora- +tively train machine learning models without sharing their local +data. FL algorithms (like FedAvg) iteratively aggregate model +updates computed by clients on their own datasets. Clients may +exhibit different levels of participation, often correlated over time +and with other clients. This paper presents the first convergence +analysis for a FedAvg-like FL algorithm under heterogeneous +and correlated client availability. Our analysis highlights how +correlation adversely affects the algorithm’s convergence rate +and how the aggregation strategy can alleviate this effect at +the cost of steering training toward a biased model. Guided +by the theoretical analysis, we propose CA-Fed, a new FL +algorithm that tries to balance the conflicting goals of maximizing +convergence speed and minimizing model bias. To this purpose, +CA-Fed dynamically adapts the weight given to each client and +may ignore clients with low availability and large correlation. Our +experimental results show that CA-Fed achieves higher time- +average accuracy and a lower standard deviation than state-of- +the-art AdaFed and F3AST, both on synthetic and real datasets. +Index Terms—Federated Learning, Distributed Optimization. +I. INTRODUCTION +The enormous amount of data generated by mobile and IoT de- +vices motivated the emergence of distributed machine learning +training paradigms [1], [2]. Federated Learning (FL) [3]–[6] +is an emerging framework where geographically distributed +devices (or clients) participate in the training of a shared +Machine Learning (ML) model without sharing their local +data. FL was proposed to reduce the overall cost of collecting +a large amount of data as well as to protect potentially +sensitive users’ private information. In the original Federated +Averaging algorithm (FedAvg) [4], a central server selects +a random subset of clients from the set of available clients +and broadcasts them the shared model. The sampled clients +perform a number of independent Stochastic Gradient Descent +(SGD) steps over their local datasets and send their local +model updates back to the server. Then, the server aggregates +the received client updates to produce a new global model, and +a new training round begins. At each iteration of FedAvg, the +server typically samples randomly a few hundred devices to +participate [7], [8]. +This research was supported by the French government through the 3IA +Cˆote d’Azur Investments in the Future project by the National Research +Agency (ANR) with reference ANR-19-P3IA-0002, and by Groupe La Poste, +sponsor of Inria Foundation, in the framework of FedMalin Inria Challenge. +A first version of this work has been accepted at IEEE INFOCOM 2023. +In real-world scenarios, the availability/activity of clients is +dictated by exogenous factors that are beyond the control of +the orchestrating server and hard to predict. For instance, only +smartphones that are idle, under charge, and connected to +broadband networks are commonly allowed to participate in +the training process [4], [9]. These eligibility requirements can +make the availability of devices correlated over time and space +[7], [10]–[12]. For example, temporal correlation may origin +from a smartphone being under charge for a few consecutive +hours and then ineligible for the rest of the day. Similarly, +the activity of a sensor powered by renewable energy may +depend on natural phenomena intrinsically correlated over +time (e.g., solar light). Spatial correlation refers instead to +correlation across different clients, which often emerges as +consequence of users’ different geographical distribution. For +instance, clients in the same time zone often exhibit similar +availability patterns, e.g., due to time-of-day effects. +Temporal correlation in the data sampling procedure is known +to negatively affect the performance of ML training even in +the centralized setting [13], [14] and can potentially lead to +catastrophic forgetting: the data used during the final training +phases can have a disproportionate effect on the final model, +“erasing” the memory of previously learned information [15], +[16]. Catastrophic forgetting has also been observed in FL, +where clients in the same geographical area have more similar +local data distributions and clients’ participation follows a +cyclic daily pattern (leading to spatial correlation) [7], [10], +[11], [17]. Despite this evidence, a theoretical study of the +convergence of FL algorithms under both temporally and +spatially correlated client participation is still missing. +This +paper +provides +the +first +convergence +analysis +of +FedAvg [4] under heterogeneous and correlated client avail- +ability. We assume that clients’ temporal and spatial availabil- +ity follows an arbitrary finite-state Markov chain: this assump- +tion models a realistic scenario in which the activity of clients +is correlated and, at the same time, still allows the analytical +tractability of the system. Our theoretical analysis (i) quantifies +the negative effect of correlation on the algorithm’s conver- +gence rate through an additional term, which depends on the +spectral properties of the Markov chain; (ii) points out a trade- +off between two conflicting objectives: slow convergence to +the optimal model, or fast convergence to a biased model, i.e., +a model that minimizes an objective function different from the +initial target. Guided by insights from the theoretical analysis, +1 +arXiv:2301.04632v1 [cs.LG] 11 Jan 2023 + +we propose CA-Fed, an algorithm which dynamically assigns +weights to clients and achieves a good trade-off between +maximizing convergence speed and minimizing model bias. +Interestingly, CA-Fed can decide to ignore clients with low +availability and high temporal correlation. Our experimental +results demonstrate that excluding such clients is a simple, but +effective approach to handle the heterogeneous and correlated +client availability in FL. Indeed, while CA-Fed achieves a +comparable maximum accuracy as the state-of-the-art methods +F3AST [18] and AdaFed [19], its test accuracy exhibits +higher time-average and smaller variability over time. +The remainder of this paper is organized as follows. Section II +describes the problem of correlated client availability in FL +and discusses the main related works. Section III provides +a convergence analysis of FedAvg under heterogeneous and +correlated client participation. CA-Fed, our correlation-aware +FL algorithm, is presented in Section IV. We evaluate CA-Fed +in Section V, comparing it with state-of-the-art methods on +synthetic and real-world data. Section VII concludes the paper. +II. BACKGROUND AND RELATED WORKS +We consider a finite set K of N clients. Each client k ∈ K +holds a local dataset Dk. Clients aim to jointly learn the +parameters w ∈ W ⊆ Rd of a global ML model (e.g., the +weights of a neural network architecture). During training, the +quality of the model with parameters w on a data sample +ξ ∈ Dk is measured by a loss function f(w; ξ). The clients +solve, under the orchestration of a central server, the following +optimization problem: +min +w∈W ⊆Rd +� +F(w) := +� +k∈K +αkFk(w) +� +, +(1) +where Fk(w) := +1 +|Dk| +� +ξ∈Dk f(w; ξ) is the average loss +computed on client k’s local dataset, and α = (αk)k∈K are +positive coefficients such that � +k αk = 1. They represent +the target importance assigned by the central server to each +client k. Typically (αk)k∈K are set proportional to the clients’ +dataset size |Dk|, such that the objective function F in (1) +coincides with the average loss computed on the union of the +clients’ local datasets D = ∪k∈KDk. +Under proper assumptions, precised in Section III, Problem (1) +admits a unique solution. We use w∗ (resp. F ∗) to denote +the minimizer (resp. the minimum value) of F. Moreover, for +k∈K, Fk admits a unique minimizer on W. We use w∗ +k (resp. +F ∗ +k ) to denote the minimizer (resp. the minimum value) of Fk. +Problem (1) is commonly solved through iterative algo- +rithms [4], [8] requiring multiple communication rounds be- +tween the server and the clients. At round t > 0, the server +broadcasts the latest estimate of the global model wt,0 to +the set of available clients (At). Client k ∈ At updates the +global model with its local data through E ≥ 1 steps of local +Stochastic Gradient Descent (SGD): +wk +t,j+1 = wk +t,j − ηt∇Fk(wk +t,j, Bk +t,j) +j = 0, . . . , E − 1, (2) +where ηt +> 0 is an appropriately chosen learning rate, +referred to as local learning rate; Bk +t,j is a random batch +sampled from client k’ local dataset at round t and step j; +∇Fk(·, B) := +1 +|B| +� +ξ∈B ∇f(·, ξ) is an unbiased estimator of +the local gradient ∇Fk. Then, each client sends its local model +update ∆k +t := wk +t,E − wk +t,0 to the server. The server computes +∆t := � +k∈At qk ·∆k +t , a weighted average of the clients’ local +updates with non-negative aggregation weights q = (qk)k∈K. +The choice of the aggregation weights defines an aggregation +strategy (we will discuss different aggregation strategies later). +The aggregated update ∆t can be interpreted as a proxy for +−∇F(wt,0); the server applies it to the global model: +wt+1,0 = ProjW (wt,0 + ηs · ∆t) +(3) +where ProjW (·) denotes the projection over the set W, and +ηs > 0 is an appropriately chosen learning rate, referred to as +the server learning rate.1 +The aggregate update ∆t is, in general, a biased estimator +of −∇F(wt,0), where each client k is taken into account +proportionally to its frequency of appearance in the set At and +to its aggregation weight qk. Indeed, under proper assumptions +specified in Section III, one can show (see Theorem 2) that the +update rule described by (2) and (3) converges to the unique +minimizer of a biased global objective FB, which depends +both on the clients’ availability (i.e., on the sequence (At)t>0) +and on the aggregation strategy (i.e., on q = (qk)k∈K): +FB(w) := �N +k=1 pkFk(w), with pk := +πkqk +�N +h=1 πhqh , +(4) +where πk := limt→∞ P(k ∈ At) is the asymptotic availability +of client k. The coefficients p = (pk)k∈K can be interpreted +as the biased importance the server is giving to each client k +during training, in general different from the target importance +α. In what follows, w∗ +B (resp. F ∗ +B) denotes the minimizer +(resp. the minimum value) of FB. +In some large-scale FL applications, like training Google +keyboard next-word prediction models, each client participates +in training at most for one round. The orchestrator usually +selects a few hundred clients at each round for a few thousand +rounds (e.g., see [5, Table 2]), but the available set of clients +may include hundreds of millions of Android devices. In this +scenario, it is difficult to address the potential bias unless there +is some a-priori information about each client’s availability. +Anyway, FL can be used by service providers with access +to a much smaller set of clients (e.g., smartphone users that +have installed a specific app). In this case, a client participates +multiple times in training: the orchestrating server may keep +track of each client’s availability and try to compensate for +the potentially dangerous heterogeneity in their participation. +Much previous effort on federated learning [4], [17]–[19], +[22]–[25] considered this problem and, under different as- +1The aggregation rule (3) has been considered also in other works, e.g., [8], +[20], [21]. In other FL algorithms, the server computes an average of clients’ +local models. This aggregation rule can be obtained with minor changes to (3). +2 + +sumptions on the clients’ availability (i.e., on (At)t>0), de- +signed aggregation strategies that unbias ∆t through an appro- +priate choice of q. Reference [22] provides the first analysis of +FedAvg on non-iid data under clients’ partial participation. +Their analysis covers both the case when active clients are +sampled uniformly at random without replacement from K and +assigned aggregation weights equal to their target importance +(as assumed in [4]), and the case when active clients are +sampled iid with replacement from K with probabilities α +and assigned equal weights (as assumed in [23]). However, +references [4], [22], [23] ignore the variance induced by the +clients stochastic availability. The authors of [24] reduce such +variance by considering only the clients with important up- +dates, as measured by the value of their norm. References [17] +and [25] reduce the aggregation variance through clustered and +soft-clustered sampling, respectively. +Some recent works [18], [19], [26] do not actively pursue the +optimization of the unbiased objective. Instead, they derive +bounds for the convergence error and propose heuristics to +minimize those bounds, potentially introducing some bias. +Our work follows a similar development: we compare our +algorithm with F3AST from [18] and AdaFed from [19]. +The novelty of our study is in considering the spatial and +temporal correlation in clients’ availability dynamics. As dis- +cussed in the introduction, such correlations are also intro- +duced by clients’ eligibility criteria, e.g., smartphones being +under charge and connected to broadband networks. The effect +of correlation has been ignored until now, probably due to the +additional complexity in studying FL algorithms’ convergence. +To the best of our knowledge, the only exception is [18], which +scratches the issue of spatial correlation by proposing two +different algorithms for the case when clients’ availabilities +are uncorrelated and for the case when they are positively +correlated (there is no smooth transition from one algorithm +to the other as a function of the degree of correlation). +The effect of temporal correlation on centralized stochastic +gradient methods has been addressed in [12]–[14], [27]: these +works study a variant of stochastic gradient descent where +samples are drawn according to a Markov chain. Refer- +ence [12] extends its analysis to a FL setting where each client +draws samples according to a Markov chain. In contrast, our +work does not assume a correlation in the data sampling but +rather in the client’s availability. Nevertheless, some of our +proof techniques are similar to those used in this line of work +and, in particular, we rely on some results in [14]. +III. ANALYSIS +A. Main assumptions +We consider a time-slotted system where a slot corresponds +to one FL communication round. We assume that clients’ +availability over the timeslots t ∈ N follows a discrete-time +Markov chain (At)t≥0.2 +2In Section III-D we will focus on the case where this chain is the +superposition of N independent Markov chains, one for each client. +Assumption 1. The Markov chain (At)t≥0 on the finite state +space [M] is time-homogeneous, irreducible, and aperiodic. It +has transition matrix P and stationary distribution π. +Markov chains have already been used in the literature to +model the dynamics of stochastic networks where some nodes +or edges in the graph can switch between active and inactive +states [28], [29]. The previous Markovian assumption, while +allowing a great degree of flexibility, still guarantees the +analytical tractability of the system. The distance dynamics +between current and stationary distribution of the Markov +process can be characterized by the spectral properties of its +transition matrix P [30]. Let λ2(P ) denote the the second +largest eigenvalue of P in absolute value. Previous works [14] +have shown that: +max +i,j∈[M] |[P t]i,j − πj| ≤ CP · λ(P )t, +for t ≥ TP , +(5) +where the parameter λ(P ) := (λ2(P ) + 1)/2, and CP , TP +are positive constants whose values are reported in [14, +Lemma 1].3 Note that λ(P ) quantifies the correlation of the +Markov process (At)t≥0: the closer λ(P ) is to one, the slower +the Markov chain converges to its stationary distribution. +In our analysis, we make the following additional assumptions. +Let w∗, w∗ +B denote the minimizers of F and FB on W, +respectively. +Assumption 2. The hypothesis class W is convex, compact, +and contains in its interior the minimizers w∗, w∗ +B, w∗ +k. +The following assumptions concern clients’ local objective +functions {Fk}k∈K. Assumptions 3 and 4 are standard in +the literature on convex optimization [31, Sections 4.1, 4.2]. +Assumption 5 is a standard hypothesis in the analysis of +federated optimization algorithms [8, Section 6.1]. +Assumption 3 (L-smoothness). The local functions {Fk}N +k=1 +have L-Lipschitz continuous gradients: Fk(v) ≤ Fk(w) + +⟨∇Fk(w), v − w⟩ + L +2 ∥v − w∥2 +2, ∀v, w ∈ W. +Assumption +4 +(Strong convexity). The local functions +{Fk}N +k=1 +are +µ-strongly +convex: +Fk(v) +≥ +Fk(w) + +⟨∇Fk(w), v − w⟩ + µ +2 ∥v − w∥2 +2 , ∀v, w ∈ W. +Assumption 5 (Bounded variance). The variance of stochastic +gradients in each device is bounded: E ∥∇Fk(wk +t,j, ξk +t,j) − +∇Fk(wk +t,j)∥2 ≤ σ2 +k, k = 1, . . . , N. +Assumptions 2–5 imply the following properties for the local +functions, described by Lemma 1 (proof in Appendix B). +Lemma 1. Under Assumptions 2–5, there exist constants D, +G, and H > 0, such that, for w ∈ W and k ∈ K, we have: +∥∇Fk(w)∥ ≤ D, +(6) +E ∥∇Fk(w, ξ)∥2 ≤ G2, +(7) +|Fk(w) − Fk(w∗ +B)| ≤ H. +(8) +3Note that (5) holds for different definitions of λ(P ) as far as λ(P ) ∈ +(λ2(P ), 1). The specific choice for λ(P ) changes the constants CP and TP . +3 + +Similarly to other works [8], [22], [23], [32], we introduce a +metric to quantify the heterogeneity of clients’ local datasets: +Γ := max +k∈K{Fk(w∗) − F ∗ +k }. +(9) +If the local datasets are identical, the local functions {Fk}k∈K +coincide among them and with F, w∗ is a minimizer of each +local function, and Γ = 0. In general, Γ is smaller the closer +the distributions the local datasets are drawn from. +B. Main theorems +Theorem 1 (proof in Appendix A) decomposes the error of +the target global objective as the sum of an optimization error +for the biased global objective and a bias error. +Theorem 1 (Decomposing the total error). Under Assump- +tions 2–4, the optimization error of the target global objective +ϵ = F(w) − F ∗ can be bounded as follows: +ϵ ≤ 2κ2(FB(w) − F ∗ +B) +� +�� +� +:=ϵopt ++ 2κ4χ2 +α∥pΓ +� +�� +� +:=ϵbias +, +(10) +where κ := L/µ, and χ2 +α∥p := �N +k=1 (αk − pk)2/pk. +Theorem 2 below proves that the optimization error ϵopt asso- +ciated to the biased objective FB, evaluated on the trajectory +determined by scheme (3), asymptotically vanishes. The non- +vanishing bias error ϵbias captures the discrepancy between +F(w) and FB(w). This latter term depends on the chi-square +divergence χ2 +α∥p between the target and biased probability +distributions α = (αk)k∈K and p = (pk)k∈K, and on +Γ, that quantifies the degree of heterogeneity of the local +functions. When all local functions are identical (Γ = 0), +the bias term ϵbias also vanishes. For Γ > 0, the bias error +can still be controlled by the aggregation weights assigned +to the devices. In particular, the bias term vanishes when +qk ∝ αk/πk, ∀k ∈ K. Since it asymptotically cancels the bias +error, we refer to this choice as unbiased aggregation strategy. +However, in practice, FL training is limited to a finite number +of iterations T (typically a few hundreds [5], [7]), and the +previous asymptotic considerations may not apply. In this +regime, the unbiased aggregation strategy can be suboptimal, +since the minimization of ϵbias not necessarily leads to the +minimization of the total error ϵ ≤ ϵopt + ϵbias. This motivates +the analysis of the optimization error ϵopt. +Theorem 2 (Convergence of the optimization error ϵopt). Let +Assumptions 1–5 hold and the constants M, L, D, G, H, Γ, +σk, CP , TP , λ(P ) be defined as above. Let Q = � +k∈K qk. +Let the stepsizes satisfy: +� +t ηt = +∞, +� +t ln(t) · η2 +t < +∞. +(11) +Let T denote the total communication rounds. For T ≥ TP , +the expected optimization error can be bounded as follows: +E[FB( ¯wT,0) − F ∗ +B] ≤ +1 +2 q⊺Σq+υ +π⊺q ++ ψ + +φ +ln(1/λ(P )) +(�T +t=1 ηt) +, +(12) +where ¯wT,0 := +�T +t=1 ηtwt,0 +�T +t=1 ηt +, and +Σ = diag(σ2 +kπk +� +t η2 +t ), +υ = 2 +E ∥w0,0 − w∗∥2 + 1 +4MQ � +t(η2 +t + 1 +t2 ), +ψ = 4L(EQ + 2)Γ � +t η2 +t + 2 +3(E − 1)(2E − 1)G2 � +t η2 +t , +Jt =min {max {⌈ln (2CP Ht)/ln (1/λ(P ))⌉ , TP } , t}, +φ = 2EDGQ � +t ln(2CP Ht)η2 +t−Jt. +Theorem 2 (proof in Appendix B) proves convergence of +the expected biased objective FB to its minimum F ∗ +B under +correlated client participation. Our bound (12) captures the +effect of correlation through the factor ln (1/λ(P )): a high +correlation worsens the convergence rate. In particular, we +found that the numerator of (12) has a quadratic-over-linear +fractional dependence on q. Minimizing ϵopt leads, in general, +to a different choice of q than minimizing ϵbias. +C. Minimizing the total error ϵ ≤ ϵopt + ϵbias +Our analysis points out a trade-off between minimizing ϵopt +or ϵbias. Our goal is to find the optimal aggregation weights q∗ +that minimize the upper bound on total error ϵ(q) in (10): +minimize +q +ϵopt(q) + ϵbias(q); +subject to +q ≥ 0, +∥q∥1 = Q. +(13) +In Appendix E we prove that (13) is a convex optimization +problem, which can be solved with the method of Lagrange +multipliers. However, the solution is not of practical utility +because the constants in (10) and (12) (e.g., L, µ, Γ, CP ) are +in general problem-dependent and difficult to estimate during +training. In particular, Γ poses particular difficulties as it is +defined in terms of the minimizer of the target objective F, but +the FL algorithm generally minimizes the biased function FB. +Moreover, the bound in (10), similarly to the bound in [32], +diverges when setting some qk equal to 0, but this is simply +an artifact of the proof technique. A result of more practical +interest is the following (proof in Appendix C): +Theorem 3 (An alternative decomposition of the total er- +ror ϵ). Under the same assumptions of Theorem 1, let Γ′ := +maxk{Fk(w∗ +B) − F ∗ +k }. The following result holds: +ϵ ≤ 2κ2(FB(w) − F ∗ +B) +� +�� +� +:=ϵopt ++ 8κ4d2 +T V (α, p)Γ′ +� +�� +� +:=ϵ′ +bias +, +(14) +where dT V (α, p) := 1 +2 +�N +k=1|αk − pk| is the total variation +distance between the probability distributions α and p. +The new constant Γ′ is defined in terms of w∗ +B, and then +it is easier to evaluate during training. However, Γ′ depends +on q, because it is evaluated at the point of minimum of FB. +This dependence makes the minimization of the right-hand +side of (14) more challenging (for example, the corresponding +problem is not convex). We study the minimization of the two +terms ϵopt and ϵ′ +bias separately and learn some insights, which +we use to design the new FL algorithm CA-Fed. +4 + +D. Minimizing ϵopt +The minimization of ϵopt is still a convex optimization problem +(Appendix D). In particular, at the optimum non-negative +weights are set accordingly to q∗ +k = a(λ∗πk − θ∗) with +a, λ∗, and θ∗ positive constants (see (29)). It follows that +clients with smaller availability get smaller weights in the +aggregation. In particular, this suggests that clients with the +smallest availability can be excluded from the aggregation, +leading to the following guideline: +Guideline A: to speed up the convergence, we can exclude, +i.e., set q∗ +k = 0, the clients with lowest availability πk. +This guideline can be justified intuitively: updates from clients +with low participation may be too sporadic to allow the FL +algorithm to keep track of their local objectives. They act as +a noise slowing down the algorithm’s convergence. It may be +advantageous to exclude these clients from participating. +We observe that the choice of the aggregation weights q does +not affect the clients’ availability process and, in particular, +λ(P ). However, if the algorithm excludes some clients, it +is possible to consider the state space of the Markov chain +that only specifies the availability state of the remaining +clients, and this Markov chain may have different spectral +properties. For the sake of concreteness, we consider here +(and in the rest of the paper) the particular case when the +availability of each client k evolves according to a two- +states Markov chain (Ak +t )t≥0 with transition probability ma- +trix Pk and these Markov chains are all independent. In +this case, the aggregate process is described by the product +Markov chain (At)t≥0 with transition matrix P = � +k∈K Pk +and λ(P ) = maxk∈K λ(Pk), where Pi +� Pj denotes the +Kronecker product between matrices Pi and Pj [30, Exer- +cise 12.6]. In this setting, it is possible to redefine the Markov +chain (At)t≥0 by taking into account the reduced state space +defined by the clients with a non-null aggregation weight, i.e., +P ′ = � +k′∈K|qk′>0 Pk′ and λ(P ′) = maxk′∈K|qk′>0 λ(Pk′), +which is potentially smaller than the case when all clients +participate to the aggregation. These considerations lead to +the following guideline: +Guideline B: to speed up the convergence, we can exclude, +i.e., set q∗ +k = 0, the clients with largest λ(Pk). +Intuition also supports this guideline. Clients with large λ(Pk) +tend to be available or unavailable for long periods of time. +Due to the well-known catastrophic forgetting problem affect- +ing gradient methods [33], [34], these clients may unfairly +steer the algorithm toward their local objective when they +appear at the final stages of the training period. Moreover, +their participation in the early stages may be useless, as their +contribution will be forgotten during their long absence. The +FL algorithm may benefit from directly neglecting such clients. +We observe that guideline B strictly applies to this specific +setting where clients’ dynamics are independent (and there +is no spatial correlation). We do not provide a corresponding +Algorithm 1: CA-Fed (Correlation-Aware FL) +Input : w0,0, α, q(0), {ηt}T +t=1, ηs, E, β, τ +1 Initialize ˆF (0), ˆF ∗, ˆΓ +′(0), ˆπ(0), and ˆλ(0); +2 for t = 1, . . . , T do +3 +Receive set of active client At, loss vector F (t); +4 +Update ˆF (t), ˆΓ +′(t), ˆπ(t), and ˆλ(t); +5 +Initialize q(t) = +α +ˆπ(t) ; +6 +q(t) ← get(q(t), α, ˆF (t), ˆF ∗, ˆΓ +′(t), ˆπ(t), ˆλ(t)); +7 +q(t) ← get(q(t), α, ˆF (t), ˆF ∗, ˆΓ +′(t), ˆπ(t), �ˆπ(t)); +8 +for client {k ∈ At; q(t) +k +> 0}, in parallel do +9 +for j = 0, . . . , E − 1 do +10 +wk +t,j+1 = wk +t,j − ηt∇Fk(wk +t,j, Bk +t,j) ; +11 +∆k +t ← wt,E − wt,0; +12 +wt+1,0 ← ProjW (wt,0 + ηs +� +k∈At q +(t) +k · ∆k +t ); +13 Function get(q, α, F , F ∗, Γ, π, ρ): +14 +K ← sort by descending order in ρ; +15 +ˆϵ ← ⟨F −F ∗, π ˜⊙q⟩ + d2 +T V (α, π ˜⊙q) · Γ; +16 +for k ∈ K do +17 +q+ +k ← 0; +18 +ˆϵ+ ← ⟨F −F ∗, π ˜⊙q+⟩ + d2 +T V (α, π ˜⊙q+) · Γ; +19 +if ˆϵ − ˆϵ+ ≥ τ then +20 +ˆϵ ← ˆϵ+; +21 +q ← q+; +22 +return q +guideline for the case when clients are spatially correlated (we +leave this task for future research). However, in this more gen- +eral setting, it is possible to ignore guideline B but still draw +on guidelines A and C, or still consider guideline B if clients +are spatially correlated (see discussion in Section VI-B). +E. Minimizing ϵ′ +bias +The bias error ϵ′ +bias in (14) vanishes when the total variation +distance between the target importance α and the biased +importance p is zero, i.e., when qk ∝ αk/πk, ∀k ∈ K. Then, +after excluding the clients that contribute the most to the +optimization error and particularly slow down the convergence +(guidelines A and B), we can assign to the remaining clients an +aggregation weight inversely proportional to their availability, +such that the bias error ϵ′ +bias is minimized. +Guideline C: to reduce the bias error, we set q∗ +k ∝ αk/πk for +the clients that are not excluded by the previous guidelines. +IV. PROPOSED ALGORITHM +Guidelines A and B in Section III suggest that the minimiza- +tion of ϵopt can lead to the exclusion of some available clients +from the aggregation step (3), in particular those with low +availability and/or high correlation. For the remaining clients, +guideline C proposes to set their aggregation weight inversely +proportional to their availability to reduce the bias error ϵ′ +bias. +Motivated by these insights, we propose CA-Fed, a client +sampling and aggregation strategy that takes into account the +problem of correlated client availability in FL, described in +5 + +Algorithm 1. CA-Fed learns during training which are the +clients to exclude and how to set the aggregation weights of the +other clients to achieve a good trade-off between ϵopt and ϵ′ +bias. +While guidelines A and B indicate which clients to remove, +the exact number of clients to remove at round t is identified +by minimizing ϵ(t) as a proxy for the bound in (14):4 +ϵ(t) := FB(wt,0)−F ∗ +B + d2 +T V (α, p)Γ′. +(15) +A. CA-Fed’s core steps +At each communication round t, the server sends the current +model wt,0 to all active clients and each client k sends back +a noisy estimate F +(t) +k +of the current loss computed on a batch +of samples Bk +t,0, i.e., F +(t) +k += +1 +|Bk +t,0| +� +ξ∈Bk +t,0 f(wt,0, ξ) (line 3). +The server uses these values and the information about the +current set of available clients At to refine its own estimates +of each client’s loss ( ˆF (t) = ( ˆF +(t) +k )k∈K), and each client’s +loss minimum value ( ˆF ∗ = ( ˆF ∗ +k )k∈K), as well as of Γ′, πk, +λk, and ϵ(t), denoted as ˆΓ +′(t), ˆπ +(t) +k , ˆλ +(t) +k , and ˆϵ(t), respectively +(possible estimators are described below) (line 4). +The server decides whether excluding clients whose avail- +ability pattern exhibits high correlation (high ˆλ +(t) +k ) (line 6). +First, the server considers all clients in descending order of +ˆλ(t) (line 14), and evaluates if, by excluding them (line 17), +ˆϵ(t) appears to be decreasing by more than a threshold τ ≥ 0 +(line 19). Then, the server considers clients in ascending order +of ˆπ(t), and repeats the same procedure to possibly exclude +some of the clients with low availability (low ˆπ +(t) +k ) (lines 7). +Once the participating clients (those with qk > 0) have +been selected, the server notifies them to proceed updating +the current models (lines 9–10) according to (2), while the +other available clients stay idle. Finally, model’s updates are +aggregated according to (3) (line 12). +B. Estimators +We now briefly discuss possible implementation of the esti- +mators ˆF +(t) +k , ˆF ∗ +k , ˆΓ +′(t), ˆπ +(t) +k , and ˆλ +(t) +k . Server’s estimates for the +clients’ local losses ( ˆF (t) = ( ˆF +(t) +k )k∈K) can be obtained from +the received active clients’ losses (F (t) = (F +(t) +k )k∈At) through +an auto-regressive filter with parameter β ∈ (0, 1]: +ˆF +(t) = (1 − β1At) ⊙ ˆF (t−1) + β1At ⊙ F +(t), +(16) +where ⊙ denotes the component-wise multiplication between +vectors, and 1At is a N-dimensions binary vector whose k-th +component equals 1 if and only if k is active at round t, i.e., +k ∈ At. The server can keep track of the clients’ loss minimum +values and estimate F ∗ +k as ˆF ∗ +k = mins∈[0,t] ˆF +(s) +k . The values of +FB(wt,0), F ∗ +B, Γ′, and ϵ(t) can be estimated as follows: +ˆF +(t) +B − ˆF ∗ +B = ⟨ ˆF (t) − ˆF ∗, ˆπ(t) ˜⊙q(t)⟩, +(17) +ˆΓ +′(t) = maxk∈K( ˆF +(t) +k +− ˆF ∗ +k ), +(18) +ˆϵ(t) = ˆF +(t) +B − ˆF ∗ +B + d2 +T V (α, ˆπ(t) ˜⊙q(t)) · ˆΓ +′(t). +(19) +4Following (14), one could reasonably introduce a hyper-parameter to +weigh the relative importance of the optimization and bias terms in the sum. +We discuss this additional optimization of CA-Fed in Section VI-A. +where π ˜⊙q ∈ RN, such that +� +π ˜⊙q +� +k = +πkqk +�N +h=1 πhqh , k ∈ K. +For ˆπ +(t) +k , the server can simply keep track of the total number +of times client k was available up to time t and compute +ˆπ +(t) +k +using a Bayesian estimator with beta prior, i.e., ˆπ +(t) +k += +(� +s≤t 1k∈As +nk)/(t+nk +mk), where nk and mk are the +initial parameters of the beta prior. +For ˆλ +(t) +k , the server can assume the client’s availability evolves +according to a Markov chain with two states (available and +unavailable), track the corresponding number of state tran- +sitions, and estimate the transition matrix +ˆP +(t) +k +through a +Bayesian estimator similarly to what done for ˆπ +(t) +k . Finally, +ˆλ +(t) +k is obtained computing the eigenvalues of ˆP +(t) +k . +C. CA-Fed’s computation/communication cost +CA-Fed aims to improve training convergence and not to +reduce its computation and communication overhead. Never- +theless, excluding some available clients reduces the overall +training cost, as we will discuss in this section referring, for +the sake of concreteness, to neural networks’ training. +The available clients not selected for training are only re- +quested to evaluate their local loss on the current model once +on a single batch instead than performing E gradient updates, +which would require roughly 2 × E − 1 more calculations +(because of the forward and backward pass). For the selected +clients, there is no extra computation cost as computing the +loss corresponds to the forward pass they should, in any case, +perform during the first local gradient update. +In terms of communication, the excluded clients only transmit +the loss, a single scalar, much smaller than the model update. +Conversely, participating clients transmit the local loss and the +model update. Still, this additional overhead is negligible and +likely fully compensated by the communication savings for +the excluded clients. +V. EXPERIMENTAL EVALUATION +A. Experimental Setup +a) Federated system simulator: In our experiments, we sim- +ulate the clients’ availability dynamics featuring different +levels of temporal correlations. We model the activity of each +client as a two-state homogeneous Markov process with state +space S = {“active”, “inactive”}. We use pk,s to denote the +probability that client k ∈ K remains in state s ∈ S. +In order to simulate the statistical heterogeneity present in the +federated learning system, we consider an experimental setting +with two disjoint groups of clients Gi, i = 1, 2, to which +we associate two different data distributions Pi, i = 1, 2, +to be precised later. Let ri = |Gi|/N, i = 1, 2 denote the +fraction of clients in group i = 1, 2. In order to simulate +the heterogeneity of clients’ availability patterns in realistic +federated systems, we split the clients of each group in two +classes uniformly at random: “more available” clients whose +steady-state probability to be active is πk,active = 1/2 + g and +“less available” clients with πk,active = 1/2 − g, where g ∈ +6 + +Inactive, +excluded +Inactive, +included +Active, +excluded +Active, +included +More Available +Less Available, Weakly Correlated +0 +20 +40 +60 +80 +100 +120 +140 +Communication round +Less Available, Correlated +Clients +Fig. 1: Clients’ activities and CA-Fed’s clients selection on the synthetic dataset. +More Available +Less Available +Correlated +Less Available +Weakly Correlated +Clients +Cumulative weight +Unbiased +CA-Fed +AdaFed +F3AST +Target +Fig. 2: Importance given to the clients by the different algorithms +throughout a whole training process on the synthetic dataset. +(0, 1/2) is a parameter controlling the heterogeneity of clients +availability. We furthermore split each class of clients in two +sub-classes uniformly at random: “correlated” clients that tend +to persist in the same state (λk = ν with values of ν close to +1), and “weakly correlated” clients that are almost as likely +to keep as to change their state (λk ∼ N(0, ε2), with ε close +to 0). In our experiments, we suppose that r1 = r2 = 1/2, +g = 0.4, ν = 0.9, and ε = 10−2. +b) Datasets and models: All experiments are performed on +a binary classification synthetic dataset (described in Ap- +pendix F) and on the real-world MNIST dataset [35], using +N = 24 clients. For MNIST dataset, we introduce statistical +heterogeneity across the two groups of clients (i.e., we make +the two distributions P1 and P2 different), following the same +approach in [36]: 1) every client is assigned a random subset +of the total training data; 2) the data of clients from the second +group is modified by randomly swapping two pairs of labels. +We maintain the original training/test data split of MNIST and +use 20% of the training dataset as validation dataset. We use a +linear classifier with a ridge penalization of parameter 10−2, +which is a strongly convex objective function, for both the +synthetic and the real-world MNIST datasets. +c) Benchmarks: +We compare CA-Fed, defined in Algo- +rithm 1, with the Unbiased aggregation strategy, where all +the active clients participate and receive a weight inversely +proportional to their availability, and with the state-of-the- +art FL algorithms discussed in Section II: F3AST [18] and +AdaFed [19]. We tuned the learning rates η, ηs via grid +search, on the grid η : {10−3, 10−2.5, 10−2, 10−1.5, 10−1}, +ηs : {10−2, 10−1.5, 10−1, 10−0.5, 100}. For CA-Fed, we used +τ = 0, β = 0.2. We assume all algorithms can access an oracle +providing the true availability parameters for each client. In +0 +20 +40 +60 +80 +100 +120 +140 +Communication round +35 +40 +45 +50 +55 +60 +65 +70 +75 +Time-average test accuracy +Unbiased +F3AST +AdaFed +CA-Fed (Ours) +(a) Synthetic +0 +20 +40 +60 +80 +100 +120 +140 +Communication round +10 +20 +30 +40 +50 +60 +Time-average test accuracy +Unbiased +F3AST +AdaFed +CA-Fed (Ours) +(b) MNIST +Fig. 3: Test accuracy vs number of communication rounds. +practice, Unbiased, AdaFed, and F3AST rely on the exact +knowledge of πk,active, and CA-Fed on πk,active and λk. 5 +B. Experimental Results +Figure 1 shows the availability of each client during a training +run on the synthetic dataset. Clients selected (resp. excluded) +by CA-Fed are highlighted in black (resp. red). We observe +that excluded clients tend to be those with low average +availability or high correlation. +Figure 2 shows the importance pk (averaged over time) given +by different algorithms to each client k during a full training +run. We observe that all the algorithms, except Unbiased, +depart from the target importance α. As suggested by guide- +lines A and B, CA-Fed tends to favor the group of “more +available” clients, at the expense of the “less available” clients. +Figure 3 shows the time-average accuracy up to round t of +the learned model averaged over three different runs. On both +datasets, CA-Fed achieves the highest accuracy, which is +about a percentage point higher than the second best algorithm +(F3AST). Table I shows for each algorithm: the average over +three runs of the maximum test accuracy achieved during train- +ing, the time-average test accuracy achieved during training, +together with its standard deviation within the second half of +the training period. Results show that while CA-Fed achieves +a maximum accuracy which is comparable to the Unbiased +baseline and state-of-the-art AdaFed and F3AST, it gets a +higher time-average accuracy (1.24 percentage points) in com- +parison to the second best (F3AST), and a smaller standard +deviation (1.5×) in comparison to the second best (F3AST). +5The authors have provided public access to their code and data at: +https://github.com/arodio/CA-Fed. +7 + +TABLE I: Maximum and time-average test accuracy, together with +their standard deviations, on the Synthetic / MNIST datasets. +TEST ACCURACY +MAXIMUM +TIME-AVERAGE +STANDARD DEVIATION +UNB I AS ED +78.94 / 64.87 +75.32 / 61.39 +0.48 / 1.09 +F3AST +78.97 / 64.91 +75.33 / 61.52 +0.40 / 0.94 +ADAFED +78.69 / 63.77 +74.81 / 60.48 +0.59 / 1.37 +CA-FE D +79.03 / 64.94 +76.22 / 62.76 +0.28 / 0.61 +VI. DISCUSSION +In this section, we discuss some general concerns and remarks +on our algorithm. +A. Controlling the number of excluded clients +Theorems 1 and 3 suggest that the condition number κ2 can +play a meaningful role in the minimization of the total error ϵ. +Our algorithm uses a proxy (ϵ(t)) of the total error. To take into +account the effect of κ2, we can introduce a hyper-parameter +that weights the relative importance of the optimization and +bias error in (15): +ϵ′(t) := FB(wt,0) − F ∗ +B + ¯κ2 · d2 +T V (α, p)Γ′. +A small value of ¯κ2 penalizes the bias term in favor of the +optimization error, resulting in a larger number of clients +excluded by CA-Fed. On the other hand, CA-Fed tends to +include more clients for a large value of ¯κ2. Asymptotically, +for ¯κ2 → +∞, CA-Fed reduces to the Unbiased baseline. +To further improve the performance of CA-Fed, a finer tuning +of the values of ¯κ2 can be performed. +B. CA-Fed in presence of spatial correlation +Although CA-Fed is mainly designed to handle temporal +correlation, it does not necessarily perform poorly in presence +of spatial correlation, as well. +Consider the following spatially-correlated scenario: clients +are grouped in clusters, each cluster c ∈ C is characterized +by an underlying Markov chain, which determines when all +clients in the cluster are available/unavailable, the Markov +chains of different clusters are independent. Let λc denote +the second largest eigenvalue in module of cluster-c’s Markov +chain. In this case, one needs to exclude all clients in the +cluster ¯c = arg maxc∈C λc to reduce the eigenvalue of the +aggregate Markov chain. +In this setting, CA-Fed would associate similar eigenvalue +estimates to all clients in the same cluster, then it would +correctly start considering for exclusion the clients in cluster +¯c and potentially remove sequentially all clients in the same +cluster. These considerations suggest that CA-Fed may still +operate correctly even in presence of spatial correlation. +C. About CA-Fed’s fairness +A strategy that excludes clients from the training phase, +such as CA-Fed, may naturally raise fairness concerns. The +concept of fairness in FL does not have a unified definition in +the literature [37, Chapter 8]: fairness goals can be captured by +a suitable choice of the target weights in (1). For example, per- +client fairness can be achieved by setting αk equal for every +client, while per-sample fairness by setting αk proportional +to the local dataset size |Dk|. If we assume that the global +objective in (1) indeed reflects also fairness concerns, then +CA-Fed is intrinsically fair, in the sense that it guarantees +that the performance objective of the learned model is as close +as possible to its minimum value. +VII. CONCLUSION +This paper presented the first convergence analysis for a +FedAvg-like FL algorithm under heterogeneous and corre- +lated client availability. The analysis quantifies how correla- +tion adversely affects the algorithm’s convergence rate and +highlights a general bias-versus-convergence-speed trade-off. +Guided by the theoretical analysis, we proposed CA-Fed, a +new FL algorithm that tries to balance the conflicting goals +of maximizing convergence speed and minimizing model bias. +Our experimental results demonstrate that adaptively excluding +clients with high temporal correlation and low availability is an +effective approach to handle the heterogeneous and correlated +client availability in FL. +APPENDIX +A. Proof of Theorem 1 +We bound the optimization error of the target objective as the +optimization error of the biased objective plus a bias term: +F(w) − F ∗ +(a) +≤ +1 +2µ ∥∇F(w)∥2 +(b) +≤ L2 +2µ ∥w − w∗∥2 +(c) +≤ L2 +µ (∥w − w∗ +B∥2 + ∥w∗ +B − w∗∥2) +(d) +≤ 2L2 +µ2 (FB(w) − F ∗ +B) +� +�� +� +:=ϵopt ++ 2L2 +µ2 (F(w∗ +B) − F ∗) +� +�� +� +:=ϵbias +, +where (a), (b), and (d) follow from the Assumptions 3, 4, +and the inequality (c) follows from (a + b)2 ≤ 2a2 + 2b2. +In particular, (b) requires ∇Fk(w∗ +k) = 0. Theorem 2 further +develops the optimization error ϵopt. We now expand ϵbias: +∥∇F(w∗ +B)∥ +(e)= +����N +k=1(αk − pk)∇Fk(w∗ +B) +��� +(f) +≤ L �N +k=1|αk − pk| ∥w∗ +B − w∗ +k∥ +(20) +(g) +≤ L +� +2 +µ +�N +k=1 +|αk−pk| +√pk +� +pk(Fk(w∗ +B) − F ∗ +k ), +where (e) uses ∇FB(w∗ +B) = 0; (f) applies first the triangle +inequality, then the L-smoothness, and (g) follows from the +µ-strong convexity. In addition, (f) requires ∇Fk(w∗ +k) = 0. +Similarly to [32], in (g) we multiply numerator and denomi- +nator by √pk. By direct calculations, it follows that: +∥∇F(w∗ +B)∥2 +(h) +≤ 2L2 +µ +� �N +k=1 +|αk−pk| +√pk +� +pk(Fk(w∗ +B) − F ∗ +k ) +�2 +(i) +≤ 2L2 +µ +� +N� +k=1 +(αk−pk)2 +pk +�� +N� +k=1 +pk(Fk(w∗ +B) − F ∗ +k ) +� +(j) +≤ 2L2 +µ χ2 +α∥pΓ, +8 + +where (i) uses the Cauchy–Schwarz inequality, and (j) used: +�N +k=1 pk(Fk(w∗ +B) − F ∗ +k ) ≤ �N +k=1 pk(Fk(w∗) − F ∗ +k ) ≤ Γ. +Finally, by strong convexity of F, we conclude that: +F(w∗ +B) − F ∗ ≤ +1 +2µ ∥∇F(w∗ +B)∥2 ≤ L2 +µ2 χ2 +α∥pΓ. +B. Proof of Theorem 2 +1) Additional notation: let wk +t,j be the model parameter vector +computed by device k at the global round t, local iteration j. +We define: +gt(At) = � +k∈At qk +�E−1 +j=0 ∇Fk(wk +t,j, ξk +t,j), +and ¯gt(At) = Eξ|At[gt(At)]. +Following (2) and (3), the update rule of CA-Fed is: +wt+1,0 = ProjW (wt,0 − ηtgt(At)). +(21) +2) Key lemmas and results: we provide useful lemmas and +results to support the proof of the main theorem. +Proof of Lemma 1. The boundedness of W gives a bound on +(wt,0)t≥0 based on the update rules in (2) and (3). From the +convexity of {Fk}k∈K, it follows that: +D := +sup +w∈W,k∈K +∥∇Fk(w)∥ < +∞. +Items (6), (8) are directly derived from the previous observa- +tion. Item (7) follows combining (6) and Assumption 5: +E ∥∇Fk(w, ξ)∥2 ≤ D2 + max +k∈K {σ2 +k} := G2. +Lemma 2 (Convergence under heterogeneous client availabil- +ity). Let the local functions {Fk}k∈K be convex, Assump- +tions 3, 5 hold. If ηt ≤ +1 +2L(EQ+1), we have: +� +t ηt E[� +k∈At qk (Fk(wt,0) − Fk(w∗ +B))] ≤ ++ 2 +E ∥w0,0 − w∗ +B∥2 + 2 �N +k=1 πkq2 +kσ2 +k +� +t η2 +t ++ 2 +3 +�N +k=1 πkqk(E − 1)(2E − 1)G2 � +t η2 +t ++ 2L(EQ + 2) �N +k=1 πkqkΓ � +t η2 +t := C1 < +∞. +Proof of Lemma 2. +∥wt+1,0 − w∗ +B∥2 = ∥ProjW (wt,0 − ηtgt) − ProjW (w∗ +B)∥2 +≤ ∥wt,0 − ηtgt − w∗ +B + ηt¯gt − ηt¯gt∥2 = A1 + A2 + A3, +where: +A1 = ∥wt,0 − w∗ +B − ηt¯gt∥2 , +A2 = 2ηt⟨wt,0 − w∗ +B − ηt¯gt, ¯gt − gt⟩, +A3 = η2 +t ∥gt − ¯gt∥2 . +Note E[A2] = 0. We bound A1, A3 using the key steps in [22]: +(1) the variance of gt(At) is bounded if the variance of the +stochastic gradients at each device is bounded: +A3 = EB|At ∥gt − ¯gt∥2 = += � +k∈At q2 +k +�E−1 +j=0 EB|At +��∇Fk(wk +t,j, ξk +t,j)−∇Fk(wk +t,j) +��2 +≤ E � +k∈At q2 +kσ2 +k; +(2) the distance of the local model wk +t,E from the global +model wt,0 is bounded since the expected squared norm of +the stochastic gradients is bounded: +EB|At +� +k∈At qk +�E−1 +j=0 +��wk +t,j − wt,0 +��2 = += EB|At +� +k∈At qk +�E−1 +j=1 η2 +t +��� +�j−1 +j′=0 ∇Fk(wk +t,j′, ξk +t,j′) +��� +2 +≤ η2 +t +� +k∈At qk +�E−1 +j=1 j �j−1 +j′=0 EB|At +��∇Fk(wk +t,j′, ξk +t,j′) +��2 +≤ η2 +t +� +k∈At qkG2 �E−1 +j=1 j2 += 1 +6η2 +t +� +k∈At qkE(E − 1)(2E − 1)G2. +Lemma 3 (Optimization error after Jt steps). Let Assump- +tions 1, 2 hold, the local functions {Fk}k∈K be convex, D, H +be defined as in (6), (8), and Jt defined as in Theorem 2. +Then: +� +t ηt E[� +k∈At qk(Fk(wt−Jt,0) − Fk(wt,0))] +≤ EDGQ � +t Jtη2 +t−Jt +�N +k=1 πkqk := +C3 +ln(1/λ(P )) < +∞. +For the proof of Lemma 3, we introduce the following results: +|Fk(v) − Fk(w)| ≤ D · ∥v − w∥ , ∀v, w ∈ W, +(22) +EBk +t,0,...,Bk +t,E−1 ∥wt+1,0 − wt,0∥ ≤ ηtGE(� +k∈At qk). (23) +Equation (22) is due to convexity of {Fk}k∈K, which gives: +⟨∇Fk(v), v − w⟩ ≤ ∥Fk(v) − Fk(w)∥ ≤ ⟨∇Fk(w), v − w⟩; +the Cauchy–Schwarz inequality concludes: +|Fk(v) − Fk(w)| ≤ max{∥∇Fk(v)∥ , ∥∇Fk(w)∥} ∥v − w∥ +≤ D · ∥v − w∥ . +Equation (23) follows combining equations (7) and (21): +EB|At ∥wt+1,0 − wt,0∥ ≤ +≤ ηt EB|At +���� +k∈At qk +�E−1 +j=0 ∇Fk(wk +t,j, ξk +t,j) +��� +≤ ηt +� +k∈At qk +�E−1 +j=0 EB|At +��∇Fk(wk +t,j, ξk +t,j) +�� +≤ ηtGE(� +k∈At qk). +Proof of Lemma 3. The evolution of the local objectives after +Jt communication rounds is bounded: +� +tηt E[� +k∈At qk(Fk(wt−Jt,0) − Fk(wt,0))] +(a) +≤ D � +t ηt E[� +k∈At qk EB ∥wt−Jt,0 − wt,0∥] +(b) +≤ D � +t ηt +�t−1 +d=t−Jt E[� +k∈At qk EB ∥wd,0 − wd+1,0∥] +(c) +≤ EDG � +t +�t−1 +d=t−Jt ηtηd E[� +k∈At qk +� +k′∈Ad qk′] +(d) +≤ EDG +2 +� +t +�t−1 +d=t−Jt(η2 +t + η2 +d) E[� +k∈At qk +� +k′∈Ad qk′] +(e) +≤ EDGQ � +t Jtη2 +t−Jt +�N +k=1 πkqk := +C3 +ln(1/λ(P )), +9 + +where (a) follows from (22); (b) applies the triangle inequal- +ity; (c) uses (23); (d) applies the Cauchy–Schwarz inequality; +(e) uses ηt < ηd ≤ ηt−Jt and �N +k=1 qk = Q. +3) Core of the proof: The proof consists in two main steps: +1. � +t ηt +�N +k=1 πkqk E[FB(wt−Jt,0) − F ∗ +B)]≤C2+ +C3 +ln(1/λ(P )); +2. � +t ηt +�N +k=1 πkqk E[FB(wt,0)−FB(wt−Jt,0)]≤ +C3 +ln(1/λ(P )). +Step 1. Combining Lemma 2 and 3, we get: +� +t ηt E[ � +k∈At +qk(Fk(wt−Jt,0) − Fk(w∗ +B))] ≤ C1 + +C3 +ln(1/λ(P )). +The constant Jt, introduced in [14], is an important parameter +for the analysis and frequently used. Combining its definition +in Theorem 2 and equation (5), it follows: +��[P Jt]i,j − πj +�� ≤ CP λ(P )Jt ≤ +1 +2Ht, +∀i, j ∈ [M]. +(24) +Assume t ≥ TP . We derive an important lower bound: +EAt|At−Jt [� +k∈At qk(Fk(wt−Jt,0) − Fk(w∗ +B))] +(a)= �M +I=1 P(At=I|At−Jt) � +k∈I qk(Fk(wt−Jt,0)−Fk(w∗ +B)) +(b)= �M +I=1 [P Jt]At−Jt,I +� +k∈I qk (Fk(wt−Jt,0) − Fk(w∗ +B)) +(c) +≥ �M +I=1 +� +π(I) − +1 +2Ht +� � +k∈I qk(Fk(wt−Jt,0) − Fk(w∗ +B)) +(d) +≥ (�N +k=1 πkqk) · (FB(wt−Jt,0) − F ∗ +B) − 1 +2tMQ, +(25) +where (a) is the definition of the conditional expectation, (b) +uses the Markov property, (c) follows from (24), and (d) is +due to (8). Taking total expectations: +( �N +k=1 πkqk) � +t ηt E[FB(wt−Jt,0) − F ∗ +B] +≤ � +t ηt E[� +k∈At qk(Fk(wt−Jt,0) − Fk(w∗ +B))] ++ 1 +4MQ � +t(η2 +t + 1 +t2 ) = C2 + +C3 +ln(1/λ(P )), +(26) +where C2 = C1 + 1 +4MQ � +t(η2 +t + 1 +t2 ). +Step 2. By direct calculation (similar to Lemma 3): +(�N +k=1 πkqk) � +t ηt E[FB(wt,0) − FB(wt−Jt,0)]≤ +C3 +ln(1/λ(P )). +Summing Step 1 and 2, and applying Jensen’s inequality: +(�T +t=1 ηt)(�N +k=1 πkqk) E[FB( ¯wT,0) − F ∗ +B] ≤ +(�N +k=1 πkqk) �T +t=1 ηt E[FB(wt,0) − F ∗ +B] ≤ C2 + +2C3 +ln(1/λ(P )), +where ¯wT,0 := +�T +t=1 ηtwt,0 +�T +t=1 ηt +, and the constants are in (12). +C. Proof of Theorem 3 +It follows the same lines of Theorem 1, developing (20) as: +∥∇F(w∗ +B)∥ ≤ L +� +2 +µ +�N +k=1|αk − pk| +� +(Fk(w∗ +B) − F ∗ +k ) +≤ 2L +� +2 +µdT V (α, p) +√ +Γ′, +where dT V (α, p) := 1 +2 +�N +k=1|αk − pk| is the total variation +distance between the probability measures α and p. +D. Minimizing ϵopt +Equation 12 defines the following optimization problem: +minimize +q +f(q) = +1 +2 q⊺Aq+B +π⊺q ++ C; +subject to +q ≥ 0, +π⊺q > 0, +∥q∥1 = Q. +Let us rewrite the problem by adding a variable s := 1/π⊺q +and then replacing y := sq. Note that the objective function is +the perspective of a convex function, and is therefore convex: +min +y,s +f(y, s) = +1 +2sy⊺Ay + Bs + C +(27a) +s.t. +y ≥ 0, s > 0, π⊺y = 1, ∥y∥1 = Qs. +(27b) +The Lagrangian function L is as follows: +L(y, s, λ, θ, µ) = +1 +2sy⊺Ay + Bs + C+ ++λ(1 − π⊺y) + θ(∥y∥1 − Qs) − µ⊺y. +(28) +Since the constraint s > 0 defines an open set, the set defined +by the constraints in (27b) is not closed. However, the solution +is never on the boundary s = 0 because L∗ → +∞ as s → 0+, +and we can consider s ≥ 0. The KKT conditions for y∗ +k read: +if y∗ +k > 0: y∗ +k = +s∗ +A[kk](λ∗πk − θ∗); y∗ +k = 0 otherwise. (29) +Since λ∗ ≥ 0, the clients with smaller πk may have q∗ +k = 0. +E. Convexity of ϵopt + ϵbias +In Appendix D, we proved that ϵopt(q) is convex. To prove +that ϵbias(q) is also convex, we need to study the convexity of +χ2 +α∥p = �N +k=1(fk ◦ gk)(q), where fk(pk) = (pk − αk)2/pk, +and gk(q) = (πkqk)/ �N +h=1 πhqh. We observe that fk(pk) +is convex, and gk(q) is a particular case of linear-fractional +function [38]. By direct inspection, it can be proved that +(fk◦gk)(q) is convex in dom(fk◦gk) = {q : ∥q∥1 = Q > 0}. +F. Synthetic dataset +Our synthetic datasets has been generated as follows: +1) For client k ∈ K, sample group identity ik from a +Bernoulli distribution of parameter 1/2; +2) Sample model parameters w∗ ∼ N(0, Id) from the d- +dimensional normal distribution; +3) For client k ∈ K and sample index j ∈ {1, . . . , 150}, +sample clients input data x(j) +k +∼ N(0, Id) from the d- +dimensional normal distribution; +4) For client k ∈ K such that ik = 0 and sample index j ∈ +{1, . . . , 150}, sample the true labels y(j) +k +from a Bernoulli +distribution with parameter equal to sigmoid(⟨w∗, x(j) +k ⟩); +5) For client k +∈ +K such that ik += +1 and sample +index j ∈ {1, . . . , 150}, sample the true labels y(j) +k +from a Bernoulli distribution with parameter equal to +0.8·sigmoid(⟨w∗, x(j) +k ⟩)+0.2·(1−sigmoid(⟨w∗, x(j) +k ⟩)). +10 + +REFERENCES +[1] J. Verbraeken, M. Wolting, J. Katzy, J. Kloppenburg, T. Verbelen, and +J. S. 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Bengio, “An +empirical investigation of catastrophic forgetting in gradient-based neu- +ral networks,” in International Conference on Learning Representations, +2013, arXiv preprint arXiv:1312.6211. +[34] R. Kemker, M. McClure, A. Abitino, T. Hayes, and C. Kanan, “Mea- +suring catastrophic forgetting in neural networks,” in Proceedings of the +AAAI Conference on Artificial Intelligence, vol. 32, no. 1, 2018. +[35] Y. LeCun and C. Cortes, “MNIST handwritten digit database,” 2010. +[36] F. Sattler, K.-R. M¨uller, and W. Samek, “Clustered federated learning: +Model-agnostic distributed multitask optimization under privacy con- +straints,” IEEE Transactions on Neural Networks and Learning Systems, +vol. 32, no. 8, pp. 3710–3722, 2020. +[37] H. Ludwig and N. Baracaldo, Federated Learning: A Comprehensive +Overview of Methods and Applications. +Springer Cham, 2022. +[38] S. Boyd and L. Vandenberghe, Convex optimization. +Cambridge +university press, 2004. +11 + diff --git a/3dE3T4oBgHgl3EQfoQqU/content/tmp_files/load_file.txt b/3dE3T4oBgHgl3EQfoQqU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..42e97230bf6c653202d78070c2637b3a9afbe399 --- /dev/null +++ b/3dE3T4oBgHgl3EQfoQqU/content/tmp_files/load_file.txt @@ -0,0 +1,843 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf,len=842 +page_content='Federated Learning under Heterogeneous and Correlated Client Availability Angelo Rodio∗, Francescomaria Faticanti∗, Othmane Marfoq∗†, Giovanni Neglia∗, Emilio Leonardi‡ ∗Inria, Universit´e Cˆote d’Azur, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Email: {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='lastname}@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='fr, †Accenture Labs, Sophia-Antipolis, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Email: {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='lastname}@accenture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='com, ‡Politecnico di Torino, Turin, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Email: {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='lastname}@polito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='it Abstract—The enormous amount of data produced by mobile and IoT devices has motivated the development of federated learning (FL), a framework allowing such devices (or clients) to collabora- tively train machine learning models without sharing their local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' FL algorithms (like FedAvg) iteratively aggregate model updates computed by clients on their own datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Clients may exhibit different levels of participation, often correlated over time and with other clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' This paper presents the first convergence analysis for a FedAvg-like FL algorithm under heterogeneous and correlated client availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our analysis highlights how correlation adversely affects the algorithm’s convergence rate and how the aggregation strategy can alleviate this effect at the cost of steering training toward a biased model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Guided by the theoretical analysis, we propose CA-Fed, a new FL algorithm that tries to balance the conflicting goals of maximizing convergence speed and minimizing model bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' To this purpose, CA-Fed dynamically adapts the weight given to each client and may ignore clients with low availability and large correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our experimental results show that CA-Fed achieves higher time- average accuracy and a lower standard deviation than state-of- the-art AdaFed and F3AST, both on synthetic and real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Index Terms—Federated Learning, Distributed Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' INTRODUCTION The enormous amount of data generated by mobile and IoT de- vices motivated the emergence of distributed machine learning training paradigms [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Federated Learning (FL) [3]–[6] is an emerging framework where geographically distributed devices (or clients) participate in the training of a shared Machine Learning (ML) model without sharing their local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' FL was proposed to reduce the overall cost of collecting a large amount of data as well as to protect potentially sensitive users’ private information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In the original Federated Averaging algorithm (FedAvg) [4], a central server selects a random subset of clients from the set of available clients and broadcasts them the shared model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The sampled clients perform a number of independent Stochastic Gradient Descent (SGD) steps over their local datasets and send their local model updates back to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Then, the server aggregates the received client updates to produce a new global model, and a new training round begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' At each iteration of FedAvg, the server typically samples randomly a few hundred devices to participate [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' This research was supported by the French government through the 3IA Cˆote d’Azur Investments in the Future project by the National Research Agency (ANR) with reference ANR-19-P3IA-0002, and by Groupe La Poste, sponsor of Inria Foundation, in the framework of FedMalin Inria Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' A first version of this work has been accepted at IEEE INFOCOM 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In real-world scenarios, the availability/activity of clients is dictated by exogenous factors that are beyond the control of the orchestrating server and hard to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For instance, only smartphones that are idle, under charge, and connected to broadband networks are commonly allowed to participate in the training process [4], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' These eligibility requirements can make the availability of devices correlated over time and space [7], [10]–[12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For example, temporal correlation may origin from a smartphone being under charge for a few consecutive hours and then ineligible for the rest of the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Similarly, the activity of a sensor powered by renewable energy may depend on natural phenomena intrinsically correlated over time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', solar light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Spatial correlation refers instead to correlation across different clients, which often emerges as consequence of users’ different geographical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For instance, clients in the same time zone often exhibit similar availability patterns, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', due to time-of-day effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Temporal correlation in the data sampling procedure is known to negatively affect the performance of ML training even in the centralized setting [13], [14] and can potentially lead to catastrophic forgetting: the data used during the final training phases can have a disproportionate effect on the final model, “erasing” the memory of previously learned information [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Catastrophic forgetting has also been observed in FL, where clients in the same geographical area have more similar local data distributions and clients’ participation follows a cyclic daily pattern (leading to spatial correlation) [7], [10], [11], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Despite this evidence, a theoretical study of the convergence of FL algorithms under both temporally and spatially correlated client participation is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' This paper provides the first convergence analysis of FedAvg [4] under heterogeneous and correlated client avail- ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We assume that clients’ temporal and spatial availabil- ity follows an arbitrary finite-state Markov chain: this assump- tion models a realistic scenario in which the activity of clients is correlated and, at the same time, still allows the analytical tractability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our theoretical analysis (i) quantifies the negative effect of correlation on the algorithm’s conver- gence rate through an additional term, which depends on the spectral properties of the Markov chain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (ii) points out a trade- off between two conflicting objectives: slow convergence to the optimal model, or fast convergence to a biased model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', a model that minimizes an objective function different from the initial target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Guided by insights from the theoretical analysis, 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='04632v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='LG] 11 Jan 2023 we propose CA-Fed, an algorithm which dynamically assigns weights to clients and achieves a good trade-off between maximizing convergence speed and minimizing model bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Interestingly, CA-Fed can decide to ignore clients with low availability and high temporal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our experimental results demonstrate that excluding such clients is a simple, but effective approach to handle the heterogeneous and correlated client availability in FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Indeed, while CA-Fed achieves a comparable maximum accuracy as the state-of-the-art methods F3AST [18] and AdaFed [19], its test accuracy exhibits higher time-average and smaller variability over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Section II describes the problem of correlated client availability in FL and discusses the main related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Section III provides a convergence analysis of FedAvg under heterogeneous and correlated client participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' CA-Fed, our correlation-aware FL algorithm, is presented in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We evaluate CA-Fed in Section V, comparing it with state-of-the-art methods on synthetic and real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Section VII concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' BACKGROUND AND RELATED WORKS We consider a finite set K of N clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Each client k ∈ K holds a local dataset Dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Clients aim to jointly learn the parameters w ∈ W ⊆ Rd of a global ML model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', the weights of a neural network architecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' During training, the quality of the model with parameters w on a data sample ξ ∈ Dk is measured by a loss function f(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The clients solve, under the orchestration of a central server, the following optimization problem: min w∈W ⊆Rd � F(w) := � k∈K αkFk(w) � , (1) where Fk(w) := 1 |Dk| � ξ∈Dk f(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' ξ) is the average loss computed on client k’s local dataset, and α = (αk)k∈K are positive coefficients such that � k αk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' They represent the target importance assigned by the central server to each client k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Typically (αk)k∈K are set proportional to the clients’ dataset size |Dk|, such that the objective function F in (1) coincides with the average loss computed on the union of the clients’ local datasets D = ∪k∈KDk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Under proper assumptions, precised in Section III, Problem (1) admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We use w∗ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' F ∗) to denote the minimizer (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' the minimum value) of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Moreover, for k∈K, Fk admits a unique minimizer on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We use w∗ k (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' F ∗ k ) to denote the minimizer (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' the minimum value) of Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Problem (1) is commonly solved through iterative algo- rithms [4], [8] requiring multiple communication rounds be- tween the server and the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' At round t > 0, the server broadcasts the latest estimate of the global model wt,0 to the set of available clients (At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Client k ∈ At updates the global model with its local data through E ≥ 1 steps of local Stochastic Gradient Descent (SGD): wk t,j+1 = wk t,j − ηt∇Fk(wk t,j, Bk t,j) j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' , E − 1, (2) where ηt > 0 is an appropriately chosen learning rate, referred to as local learning rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Bk t,j is a random batch sampled from client k’ local dataset at round t and step j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' ∇Fk(·, B) := 1 |B| � ξ∈B ∇f(·, ξ) is an unbiased estimator of the local gradient ∇Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Then, each client sends its local model update ∆k t := wk t,E − wk t,0 to the server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The server computes ∆t := � k∈At qk ·∆k t , a weighted average of the clients’ local updates with non-negative aggregation weights q = (qk)k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The choice of the aggregation weights defines an aggregation strategy (we will discuss different aggregation strategies later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The aggregated update ∆t can be interpreted as a proxy for −∇F(wt,0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' the server applies it to the global model: wt+1,0 = ProjW (wt,0 + ηs · ∆t) (3) where ProjW (·) denotes the projection over the set W, and ηs > 0 is an appropriately chosen learning rate, referred to as the server learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='1 The aggregate update ∆t is, in general, a biased estimator of −∇F(wt,0), where each client k is taken into account proportionally to its frequency of appearance in the set At and to its aggregation weight qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Indeed, under proper assumptions specified in Section III, one can show (see Theorem 2) that the update rule described by (2) and (3) converges to the unique minimizer of a biased global objective FB, which depends both on the clients’ availability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', on the sequence (At)t>0) and on the aggregation strategy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', on q = (qk)k∈K): FB(w) := �N k=1 pkFk(w), with pk := πkqk �N h=1 πhqh , (4) where πk := limt→∞ P(k ∈ At) is the asymptotic availability of client k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The coefficients p = (pk)k∈K can be interpreted as the biased importance the server is giving to each client k during training, in general different from the target importance α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In what follows, w∗ B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' F ∗ B) denotes the minimizer (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' the minimum value) of FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In some large-scale FL applications, like training Google keyboard next-word prediction models, each client participates in training at most for one round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The orchestrator usually selects a few hundred clients at each round for a few thousand rounds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', see [5, Table 2]), but the available set of clients may include hundreds of millions of Android devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In this scenario, it is difficult to address the potential bias unless there is some a-priori information about each client’s availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Anyway, FL can be used by service providers with access to a much smaller set of clients (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', smartphone users that have installed a specific app).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In this case, a client participates multiple times in training: the orchestrating server may keep track of each client’s availability and try to compensate for the potentially dangerous heterogeneity in their participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Much previous effort on federated learning [4], [17]–[19], [22]–[25] considered this problem and, under different as- 1The aggregation rule (3) has been considered also in other works, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', [8], [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In other FL algorithms, the server computes an average of clients’ local models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' This aggregation rule can be obtained with minor changes to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 2 sumptions on the clients’ availability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', on (At)t>0), de- signed aggregation strategies that unbias ∆t through an appro- priate choice of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Reference [22] provides the first analysis of FedAvg on non-iid data under clients’ partial participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Their analysis covers both the case when active clients are sampled uniformly at random without replacement from K and assigned aggregation weights equal to their target importance (as assumed in [4]), and the case when active clients are sampled iid with replacement from K with probabilities α and assigned equal weights (as assumed in [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' However, references [4], [22], [23] ignore the variance induced by the clients stochastic availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The authors of [24] reduce such variance by considering only the clients with important up- dates, as measured by the value of their norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' References [17] and [25] reduce the aggregation variance through clustered and soft-clustered sampling, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Some recent works [18], [19], [26] do not actively pursue the optimization of the unbiased objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Instead, they derive bounds for the convergence error and propose heuristics to minimize those bounds, potentially introducing some bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our work follows a similar development: we compare our algorithm with F3AST from [18] and AdaFed from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The novelty of our study is in considering the spatial and temporal correlation in clients’ availability dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' As dis- cussed in the introduction, such correlations are also intro- duced by clients’ eligibility criteria, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', smartphones being under charge and connected to broadband networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The effect of correlation has been ignored until now, probably due to the additional complexity in studying FL algorithms’ convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' To the best of our knowledge, the only exception is [18], which scratches the issue of spatial correlation by proposing two different algorithms for the case when clients’ availabilities are uncorrelated and for the case when they are positively correlated (there is no smooth transition from one algorithm to the other as a function of the degree of correlation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The effect of temporal correlation on centralized stochastic gradient methods has been addressed in [12]–[14], [27]: these works study a variant of stochastic gradient descent where samples are drawn according to a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Refer- ence [12] extends its analysis to a FL setting where each client draws samples according to a Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In contrast, our work does not assume a correlation in the data sampling but rather in the client’s availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Nevertheless, some of our proof techniques are similar to those used in this line of work and, in particular, we rely on some results in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' ANALYSIS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Main assumptions We consider a time-slotted system where a slot corresponds to one FL communication round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We assume that clients’ availability over the timeslots t ∈ N follows a discrete-time Markov chain (At)t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='2 2In Section III-D we will focus on the case where this chain is the superposition of N independent Markov chains, one for each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The Markov chain (At)t≥0 on the finite state space [M] is time-homogeneous, irreducible, and aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' It has transition matrix P and stationary distribution π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Markov chains have already been used in the literature to model the dynamics of stochastic networks where some nodes or edges in the graph can switch between active and inactive states [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The previous Markovian assumption, while allowing a great degree of flexibility, still guarantees the analytical tractability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The distance dynamics between current and stationary distribution of the Markov process can be characterized by the spectral properties of its transition matrix P [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let λ2(P ) denote the the second largest eigenvalue of P in absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Previous works [14] have shown that: max i,j∈[M] |[P t]i,j − πj| ≤ CP · λ(P )t, for t ≥ TP , (5) where the parameter λ(P ) := (λ2(P ) + 1)/2, and CP , TP are positive constants whose values are reported in [14, Lemma 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='3 Note that λ(P ) quantifies the correlation of the Markov process (At)t≥0: the closer λ(P ) is to one, the slower the Markov chain converges to its stationary distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In our analysis, we make the following additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let w∗, w∗ B denote the minimizers of F and FB on W, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The hypothesis class W is convex, compact, and contains in its interior the minimizers w∗, w∗ B, w∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The following assumptions concern clients’ local objective functions {Fk}k∈K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Assumptions 3 and 4 are standard in the literature on convex optimization [31, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Assumption 5 is a standard hypothesis in the analysis of federated optimization algorithms [8, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Assumption 3 (L-smoothness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The local functions {Fk}N k=1 have L-Lipschitz continuous gradients: Fk(v) ≤ Fk(w) + ⟨∇Fk(w), v − w⟩ + L 2 ∥v − w∥2 2, ∀v, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Assumption 4 (Strong convexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The local functions {Fk}N k=1 are µ-strongly convex: Fk(v) ≥ Fk(w) + ⟨∇Fk(w), v − w⟩ + µ 2 ∥v − w∥2 2 , ∀v, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Assumption 5 (Bounded variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The variance of stochastic gradients in each device is bounded: E ∥∇Fk(wk t,j, ξk t,j) − ∇Fk(wk t,j)∥2 ≤ σ2 k, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Assumptions 2–5 imply the following properties for the local functions, described by Lemma 1 (proof in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Under Assumptions 2–5, there exist constants D, G, and H > 0, such that, for w ∈ W and k ∈ K, we have: ∥∇Fk(w)∥ ≤ D, (6) E ∥∇Fk(w, ξ)∥2 ≤ G2, (7) |Fk(w) − Fk(w∗ B)| ≤ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (8) 3Note that (5) holds for different definitions of λ(P ) as far as λ(P ) ∈ (λ2(P ), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The specific choice for λ(P ) changes the constants CP and TP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 3 Similarly to other works [8], [22], [23], [32], we introduce a metric to quantify the heterogeneity of clients’ local datasets: Γ := max k∈K{Fk(w∗) − F ∗ k }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (9) If the local datasets are identical, the local functions {Fk}k∈K coincide among them and with F, w∗ is a minimizer of each local function, and Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In general, Γ is smaller the closer the distributions the local datasets are drawn from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Main theorems Theorem 1 (proof in Appendix A) decomposes the error of the target global objective as the sum of an optimization error for the biased global objective and a bias error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Theorem 1 (Decomposing the total error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Under Assump- tions 2–4, the optimization error of the target global objective ϵ = F(w) − F ∗ can be bounded as follows: ϵ ≤ 2κ2(FB(w) − F ∗ B) � �� � :=ϵopt + 2κ4χ2 α∥pΓ � �� � :=ϵbias , (10) where κ := L/µ, and χ2 α∥p := �N k=1 (αk − pk)2/pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Theorem 2 below proves that the optimization error ϵopt asso- ciated to the biased objective FB, evaluated on the trajectory determined by scheme (3), asymptotically vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The non- vanishing bias error ϵbias captures the discrepancy between F(w) and FB(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' This latter term depends on the chi-square divergence χ2 α∥p between the target and biased probability distributions α = (αk)k∈K and p = (pk)k∈K, and on Γ, that quantifies the degree of heterogeneity of the local functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' When all local functions are identical (Γ = 0), the bias term ϵbias also vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For Γ > 0, the bias error can still be controlled by the aggregation weights assigned to the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In particular, the bias term vanishes when qk ∝ αk/πk, ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Since it asymptotically cancels the bias error, we refer to this choice as unbiased aggregation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' However, in practice, FL training is limited to a finite number of iterations T (typically a few hundreds [5], [7]), and the previous asymptotic considerations may not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In this regime, the unbiased aggregation strategy can be suboptimal, since the minimization of ϵbias not necessarily leads to the minimization of the total error ϵ ≤ ϵopt + ϵbias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' This motivates the analysis of the optimization error ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Theorem 2 (Convergence of the optimization error ϵopt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let Assumptions 1–5 hold and the constants M, L, D, G, H, Γ, σk, CP , TP , λ(P ) be defined as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let Q = � k∈K qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let the stepsizes satisfy: � t ηt = +∞, � t ln(t) · η2 t < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (11) Let T denote the total communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For T ≥ TP , the expected optimization error can be bounded as follows: E[FB( ¯wT,0) − F ∗ B] ≤ 1 2 q⊺Σq+υ π⊺q + ψ + φ ln(1/λ(P )) (�T t=1 ηt) , (12) where ¯wT,0 := �T t=1 ηtwt,0 �T t=1 ηt , and Σ = diag(σ2 kπk � t η2 t ), υ = 2 E ∥w0,0 − w∗∥2 + 1 4MQ � t(η2 t + 1 t2 ), ψ = 4L(EQ + 2)Γ � t η2 t + 2 3(E − 1)(2E − 1)G2 � t η2 t , Jt =min {max {⌈ln (2CP Ht)/ln (1/λ(P ))⌉ , TP } , t}, φ = 2EDGQ � t ln(2CP Ht)η2 t−Jt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Theorem 2 (proof in Appendix B) proves convergence of the expected biased objective FB to its minimum F ∗ B under correlated client participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our bound (12) captures the effect of correlation through the factor ln (1/λ(P )): a high correlation worsens the convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In particular, we found that the numerator of (12) has a quadratic-over-linear fractional dependence on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Minimizing ϵopt leads, in general, to a different choice of q than minimizing ϵbias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Minimizing the total error ϵ ≤ ϵopt + ϵbias Our analysis points out a trade-off between minimizing ϵopt or ϵbias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our goal is to find the optimal aggregation weights q∗ that minimize the upper bound on total error ϵ(q) in (10): minimize q ϵopt(q) + ϵbias(q);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' subject to q ≥ 0, ∥q∥1 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (13) In Appendix E we prove that (13) is a convex optimization problem, which can be solved with the method of Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' However, the solution is not of practical utility because the constants in (10) and (12) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', L, µ, Γ, CP ) are in general problem-dependent and difficult to estimate during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In particular, Γ poses particular difficulties as it is defined in terms of the minimizer of the target objective F, but the FL algorithm generally minimizes the biased function FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Moreover, the bound in (10), similarly to the bound in [32], diverges when setting some qk equal to 0, but this is simply an artifact of the proof technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' A result of more practical interest is the following (proof in Appendix C): Theorem 3 (An alternative decomposition of the total er- ror ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Under the same assumptions of Theorem 1, let Γ′ := maxk{Fk(w∗ B) − F ∗ k }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The following result holds: ϵ ≤ 2κ2(FB(w) − F ∗ B) � �� � :=ϵopt + 8κ4d2 T V (α, p)Γ′ � �� � :=ϵ′ bias , (14) where dT V (α, p) := 1 2 �N k=1|αk − pk| is the total variation distance between the probability distributions α and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The new constant Γ′ is defined in terms of w∗ B, and then it is easier to evaluate during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' However, Γ′ depends on q, because it is evaluated at the point of minimum of FB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' This dependence makes the minimization of the right-hand side of (14) more challenging (for example, the corresponding problem is not convex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We study the minimization of the two terms ϵopt and ϵ′ bias separately and learn some insights, which we use to design the new FL algorithm CA-Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Minimizing ϵopt The minimization of ϵopt is still a convex optimization problem (Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In particular, at the optimum non-negative weights are set accordingly to q∗ k = a(λ∗πk − θ∗) with a, λ∗, and θ∗ positive constants (see (29)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' It follows that clients with smaller availability get smaller weights in the aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In particular, this suggests that clients with the smallest availability can be excluded from the aggregation, leading to the following guideline: Guideline A: to speed up the convergence, we can exclude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', set q∗ k = 0, the clients with lowest availability πk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' This guideline can be justified intuitively: updates from clients with low participation may be too sporadic to allow the FL algorithm to keep track of their local objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' They act as a noise slowing down the algorithm’s convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' It may be advantageous to exclude these clients from participating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We observe that the choice of the aggregation weights q does not affect the clients’ availability process and, in particular, λ(P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' However, if the algorithm excludes some clients, it is possible to consider the state space of the Markov chain that only specifies the availability state of the remaining clients, and this Markov chain may have different spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For the sake of concreteness, we consider here (and in the rest of the paper) the particular case when the availability of each client k evolves according to a two- states Markov chain (Ak t )t≥0 with transition probability ma- trix Pk and these Markov chains are all independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In this case, the aggregate process is described by the product Markov chain (At)t≥0 with transition matrix P = � k∈K Pk and λ(P ) = maxk∈K λ(Pk), where Pi � Pj denotes the Kronecker product between matrices Pi and Pj [30, Exer- cise 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In this setting, it is possible to redefine the Markov chain (At)t≥0 by taking into account the reduced state space defined by the clients with a non-null aggregation weight, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', P ′ = � k′∈K|qk′>0 Pk′ and λ(P ′) = maxk′∈K|qk′>0 λ(Pk′), which is potentially smaller than the case when all clients participate to the aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' These considerations lead to the following guideline: Guideline B: to speed up the convergence, we can exclude, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', set q∗ k = 0, the clients with largest λ(Pk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Intuition also supports this guideline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Clients with large λ(Pk) tend to be available or unavailable for long periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Due to the well-known catastrophic forgetting problem affect- ing gradient methods [33], [34], these clients may unfairly steer the algorithm toward their local objective when they appear at the final stages of the training period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Moreover, their participation in the early stages may be useless, as their contribution will be forgotten during their long absence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The FL algorithm may benefit from directly neglecting such clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We observe that guideline B strictly applies to this specific setting where clients’ dynamics are independent (and there is no spatial correlation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We do not provide a corresponding Algorithm 1: CA-Fed (Correlation-Aware FL) Input : w0,0, α, q(0), {ηt}T t=1, ηs, E, β, τ 1 Initialize ˆF (0), ˆF ∗, ˆΓ ′(0), ˆπ(0), and ˆλ(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 2 for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' , T do 3 Receive set of active client At, loss vector F (t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 4 Update ˆF (t), ˆΓ ′(t), ˆπ(t), and ˆλ(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 5 Initialize q(t) = α ˆπ(t) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 6 q(t) ← get(q(t), α, ˆF (t), ˆF ∗, ˆΓ ′(t), ˆπ(t), ˆλ(t));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 7 q(t) ← get(q(t), α, ˆF (t), ˆF ∗, ˆΓ ′(t), ˆπ(t), �ˆπ(t));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 8 for client {k ∈ At;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' q(t) k > 0}, in parallel do 9 for j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' , E − 1 do 10 wk t,j+1 = wk t,j − ηt∇Fk(wk t,j, Bk t,j) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 11 ∆k t ← wt,E − wt,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 12 wt+1,0 ← ProjW (wt,0 + ηs � k∈At q (t) k · ∆k t );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 13 Function get(q, α, F , F ∗, Γ, π, ρ): 14 K ← sort by descending order in ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 15 ˆϵ ← ⟨F −F ∗, π ˜⊙q⟩ + d2 T V (α, π ˜⊙q) · Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 16 for k ∈ K do 17 q+ k ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 18 ˆϵ+ ← ⟨F −F ∗, π ˜⊙q+⟩ + d2 T V (α, π ˜⊙q+) · Γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 19 if ˆϵ − ˆϵ+ ≥ τ then 20 ˆϵ ← ˆϵ+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 21 q ← q+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 22 return q guideline for the case when clients are spatially correlated (we leave this task for future research).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' However, in this more gen- eral setting, it is possible to ignore guideline B but still draw on guidelines A and C, or still consider guideline B if clients are spatially correlated (see discussion in Section VI-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Minimizing ϵ′ bias The bias error ϵ′ bias in (14) vanishes when the total variation distance between the target importance α and the biased importance p is zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', when qk ∝ αk/πk, ∀k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Then, after excluding the clients that contribute the most to the optimization error and particularly slow down the convergence (guidelines A and B), we can assign to the remaining clients an aggregation weight inversely proportional to their availability, such that the bias error ϵ′ bias is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Guideline C: to reduce the bias error, we set q∗ k ∝ αk/πk for the clients that are not excluded by the previous guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' PROPOSED ALGORITHM Guidelines A and B in Section III suggest that the minimiza- tion of ϵopt can lead to the exclusion of some available clients from the aggregation step (3), in particular those with low availability and/or high correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For the remaining clients, guideline C proposes to set their aggregation weight inversely proportional to their availability to reduce the bias error ϵ′ bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Motivated by these insights, we propose CA-Fed, a client sampling and aggregation strategy that takes into account the problem of correlated client availability in FL, described in 5 Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' CA-Fed learns during training which are the clients to exclude and how to set the aggregation weights of the other clients to achieve a good trade-off between ϵopt and ϵ′ bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' While guidelines A and B indicate which clients to remove, the exact number of clients to remove at round t is identified by minimizing ϵ(t) as a proxy for the bound in (14):4 ϵ(t) := FB(wt,0)−F ∗ B + d2 T V (α, p)Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (15) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' CA-Fed’s core steps At each communication round t, the server sends the current model wt,0 to all active clients and each client k sends back a noisy estimate F (t) k of the current loss computed on a batch of samples Bk t,0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', F (t) k = 1 |Bk t,0| � ξ∈Bk t,0 f(wt,0, ξ) (line 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The server uses these values and the information about the current set of available clients At to refine its own estimates of each client’s loss ( ˆF (t) = ( ˆF (t) k )k∈K), and each client’s loss minimum value ( ˆF ∗ = ( ˆF ∗ k )k∈K), as well as of Γ′, πk, λk, and ϵ(t), denoted as ˆΓ ′(t), ˆπ (t) k , ˆλ (t) k , and ˆϵ(t), respectively (possible estimators are described below) (line 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The server decides whether excluding clients whose avail- ability pattern exhibits high correlation (high ˆλ (t) k ) (line 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' First, the server considers all clients in descending order of ˆλ(t) (line 14), and evaluates if, by excluding them (line 17), ˆϵ(t) appears to be decreasing by more than a threshold τ ≥ 0 (line 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Then, the server considers clients in ascending order of ˆπ(t), and repeats the same procedure to possibly exclude some of the clients with low availability (low ˆπ (t) k ) (lines 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Once the participating clients (those with qk > 0) have been selected, the server notifies them to proceed updating the current models (lines 9–10) according to (2), while the other available clients stay idle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Finally, model’s updates are aggregated according to (3) (line 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Estimators We now briefly discuss possible implementation of the esti- mators ˆF (t) k , ˆF ∗ k , ˆΓ ′(t), ˆπ (t) k , and ˆλ (t) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Server’s estimates for the clients’ local losses ( ˆF (t) = ( ˆF (t) k )k∈K) can be obtained from the received active clients’ losses (F (t) = (F (t) k )k∈At) through an auto-regressive filter with parameter β ∈ (0, 1]: ˆF (t) = (1 − β1At) ⊙ ˆF (t−1) + β1At ⊙ F (t), (16) where ⊙ denotes the component-wise multiplication between vectors, and 1At is a N-dimensions binary vector whose k-th component equals 1 if and only if k is active at round t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', k ∈ At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The server can keep track of the clients’ loss minimum values and estimate F ∗ k as ˆF ∗ k = mins∈[0,t] ˆF (s) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The values of FB(wt,0), F ∗ B, Γ′, and ϵ(t) can be estimated as follows: ˆF (t) B − ˆF ∗ B = ⟨ ˆF (t) − ˆF ∗, ˆπ(t) ˜⊙q(t)⟩, (17) ˆΓ ′(t) = maxk∈K( ˆF (t) k − ˆF ∗ k ), (18) ˆϵ(t) = ˆF (t) B − ˆF ∗ B + d2 T V (α, ˆπ(t) ˜⊙q(t)) · ˆΓ ′(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (19) 4Following (14), one could reasonably introduce a hyper-parameter to weigh the relative importance of the optimization and bias terms in the sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We discuss this additional optimization of CA-Fed in Section VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' where π ˜⊙q ∈ RN, such that � π ˜⊙q � k = πkqk �N h=1 πhqh , k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For ˆπ (t) k , the server can simply keep track of the total number of times client k was available up to time t and compute ˆπ (t) k using a Bayesian estimator with beta prior, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', ˆπ (t) k = (� s≤t 1k∈As +nk)/(t+nk +mk), where nk and mk are the initial parameters of the beta prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For ˆλ (t) k , the server can assume the client’s availability evolves according to a Markov chain with two states (available and unavailable), track the corresponding number of state tran- sitions, and estimate the transition matrix ˆP (t) k through a Bayesian estimator similarly to what done for ˆπ (t) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Finally, ˆλ (t) k is obtained computing the eigenvalues of ˆP (t) k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' CA-Fed’s computation/communication cost CA-Fed aims to improve training convergence and not to reduce its computation and communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Never- theless, excluding some available clients reduces the overall training cost, as we will discuss in this section referring, for the sake of concreteness, to neural networks’ training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The available clients not selected for training are only re- quested to evaluate their local loss on the current model once on a single batch instead than performing E gradient updates, which would require roughly 2 × E − 1 more calculations (because of the forward and backward pass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For the selected clients, there is no extra computation cost as computing the loss corresponds to the forward pass they should, in any case, perform during the first local gradient update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In terms of communication, the excluded clients only transmit the loss, a single scalar, much smaller than the model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Conversely, participating clients transmit the local loss and the model update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Still, this additional overhead is negligible and likely fully compensated by the communication savings for the excluded clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' EXPERIMENTAL EVALUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Experimental Setup a) Federated system simulator: In our experiments, we sim- ulate the clients’ availability dynamics featuring different levels of temporal correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We model the activity of each client as a two-state homogeneous Markov process with state space S = {“active”, “inactive”}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We use pk,s to denote the probability that client k ∈ K remains in state s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In order to simulate the statistical heterogeneity present in the federated learning system, we consider an experimental setting with two disjoint groups of clients Gi, i = 1, 2, to which we associate two different data distributions Pi, i = 1, 2, to be precised later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let ri = |Gi|/N, i = 1, 2 denote the fraction of clients in group i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In order to simulate the heterogeneity of clients’ availability patterns in realistic federated systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' we split the clients of each group in two classes uniformly at random: “more available” clients whose steady-state probability to be active is πk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='active = 1/2 + g and “less available” clients with πk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='active = 1/2 − g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' where g ∈ 6 Inactive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' excluded Inactive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' included Active,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' excluded Active,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' included More Available Less Available,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Weakly Correlated 0 20 40 60 80 100 120 140 Communication round Less Available,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Correlated Clients Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 1: Clients’ activities and CA-Fed’s clients selection on the synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' More Available Less Available Correlated Less Available Weakly Correlated Clients Cumulative weight Unbiased CA-Fed AdaFed F3AST Target Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 2: Importance given to the clients by the different algorithms throughout a whole training process on the synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (0, 1/2) is a parameter controlling the heterogeneity of clients availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We furthermore split each class of clients in two sub-classes uniformly at random: “correlated” clients that tend to persist in the same state (λk = ν with values of ν close to 1), and “weakly correlated” clients that are almost as likely to keep as to change their state (λk ∼ N(0, ε2), with ε close to 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In our experiments, we suppose that r1 = r2 = 1/2, g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='4, ν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='9, and ε = 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' b) Datasets and models: All experiments are performed on a binary classification synthetic dataset (described in Ap- pendix F) and on the real-world MNIST dataset [35], using N = 24 clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For MNIST dataset, we introduce statistical heterogeneity across the two groups of clients (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=', we make the two distributions P1 and P2 different), following the same approach in [36]: 1) every client is assigned a random subset of the total training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 2) the data of clients from the second group is modified by randomly swapping two pairs of labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We maintain the original training/test data split of MNIST and use 20% of the training dataset as validation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We use a linear classifier with a ridge penalization of parameter 10−2, which is a strongly convex objective function, for both the synthetic and the real-world MNIST datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' c) Benchmarks: We compare CA-Fed, defined in Algo- rithm 1, with the Unbiased aggregation strategy, where all the active clients participate and receive a weight inversely proportional to their availability, and with the state-of-the- art FL algorithms discussed in Section II: F3AST [18] and AdaFed [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We tuned the learning rates η, ηs via grid search, on the grid η : {10−3, 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='5, 10−2, 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='5, 10−1}, ηs : {10−2, 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='5, 10−1, 10−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='5, 100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For CA-Fed, we used τ = 0, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We assume all algorithms can access an oracle providing the true availability parameters for each client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In 0 20 40 60 80 100 120 140 Communication round 35 40 45 50 55 60 65 70 75 Time-average test accuracy Unbiased F3AST AdaFed CA-Fed (Ours) (a) Synthetic 0 20 40 60 80 100 120 140 Communication round 10 20 30 40 50 60 Time-average test accuracy Unbiased F3AST AdaFed CA-Fed (Ours) (b) MNIST Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 3: Test accuracy vs number of communication rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' practice, Unbiased, AdaFed, and F3AST rely on the exact knowledge of πk,active, and CA-Fed on πk,active and λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 5 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Experimental Results Figure 1 shows the availability of each client during a training run on the synthetic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Clients selected (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' excluded) by CA-Fed are highlighted in black (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We observe that excluded clients tend to be those with low average availability or high correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Figure 2 shows the importance pk (averaged over time) given by different algorithms to each client k during a full training run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We observe that all the algorithms, except Unbiased, depart from the target importance α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' As suggested by guide- lines A and B, CA-Fed tends to favor the group of “more available” clients, at the expense of the “less available” clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Figure 3 shows the time-average accuracy up to round t of the learned model averaged over three different runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' On both datasets, CA-Fed achieves the highest accuracy, which is about a percentage point higher than the second best algorithm (F3AST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Table I shows for each algorithm: the average over three runs of the maximum test accuracy achieved during train- ing, the time-average test accuracy achieved during training, together with its standard deviation within the second half of the training period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Results show that while CA-Fed achieves a maximum accuracy which is comparable to the Unbiased baseline and state-of-the-art AdaFed and F3AST, it gets a higher time-average accuracy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='24 percentage points) in com- parison to the second best (F3AST), and a smaller standard deviation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='5×) in comparison to the second best (F3AST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 5The authors have provided public access to their code and data at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='com/arodio/CA-Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 7 TABLE I: Maximum and time-average test accuracy, together with their standard deviations, on the Synthetic / MNIST datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' TEST ACCURACY MAXIMUM TIME-AVERAGE STANDARD DEVIATION UNB I AS ED 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='94 / 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='87 75.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='40 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='94 ADAFED 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='69 / 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='77 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='81 / 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='59 / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='37 CA-FE D 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='03 / 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='94 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='22 / 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='28 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='61 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' DISCUSSION In this section, we discuss some general concerns and remarks on our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Controlling the number of excluded clients Theorems 1 and 3 suggest that the condition number κ2 can play a meaningful role in the minimization of the total error ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our algorithm uses a proxy (ϵ(t)) of the total error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' To take into account the effect of κ2, we can introduce a hyper-parameter that weights the relative importance of the optimization and bias error in (15): ϵ′(t) := FB(wt,0) − F ∗ B + ¯κ2 · d2 T V (α, p)Γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' A small value of ¯κ2 penalizes the bias term in favor of the optimization error, resulting in a larger number of clients excluded by CA-Fed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' On the other hand, CA-Fed tends to include more clients for a large value of ¯κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Asymptotically, for ¯κ2 → +∞, CA-Fed reduces to the Unbiased baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' To further improve the performance of CA-Fed, a finer tuning of the values of ¯κ2 can be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' CA-Fed in presence of spatial correlation Although CA-Fed is mainly designed to handle temporal correlation, it does not necessarily perform poorly in presence of spatial correlation, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Consider the following spatially-correlated scenario: clients are grouped in clusters, each cluster c ∈ C is characterized by an underlying Markov chain, which determines when all clients in the cluster are available/unavailable, the Markov chains of different clusters are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let λc denote the second largest eigenvalue in module of cluster-c’s Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In this case, one needs to exclude all clients in the cluster ¯c = arg maxc∈C λc to reduce the eigenvalue of the aggregate Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In this setting, CA-Fed would associate similar eigenvalue estimates to all clients in the same cluster, then it would correctly start considering for exclusion the clients in cluster ¯c and potentially remove sequentially all clients in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' These considerations suggest that CA-Fed may still operate correctly even in presence of spatial correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' About CA-Fed’s fairness A strategy that excludes clients from the training phase, such as CA-Fed, may naturally raise fairness concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The concept of fairness in FL does not have a unified definition in the literature [37, Chapter 8]: fairness goals can be captured by a suitable choice of the target weights in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For example, per- client fairness can be achieved by setting αk equal for every client, while per-sample fairness by setting αk proportional to the local dataset size |Dk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' If we assume that the global objective in (1) indeed reflects also fairness concerns, then CA-Fed is intrinsically fair, in the sense that it guarantees that the performance objective of the learned model is as close as possible to its minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' CONCLUSION This paper presented the first convergence analysis for a FedAvg-like FL algorithm under heterogeneous and corre- lated client availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The analysis quantifies how correla- tion adversely affects the algorithm’s convergence rate and highlights a general bias-versus-convergence-speed trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Guided by the theoretical analysis, we proposed CA-Fed, a new FL algorithm that tries to balance the conflicting goals of maximizing convergence speed and minimizing model bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Our experimental results demonstrate that adaptively excluding clients with high temporal correlation and low availability is an effective approach to handle the heterogeneous and correlated client availability in FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Proof of Theorem 1 We bound the optimization error of the target objective as the optimization error of the biased objective plus a bias term: F(w) − F ∗ (a) ≤ 1 2µ ∥∇F(w)∥2 (b) ≤ L2 2µ ∥w − w∗∥2 (c) ≤ L2 µ (∥w − w∗ B∥2 + ∥w∗ B − w∗∥2) (d) ≤ 2L2 µ2 (FB(w) − F ∗ B) � �� � :=ϵopt + 2L2 µ2 (F(w∗ B) − F ∗) � �� � :=ϵbias , where (a), (b), and (d) follow from the Assumptions 3, 4, and the inequality (c) follows from (a + b)2 ≤ 2a2 + 2b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In particular, (b) requires ∇Fk(w∗ k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Theorem 2 further develops the optimization error ϵopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We now expand ϵbias: ∥∇F(w∗ B)∥ (e)= ����N k=1(αk − pk)∇Fk(w∗ B) ��� (f) ≤ L �N k=1|αk − pk| ∥w∗ B − w∗ k∥ (20) (g) ≤ L � 2 µ �N k=1 |αk−pk| √pk � pk(Fk(w∗ B) − F ∗ k ), where (e) uses ∇FB(w∗ B) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (f) applies first the triangle inequality, then the L-smoothness, and (g) follows from the µ-strong convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' In addition, (f) requires ∇Fk(w∗ k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Similarly to [32], in (g) we multiply numerator and denomi- nator by √pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' By direct calculations, it follows that: ∥∇F(w∗ B)∥2 (h) ≤ 2L2 µ � �N k=1 |αk−pk| √pk � pk(Fk(w∗ B) − F ∗ k ) �2 (i) ≤ 2L2 µ � N� k=1 (αk−pk)2 pk �� N� k=1 pk(Fk(w∗ B) − F ∗ k ) � (j) ≤ 2L2 µ χ2 α∥pΓ, 8 where (i) uses the Cauchy–Schwarz inequality, and (j) used: �N k=1 pk(Fk(w∗ B) − F ∗ k ) ≤ �N k=1 pk(Fk(w∗) − F ∗ k ) ≤ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Finally, by strong convexity of F, we conclude that: F(w∗ B) − F ∗ ≤ 1 2µ ∥∇F(w∗ B)∥2 ≤ L2 µ2 χ2 α∥pΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Proof of Theorem 2 1) Additional notation: let wk t,j be the model parameter vector computed by device k at the global round t, local iteration j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We define: gt(At) = � k∈At qk �E−1 j=0 ∇Fk(wk t,j, ξk t,j), and ¯gt(At) = Eξ|At[gt(At)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Following (2) and (3), the update rule of CA-Fed is: wt+1,0 = ProjW (wt,0 − ηtgt(At)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (21) 2) Key lemmas and results: we provide useful lemmas and results to support the proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The boundedness of W gives a bound on (wt,0)t≥0 based on the update rules in (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' From the convexity of {Fk}k∈K, it follows that: D := sup w∈W,k∈K ∥∇Fk(w)∥ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Items (6), (8) are directly derived from the previous observa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Item (7) follows combining (6) and Assumption 5: E ∥∇Fk(w, ξ)∥2 ≤ D2 + max k∈K {σ2 k} := G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Lemma 2 (Convergence under heterogeneous client availabil- ity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let the local functions {Fk}k∈K be convex, Assump- tions 3, 5 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' If ηt ≤ 1 2L(EQ+1), we have: � t ηt E[� k∈At qk (Fk(wt,0) − Fk(w∗ B))] ≤ + 2 E ∥w0,0 − w∗ B∥2 + 2 �N k=1 πkq2 kσ2 k � t η2 t + 2 3 �N k=1 πkqk(E − 1)(2E − 1)G2 � t η2 t + 2L(EQ + 2) �N k=1 πkqkΓ � t η2 t := C1 < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' ∥wt+1,0 − w∗ B∥2 = ∥ProjW (wt,0 − ηtgt) − ProjW (w∗ B)∥2 ≤ ∥wt,0 − ηtgt − w∗ B + ηt¯gt − ηt¯gt∥2 = A1 + A2 + A3, where: A1 = ∥wt,0 − w∗ B − ηt¯gt∥2 , A2 = 2ηt⟨wt,0 − w∗ B − ηt¯gt, ¯gt − gt⟩, A3 = η2 t ∥gt − ¯gt∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Note E[A2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We bound A1, A3 using the key steps in [22]: (1) the variance of gt(At) is bounded if the variance of the stochastic gradients at each device is bounded: A3 = EB|At ∥gt − ¯gt∥2 = = � k∈At q2 k �E−1 j=0 EB|At ��∇Fk(wk t,j, ξk t,j)−∇Fk(wk t,j) ��2 ≤ E � k∈At q2 kσ2 k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (2) the distance of the local model wk t,E from the global model wt,0 is bounded since the expected squared norm of the stochastic gradients is bounded: EB|At � k∈At qk �E−1 j=0 ��wk t,j − wt,0 ��2 = = EB|At � k∈At qk �E−1 j=1 η2 t ��� �j−1 j′=0 ∇Fk(wk t,j′, ξk t,j′) ��� 2 ≤ η2 t � k∈At qk �E−1 j=1 j �j−1 j′=0 EB|At ��∇Fk(wk t,j′, ξk t,j′) ��2 ≤ η2 t � k∈At qkG2 �E−1 j=1 j2 = 1 6η2 t � k∈At qkE(E − 1)(2E − 1)G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Lemma 3 (Optimization error after Jt steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let Assump- tions 1, 2 hold, the local functions {Fk}k∈K be convex, D, H be defined as in (6), (8), and Jt defined as in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Then: � t ηt E[� k∈At qk(Fk(wt−Jt,0) − Fk(wt,0))] ≤ EDGQ � t Jtη2 t−Jt �N k=1 πkqk := C3 ln(1/λ(P )) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' For the proof of Lemma 3, we introduce the following results: |Fk(v) − Fk(w)| ≤ D · ∥v − w∥ , ∀v, w ∈ W, (22) EBk t,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=',Bk t,E−1 ∥wt+1,0 − wt,0∥ ≤ ηtGE(� k∈At qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (23) Equation (22) is due to convexity of {Fk}k∈K, which gives: ⟨∇Fk(v), v − w⟩ ≤ ∥Fk(v) − Fk(w)∥ ≤ ⟨∇Fk(w), v − w⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' the Cauchy–Schwarz inequality concludes: |Fk(v) − Fk(w)| ≤ max{∥∇Fk(v)∥ , ∥∇Fk(w)∥} ∥v − w∥ ≤ D · ∥v − w∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Equation (23) follows combining equations (7) and (21): EB|At ∥wt+1,0 − wt,0∥ ≤ ≤ ηt EB|At ���� k∈At qk �E−1 j=0 ∇Fk(wk t,j, ξk t,j) ��� ≤ ηt � k∈At qk �E−1 j=0 EB|At ��∇Fk(wk t,j, ξk t,j) �� ≤ ηtGE(� k∈At qk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The evolution of the local objectives after Jt communication rounds is bounded: � tηt E[� k∈At qk(Fk(wt−Jt,0) − Fk(wt,0))] (a) ≤ D � t ηt E[� k∈At qk EB ∥wt−Jt,0 − wt,0∥] (b) ≤ D � t ηt �t−1 d=t−Jt E[� k∈At qk EB ∥wd,0 − wd+1,0∥] (c) ≤ EDG � t �t−1 d=t−Jt ηtηd E[� k∈At qk � k′∈Ad qk′] (d) ≤ EDG 2 � t �t−1 d=t−Jt(η2 t + η2 d) E[� k∈At qk � k′∈Ad qk′] (e) ≤ EDGQ � t Jtη2 t−Jt �N k=1 πkqk := C3 ln(1/λ(P )), 9 where (a) follows from (22);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (b) applies the triangle inequal- ity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (c) uses (23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (d) applies the Cauchy–Schwarz inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (e) uses ηt < ηd ≤ ηt−Jt and �N k=1 qk = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 3) Core of the proof: The proof consists in two main steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' � t ηt �N k=1 πkqk E[FB(wt−Jt,0) − F ∗ B)]≤C2+ C3 ln(1/λ(P ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' � t ηt �N k=1 πkqk E[FB(wt,0)−FB(wt−Jt,0)]≤ C3 ln(1/λ(P )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Combining Lemma 2 and 3, we get: � t ηt E[ � k∈At qk(Fk(wt−Jt,0) − Fk(w∗ B))] ≤ C1 + C3 ln(1/λ(P )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The constant Jt, introduced in [14], is an important parameter for the analysis and frequently used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Combining its definition in Theorem 2 and equation (5), it follows: ��[P Jt]i,j − πj �� ≤ CP λ(P )Jt ≤ 1 2Ht, ∀i, j ∈ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (24) Assume t ≥ TP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We derive an important lower bound: EAt|At−Jt [� k∈At qk(Fk(wt−Jt,0) − Fk(w∗ B))] (a)= �M I=1 P(At=I|At−Jt) � k∈I qk(Fk(wt−Jt,0)−Fk(w∗ B)) (b)= �M I=1 [P Jt]At−Jt,I � k∈I qk (Fk(wt−Jt,0) − Fk(w∗ B)) (c) ≥ �M I=1 � π(I) − 1 2Ht � � k∈I qk(Fk(wt−Jt,0) − Fk(w∗ B)) (d) ≥ (�N k=1 πkqk) · (FB(wt−Jt,0) − F ∗ B) − 1 2tMQ, (25) where (a) is the definition of the conditional expectation, (b) uses the Markov property, (c) follows from (24), and (d) is due to (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Taking total expectations: ( �N k=1 πkqk) � t ηt E[FB(wt−Jt,0) − F ∗ B] ≤ � t ηt E[� k∈At qk(Fk(wt−Jt,0) − Fk(w∗ B))] + 1 4MQ � t(η2 t + 1 t2 ) = C2 + C3 ln(1/λ(P )), (26) where C2 = C1 + 1 4MQ � t(η2 t + 1 t2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' By direct calculation (similar to Lemma 3): (�N k=1 πkqk) � t ηt E[FB(wt,0) − FB(wt−Jt,0)]≤ C3 ln(1/λ(P )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Summing Step 1 and 2, and applying Jensen’s inequality: (�T t=1 ηt)(�N k=1 πkqk) E[FB( ¯wT,0) − F ∗ B] ≤ (�N k=1 πkqk) �T t=1 ηt E[FB(wt,0) − F ∗ B] ≤ C2 + 2C3 ln(1/λ(P )), where ¯wT,0 := �T t=1 ηtwt,0 �T t=1 ηt , and the constants are in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Proof of Theorem 3 It follows the same lines of Theorem 1, developing (20) as: ∥∇F(w∗ B)∥ ≤ L � 2 µ �N k=1|αk − pk| � (Fk(w∗ B) − F ∗ k ) ≤ 2L � 2 µdT V (α, p) √ Γ′, where dT V (α, p) := 1 2 �N k=1|αk − pk| is the total variation distance between the probability measures α and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Minimizing ϵopt Equation 12 defines the following optimization problem: minimize q f(q) = 1 2 q⊺Aq+B π⊺q + C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' subject to q ≥ 0, π⊺q > 0, ∥q∥1 = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Let us rewrite the problem by adding a variable s := 1/π⊺q and then replacing y := sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Note that the objective function is the perspective of a convex function, and is therefore convex: min y,s f(y, s) = 1 2sy⊺Ay + Bs + C (27a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' y ≥ 0, s > 0, π⊺y = 1, ∥y∥1 = Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (27b) The Lagrangian function L is as follows: L(y, s, λ, θ, µ) = 1 2sy⊺Ay + Bs + C+ +λ(1 − π⊺y) + θ(∥y∥1 − Qs) − µ⊺y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (28) Since the constraint s > 0 defines an open set, the set defined by the constraints in (27b) is not closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' However, the solution is never on the boundary s = 0 because L∗ → +∞ as s → 0+, and we can consider s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' The KKT conditions for y∗ k read: if y∗ k > 0: y∗ k = s∗ A[kk](λ∗πk − θ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' y∗ k = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' (29) Since λ∗ ≥ 0, the clients with smaller πk may have q∗ k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Convexity of ϵopt + ϵbias In Appendix D, we proved that ϵopt(q) is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' To prove that ϵbias(q) is also convex, we need to study the convexity of χ2 α∥p = �N k=1(fk ◦ gk)(q), where fk(pk) = (pk − αk)2/pk, and gk(q) = (πkqk)/ �N h=1 πhqh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' We observe that fk(pk) is convex, and gk(q) is a particular case of linear-fractional function [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' By direct inspection, it can be proved that (fk◦gk)(q) is convex in dom(fk◦gk) = {q : ∥q∥1 = Q > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Synthetic dataset Our synthetic datasets has been generated as follows: 1) For client k ∈ K, sample group identity ik from a Bernoulli distribution of parameter 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 2) Sample model parameters w∗ ∼ N(0, Id) from the d- dimensional normal distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 3) For client k ∈ K and sample index j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' , 150}, sample clients input data x(j) k ∼ N(0, Id) from the d- dimensional normal distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 4) For client k ∈ K such that ik = 0 and sample index j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' , 150}, sample the true labels y(j) k from a Bernoulli distribution with parameter equal to sigmoid(⟨w∗, x(j) k ⟩);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 5) For client k ∈ K such that ik = 1 and sample index j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' , 150}, sample the true labels y(j) k from a Bernoulli distribution with parameter equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='8·sigmoid(⟨w∗, x(j) k ⟩)+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content='2·(1−sigmoid(⟨w∗, x(j) k ⟩)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' 10 REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Verbraeken, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Wolting, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Katzy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE3T4oBgHgl3EQfoQqU/content/2301.04632v1.pdf'} +page_content=' Kloppenburg, T.' metadata={'source': 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b/4dFKT4oBgHgl3EQf9S5d/content/tmp_files/2301.11953v1.pdf.txt @@ -0,0 +1,681 @@ +FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS OF SURFACES +C. CASAGRANDE +Dedicated to Lorenzo, Sabrina, and Fabrizio +Abstract. Let X be a smooth, complex Fano 4-fold, and ρX its Picard num- +ber. We show that if ρX > 12, then X is a product of del Pezzo surfaces. The +proof relies on a careful study of divisorial elementary contractions f : X → Y +such that dim f(Exc(f)) = 2, together with the author’s previous work on Fano +4-folds. In particular, given f : X → Y as above, under suitable assumptions we +show that S := f(Exc(f)) is a smooth del Pezzo surface with −KS = (−KY )|S. +1. Introduction +Smooth, complex Fano varieties have been classically intensively studied, and +have attracted a lot of attention also in the last decades, due to their role in +the framework of the Minimal Model Program. The Fano condition is a natural +positivity condition of the tangent bundle, and it ensures a rich geometry, from +both the points of view of birational geometry and of families of rational curves. +It has been known since the 90’s that Fano varieties form a bounded family in +each dimension. Del Pezzo surfaces are known classically, and the classification +of Fano 3-folds have been in achieved in the 80’s, there are 105 families. +Starting from dimension 4, there are probably too many families to get a com- +plete classification; still we aim to better understand and describe the behavior +and properties of these varieties. In this paper we focus on Fano 4-folds X with +“large” Picard number ρX; let us recall that since X is Fano, ρX is equal to the +second Betti number b2(X). We show the following result. +Theorem 1.1. Let X be a smooth Fano 4-fold with ρX > 12. Then X ∼= S1 ×S2, +where Si are del Pezzo surfaces. +To the author’s knowledge, all known examples of Fano 4-folds which are not +products of surfaces have ρ ≤ 9, so that we do not know whether the condition +ρ > 12 in Th. 1.1 is sharp. We refer the reader to [Cas22b, §6] for an overview +of known Fano 4-folds with ρ ≥ 6; there are few examples and it is an interesting +problem to construct new ones. +As ρS1×S2 = ρS1 + ρS2, and del Pezzo surfaces have ρ ≤ 9, Th. 1.1 implies the +following. +Corollary 1.2. Let X be a smooth Fano 4-fold. Then ρX ≤ 18. +2020 Mathematics Subject Classification. 14J45,14J35,14E30. +1 +arXiv:2301.11953v1 [math.AG] 27 Jan 2023 + +2 +C. CASAGRANDE +Let us note that Th. 1.1 and Cor. 1.2 generalize to dimension 4 the analogous +result for Fano 3-folds, established by Mori and Mukai in the 80’s: +Theorem 1.3 ([MM86], Th. 1.2). Let X be a smooth Fano 3-fold with ρX > 5. +Then X ∼= S × P1 where S is a del Pezzo surface. In particular ρX ≤ 10. +The proof of Th. 1.1 relies on a careful study of elementary contractions of X of +type (3, 2), together with the author’s previous work on Fano 4-folds. To explain +this, let us introduce some notation. +Let X be a Fano 4-fold. A contraction is a surjective morphism f : X → Y , with +connected fibers, where Y is normal and projective; f is elementary if ρX−ρY = 1. +As usual, an elementary contraction can be of fiber type, divisorial, or small. +We say that an elementary contraction f : X → Y is of type (3, 2) if it is +divisorial with dim S = 2, where E := Exc(f) and S := f(E) ⊂ Y . Such f +can have at most finitely many 2-dimensional fibers; outside the images of these +fibers, Y and S are smooth, and f is just the blow-up of the surface S. If y0 ∈ S is +the image of a two-dimensional fiber, then either Y or S are singular at y0; these +singularities have been described by Andreatta and Wi´sniewski, see Th. 2.1. In +any case, Y has at most isolated locally factorial and terminal singularities, while +S can be not normal. +We denote by N1(X) the real vector space of one-cycles with real coefficients, +modulo numerical equivalence; we have dim N1(X) = ρX. For any closed subset +Z ⊂ X, we set +N1(Z, X) := ι∗(N1(Z)) ⊂ N1(X) +where ι: Z �→ X is the inclusion, so that N1(Z, X) is the subspace of N1(X) +spanned by classes of curves in Z, and dim N1(Z, X) ≤ ρZ. +We study an elementary contraction f : X → Y of type (3, 2) under the hy- +pothesis that: +dim N1(E, X) ≥ 4. +In particular this implies that Y is Fano too (Lemma 2.2). +We would like to compare (−KY )|S to −KS, but since S may be singular, +we consider the minimal resolution of singularities µ: S′ → S and set L := +µ∗((−KY )|S), a nef and big divisor class on S′. We show that KS′+L is semiample +(Lemma 3.1). Then our strategy is to look for curves in S′ on which KS′ + L is +trivial, using other elementary contractions of X of type (3, 2) whose exceptional +divisor intersects E in a suitable way. +Hence let us assume that X has another elementary contraction g1 of type (3, 2) +whose exceptional divisor E1 intersects E, and such that E ·Γ1 = 0 for a curve Γ1 +contracted by g1. Set D := f(E1) ⊂ Y . We show that an irreducible component +C1 of D ∩ S is a (−1)-curve contained in the smooth locus Sreg, and such that +−KY · C1 = 1 (Lemma 3.2, see Fig. 3.1 on p. 7). If C′ +1 ⊂ S′ is the transform of +C1, we have (KS′ + L) · C′ +1 = 0. +Finally let us assume that X has three elementary contractions g1, g2, g3, all of +type (3, 2), satisfying the same assumptions as g1 above. We also assume that + +FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS +3 +E1 · Γ2 > 0 and E1 · Γ3 > 0, where E1 = Exc(g1) and Γ2, Γ3 are curves contracted +by g2, g3 respectively. Then we show that S is a smooth del Pezzo surface with +−KS = (−KY )|S (Th. 3.6 and Prop. 3.10); let us give an overview of the proof. +The previous construction yields three distinct (−1)-curves C′ +1, C′ +2, C′ +3 ⊂ S′ such +that (KS′ + L) · C′ +i = 0 and C′ +1 intersects both C′ +2 and C′ +3. This shows that the +contraction of S′ given by KS′ +L cannot be birational, namely KS′ +L is not big. +We also rule out the possibility of a contraction onto a curve, and conclude that +KS′ + L ≡ 0. Finally we show that ωS ∼= OY (KY )|S, where ωS is the dualizing +sheaf of S, and conclude that S is smooth and del Pezzo. +We believe that these results can be useful in the study of Fano 4-folds besides +their use in the present work. It would be interesting to generalize this technique +to higher dimensions. +Let us now explain how we use these results to prove Th. 1.1. We define the +Lefschetz defect of X as: +δX := max +� +codim N1(D, X) | D ⊂ X a prime divisor +� +. +This invariant, introduced in [Cas12], measures the difference between the Picard +number of X and that of its prime divisors; we refer the reader to [Cas22b] for a +survey on δX. +Fano 4-folds with δX ≥ 3 are classified, as follows. +Theorem 1.4 ([Cas12], Th. 3.3). Let X be a smooth Fano 4-fold. If δX ≥ 4, +then X ∼= S1 × S2 where Si are del Pezzo surfaces, and δX = maxi ρSi − 1. +Theorem 1.5 ([CRS22], Prop. 1.5). Smooth Fano 4-folds with δX = 3 are clas- +sified. They have 5 ≤ ρX ≤ 8, and if ρX ∈ {7, 8} then X is a product of surfaces. +Therefore in our study of Fano 4-folds we can assume that δX ≤ 2, that is, +codim N1(D, X) ≤ 2 for every prime divisor D ⊂ X. To prove that ρX ≤ 12, we +look for a prime divisor D ⊂ X with dim N1(D, X) ≤ 10. +To produce such a divisor, we look at contractions of X. If X has an elementary +contraction of fiber type, or a divisorial elementary contraction f : X → Y with +dim f(Exc(f)) ≤ 1, it is not difficult to find a prime divisor D ⊂ X such that +dim N1(D, X) ≤ 3, hence ρX ≤ 5 (Lemmas 2.5 and 2.6). +The case where X has a small elementary contraction is much harder and is +treated in [Cas22a], where the following result is proven. +Theorem 1.6 ([Cas22a], Th. 1.1). Let X be a smooth Fano 4-fold. If X has a +small elementary contraction, then ρX ≤ 12. +We are left with the case where every elementary contraction f : X → Y is of +type (3, 2). In this case we show (Th. 4.1) that, if ρX ≥ 8, we can apply our +previous study of elementary contractions of type (3, 2), so that if E := Exc(f) +and S := f(E) ⊂ Y , then S is a smooth del Pezzo surface. This implies that +dim N1(S, Y ) ≤ ρS ≤ 9, dim N1(E, X) = dim N1(S, Y ) + 1 ≤ 10, and finally that +ρX ≤ 12, proving Th. 1.1. + +4 +C. CASAGRANDE +The structure of the paper is as follows. In §2 we gather some preliminary +results. +Then in §3 we develop our study of elementary contractions of type +(3, 2), while in §4 we prove Th. 1.1. +1.1. Notation +We work over the field of complex numbers. Let X be a projective variety. +We denote by N1(X) (respectively, N 1(X)) the real vector space of one-cycles +(respectively, Cartier divisors) with real coefficients, modulo numerical equiva- +lence; dim N1(X) = dim N 1(X) = ρX is the Picard number of X. +Let C be a one-cycle of X, and D a Cartier divisor. We denote by [C] (respec- +tively, [D]) the numerical equivalence class in N1(X) (respectively, N 1(X)). We +also denote by D⊥ ⊂ N1(X) the orthogonal hyperplane to the class [D]. +The symbol ≡ stands for numerical equivalence (for both one-cycles and divi- +sors), and ∼ stands for linear equivalence of divisors. +NE(X) ⊂ N1(X) is the convex cone generated by classes of effective curves, +and NE(X) is its closure. An extremal ray R is a one-dimensional face of NE(X). +If D is a Cartier divisor in X, we write D·R > 0, D·R = 0, and so on, if D·γ > 0, +D · γ = 0, and so on, for a non-zero class γ ∈ R. We say that R is K-negative if +KX · R < 0. +Suppose that X has terminal and locally factorial singularities, and is Fano. +Then NE(X) is a convex polyhedral cone. Given a contraction f : X → Y , we +denote by NE(f) the convex subcone of NE(X) generated by classes of curves +contracted by f; we recall that there is a bijection between contractions of X +and faces of NE(X), given by f �→ NE(f). Moreover dim NE(f) = ρX − ρY , in +particular f is elementary if and only if NE(f) is an extremal ray. +When dim X = 4, we say that an extremal ray R is of type (3, 2) if the as- +sociated elementary contraction f is of type (3, 2), namely if f is divisorial with +dim f(Exc(f)) = 2. We also set ER := Exc(f) and denote by CR ⊂ ER a general +fiber of f|ER; note that ER · CR = −1. +We will also consider the cones Eff(X) ⊂ N 1(X) of classes of effective divisors, +and mov(X) ⊂ N1(X) of classes of curves moving in a family covering X. Since +X is Fano, both cones are polyhedral; we have the duality relation Eff(X) = +mov(X)∨. +2. Preliminaries +In this section we gather some preliminary results that will be used in the sequel. +Andreatta and Wi´sniewski have classified the possible 2-dimensional fibers of +an elementary contraction of type (3, 2) of a smooth Fano 4-fold. In doing this, +they also describe precisely the singularities both of the target, and of the image +of the exceptional divisor, as follows. +Theorem 2.1 ([AW98], Theorem on p. 256). Let X be a smooth Fano 4-fold and +f : X → Y an elementary contraction of type (3, 2). Set S := f(Exc(f)). + +FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS +5 +Then f can have at most finitely many 2-dimensional fibers. Outside the images +of these fibers, Y and S are smooth, and f is the blow-up of S. +Let y0 ∈ S ⊂ Y be the image of a 2-dimensional fiber; then one of the following +holds: +(i) S is smooth at y0, while Y has an ordinary double point at y0, locally factorial +and terminal; +(ii) Y is smooth at y0, while S is singular at y0. More precisely either S is not +normal at y0, or it has a singularity of type 1 +3(1, 1) at y0 (as the cone over +a twisted cubic). +In particular the singularities of Y are at most isolated, locally factorial, and +terminal. +Now we give some simple preliminary results on extremal rays of type (3, 2). +Lemma 2.2. Let X be a smooth Fano 4-fold and f : X → Y an elementary +contraction of type (3, 2); set E := Exc(f). If dim N1(E, X) ≥ 4, then E · R ≥ 0 +for every extremal ray R of X different from NE(f), and Y is Fano. +Proof. It follows from [Cas17, Lemma 2.16 and Rem. 2.17] that NE(f) is the +unique extremal ray of X having negative intersection with E, −KX + E = +f ∗(−KY ) is nef, and (−KX + E)⊥ ∩ NE(X) = NE(f), so that −KY is ample. +■ +Lemma 2.3. Let X be a smooth Fano 4-fold and R1, R2 extremal rays of X of +type (3, 2) such that dim N1(ER1, X) ≥ 4 and ER1 · R2 = 0. +Then ER2 ·R1 = 0 and R1+R2 is a face of NE(X) whose associated contraction +is birational, with exceptional locus ER1 ∪ ER2. +Proof. Let H be a nef divisor on X such that H⊥ ∩ NE(X) = R2, and set H′ := +H + (H · CR1)ER1. Then H′ · CR1 = H′ · CR2 = 0, and if R3 is an extremal ray +of NE(X) different from R1 and R2, we have ER1 · R3 ≥ 0 by Lemma 2.2, hence +H′·R3 > 0. Therefore H′ is nef and (H′)⊥∩NE(X) = R1+R2 is a face of NE(X). +If Γ ⊂ X is an irreducible curve with [Γ] ∈ R1 + R2, then H′ · Γ = 0, so that +either ER1 · Γ < 0 and Γ ⊂ ER1, or H · Γ = 0, [Γ] ∈ R2 and Γ ⊂ ER2. This shows +that the contraction of R1 + R2 is birational with exceptional locus ER1 ∪ ER2. +Finally we have ER2 · R1 = 0 by [Cas13b, Lemma 2.2(b) and its proof]. +■ +Lemma 2.4. Let X be a smooth Fano 4-fold and R1, R2 distinct extremal rays of +X of type (3, 2) with dim N1(ERi, X) ≥ 4 for i = 1, 2. If there exists a birational +contraction g: X → Z with R1, R2 ⊂ NE(g), then ER1 · R2 = ER2 · R1 = 0. +Proof. We note first of all that ERi · Rj ≥ 0 for i ̸= j by Lemma 2.2. Suppose +that ER1 · R2 > 0. Then ER1 · (CR1 + CR2) = ER1 · CR2 − 1 ≥ 0. Moreover +ER2 · R1 > 0 by Lemma 2.3, so that ER2 · (CR1 + CR2) ≥ 0. On the other hand +for every prime divisor D different from ER1, ER2 we have D · (CR1 + CR2) ≥ 0, +therefore [CR1 + CR2] ∈ Eff(X)∨ = mov(X). Since [CR1 + CR2] ∈ NE(g), g should +be of fiber type, a contradiction. +■ + +6 +C. CASAGRANDE +Lemma 2.5. Let X be a smooth Fano 4-fold with δX ≤ 2, and g: X → Z a +contraction of fiber type. Then ρZ ≤ 4. +Proof. This follows from [Cas12]; for the reader’s convenience we report the proof. +If dim Z ≤ 1, then ρZ ≤ 1. If Z is a surface, take any prime divisor D ⊂ X such +that g(D) ⊊ Z, so that N1(g(D), Z) = {0} if g(D) = {pt}, and N1(g(D), Z) = +R[g(D)] if g(D) is a curve. Consider the pushforward of one-cycles g∗ : N1(X) → +N1(Z), and note that dim ker g∗ = ρX−ρZ. We have g∗(N1(D, X)) = N1(g(D), Z) +and dim N1(g(D), Z) ≤ 1, thus codim N1(D, X) ≥ ρZ − 1, and δX ≤ 2 yields +ρZ ≤ 3. +If dim Z = 3, then as in [Cas12, proof of Cor. 1.6] one shows that there exists +a prime divisor D ⊂ X such that dim N1(g(D), Z) ≤ 2, and reasoning as before +we get ρZ ≤ 4. +■ +Lemma 2.6 ([Cas17], Rem. 2.17(1)). Let X be a smooth Fano 4-fold. If X has +a divisorial elementary contraction not of type (3, 2), then ρX ≤ 5. +3. Showing that S is a del Pezzo surface +In this section we study elementary contractions of type (3, 2) of a Fano 4-fold. We +focus on the surface S which is the image of the exceptional divisor; as explained +in the Introduction, our goal is to show that under suitable assumptions, S is a +smooth del Pezzo surface. +Recall that S has isolated singularities by Th. 2.1. +Lemma 3.1. Let X be a smooth Fano 4-fold and f : X → Y an elementary +contraction of type (3, 2). Set E := Exc(f) and S := f(E), and assume that +dim N1(E, X) ≥ 4. +Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S). +Then KS′ + L is semiample. +Proof. Note that −KY is Cartier by Th. 2.1, and ample by Lemma 2.2, so that +L is nef and big on S′, and for every irreducible curve Γ ⊂ S′, we have L · Γ = 0 +if and only if Γ is µ-exceptional. +Consider the pushforward of one-cycles f∗ : N1(X) → N1(Y ). Then f∗(N1(E, X)) = +N1(S, Y ), therefore ρS′ ≥ ρS ≥ dim N1(S, Y ) ≥ 3. +Let R be a KS′-negative extremal ray of NE(S′). The contraction associated +to R can be onto a point (if S′ ∼= P2), onto a curve (so that ρS′ = 2), or the +blow-up of a smooth point (see for instance [Mat02, Th. 1-4-8]). Since ρS′ > 2, R +is generated by the class of a (−1)-curve Γ, that cannot be µ-exceptional, because +µ is minimal. Then L · Γ > 0 and (KS′ + L) · Γ = L · Γ − 1 ≥ 0. +Moreover, if γ ∈ NE(S′)KS′≥0, then (KS′ + L) · γ = KS′ · γ + L · γ ≥ 0. +By the Cone Theorem, we conclude that KS′+L is nef on S′, and also semiample +by the Base-Point-Free Theorem. +■ +Lemma 3.2. Let X be a smooth Fano 4-fold and f : X → Y an elementary +contraction of type (3, 2). Set E := Exc(f) and S := f(E), and assume that + +FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS +7 +Figure 3.1. The varieties in Lemma 3.2. +g +E +X +ER1 +T1 +Y +S +C1 +D = f(ER1) +f +Z +h(E) +h(ER1) +h +dim N1(E, X) ≥ 4. Let µ: S′ → S be the minimal resolution of singularities, and +set L := µ∗((−KY )|S). +Suppose that X has an extremal ray R1 of type (3, 2) such that: +E · R1 = 0 +and +E ∩ ER1 ̸= ∅. +Set D := f(ER1) ⊂ Y . +Then D|S = C1 + · · · + Cr where Ci are pairwise disjoint (−1)-curves contained +in Sreg, ER1 = f ∗(D), and f∗(CR1) ≡Y Ci. Moreover if C′ +i ⊂ S′ is the transform +of Ci, we have (KS′ + L) · C′ +i = 0 for every i = 1, . . . , r. +Proof. By Lemma 2.3 we have ER1·NE(f) = 0 and NE(f)+R1 is a face of NE(X), +whose associated contraction h: X → Z is birational with Exc(h) = E ∪ER1. We +have a diagram (see Fig. 3.1): +(3.4) +X +f +� +h +� +Y +g +� Z +where g is an elementary, K-negative, divisorial contraction, with Exc(g) = D +(recall that Y is is locally factorial by Th. 2.1, and Fano by Lemma 2.2). +Since ER1·NE(f) = E·R1 = 0, both h(E) and h(ER1) are surfaces in Z, and the +general fiber of h over these surfaces is one-dimensional. Moreover h(E) ∩ h(ER1) +is finite, and the connected components of E ∩ ER1 are 2-dimensional fibers of h +over these points. +Using the classification of the possible 2-dimensional fibers of h in [AW98], as +in [Cas22a, Lemma 4.15] we see that every connected component Ti of E ∩ ER1 + +8 +C. CASAGRANDE +(which is non-empty by assumption) is isomorphic to P1 ×P1 with normal bundle +O(−1, 0) ⊕ O(0, −1), for i = 1, . . . , r. Set Ci := f(Ti), so that D ∩ S = f(E ∩ +ER1) = f(∪iTi) = ∪iCi. Then Ci ∼= P1, Ci ∩ Cj = ∅ if i ̸= j, and f has fibers of +dimension one over Ci, therefore Ci ⊂ Sreg and Ci ⊂ Yreg by Th. 2.1. +Moreover g(D) = h(ER1) is a surface, namely g is of type (3, 2), and Ci is a +one-dimensional fiber of g contained in Yreg, hence KY · Ci = D · Ci = −1. We +also have ER1 = f ∗(D) and f∗(CR1) ≡Y Ci. +Since Ci ⊂ Sreg, it is a Cartier divisor in S, and we can write D|S = m1C1 + +· · · + mrCr with mi ∈ Z>0 for every i = 1, . . . , r. In S we have Ci · Cj = 0 for +i ̸= j, hence for i ∈ {1, . . . , r} we get +−1 = D · Ci = (m1C1 + · · · + mrCr) · Ci = miC2 +i +and we conclude that mi = 1 and C2 +i = −1, so that Ci is a (−1)-curve in S. +Finally −KS · Ci = −KY · Ci = 1, hence if C′ +i ⊂ S′ is the transform of Ci, we +have (KS′ + L) · C′ +i = 0. +■ +Corollary 3.5. Let X be a smooth Fano 4-fold and f : X → Y an elementary +contraction of type (3, 2). Set E := Exc(f), and assume that dim N1(E, X) ≥ 4. +Suppose that X has an extremal ray R1 of type (3, 2) such that E · R1 = 0. +Then R′ +1 := f∗(R1) is an extremal ray of Y of type (3, 2), and ER1 = f ∗(ER′ +1). +Proof. If E∩ER1 ̸= ∅, we are in the setting of Lemma 3.2; consider the elementary +contraction g: Y → Z as in (3.4). Then NE(g) = f∗(R1) = R′ +1 is an extremal ray +of Y of type (3, 2), and f ∗(ER′ +1) = ER1. +If E ∩ ER1 = ∅, then we still have a diagram as (3.4), where g is locally +isomorphic to the contraction of R1 in X, and the statement is clear. +■ +Theorem 3.6. Let X be a smooth Fano 4-fold and f : X → Y an elementary +contraction of type (3, 2). Set E := Exc(f) and S := f(E), and assume that +dim N1(E, X) ≥ 4. +Suppose that X has two extremal rays R1, R2 of type (3, 2) such that: +ER1 · R2 > 0 and E · Ri = 0, E ∩ ERi ̸= ∅ for i = 1, 2. +Then one of the following holds: +(i) S is a smooth del Pezzo surface and −KS = (−KY )|S; +(ii) ER1 · CR2 = ER2 · CR1 = 1. +Proof. We apply Lemma 3.2 to f, R1 and to f, R2. Write f(ER1)|S = C1+· · ·+Cr, +and let Γ2 be an irreducible component of f(ER2)|S, so that C1, . . . , Cr, Γ2 are +(−1)-curves contained in Sreg, and Γ2 ≡ f∗(CR2). Then +(3.7) +0 < ER1 · CR2 = f ∗(f(ER1)) · CR2 = f(ER1) · Γ2 = (C1 + · · · + Cr) · Γ2, +hence Ci · Γ2 > 0 for some i, say i = 1. +Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S). +Moreover let Γ′ +2 and C′ +1 in S′ be the transforms of Γ2 and C1 respectively; + +FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS +9 +then Γ′ +2 and C′ +1 are disjoint from the µ-exceptional locus, are (−1)-curves in +S′, (KS′ + L) · C′ +1 = (KS′ + L) · Γ′ +2 = 0, and C′ +1 · Γ′ +2 > 0. +Recall that KS′ + L is semiample by Lemma 3.1. In particular, the face (KS′ + +L)⊥ ∩ NE(S′) contains the classes of two distinct (−1)-curves which meet. This +means that the associated contraction cannot be birational, and we have two +possibilities: either KS′ + L ≡ 0, or the contraction associated to KS′ + L is onto +a curve. We show that these two cases yield respectively (i) and (ii). +Suppose first that KS′ + L ≡ 0; in particular −KS′ is nef and big, namely S′ is +a weak del Pezzo surface. +Set for simplicity F := OY (KY )|S, invertible sheaf on S, and let ωS be the +dualizing sheaf of S. We have KS′ ≡ µ∗(F), and since S′ is rational, we also have +OS′(KS′) ∼= µ∗(F). By restricting to the open subset µ−1(Sreg), we conclude that +(ωS)|Sreg ∼= F|Sreg. Now we use the following. +Lemma 3.8. Let S be a reduced and irreducible projective surface with isolated +singularities, and ωS its dualizing sheaf. If there exists an invertible sheaf F on +S such that (ωS)|Sreg ∼= F|Sreg, then S is normal and ωS ∼= F. +This should be well-known to experts, we include a proof for lack of references. +We postpone the proof of Lemma 3.8 and carry on with the proof of Th. 3.6. +By Lemma 3.8 we have that S is normal and ωS ∼= F, in particular ωS is +locally free. If y0 is a singular point of S, then by Th. 2.1 y0 is a singularity of +type 1 +3(1, 1), but this contradicts the fact that ωS is locally free. We conclude +that S is smooth, and finally that −KS = (−KY )|S is ample, so that S is a del +Pezzo surface, and we have (i). +Assume now that KS′ +L yields a contraction g: S′ → B onto a smooth curve. +Let F ⊂ S′ be a general fiber F of g, so that −KS′ · F = L · F. Since F is +not µ-exceptional, we have L · F > 0 and hence −KS′ · F > 0. Thus there is +a non-empty open subset B0 ⊆ B such that (−KS′)|g−1(B0) is g-ample, therefore +g|g−1(B0) : g−1(B0) → B0 is a conic bundle, F ∼= P1, and −KS′ · F = 2. +The curves C′ +1 and Γ′ +2 are components of the same fiber F0 of g, and −KS′ ·F0 = +2 = −KS′ · (C′ +1 + Γ′ +2). For any irreducible curve C0 contained in F0 we have +−KS′ · C0 = L · C0 ≥ 0, so that if C0 is different from C′ +1 and Γ′ +2, we must have +−KS′ · C0 = L · C0 = 0 and C0 is µ-exceptional. Thus C0 ∩ (C′ +1 ∪ Γ′ +2) = ∅, and +since F0 is connected, we conclude that F0 = C′ +1 + Γ′ +2 and F0 ⊂ g−1(B0), hence +F0 is isomorphic to a reducible conic. +This also shows that C′ +i for i > 1 are contained in different fibers of g, so that +C1 · Γ2 = Γ2 · C1 = 1 +and +Ci · Γ2 = 0 +for every i = 2, . . . , r, +and finally using (3.7) +ER1 · CR2 = (C1 + · · · + Cr) · Γ2 = 1. +Similarly we conclude that ER2 · CR1 = 1. +■ + +10 +C. CASAGRANDE +Remark 3.9. In the setting of Th. 3.6(i), we cannot conclude that Y is smooth. +A priori Y could have isolated singularities at some y0 ∈ S; by [AW98] in this +case f −1(y0) ∼= P2. +Proof of Lemma 3.8. Recall that S has isolated singularities. The surface S is +reduced, thus it satisfies condition (S1), namely +depth OS,y ≥ 1 +for every y ∈ S. +Then by [Har07, Lemma 1.3] the dualizing sheaf ωS satisfies condition (S2): +depth ωS,y ≥ 2 +for every y ∈ S, +where depth ωS,y is the depth of the stalk ωS,y as an OS,y-module. +Then, for every open subset U ⊂ S such that S ∖ U is finite, we have ωS = +j∗((ωS)|U), where j : U �→ S is the inclusion, see [Har07, Rem. 1.8]. +This is +analogous to the properties of reflexive sheaves on normal varieties, see [Har80, +Propositions 1.3 and 1.6], and can be proved using local cohomology [Gro67]. +Hence we have ωS = j∗((ωS)|Sreg), where j : Sreg �→ S is the inclusion. Since F +is locally free, we get +ωS = j∗((ωS)|Sreg) ∼= j∗(F|Sreg) = F, +in particular ωS is an invertible sheaf and for every y ∈ Y we have ωS,y ∼= OS,y +as an OS,y-module, thus depth OS,y = 2. Therefore S has property (S2), and it is +normal by Serre’s criterion. +■ +Proposition 3.10. Let X be a smooth Fano 4-fold and f : X → Y an elementary +contraction of type (3, 2). Set E := Exc(f) and S := f(E), and assume that +dim N1(E, X) ≥ 4. +Suppose that X has three distinct extremal rays R1, R2, R3 of type (3, 2) such +that: +E · Ri = 0, E ∩ ERi ̸= ∅ for i = 1, 2, 3, and ER1 · Rj > 0 for j = 2, 3. +Then S is a smooth del Pezzo surface and −KS = (−KY )|S. +Proof. We apply Th. 3.6 to f, R1, R2 and to f, R1, R3. +Let us keep the same +notation as in the proof of Th. 3.6; moreover we denote by Γ3 an irreducible +component of f(ER3)|S and Γ′ +3 ⊂ S′ its transform. We show that KS′ + L ≡ 0, +which yields the statement by the proof of Th. 3.6. +Otherwise, KS′ + L yields a contraction g: S′ → B onto a curve, and F0 = +C′ +1 + Γ′ +2 is a fiber of g. On the other hand also Γ′ +3 is contained in a fiber of g, it +is different from C′ +1 and Γ′ +2, and C′ +1 · Γ′ +3 > 0, which is impossible. +■ +Corollary 3.11. Let X be a smooth Fano 4-fold with δX ≤ 2. Suppose that X +has four distinct extremal rays R0, R1, R2, R3 of type (3, 2) such that: +ER0 · Ri = 0 for i = 1, 2, 3, and ER1 · Rj > 0 for j = 2, 3. +Then one of the following holds: +(i) dim N1(ERi, X) ≤ 3 for some i ∈ {0, 1, 2, 3}, in particular ρX ≤ 5; + +FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS +11 +(ii) dim N1(ER0, X) ≤ 10, in particular ρX ≤ 12. +Moreover if f : X → Y is the contraction of R0 and S := f(ER0), then S is +a smooth del Pezzo surface and −KS = (−KY )|S. +Proof. We assume that dim N1(ERi, X) ≥ 4 for every i = 0, 1, 2, 3, and prove (ii). +We show that ER0 ∩ ERi ̸= ∅ for every i = 1, 2, 3. +If ER0 ∩ ERi = ∅ for +some i ∈ {1, 2, 3}, then for every curve C ⊂ ER0 we have ERi · C = 0, so that +[C] ∈ (ERi)⊥, and N1(ER0, X) ⊂ (ERi)⊥. +Since the classes [ER1], [ER2], [ER3] ∈ N 1(X) generate distinct one dimensional +faces of Eff(X) (see [Cas13a, Rem. 2.19]), they are linearly independent, hence in +N1(X) we have +codim +� +(ER1)⊥ ∩ (ER2)⊥ ∩ (ER3)⊥� += 3. +On the other hand codim N1(ER0, X) ≤ δX ≤ 2, thus N1(ER0, X) cannot be +contained in the above intersection. Then N1(ER0, X) ̸⊂ (ERh)⊥ for some h ∈ +{1, 2, 3}, hence ER0 ∩ ERh ̸= ∅. In particular, since ER0 · Rh = 0, there exists an +irreducible curve C ⊂ ER0 with [C] ∈ Rh. +For j = 2, 3 we have ER1 · Rj > 0, and by Lemma 2.3 also ERj · R1 > 0. This +implies that ER0 ∩ ERi ̸= ∅ for every i = 1, 2, 3. For instance say h = 3: then +ER1 · R3 > 0 yields ER1 ∩ C ̸= ∅, hence ER0 ∩ ER1 ̸= ∅. Then there exists an +irreducible curve C′ ⊂ ER0 with [C′] ∈ R1, and ER2 ·R1 > 0 yields ER0 ∩ER2 ̸= ∅. +Finally we apply Prop. 3.10 to get that S is a smooth del Pezzo surface and +−KS = (−KY )|S. +Therefore dim N1(S, Y ) ≤ ρS ≤ 9 and dim N1(ER0, X) = +dim N1(S, X) + 1 ≤ 10, so we get (ii). +■ +4. Proof of Th. 1.1 +In this section we show how to apply the results of §3 to bound ρX; the following +is our main result. +Theorem 4.1. Let X be a smooth Fano 4-fold with δX ≤ 2 and ρX ≥ 8, and with +no small elementary contraction. +Then ρX ≤ δX + 10 ≤ 12. Moreover every elementary contraction f : X → Y +is of type (3, 2), and S := f(Exc(f)) ⊂ Y is a smooth del Pezzo surface with +−KS = (−KY )|S. +In the proof we will use the following terminology: if R1, R2 are distinct one- +dimensional faces of a convex polyhedral cone C, we say that R1 and R2 are +adjacent if R1 + R2 is a face of C. A facet of C is a face of codimension one, and +RC is the linear span of C. We will also need the following elementary fact. +Lemma 4.2 ([Ewa96], Lemma II.2.6). Let C be a convex polyhedral cone not +containing non-zero linear subspaces, and R0 a one-dimensional face of C. Let +R1, . . . , Rm be the one-dimensional faces of C that are adjacent to R0. Then the +linear span of R0, R1, . . . , Rm is RC. + +12 +C. CASAGRANDE +Proof of Th. 4.1. Let f : X → Y be an elementary contraction; note that ρY = +ρX − 1 ≥ 7. +Then f is not of fiber type by Lemma 2.5, and not small by +assumption, so that f is divisorial. Moreover f is of type (3, 2) by Lemma 2.6. +Set E := Exc(f) and S := f(E) ⊂ Y ; we have dim N1(E, X) ≥ ρX − δX ≥ 6, +and if R′ ̸= NE(f) is another extremal ray of X, we have E · R′ ≥ 0 by Lemma +2.2. Moreover, if R′ is adjacent to NE(f), then E ·R′ = 0. Indeed the contraction +g: X → Z of the face R′ + NE(f) cannot be of fiber type by Lemma 2.5, thus it +is birational and we apply Lemma 2.4. +We are going to show that there exists three extremal rays R′ +1, R′ +2, R′ +3 adjacent +to NE(f) such that ER′ +1 · R′ +j > 0 for j = 2, 3, and then apply Cor. 3.11. +Let us consider the cone NE(Y ). It is a convex polyhedral cone whose extremal +rays R are in bijection with the extremal rays R′ of X adjacent to NE(f), via +R = f∗(R′), see [Cas08, §2.5]. +By Cor. 3.5, R is still of type (3, 2), and f ∗(ER) = ER′. Thus for every pair +R1, R2 of distinct extremal rays of Y , with Ri = f∗(R′ +i) for i = 1, 2, we have +ER1 · R2 = ER′ +1 · R′ +2 ≥ 0. +If R1 and R2 are adjacent, we show that ER1·R2 = ER2·R1 = 0. Indeed consider +the contraction Y → Z of the face R1 + R2 and the composition g: X → Z, +which contracts R′ +1 and R′ +2. Again g cannot be of fiber type by Lemma 2.5, thus +it is birational and we apply Lemma 2.4 to get ER′ +1 · R′ +2 = ER′ +2 · R′ +1 = 0, thus +ER1 · R2 = ER2 · R1 = 0. +Fix an extremal ray R1 of Y . We show that there exist two distinct extremal +rays R2, R3 of Y with ER1 · Rj > 0 for j = 2, 3. +Indeed since ER1 is an effective divisor, there exists some curve C ⊂ Y with +ER1 · C > 0, hence there exists some extremal ray R2 with ER1 · R2 > 0. +By contradiction, let us assume that ER1 · R = 0 for every extremal ray R of Y +different from R1, R2. This means that the cone NE(Y ) has the extremal ray R1 +in the halfspace N1(Y )ER1<0, the extremal ray R2 in the halfspace N1(Y )ER1>0, +and all other extremal rays in the hyperplane (ER1)⊥. +Fix R ̸= R1, R2, and let τ be a facet of NE(Y ) containing R and not R1. Note +that Rτ ̸= (ER1)⊥, as ER1 and −ER1 are not nef. By Lemma 4.2 the rays adjacent +to R in τ cannot be all contained in (ER1)⊥. We conclude that R2 is adjacent to +R, therefore ER2 · R = 0, namely R ⊂ (ER2)⊥. +Summing up, we have shown that every extremal ray R ̸= R1, R2 of Y is +contained in both (ER1)⊥ and (ER2)⊥. On the other hand these rays include all +the rays adjacent to R1, so by Lemma 4.2 their linear span must be at least a +hyperplane. Therefore (ER1)⊥ = (ER2)⊥ and the classes [ER1], [ER2] ∈ N 1(Y ) are +proportional, which is impossible, because they generate distinct one dimensional +faces of the cone Eff(Y ) (see [Cas13a, Rem. 2.19]). +We conclude that there exist two distinct extremal rays R2, R3 of Y with ER1 · +Rj > 0 for j = 2, 3. + +FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS +13 +For i = 1, 2, 3 we have Ri = f∗(R′ +i) where R′ +i is an extremal ray of X adjacent to +NE(f), so that E · R′ +i = 0. Moreover for j = 2, 3 we have ER′ +1 · R′ +j = ER1 · Rj > 0. +We apply Cor. 3.11 to NE(f), R′ +1, R′ +2, R′ +3. We have already excluded (i), and +(ii) yields the statement. +■ +We can finally prove the following more detailed version of Th. 1.1. +Theorem 4.3. Let X be a smooth Fano 4-fold which is not a product of surfaces. +Then ρX ≤ 12, and if ρX = 12, then there exist X +ϕ +��� X′ +g→ Z where ϕ is a +finite sequence of flips, X′ is smooth, g is a contraction, and dim Z = 3. +Proof. Since X is not a product of surfaces, we have δX ≤ 3 by Th. 1.4. Moreover +δX = 3 yields ρX ≤ 6 by Th. 1.5, while δX ≤ 2 yields ρX ≤ 12 by Theorems 1.6 +and 4.1. +If ρX = 12, the statement follows from [Cas22a, Theorems 2.7 and 9.1]. +■ +References +[AW98] +M. Andreatta and J.A. Wi´sniewski, On contractions of smooth varieties, J. Algebraic +Geom. 7 (1998), 253–312. +[Cas08] +C. Casagrande, Quasi-elementary contractions of Fano manifolds, Compos. Math. +144 (2008), 1429–1460. +[Cas12] +, +On +the +Picard +number +of +divisors +in +Fano +manifolds, +Ann. Sci. ´Ec. Norm. Sup´er. 45 (2012), 363–403. +[Cas13a] +, On the birational geometry of Fano 4-folds, Math. Ann. 355 (2013), 585–628. +[Cas13b] +, Numerical invariants of Fano 4-folds, Math. Nachr. 286 (2013), 1107–1113. +[Cas17] +, Fano 4-folds, flips, and blow-ups of points, J. Algebra 483 (2017), 362–414. +[Cas22a] +, Fano 4-folds with a small contraction, Adv. Math. 405 (2022), 1–55, paper +no. 108492. +[Cas22b] +, The Lefschetz defect of Fano varieties, Rend. Circ. Mat. Palermo (2), pub- +lished online 19 December, 2022. +[CRS22] C. Casagrande, E.A. Romano, and S.A.Secci, Fano manifolds with Lefschetz defect 3, +J. Math. Pures Appl. 163 (2022), 625–653, Corrigendum: 168 (2022), 108–109. +[Ewa96] G. Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Math- +ematics, vol. 168, Springer-Verlag, 1996. +[Gro67] +A. Grothendieck, Local cohomology, Lecture Notes in Math., vol. 41, Springer-Verlag, +1967. +[Har80] +R. Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), 121–176. +[Har07] +, Generalized divisors and biliaison, Illinois J. Math. 51 (2007), 83–98. +[Mat02] +K. Matsuki, Introduction to the Mori program, Universitext, Springer-Verlag, 2002. +[MM86] +S. Mori and S. Mukai, Classification of Fano 3-folds with b2 ≥ 2, I, Algebraic and +Topological Theories – to the memory of Dr. Takehiko Miyata (Kinosaki, 1984), Ki- +nokuniya, Tokyo, 1986, pp. 496–545. +Universit`a di Torino, Dipartimento di Matematica, via Carlo Alberto 10, 10123 +Torino - Italy +Email address: cinzia.casagrande@unito.it + diff --git a/4dFKT4oBgHgl3EQf9S5d/content/tmp_files/load_file.txt b/4dFKT4oBgHgl3EQf9S5d/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..21a88d9798e4f0efaefbe9ac231a5bfd46d21e54 --- /dev/null +++ b/4dFKT4oBgHgl3EQf9S5d/content/tmp_files/load_file.txt @@ -0,0 +1,622 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf,len=621 +page_content='FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS OF SURFACES C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Dedicated to Lorenzo, Sabrina, and Fabrizio Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth, complex Fano 4-fold, and ρX its Picard num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that if ρX > 12, then X is a product of del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The proof relies on a careful study of divisorial elementary contractions f : X → Y such that dim f(Exc(f)) = 2, together with the author’s previous work on Fano 4-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular, given f : X → Y as above, under suitable assumptions we show that S := f(Exc(f)) is a smooth del Pezzo surface with −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Introduction Smooth, complex Fano varieties have been classically intensively studied, and have attracted a lot of attention also in the last decades, due to their role in the framework of the Minimal Model Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The Fano condition is a natural positivity condition of the tangent bundle, and it ensures a rich geometry, from both the points of view of birational geometry and of families of rational curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' It has been known since the 90’s that Fano varieties form a bounded family in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Del Pezzo surfaces are known classically, and the classification of Fano 3-folds have been in achieved in the 80’s, there are 105 families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Starting from dimension 4, there are probably too many families to get a com- plete classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' still we aim to better understand and describe the behavior and properties of these varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In this paper we focus on Fano 4-folds X with “large” Picard number ρX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' let us recall that since X is Fano, ρX is equal to the second Betti number b2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold with ρX > 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then X ∼= S1 ×S2, where Si are del Pezzo surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' To the author’s knowledge, all known examples of Fano 4-folds which are not products of surfaces have ρ ≤ 9, so that we do not know whether the condition ρ > 12 in Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We refer the reader to [Cas22b, §6] for an overview of known Fano 4-folds with ρ ≥ 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' there are few examples and it is an interesting problem to construct new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' As ρS1×S2 = ρS1 + ρS2, and del Pezzo surfaces have ρ ≤ 9, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ρX ≤ 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 14J45,14J35,14E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='11953v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='AG] 27 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Let us note that Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 and Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 generalize to dimension 4 the analogous result for Fano 3-folds, established by Mori and Mukai in the 80’s: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3 ([MM86], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 3-fold with ρX > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then X ∼= S × P1 where S is a del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular ρX ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 relies on a careful study of elementary contractions of X of type (3, 2), together with the author’s previous work on Fano 4-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' To explain this, let us introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' A contraction is a surjective morphism f : X → Y , with connected fibers, where Y is normal and projective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' f is elementary if ρX−ρY = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' As usual, an elementary contraction can be of fiber type, divisorial, or small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We say that an elementary contraction f : X → Y is of type (3, 2) if it is divisorial with dim S = 2, where E := Exc(f) and S := f(E) ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Such f can have at most finitely many 2-dimensional fibers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' outside the images of these fibers, Y and S are smooth, and f is just the blow-up of the surface S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If y0 ∈ S is the image of a two-dimensional fiber, then either Y or S are singular at y0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' these singularities have been described by Andreatta and Wi´sniewski, see Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In any case, Y has at most isolated locally factorial and terminal singularities, while S can be not normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We denote by N1(X) the real vector space of one-cycles with real coefficients, modulo numerical equivalence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we have dim N1(X) = ρX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' For any closed subset Z ⊂ X, we set N1(Z, X) := ι∗(N1(Z)) ⊂ N1(X) where ι: Z �→ X is the inclusion, so that N1(Z, X) is the subspace of N1(X) spanned by classes of curves in Z, and dim N1(Z, X) ≤ ρZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We study an elementary contraction f : X → Y of type (3, 2) under the hy- pothesis that: dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular this implies that Y is Fano too (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We would like to compare (−KY )|S to −KS, but since S may be singular, we consider the minimal resolution of singularities µ: S′ → S and set L := µ∗((−KY )|S), a nef and big divisor class on S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that KS′+L is semiample (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then our strategy is to look for curves in S′ on which KS′ + L is trivial, using other elementary contractions of X of type (3, 2) whose exceptional divisor intersects E in a suitable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Hence let us assume that X has another elementary contraction g1 of type (3, 2) whose exceptional divisor E1 intersects E, and such that E ·Γ1 = 0 for a curve Γ1 contracted by g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set D := f(E1) ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that an irreducible component C1 of D ∩ S is a (−1)-curve contained in the smooth locus Sreg, and such that −KY · C1 = 1 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If C′ 1 ⊂ S′ is the transform of C1, we have (KS′ + L) · C′ 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally let us assume that X has three elementary contractions g1, g2, g3, all of type (3, 2), satisfying the same assumptions as g1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also assume that FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 3 E1 · Γ2 > 0 and E1 · Γ3 > 0, where E1 = Exc(g1) and Γ2, Γ3 are curves contracted by g2, g3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then we show that S is a smooth del Pezzo surface with −KS = (−KY )|S (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' let us give an overview of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The previous construction yields three distinct (−1)-curves C′ 1, C′ 2, C′ 3 ⊂ S′ such that (KS′ + L) · C′ i = 0 and C′ 1 intersects both C′ 2 and C′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This shows that the contraction of S′ given by KS′ +L cannot be birational, namely KS′ +L is not big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also rule out the possibility of a contraction onto a curve, and conclude that KS′ + L ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally we show that ωS ∼= OY (KY )|S, where ωS is the dualizing sheaf of S, and conclude that S is smooth and del Pezzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We believe that these results can be useful in the study of Fano 4-folds besides their use in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' It would be interesting to generalize this technique to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let us now explain how we use these results to prove Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We define the Lefschetz defect of X as: δX := max � codim N1(D, X) | D ⊂ X a prime divisor � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This invariant, introduced in [Cas12], measures the difference between the Picard number of X and that of its prime divisors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we refer the reader to [Cas22b] for a survey on δX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Fano 4-folds with δX ≥ 3 are classified, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4 ([Cas12], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If δX ≥ 4, then X ∼= S1 × S2 where Si are del Pezzo surfaces, and δX = maxi ρSi − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5 ([CRS22], Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Smooth Fano 4-folds with δX = 3 are clas- sified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' They have 5 ≤ ρX ≤ 8, and if ρX ∈ {7, 8} then X is a product of surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore in our study of Fano 4-folds we can assume that δX ≤ 2, that is, codim N1(D, X) ≤ 2 for every prime divisor D ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' To prove that ρX ≤ 12, we look for a prime divisor D ⊂ X with dim N1(D, X) ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' To produce such a divisor, we look at contractions of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If X has an elementary contraction of fiber type, or a divisorial elementary contraction f : X → Y with dim f(Exc(f)) ≤ 1, it is not difficult to find a prime divisor D ⊂ X such that dim N1(D, X) ≤ 3, hence ρX ≤ 5 (Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The case where X has a small elementary contraction is much harder and is treated in [Cas22a], where the following result is proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 ([Cas22a], Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If X has a small elementary contraction, then ρX ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We are left with the case where every elementary contraction f : X → Y is of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In this case we show (Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1) that, if ρX ≥ 8, we can apply our previous study of elementary contractions of type (3, 2), so that if E := Exc(f) and S := f(E) ⊂ Y , then S is a smooth del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This implies that dim N1(S, Y ) ≤ ρS ≤ 9, dim N1(E, X) = dim N1(S, Y ) + 1 ≤ 10, and finally that ρX ≤ 12, proving Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE The structure of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In §2 we gather some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then in §3 we develop our study of elementary contractions of type (3, 2), while in §4 we prove Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Notation We work over the field of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We denote by N1(X) (respectively, N 1(X)) the real vector space of one-cycles (respectively, Cartier divisors) with real coefficients, modulo numerical equiva- lence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' dim N1(X) = dim N 1(X) = ρX is the Picard number of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let C be a one-cycle of X, and D a Cartier divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We denote by [C] (respec- tively, [D]) the numerical equivalence class in N1(X) (respectively, N 1(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also denote by D⊥ ⊂ N1(X) the orthogonal hyperplane to the class [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The symbol ≡ stands for numerical equivalence (for both one-cycles and divi- sors), and ∼ stands for linear equivalence of divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' NE(X) ⊂ N1(X) is the convex cone generated by classes of effective curves, and NE(X) is its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' An extremal ray R is a one-dimensional face of NE(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If D is a Cartier divisor in X, we write D·R > 0, D·R = 0, and so on, if D·γ > 0, D · γ = 0, and so on, for a non-zero class γ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We say that R is K-negative if KX · R < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has terminal and locally factorial singularities, and is Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then NE(X) is a convex polyhedral cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Given a contraction f : X → Y , we denote by NE(f) the convex subcone of NE(X) generated by classes of curves contracted by f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we recall that there is a bijection between contractions of X and faces of NE(X), given by f �→ NE(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover dim NE(f) = ρX − ρY , in particular f is elementary if and only if NE(f) is an extremal ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' When dim X = 4, we say that an extremal ray R is of type (3, 2) if the as- sociated elementary contraction f is of type (3, 2), namely if f is divisorial with dim f(Exc(f)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also set ER := Exc(f) and denote by CR ⊂ ER a general fiber of f|ER;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' note that ER · CR = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We will also consider the cones Eff(X) ⊂ N 1(X) of classes of effective divisors, and mov(X) ⊂ N1(X) of classes of curves moving in a family covering X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since X is Fano, both cones are polyhedral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we have the duality relation Eff(X) = mov(X)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Preliminaries In this section we gather some preliminary results that will be used in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Andreatta and Wi´sniewski have classified the possible 2-dimensional fibers of an elementary contraction of type (3, 2) of a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In doing this, they also describe precisely the singularities both of the target, and of the image of the exceptional divisor, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 ([AW98], Theorem on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 256).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set S := f(Exc(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 5 Then f can have at most finitely many 2-dimensional fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Outside the images of these fibers, Y and S are smooth, and f is the blow-up of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let y0 ∈ S ⊂ Y be the image of a 2-dimensional fiber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' then one of the following holds: (i) S is smooth at y0, while Y has an ordinary double point at y0, locally factorial and terminal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' (ii) Y is smooth at y0, while S is singular at y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' More precisely either S is not normal at y0, or it has a singularity of type 1 3(1, 1) at y0 (as the cone over a twisted cubic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular the singularities of Y are at most isolated, locally factorial, and terminal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Now we give some simple preliminary results on extremal rays of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' set E := Exc(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If dim N1(E, X) ≥ 4, then E · R ≥ 0 for every extremal ray R of X different from NE(f), and Y is Fano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' It follows from [Cas17, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='16 and Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='17] that NE(f) is the unique extremal ray of X having negative intersection with E, −KX + E = f ∗(−KY ) is nef, and (−KX + E)⊥ ∩ NE(X) = NE(f), so that −KY is ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and R1, R2 extremal rays of X of type (3, 2) such that dim N1(ER1, X) ≥ 4 and ER1 · R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ER2 ·R1 = 0 and R1+R2 is a face of NE(X) whose associated contraction is birational, with exceptional locus ER1 ∪ ER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let H be a nef divisor on X such that H⊥ ∩ NE(X) = R2, and set H′ := H + (H · CR1)ER1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then H′ · CR1 = H′ · CR2 = 0, and if R3 is an extremal ray of NE(X) different from R1 and R2, we have ER1 · R3 ≥ 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2, hence H′·R3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore H′ is nef and (H′)⊥∩NE(X) = R1+R2 is a face of NE(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If Γ ⊂ X is an irreducible curve with [Γ] ∈ R1 + R2, then H′ · Γ = 0, so that either ER1 · Γ < 0 and Γ ⊂ ER1, or H · Γ = 0, [Γ] ∈ R2 and Γ ⊂ ER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This shows that the contraction of R1 + R2 is birational with exceptional locus ER1 ∪ ER2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally we have ER2 · R1 = 0 by [Cas13b, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2(b) and its proof].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and R1, R2 distinct extremal rays of X of type (3, 2) with dim N1(ERi, X) ≥ 4 for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If there exists a birational contraction g: X → Z with R1, R2 ⊂ NE(g), then ER1 · R2 = ER2 · R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We note first of all that ERi · Rj ≥ 0 for i ̸= j by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that ER1 · R2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ER1 · (CR1 + CR2) = ER1 · CR2 − 1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover ER2 · R1 > 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3, so that ER2 · (CR1 + CR2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' On the other hand for every prime divisor D different from ER1, ER2 we have D · (CR1 + CR2) ≥ 0, therefore [CR1 + CR2] ∈ Eff(X)∨ = mov(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since [CR1 + CR2] ∈ NE(g), g should be of fiber type, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold with δX ≤ 2, and g: X → Z a contraction of fiber type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ρZ ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This follows from [Cas12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' for the reader’s convenience we report the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If dim Z ≤ 1, then ρZ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If Z is a surface, take any prime divisor D ⊂ X such that g(D) ⊊ Z, so that N1(g(D), Z) = {0} if g(D) = {pt}, and N1(g(D), Z) = R[g(D)] if g(D) is a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Consider the pushforward of one-cycles g∗ : N1(X) → N1(Z), and note that dim ker g∗ = ρX−ρZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We have g∗(N1(D, X)) = N1(g(D), Z) and dim N1(g(D), Z) ≤ 1, thus codim N1(D, X) ≥ ρZ − 1, and δX ≤ 2 yields ρZ ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If dim Z = 3, then as in [Cas12, proof of Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6] one shows that there exists a prime divisor D ⊂ X such that dim N1(g(D), Z) ≤ 2, and reasoning as before we get ρZ ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 ([Cas17], Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='17(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If X has a divisorial elementary contraction not of type (3, 2), then ρX ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Showing that S is a del Pezzo surface In this section we study elementary contractions of type (3, 2) of a Fano 4-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We focus on the surface S which is the image of the exceptional divisor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' as explained in the Introduction, our goal is to show that under suitable assumptions, S is a smooth del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Recall that S has isolated singularities by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E), and assume that dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then KS′ + L is semiample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Note that −KY is Cartier by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1, and ample by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2, so that L is nef and big on S′, and for every irreducible curve Γ ⊂ S′, we have L · Γ = 0 if and only if Γ is µ-exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Consider the pushforward of one-cycles f∗ : N1(X) → N1(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then f∗(N1(E, X)) = N1(S, Y ), therefore ρS′ ≥ ρS ≥ dim N1(S, Y ) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let R be a KS′-negative extremal ray of NE(S′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The contraction associated to R can be onto a point (if S′ ∼= P2), onto a curve (so that ρS′ = 2), or the blow-up of a smooth point (see for instance [Mat02, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1-4-8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since ρS′ > 2, R is generated by the class of a (−1)-curve Γ, that cannot be µ-exceptional, because µ is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then L · Γ > 0 and (KS′ + L) · Γ = L · Γ − 1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover, if γ ∈ NE(S′)KS′≥0, then (KS′ + L) · γ = KS′ · γ + L · γ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By the Cone Theorem, we conclude that KS′+L is nef on S′, and also semiample by the Base-Point-Free Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E), and assume that FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The varieties in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' g E X ER1 T1 Y S C1 D = f(ER1) f Z h(E) h(ER1) h dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has an extremal ray R1 of type (3, 2) such that: E · R1 = 0 and E ∩ ER1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set D := f(ER1) ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then D|S = C1 + · · · + Cr where Ci are pairwise disjoint (−1)-curves contained in Sreg, ER1 = f ∗(D), and f∗(CR1) ≡Y Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover if C′ i ⊂ S′ is the transform of Ci, we have (KS′ + L) · C′ i = 0 for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3 we have ER1·NE(f) = 0 and NE(f)+R1 is a face of NE(X), whose associated contraction h: X → Z is birational with Exc(h) = E ∪ER1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We have a diagram (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4) X f � h � Y g � Z where g is an elementary, K-negative, divisorial contraction, with Exc(g) = D (recall that Y is is locally factorial by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1, and Fano by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since ER1·NE(f) = E·R1 = 0, both h(E) and h(ER1) are surfaces in Z, and the general fiber of h over these surfaces is one-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover h(E) ∩ h(ER1) is finite, and the connected components of E ∩ ER1 are 2-dimensional fibers of h over these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Using the classification of the possible 2-dimensional fibers of h in [AW98], as in [Cas22a, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='15] we see that every connected component Ti of E ∩ ER1 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE (which is non-empty by assumption) is isomorphic to P1 ×P1 with normal bundle O(−1, 0) ⊕ O(0, −1), for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set Ci := f(Ti), so that D ∩ S = f(E ∩ ER1) = f(∪iTi) = ∪iCi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then Ci ∼= P1, Ci ∩ Cj = ∅ if i ̸= j, and f has fibers of dimension one over Ci, therefore Ci ⊂ Sreg and Ci ⊂ Yreg by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover g(D) = h(ER1) is a surface, namely g is of type (3, 2), and Ci is a one-dimensional fiber of g contained in Yreg, hence KY · Ci = D · Ci = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We also have ER1 = f ∗(D) and f∗(CR1) ≡Y Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since Ci ⊂ Sreg, it is a Cartier divisor in S, and we can write D|S = m1C1 + · · + mrCr with mi ∈ Z>0 for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In S we have Ci · Cj = 0 for i ̸= j, hence for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r} we get −1 = D · Ci = (m1C1 + · · · + mrCr) · Ci = miC2 i and we conclude that mi = 1 and C2 i = −1, so that Ci is a (−1)-curve in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally −KS · Ci = −KY · Ci = 1, hence if C′ i ⊂ S′ is the transform of Ci, we have (KS′ + L) · C′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f), and assume that dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has an extremal ray R1 of type (3, 2) such that E · R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then R′ 1 := f∗(R1) is an extremal ray of Y of type (3, 2), and ER1 = f ∗(ER′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If E∩ER1 ̸= ∅, we are in the setting of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' consider the elementary contraction g: Y → Z as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then NE(g) = f∗(R1) = R′ 1 is an extremal ray of Y of type (3, 2), and f ∗(ER′ 1) = ER1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If E ∩ ER1 = ∅, then we still have a diagram as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4), where g is locally isomorphic to the contraction of R1 in X, and the statement is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E), and assume that dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has two extremal rays R1, R2 of type (3, 2) such that: ER1 · R2 > 0 and E · Ri = 0, E ∩ ERi ̸= ∅ for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then one of the following holds: (i) S is a smooth del Pezzo surface and −KS = (−KY )|S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' (ii) ER1 · CR2 = ER2 · CR1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 to f, R1 and to f, R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Write f(ER1)|S = C1+· · ·+Cr, and let Γ2 be an irreducible component of f(ER2)|S, so that C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , Cr, Γ2 are (−1)-curves contained in Sreg, and Γ2 ≡ f∗(CR2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='7) 0 < ER1 · CR2 = f ∗(f(ER1)) · CR2 = f(ER1) · Γ2 = (C1 + · · · + Cr) · Γ2, hence Ci · Γ2 > 0 for some i, say i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let µ: S′ → S be the minimal resolution of singularities, and set L := µ∗((−KY )|S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover let Γ′ 2 and C′ 1 in S′ be the transforms of Γ2 and C1 respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 9 then Γ′ 2 and C′ 1 are disjoint from the µ-exceptional locus, are (−1)-curves in S′, (KS′ + L) · C′ 1 = (KS′ + L) · Γ′ 2 = 0, and C′ 1 · Γ′ 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Recall that KS′ + L is semiample by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular, the face (KS′ + L)⊥ ∩ NE(S′) contains the classes of two distinct (−1)-curves which meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This means that the associated contraction cannot be birational, and we have two possibilities: either KS′ + L ≡ 0, or the contraction associated to KS′ + L is onto a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that these two cases yield respectively (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose first that KS′ + L ≡ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' in particular −KS′ is nef and big, namely S′ is a weak del Pezzo surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set for simplicity F := OY (KY )|S, invertible sheaf on S, and let ωS be the dualizing sheaf of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We have KS′ ≡ µ∗(F), and since S′ is rational, we also have OS′(KS′) ∼= µ∗(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By restricting to the open subset µ−1(Sreg), we conclude that (ωS)|Sreg ∼= F|Sreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Now we use the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let S be a reduced and irreducible projective surface with isolated singularities, and ωS its dualizing sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If there exists an invertible sheaf F on S such that (ωS)|Sreg ∼= F|Sreg, then S is normal and ωS ∼= F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This should be well-known to experts, we include a proof for lack of references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We postpone the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8 and carry on with the proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8 we have that S is normal and ωS ∼= F, in particular ωS is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If y0 is a singular point of S, then by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 y0 is a singularity of type 1 3(1, 1), but this contradicts the fact that ωS is locally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We conclude that S is smooth, and finally that −KS = (−KY )|S is ample, so that S is a del Pezzo surface, and we have (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Assume now that KS′ +L yields a contraction g: S′ → B onto a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let F ⊂ S′ be a general fiber F of g, so that −KS′ · F = L · F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since F is not µ-exceptional, we have L · F > 0 and hence −KS′ · F > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Thus there is a non-empty open subset B0 ⊆ B such that (−KS′)|g−1(B0) is g-ample, therefore g|g−1(B0) : g−1(B0) → B0 is a conic bundle, F ∼= P1, and −KS′ · F = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The curves C′ 1 and Γ′ 2 are components of the same fiber F0 of g, and −KS′ ·F0 = 2 = −KS′ · (C′ 1 + Γ′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' For any irreducible curve C0 contained in F0 we have −KS′ · C0 = L · C0 ≥ 0, so that if C0 is different from C′ 1 and Γ′ 2, we must have −KS′ · C0 = L · C0 = 0 and C0 is µ-exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Thus C0 ∩ (C′ 1 ∪ Γ′ 2) = ∅, and since F0 is connected, we conclude that F0 = C′ 1 + Γ′ 2 and F0 ⊂ g−1(B0), hence F0 is isomorphic to a reducible conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This also shows that C′ i for i > 1 are contained in different fibers of g, so that C1 · Γ2 = Γ2 · C1 = 1 and Ci · Γ2 = 0 for every i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , r, and finally using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='7) ER1 · CR2 = (C1 + · · · + Cr) · Γ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Similarly we conclude that ER2 · CR1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In the setting of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6(i), we cannot conclude that Y is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' A priori Y could have isolated singularities at some y0 ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' by [AW98] in this case f −1(y0) ∼= P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Recall that S has isolated singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' The surface S is reduced, thus it satisfies condition (S1), namely depth OS,y ≥ 1 for every y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then by [Har07, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3] the dualizing sheaf ωS satisfies condition (S2): depth ωS,y ≥ 2 for every y ∈ S, where depth ωS,y is the depth of the stalk ωS,y as an OS,y-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then, for every open subset U ⊂ S such that S ∖ U is finite, we have ωS = j∗((ωS)|U), where j : U �→ S is the inclusion, see [Har07, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This is analogous to the properties of reflexive sheaves on normal varieties, see [Har80, Propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6], and can be proved using local cohomology [Gro67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Hence we have ωS = j∗((ωS)|Sreg), where j : Sreg �→ S is the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since F is locally free, we get ωS = j∗((ωS)|Sreg) ∼= j∗(F|Sreg) = F, in particular ωS is an invertible sheaf and for every y ∈ Y we have ωS,y ∼= OS,y as an OS,y-module, thus depth OS,y = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore S has property (S2), and it is normal by Serre’s criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold and f : X → Y an elementary contraction of type (3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E), and assume that dim N1(E, X) ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has three distinct extremal rays R1, R2, R3 of type (3, 2) such that: E · Ri = 0, E ∩ ERi ̸= ∅ for i = 1, 2, 3, and ER1 · Rj > 0 for j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then S is a smooth del Pezzo surface and −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We apply Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 to f, R1, R2 and to f, R1, R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let us keep the same notation as in the proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' moreover we denote by Γ3 an irreducible component of f(ER3)|S and Γ′ 3 ⊂ S′ its transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that KS′ + L ≡ 0, which yields the statement by the proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Otherwise, KS′ + L yields a contraction g: S′ → B onto a curve, and F0 = C′ 1 + Γ′ 2 is a fiber of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' On the other hand also Γ′ 3 is contained in a fiber of g, it is different from C′ 1 and Γ′ 2, and C′ 1 · Γ′ 3 > 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold with δX ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Suppose that X has four distinct extremal rays R0, R1, R2, R3 of type (3, 2) such that: ER0 · Ri = 0 for i = 1, 2, 3, and ER1 · Rj > 0 for j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then one of the following holds: (i) dim N1(ERi, X) ≤ 3 for some i ∈ {0, 1, 2, 3}, in particular ρX ≤ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 11 (ii) dim N1(ER0, X) ≤ 10, in particular ρX ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover if f : X → Y is the contraction of R0 and S := f(ER0), then S is a smooth del Pezzo surface and −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We assume that dim N1(ERi, X) ≥ 4 for every i = 0, 1, 2, 3, and prove (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that ER0 ∩ ERi ̸= ∅ for every i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If ER0 ∩ ERi = ∅ for some i ∈ {1, 2, 3}, then for every curve C ⊂ ER0 we have ERi · C = 0, so that [C] ∈ (ERi)⊥, and N1(ER0, X) ⊂ (ERi)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since the classes [ER1], [ER2], [ER3] ∈ N 1(X) generate distinct one dimensional faces of Eff(X) (see [Cas13a, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='19]), they are linearly independent, hence in N1(X) we have codim � (ER1)⊥ ∩ (ER2)⊥ ∩ (ER3)⊥� = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' On the other hand codim N1(ER0, X) ≤ δX ≤ 2, thus N1(ER0, X) cannot be contained in the above intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then N1(ER0, X) ̸⊂ (ERh)⊥ for some h ∈ {1, 2, 3}, hence ER0 ∩ ERh ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In particular, since ER0 · Rh = 0, there exists an irreducible curve C ⊂ ER0 with [C] ∈ Rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' For j = 2, 3 we have ER1 · Rj > 0, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3 also ERj · R1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This implies that ER0 ∩ ERi ̸= ∅ for every i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' For instance say h = 3: then ER1 · R3 > 0 yields ER1 ∩ C ̸= ∅, hence ER0 ∩ ER1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then there exists an irreducible curve C′ ⊂ ER0 with [C′] ∈ R1, and ER2 ·R1 > 0 yields ER0 ∩ER2 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Finally we apply Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='10 to get that S is a smooth del Pezzo surface and −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore dim N1(S, Y ) ≤ ρS ≤ 9 and dim N1(ER0, X) = dim N1(S, X) + 1 ≤ 10, so we get (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1 In this section we show how to apply the results of §3 to bound ρX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' the following is our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold with δX ≤ 2 and ρX ≥ 8, and with no small elementary contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ρX ≤ δX + 10 ≤ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover every elementary contraction f : X → Y is of type (3, 2), and S := f(Exc(f)) ⊂ Y is a smooth del Pezzo surface with −KS = (−KY )|S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' In the proof we will use the following terminology: if R1, R2 are distinct one- dimensional faces of a convex polyhedral cone C, we say that R1 and R2 are adjacent if R1 + R2 is a face of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' A facet of C is a face of codimension one, and RC is the linear span of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We will also need the following elementary fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 ([Ewa96], Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let C be a convex polyhedral cone not containing non-zero linear subspaces, and R0 a one-dimensional face of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , Rm be the one-dimensional faces of C that are adjacent to R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then the linear span of R0, R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' , Rm is RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' CASAGRANDE Proof of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let f : X → Y be an elementary contraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' note that ρY = ρX − 1 ≥ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then f is not of fiber type by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, and not small by assumption, so that f is divisorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover f is of type (3, 2) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Set E := Exc(f) and S := f(E) ⊂ Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' we have dim N1(E, X) ≥ ρX − δX ≥ 6, and if R′ ̸= NE(f) is another extremal ray of X, we have E · R′ ≥ 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover, if R′ is adjacent to NE(f), then E ·R′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Indeed the contraction g: X → Z of the face R′ + NE(f) cannot be of fiber type by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, thus it is birational and we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We are going to show that there exists three extremal rays R′ 1, R′ 2, R′ 3 adjacent to NE(f) such that ER′ 1 · R′ j > 0 for j = 2, 3, and then apply Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let us consider the cone NE(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' It is a convex polyhedral cone whose extremal rays R are in bijection with the extremal rays R′ of X adjacent to NE(f), via R = f∗(R′), see [Cas08, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, R is still of type (3, 2), and f ∗(ER) = ER′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Thus for every pair R1, R2 of distinct extremal rays of Y , with Ri = f∗(R′ i) for i = 1, 2, we have ER1 · R2 = ER′ 1 · R′ 2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If R1 and R2 are adjacent, we show that ER1·R2 = ER2·R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Indeed consider the contraction Y → Z of the face R1 + R2 and the composition g: X → Z, which contracts R′ 1 and R′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Again g cannot be of fiber type by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, thus it is birational and we apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4 to get ER′ 1 · R′ 2 = ER′ 2 · R′ 1 = 0, thus ER1 · R2 = ER2 · R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Fix an extremal ray R1 of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We show that there exist two distinct extremal rays R2, R3 of Y with ER1 · Rj > 0 for j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Indeed since ER1 is an effective divisor, there exists some curve C ⊂ Y with ER1 · C > 0, hence there exists some extremal ray R2 with ER1 · R2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By contradiction, let us assume that ER1 · R = 0 for every extremal ray R of Y different from R1, R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' This means that the cone NE(Y ) has the extremal ray R1 in the halfspace N1(Y )ER1<0, the extremal ray R2 in the halfspace N1(Y )ER1>0, and all other extremal rays in the hyperplane (ER1)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Fix R ̸= R1, R2, and let τ be a facet of NE(Y ) containing R and not R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Note that Rτ ̸= (ER1)⊥, as ER1 and −ER1 are not nef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 the rays adjacent to R in τ cannot be all contained in (ER1)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We conclude that R2 is adjacent to R, therefore ER2 · R = 0, namely R ⊂ (ER2)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Summing up, we have shown that every extremal ray R ̸= R1, R2 of Y is contained in both (ER1)⊥ and (ER2)⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' On the other hand these rays include all the rays adjacent to R1, so by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='2 their linear span must be at least a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Therefore (ER1)⊥ = (ER2)⊥ and the classes [ER1], [ER2] ∈ N 1(Y ) are proportional, which is impossible, because they generate distinct one dimensional faces of the cone Eff(Y ) (see [Cas13a, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We conclude that there exist two distinct extremal rays R2, R3 of Y with ER1 · Rj > 0 for j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' FANO 4-FOLDS WITH b2 > 12 ARE PRODUCTS 13 For i = 1, 2, 3 we have Ri = f∗(R′ i) where R′ i is an extremal ray of X adjacent to NE(f), so that E · R′ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover for j = 2, 3 we have ER′ 1 · R′ j = ER1 · Rj > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We apply Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='11 to NE(f), R′ 1, R′ 2, R′ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' We have already excluded (i), and (ii) yields the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ We can finally prove the following more detailed version of Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Let X be a smooth Fano 4-fold which is not a product of surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Then ρX ≤ 12, and if ρX = 12, then there exist X ϕ ��� X′ g→ Z where ϕ is a finite sequence of flips, X′ is smooth, g is a contraction, and dim Z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Since X is not a product of surfaces, we have δX ≤ 3 by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Moreover δX = 3 yields ρX ≤ 6 by Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='5, while δX ≤ 2 yields ρX ≤ 12 by Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' If ρX = 12, the statement follows from [Cas22a, Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='7 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ■ References [AW98] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Andreatta and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Wi´sniewski, On contractions of smooth varieties, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Algebraic Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 7 (1998), 253–312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas08] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Casagrande, Quasi-elementary contractions of Fano manifolds, Compos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 144 (2008), 1429–1460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas12] , On the Picard number of divisors in Fano manifolds, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' ´Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Sup´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 45 (2012), 363–403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas13a] , On the birational geometry of Fano 4-folds, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 355 (2013), 585–628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas13b] , Numerical invariants of Fano 4-folds, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 286 (2013), 1107–1113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas17] , Fano 4-folds, flips, and blow-ups of points, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Algebra 483 (2017), 362–414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas22a] , Fano 4-folds with a small contraction, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 405 (2022), 1–55, paper no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 108492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Cas22b] , The Lefschetz defect of Fano varieties, Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Circ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Palermo (2), pub- lished online 19 December, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [CRS22] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Casagrande, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Romano, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='Secci, Fano manifolds with Lefschetz defect 3, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Pures Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 163 (2022), 625–653, Corrigendum: 168 (2022), 108–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' [Ewa96] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Ewald, Combinatorial convexity and algebraic geometry, Graduate Texts in Math- ematics, vol.' metadata={'source': 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+page_content=' Takehiko Miyata (Kinosaki, 1984), Ki- nokuniya, Tokyo, 1986, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' 496–545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content=' Universit`a di Torino, Dipartimento di Matematica, via Carlo Alberto 10, 10123 Torino - Italy Email address: cinzia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='casagrande@unito.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} +page_content='it' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dFKT4oBgHgl3EQf9S5d/content/2301.11953v1.pdf'} diff --git a/4tAzT4oBgHgl3EQfEPpQ/content/tmp_files/2301.00989v1.pdf.txt b/4tAzT4oBgHgl3EQfEPpQ/content/tmp_files/2301.00989v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf119a5f214c0e7ef2ee7568ba7972a8f7a4a20a --- /dev/null +++ b/4tAzT4oBgHgl3EQfEPpQ/content/tmp_files/2301.00989v1.pdf.txt @@ -0,0 +1,763 @@ +arXiv:2301.00989v1 [cs.CV] 3 Jan 2023 +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +1 +A New Perspective to Boost Vision +Transformer for Medical Image Classification +Yuexiang Li +vicyxli@tencent.com +Yawen Huang +yawenhuang@tencent.com +Nanjun He +nanjunhe@tencent.com +Kai Ma +kylekma@tencent.com +Yefeng Zheng +yefengzheng@tencent.com +Tencent Jarvis Lab +Shenzhen +China +Abstract +Transformer has achieved impressive successes for various computer vision tasks. +However, most of existing studies require to pretrain the Transformer backbone on a +large-scale labeled dataset (e.g., ImageNet) for achieving satisfactory performance, which +is usually unavailable for medical images. Additionally, due to the gap between medical +and natural images, the improvement generated by the ImageNet pretrained weights sig- +nificantly degrades while transferring the weights to medical image processing tasks. In +this paper, we propose Bootstrap Own Latent of Transformer (BOLT), a self-supervised +learning approach specifically for medical image classification with the Transformer +backbone. Our BOLT consists of two networks, namely online and target branches, +for self-supervised representation learning. Concretely, the online network is trained to +predict the target network representation of the same patch embedding tokens with a dif- +ferent perturbation. To maximally excavate the impact of Transformer from limited med- +ical data, we propose an auxiliary difficulty ranking task. The Transformer is enforced +to identify which branch (i.e., online/target) is processing the more difficult perturbed +tokens. Overall, the Transformer endeavours itself to distill the transformation-invariant +features from the perturbed tokens to simultaneously achieve difficulty measurement and +maintain the consistency of self-supervised representations. The proposed BOLT is eval- +uated on three medical image processing tasks, i.e., skin lesion classification, knee fatigue +fracture grading and diabetic retinopathy grading. The experimental results validate the +superiority of our BOLT for medical image classification, compared to ImageNet pre- +trained weights and state-of-the-art self-supervised learning approaches. +1 +Introduction +Recently, vision Transformer (ViT) [10] and its variants [23, 32, 36] has been introduced +for various computer vision tasks (e.g., image classification [10, 18], object detection [9, +© 2022. The copyright of this document resides with its authors. +It may be distributed unchanged freely in print or electronic forms. + +2 +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +41], semantic segmentation [34, 39] and medical image processing [11, 15, 16, 31, 38]) +and gained increasing attentions from the community. The common ViT usually requires +pretrainig on large-scale natural image datasets, e.g., ImageNet, to achieve the satisfactory +performance. For natural images, the labels for pretraining dataset can be efficiently obtained +by crowdsourcing, as even ordinary people possess the ability to effectively identify and label +objects in natural images. However, the same strategy cannot be adopted for medical images, +as professional expertise is mandatory for high-quality medical image annotations. Hence, +the limited amount of annotated medical data is the major obstacle for the improvement of +diagnosis accuracy even with the powerful vision Transformer. +Self-supervised learning (SSL) approach is a potential solution to tackle the challenge of +insufficient annotated data. The typical self-supervised learning formulates a proxy task to +extract representative features from unlabeled data, which can boost the accuracy of subse- +quent target task. Existing studies have proposed various proxy tasks, including grayscale +image colorization [19], patch re-ordering [25], and context restoration [27]. The SSL was +firstly brought to medical image processing by Zhang et al. [37]. Concretely, the neural +network was pretrained with a proxy task that sorted the 2D slices from the conventional +3D medical volumes for the subsequent fine-grained body part recognition. Zhu et al. [40] +enforced 3D networks to play a Rubik’s cube game for pretraining, which can be seen as an +extension of 2D Jigsaw puzzles [24]. Contrastive learning [13] has been recently popular- +ized for self-supervised representation learning. These approaches enforce neural networks +to spontaneously exploit useful information from pairs of positive and negative samples, +instead of permuting the contextual information of images for self-supervised signal for- +mulation. He et al. [14] firstly introduced the idea of contrastive learning into the area +of self-supervised learning. They proposed an approach, namely MoCo, which addressed +the problem of large number of negative samples for contrastive learning by maintaining a +memory bank of negative samples. Following the direction, various contrastive-learning- +based self-supervised approaches have been proposed [4, 6, 7, 12, 26, 33, 35]. Inspired by +the success of self-supervised learning for CNNs, researchers began to make their efforts to +ViT. Atito et al. [1] directly utilized the existing SSL approaches, including rotation pre- +diction, contrastive learning and image restoration, to pretrain vision Transformers. Several +studies [2, 3] have been proposed along this direction. However, taking the architecture dif- +ference between CNN and ViT into account, i.e., CNN takes the whole image as input, while +the input of ViT is the embedding tokens of image tiles, the self-supervised learning approach +specifically for ViT is worthwhile to develop. +In the recent study, Chen et al. [7] proposed MoCo V3 as a token-based constrastive +learning approach, specifically for ViT to extract self-supervised features from raw data. +The network pretrained with MoCo V3 outperformed the ImageNet-pretrained one, which +demonstrated the effectiveness of token-based self-supervised learning. In this paper, we +follow the direction and propose a token-wise perturbation based self-supervised learning +framework specifically for medical image classification with vision Transformer, namely +Bootstrap Own Latent of Transformer (BOLT). Similar to the existing Bootstrap Your Own +Latent (BYOL) [12], our BOLT consists of two networks, namely online and target branches, +for self-supervised representation learning. Instead of image-wise transformation adopted by +BYOL, the online network of our BOLT is trained to predict the target network representa- +tion of the same patch embedding tokens with a different perturbation. Moreover, to encour- +age the vision Transformer to deeply exploit useful information from limited medical data, +we propose an auxiliary difficulty ranking task. The difference between the original patch +embedding tokens and the perturbed ones is measured as the difficulty (i.e., the larger dif- + +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +3 +Linear Projection of Flattened Patches +Permutation +Linear Projection +Split +Sliding Window +Token Permutation Module +x +Vision Transformer �� +Vision Transformer �� +Patch +Embedding +Token +Perturbation +Module +Token +Perturbation +Module +�� +�� +�� +�� +��(��) +�� (��) +Difficulty-awareness Loss +Exponential Moving Average +�� +�� +Similarity Loss +Online +Target +Embedded Token �� +Content perturbed Token �� +Long Token �� +Figure 1: The architecture of our BOLT framework. Compared to the original BYOL, our +BOLT consists of two main revisions: 1) The proposed BOLT generates two views of em- +bedding tokens for self-supervised learning; 2) A novel difficulty-awareness loss is proposed +to encourage the ViT to deeply exploit useful information from raw data. sg(.) means stop- +gradient. +ference means more difficult for the vision Transformer to process), which is then adopted +as the supervision signal. In other words, the vision Transformer is required to identify +which branch (online/target) is processing the more difficult perturbed tokens. Under the +co-supervision of the two tasks, the vision Transformer is encouraged to endeavour itself to +distill the transformation-invariantfeatures from the perturbed tokens, which should be capa- +ble for simultaneous difficulty measurement and maintain the consistency of self-supervised +representations. In summary, the main contributions of our work can be concluded into +four-fold: +• A token perturbation based self-supervised learning approach, namely BOLT, specif- +ically designed for vision Transformer is proposed. A token perturbation module is +integrated to the existing BYOL framework for the more effective ViT pretraining. +• An auxiliary self-supervised task, i.e., difficulty ranking, is proposed to encourage +ViTs to deeply exploit useful information from limited medical data. The self-supervised +signal of this auxiliary task also derives from the perturbed tokens generated by our +perturbation module. To our best knowledge, this is the first SSL framework based on +the difficulty-awareness paradigm. +• The proposed BOLT is evaluated on three medical image processing tasks, i.e., skin +lesion classification, knee fatigue fracture grading and diabetic retinopathy grading. +The experimental results demonstrate the superiority of our BOLT, compared to the +widely-used ImageNet pretrained weights. +• Last but not least, we pretrain the ViT using different self-supervised learning ap- +proaches on a large-scale private fundus image dataset captured from a collaborating +hospital for diabetic retinopathy grading task. The dataset consists of 350,000 fundus +images of normal cohort and patients with various diseases, which may be the largest +fundus image dataset in the worldwide. The pretraining on our private large-scale +dataset is verified to benefit the related downstream target task. To advance the de- +velopment of automated fundus image processing, we will release the ViT pretrained +models to the community. + +4 +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +Linear Projection of Flattened Patches +1 +2 +3 +4 +5 +6 +7 +8 +9 +6 +1 +5 +7 +3 +9 +8 +2 +4 +Permutation +6 1 5 +7 3 9 +8 2 4 +Linear Projection +1 +2 +3 +Split +6 +1 +5 +7 +3 +9 +8 +2 +4 +Sliding Window +Token Permutation Module +Vision Transformer �� +Vision Transformer �� +Patch +Embedding +Token +Permutation +Module +Token +Permutation +Module +�� +�� +�� +�� +�� �� +�� �� +Difficulty awareness Loss +Exponential Moving Average +�� +�� +Similarity Loss +Online +Target +Embedded Token �� +Content-perturbed Token �� +Long Token �� +Figure 2: The architecture of the proposed token perturbation module. The module consists +of three operations (i.e., permutation, linear projection and split) to perturb the order and +content of embedded tokens. Note that nine embedding tokens in this figure are taken as an +example. The exact number (N) of embedding tokens is decided by HW +P2 , where H and W are +the height and width of the original image, respectively, and (P, P) is the size of each image +patch. +2 +Method +In this section, we introduce the proposed BOLT framework in details. The pipeline of our +Bootstrap Own Latent of Transformer (BOLT) is illustrated in Fig. 1. Similar to BYOL, the +proposed BOLT adopts two branches to extract useful information from raw data, i.e., the +online and target branches. The online branch consists of a set of weights θ, including a +vision Transformer fθ, a projector gθ and a predictor qθ. The target branch is of the same +architecture with a different set of weights ξ. The target branch generates the regression +targets for the online branch to learn, and its parameters ξ are an exponential moving average +of the online branch parameters θ, which can be defined as: +ξ ← τξ + (1 − τ)θ +(1) +where τ ∈ [0,1] is the decay rate. +Compared to the existing BYOL [12], the proposed BOLT has two differences: First, +instead of image-based perturbation, we implement a token-based perturbation module for +the constrastive learning. The underlying reason for the token-based perturbation is that the +vision Transformer is insensitive to the order of input embedded tokens due to the mechanism +of self-attention, which neutralizes the effectiveness of typical image-based transformation +(e.g., Jigsaw puzzle permutation [24]) made to the self-supervised learning of ViT. Inspired +by recent studies [8, 36], our token perturbation module involves permutation, fusion and +split operations to simultaneously disarrange the order and content of tokens. Second, since +the recent study [29] demonstrated the difficulty-awareness can boost the performance of +CNNs, a difficulty-awareness auxiliary task, i.e., requiring the ViT to identify which branch +(online/target) is processing the more difficult perturbed tokens, is integrated to the existing +BYOL framework. + +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +5 +2.1 +Token Perturbation Module +Instead of permuting the image content, we propose a token perturbation module to per- +turb the order and content of embedded tokens for the self-supervised learning of a vision +Transformer. The architecture of our token perturbation module is presented in Fig. 2, which +involves three operations, i.e., permutation, linear projection and split. +Permutation. Similar to the typical vision Transformer, the input image x ∈ RH×W×C is +cropped into a sequence of flattened 2D patches xp ∈ RN×(P2C), where H and W are the +height and width of the original image, respectively, C is the number of channels, (P, P) is +the size of each image patch, and N = HW +P2 is the resulting number of patches. Therefore, the +embedded tokens zo can be written as: +zo = [x1 +pE;x2 +pE;··· ;xN +p E], +(2) +where E ∈ R(P2C)×D is a trainable linear projection (D is the latent vector size of the vision +Transformer). Then, the permuted tokens zp are obtained using a permutation operation +(Perm(·)), which randomly disarranges the order of zo: zp = Perm(zo). Fig. 2 shows an +example, the order of zo is disarranged to [z6 +o;z1 +o;z5 +o;z7 +o;z3 +o;z9 +o;z8 +o;z2 +o;z4 +o]. +Linear Projection. After the permutation, we concatenate M adjacent tokens using a sliding +window with a stride S = W +P , which results in K = N +S long tokens (z′ +p) with the length of +M × D. The obtained tokens are then fed to a linear projection layer (Efuse ∈ RMD×SD) for +information fusion, which yields K content-perturbed long tokens (zl): +zl = z′ +pEfuse. +(3) +Split. As previously mentioned, the typical vision Transformer uses the constant latent vec- +tor size D through all of its layers; hence, the fused tokens with the length of S×D need to be +reshaped back to the length of D to fulfill the input requirement of ViT. To achieve that, the +proposed token perturbation module adopts a split operation to separate each long token into +S D-length tokens. The splitted tokens (zs) is then fed to ViT for self-supervised learning. +2.2 +Loss Function +As shown in Fig. 1, our BOLT is jointly supervised by two loss functions, i.e., similarity +loss and difficulty-awareness loss. The similarity loss is consistent to the existing BYOL +framework. Concretely, for a set of embedded tokens zo, our BOLT produces two augmented +perturbed tokens zt and z′ +t for online and target branches, respectively. The perturbed tokens +zt are then fed to a ViT fθ, which yields a representation yθ = fθ(zt) and a projection zθ = +gθ(yθ). For the perturbed tokens for the target branch, a representation yξ = fξ(z′ +t) and a +projection zξ = gξ(yξ) are accordingly generated. Consistent to BYOL, a prediction network +qθ(.) is adopted to yield the prediction of zξ and l2-norm is calculated for network training: +Lθ = +��qθ(zθ)− zξ +��2 +2 +(4) +where θ denotes the network weights of the online branch including fθ, gθ and qθ. The loss +LBOLT +θ += Lθ + ˜Lθ only optimizes the weights of online branch θ, where ˜Lθ is the symmetric +loss of Lθ by feeding z′ +t and zt to online and target branches, respectively. + +6 +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +Difficulty-awareness Loss. Apart from the similarity loss, inspired by the curriculum learn- +ing [17], we propose an auxiliary task—identifying which branch is processing the tokens +with a larger level of perturbation. Such an auxiliary task can drive ViTs to self-adaptively +pay more attention on the hard case and accordingly better exploit the semantic information +from the embedded tokens, since they are required to understand the content of tokens for +the accurate difficulty ranking. +To formulate the auxiliary task, the self-supervised signal needs to be first generated. As- +suming the perturbed tokens feeding to online and target branches as zt and z′ +t, respectively, +the self-supervised signal ysel f can be defined as: +ysel f = +� +0, +MSE(Perm−1 +zt (zt)−zo) < MSE(Perm−1 +z′t (z′ +t)−zo) +1, +MSE(Perm−1 +zt (zt)−zo) ⩾ MSE(Perm−1 +z′t (z′ +t)−zo) +(5) +where MSE(.) is the mean squared error function; Perm−1(.) is the inverse permutation +operation rearranging the perturbed tokens back to the original order. +After the self-supervision is obtained, the features extracted by the online and target +ViTs (i.e., yθ and yξ) are concatenated (Cat(.)) and sent to a fully-connected layer (FC(.)) +for difficulty classification. Specifically, the process can be written as: +LDif f +fθ += −ysel f ∗ log(p)− (1 − ysel f)∗ log(1 − p)) +(6) +where p = FC(Cat(yθ,yξ))) is the probability of ysel f = 1. Similar to LBOLT +θ,ξ +, the difficulty- +awareness loss only optimizes the online branch (fθ). +We notice that the recent study [29] has already proposed a difficulty-awareness loss for +scleral spur localization. Hence, it is worthwhile to emphasize the difference between it and +our loss function. Concretely, Tao et al. [29] explicitly enforced networks to predict the Dice +score of input images using segmentation ground truth to achieve difficulty-awareness. Due +to the lack of manual annotations, few study introduces the idea of difficulty-awareness for +self-supervised learning (SSL). In this study, we obtain the difficulty-related information in +a self-supervised manner using the token perturbation module, and implicitly formulate the +difficulty-ranking proxy task. To our best knowledge, this is the first SSL framework based +on the difficulty-awareness paradigm. +Overall Objective. Combining the aforementioned loss functions (LBOLT and LDif f ), the +full objective L for the optimization of the online branch can be written as: +L = LBOLT +θ ++ αLDif f +fθ +(7) +where α = 0.1 is the loss weight of LDif f +fθ +. According to Eq. (1), the weights of target branch +ξ are updated via exponential moving average. +3 +Experiments +We evaluate the proposed BOLT on three target tasks, including skin lesion classification, +knee fatigue grading and diabetic retinopathy grading, using publicly available and private +datasets. Conventional self-supervised learning approaches often pretrain the models on +a large-scale unlabeled dataset (i.e., proxy set), and then finetune them on the relatively +smaller target set. In this paper, three different medical image processing tasks are involved + +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +7 +for performance evaluation and the corresponding proxy and target datasets (example images +are shown in Supplementary Material) for each task are introduced in the followings: +Skin Lesion Classification. The publicly available ISIC 2019 dataset1 is used to validate +the effectiveness of the proposed BOLT. Specifically, the dataset [30] is provided by the +ISIC 2019 challenge, which encourages researchers to develop the automated systems pre- +dicting eight skin disease categories with dermoscopic images, i.e., squamous cell carci- +noma, melanocytic nevus, benign keratosis, actinic keratosis, dermatofibroma, basal cell +carcinoma, vascular lesion, and melanoma. The whole ISIC 2019 dataset, consisting of over +20,000 dermoscopic images, is adopted as the proxy set. Due to the class imbalance prob- +lem of original ISIC dataset, consistent to [21], 628 images are randomly sampled from each +class to establish a balanced target set. It is worthwhile to mention that the images from the +two classes consisting of fewer than 628 images are all taken into the target set. After that, +the balanced target set with 4,260 images is randomly separated into training, validation and +test sets based on the ratio of 70:10:20. Note that the ViT is first pretrained on the proxy set +and finetuned on the training and validation sets, and then evaluated on the test set. +Knee Fatigue Grading. The publicly available MURA dataset2 (musculoskeletal radio- +graphs) [28], which is a large dataset of bone X-rays (over 40,000 images), is adopted as the +proxy set to pretrain ViTs for the subsequent target task (i.e., knee fatigue grading). For the +knee fatigue grading, 2,725 X-ray images are collected from a collaborating hospital as the +target set [20]. The positions of fatigue fracture are different, i.e., navicular bone, tibia and +fibula. Each X-ray image is labeled by three physicians, and the final grade is decided via +majority-voting. In particular, the target set has 1,785 normal, 190 grade-1, 452 grade-2, 196 +grade-3 and 102 grade-4 cases, respectively. For the evaluation on our private knee fatigue +grading dataset, the target set is divided to training, validation and test sets according to the +ratio of 70:10:20. Similar to [20], due to the imbalance problem (normal vs. fatigue fracture +and grade-2 vs. other fracture grades), an equal number (20) of test images from each cate- +gory are randomly sampled to form an uniform-distribution set for performance evaluation, +instead of using the whole test set. +Diabetic Retinopathy Grading. For the diabetic retinopathy grading task, we pretrain the +ViT on a large-scale private dataset captured from a collaborating hospital (proxy set), with +approval obtained from the institutional review board of the hospital. The dataset consists of +350,000 fundus images of normal cohort and patients with various diseases. Then, the pre- +trained ViT is finetuned on the publicly available APTOS 2019 blindness detection dataset +(target set) for performance evaluation.3 In particular, there are 3,662 fundus images con- +tained in the target set and the severity of diabetic retinopathy (DR) can be classified to +four grades, i.e., normal (1,805), mild DR (370), moderate DR (999), severe DR (193) and +proliferative DR (295). Consistent to [22], a five fold cross-validation is conducted on this +dataset.4 +Baselines & Evaluation Criterion. To demonstrate the effectiveness of our BOLT pretrain- +ing, we finetune ViTs with ImageNet pretrained weights on the target tasks and evaluate +1https://challenge2019.isic-archive.com/ +2https://stanfordmlgroup.github.io/competitions/mura/ +3https://www.kaggle.com/c/aptos2019-blindness-detection +4The ViT pretrained on our private large-scale dataset may benefit the related downstream target tasks. To +advance the development of automated fundus image processing, we will release the ViT pretrained models to the +community soon. + +8 +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +Table 1: The classification accuracy (ACC) presented in percentage (%) of ViTs using dif- +ferent training strategies with different amounts of training data on the ISIC 2019 test set. +100% +50% +10% +Train-from-scratch +39.4 +35.2 +31.3 +ImageNet Pretrained +80.5 +76.1 +62.1 +SimSam [5] +79.9 +75.9 +61.2 +BYOL [12] +80.1 +75.4 +61.3 +MoCo V3 [7] +80.3 +75.2 +61.2 +BOLT w./o. LDi f f +80.8 +75.8 +62.1 +BOLT (ours) +81.5 +76.6 +62.4 +ImageNet Pretrained ResNet-50 +75.7 +72.5 +61.2 +their performances on the test set. Consistent to MoCo V3 [7], the basic ViT-B/16 is adopted +as backbone. The original BYOL [12], state-of-the-art self-supervised learning approach +SimSam [5] and token-based self-supervised learning approach MoCo V3 [7] are assessed +for comparison. It is worthwhile to mention that the backbones of representation networks +of BYOL and SimSam implemented in this study are ViT-B/16. The average classification +accuracy (ACC) is adopted as metric for the performance evaluation. +3.1 +Performance Evaluation +In this section, we evaluate the effectiveness of different training strategies on different +datasets and present the experimental results. The widely-used ImageNet pretrained ResNet- +50 is also adopted as a baseline for comparison. Some detailed discussions are presented in +Supplementary Material. +Skin Lesion Classification. First, the different training strategies are evaluated on the pub- +licly available ISIC 2019 dataset. The evaluation results of models finetuned with all train- +ing data (100%) on the test set are listed in Table 1. The ImageNet pretrained ViT is ob- +served to surpass the ImageNet pretrained ResNet-50 by a large margin (i.e., +4.8%), which +demonstrates the superiority of ViT for medical image classification. Compared to the state- +of-the-art self-supervised learning approaches (i.e., SimSam, BYOL and MoCo V3), our +token-based BOLT achieves a higher ACC (80.8%). By using the difficulty-awareness loss +(LDif f ), the ACC of BOLT can be further improved to 81.5%, which outperforms the runner- +up (MoCo V3) by a margin of +1.2%. +The goal of self-supervised learning approach primarily is to deal with the insufficient +training data. Hence, to better verify the superiority of our BOLT approach, we conduct +an experiment to assess the performance of BOLT pretrained ViTs with different numbers +of labeled samples used for finetuning (i.e., 10% and 50% in Table 1). It can be observed +that our BOLT can effectively tackle the situation with few labeled training samples—the +proposed BOLT with difficulty-awareness loss achieves the best ACC under both 50% and +10% settings. +Knee Fatigue Grading. Consistent to the previous study [20], apart from classification ac- +curacy, the F1 score is also adopted for performance evaluation. The experimental results on +the uniform test set are listed in Table 2. As shown, the ViT pretrained with the proposed +BOLT outperforms the ones using existing self-supervised learning approaches and the Ima- +geNet pretrained weights, i.e., an ACC of 54.0% is achieved (+2.0% higher than the runner- + +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +9 +Table 2: The accuracy (ACC and F1 score) presented in percentage (%) of different training +strategies on knee fatigue grading and diabetic retinopathy grading tasks. +Knee Fatigue Grading +Diabetic Retinopathy Grading +ACC +F1 +ACC +F1 +Train-from-scratch +30.0 +23.1 +71.0 +65.3 +ImageNet Pretrained +51.0 +49.4 +83.6 +83.2 +SimSam [5] +52.0 +51.1 +84.5 +84.3 +BYOL [12] +51.0 +50.2 +84.8 +84.7 +MoCo V3 [7] +52.0 +51.2 +84.7 +84.3 +BOLT w./o. LDif f +52.0 +51.2 +85.4 +85.3 +BOLT (ours) +54.0 +53.6 +85.9 +85.8 +ImageNet Pretrained ResNet-50 +36.0 +31.7 +81.7 +82.0 +up). Similar trend to ISIC 2019 is observed—the ACC of ImageNet pretrained ViT (51%) +is significantly higher than that of ImageNet pretrained ResNet-50 (36%), demonstrating +the effectiveness of ViT backbone. We notice that the improvements to train-from-scratch +yielded by pretraining are more obvious on our knee fatigue grading dataset (over +20%), +compared to the skin lesion classification task. The reason may be that the target set of knee +fatigue grading contains less training samples (around 1,000 X-ray images); thus, it is more +difficult to well train the model from scratch, compared to the skin lesion classification task +with a target set of 4,260 images. +Diabetic Retinopathy Grading. Consistent to [22], we split the APTOS 2019 dataset into +five folds for cross-validation and adopt the F1 score for performance evaluation. The grad- +ing accuracy of models using different training strategies is shown in Table 2. The proposed +BOLT pretrained ViT achieves the best ACC (85.9%) and F1 score (85.8%) among the listed +approaches, which are +1.1% and +1.1% higher than the original BYOL, respectively. +4 +Conclusion +In this paper, a self-supervised learning approach, termed Boostrap Own Latent of Trans- +former (BOLT), was proposed specifically for medical image classification with the vision +Transformer backbone. The proposed BOLT involved online and target branches, which ex- +tracted the self-supervised representation from raw data via contrastive learning. Concretely, +the online network was trained to predict the target network representation of the same patch +embedding tokens with a different perturbation. Furthermore, we proposed an auxiliary dif- +ficulty ranking task to enable the vision Transformer to exploit diverse information from the +limited medical data. The difference between the original patch embedding tokens and the +perturbed ones was calculated as the difficulty measurement (i.e., the larger difference means +more difficult for the vision Transformer to process), which was then adopted as the supervi- +sion signal for self-supervised learning. The vision Transformer was trained to identify the +branch (online/target) processing for the more difficult perturbed tokens, which enabled it +to distill the transformation-invariant features from the perturbed tokens. The experimental +results on three medical image classification tasks (i.e., skin lesion classification, knee fa- +tigue fracture grading and dabetic retinopathy grading) demonstrated the effectiveness of the +proposed BOLT. We notice several limitations of this study and plan to address them in the +future works: +Extension to Medical Image Segmentation Task. The proposed BOLT can be easily ex- + +10 +STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES +tended to medical image segmentation in a similar way like [40], i.e., pretraining the encoder +and using a random initialization for the decoder. Yet, the randomly initialized decoder may +neutralize the performance improvement. Therefore, we plan to explore a more effective +way extending our pretrained ViTs for medical image segmentation task in the future. +Pretrained Weights for ViT Variants. Recently, many powerful ViT-based backbones, such +as Swin Transformer [23], have been proposed. The weights of these ViT variants pretrained +on our large-scale fundus image dataset will be continuously provided in the future. +References +[1] Sara Atito, Muhammad Awais, and Josef Kittler. SiT: Self-supervised vision Trans- +former. arXiv preprint arXiv:2104.03602, 2021. +[2] H. Bao, L. Dong, and F. Wei. 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Medical Image Analysis, 64:101746, 2020. +[41] Xizhou Zhu, Weijie Su, Lewei Lu, Bin Li, Xiaogang Wang, and Jifeng Dai. +De- +formable DETR: Deformable Transformers for end-to-end object detection. +arXiv +preprint arXiv:2010.04159, 2020. + diff --git a/4tAzT4oBgHgl3EQfEPpQ/content/tmp_files/load_file.txt b/4tAzT4oBgHgl3EQfEPpQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bddffa4696561ca0bf8f880a931721f8e29bbd39 --- /dev/null +++ b/4tAzT4oBgHgl3EQfEPpQ/content/tmp_files/load_file.txt @@ -0,0 +1,642 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf,len=641 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='00989v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='CV] 3 Jan 2023 STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES 1 A New Perspective to Boost Vision Transformer for Medical Image Classification Yuexiang Li vicyxli@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='com Yawen Huang yawenhuang@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='com Nanjun He nanjunhe@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='com Kai Ma kylekma@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='com Yefeng Zheng yefengzheng@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='com Tencent Jarvis Lab Shenzhen China Abstract Transformer has achieved impressive successes for various computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' However, most of existing studies require to pretrain the Transformer backbone on a large-scale labeled dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', ImageNet) for achieving satisfactory performance, which is usually unavailable for medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Additionally, due to the gap between medical and natural images, the improvement generated by the ImageNet pretrained weights sig- nificantly degrades while transferring the weights to medical image processing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' In this paper, we propose Bootstrap Own Latent of Transformer (BOLT), a self-supervised learning approach specifically for medical image classification with the Transformer backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Our BOLT consists of two networks, namely online and target branches, for self-supervised representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Concretely, the online network is trained to predict the target network representation of the same patch embedding tokens with a dif- ferent perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' To maximally excavate the impact of Transformer from limited med- ical data, we propose an auxiliary difficulty ranking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The Transformer is enforced to identify which branch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', online/target) is processing the more difficult perturbed tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Overall, the Transformer endeavours itself to distill the transformation-invariant features from the perturbed tokens to simultaneously achieve difficulty measurement and maintain the consistency of self-supervised representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The proposed BOLT is eval- uated on three medical image processing tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', skin lesion classification, knee fatigue fracture grading and diabetic retinopathy grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The experimental results validate the superiority of our BOLT for medical image classification, compared to ImageNet pre- trained weights and state-of-the-art self-supervised learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 1 Introduction Recently, vision Transformer (ViT) [10] and its variants [23, 32, 36] has been introduced for various computer vision tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', image classification [10, 18], object detection [9, © 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The copyright of this document resides with its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' It may be distributed unchanged freely in print or electronic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 2 STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES 41], semantic segmentation [34, 39] and medical image processing [11, 15, 16, 31, 38]) and gained increasing attentions from the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The common ViT usually requires pretrainig on large-scale natural image datasets, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', ImageNet, to achieve the satisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' For natural images, the labels for pretraining dataset can be efficiently obtained by crowdsourcing, as even ordinary people possess the ability to effectively identify and label objects in natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' However, the same strategy cannot be adopted for medical images, as professional expertise is mandatory for high-quality medical image annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Hence, the limited amount of annotated medical data is the major obstacle for the improvement of diagnosis accuracy even with the powerful vision Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Self-supervised learning (SSL) approach is a potential solution to tackle the challenge of insufficient annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The typical self-supervised learning formulates a proxy task to extract representative features from unlabeled data, which can boost the accuracy of subse- quent target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Existing studies have proposed various proxy tasks, including grayscale image colorization [19], patch re-ordering [25], and context restoration [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The SSL was firstly brought to medical image processing by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Concretely, the neural network was pretrained with a proxy task that sorted the 2D slices from the conventional 3D medical volumes for the subsequent fine-grained body part recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' [40] enforced 3D networks to play a Rubik’s cube game for pretraining, which can be seen as an extension of 2D Jigsaw puzzles [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Contrastive learning [13] has been recently popular- ized for self-supervised representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' These approaches enforce neural networks to spontaneously exploit useful information from pairs of positive and negative samples, instead of permuting the contextual information of images for self-supervised signal for- mulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' [14] firstly introduced the idea of contrastive learning into the area of self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' They proposed an approach, namely MoCo, which addressed the problem of large number of negative samples for contrastive learning by maintaining a memory bank of negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Following the direction, various contrastive-learning- based self-supervised approaches have been proposed [4, 6, 7, 12, 26, 33, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Inspired by the success of self-supervised learning for CNNs, researchers began to make their efforts to ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Atito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' [1] directly utilized the existing SSL approaches, including rotation pre- diction, contrastive learning and image restoration, to pretrain vision Transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Several studies [2, 3] have been proposed along this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' However, taking the architecture dif- ference between CNN and ViT into account, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', CNN takes the whole image as input, while the input of ViT is the embedding tokens of image tiles, the self-supervised learning approach specifically for ViT is worthwhile to develop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' In the recent study, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' [7] proposed MoCo V3 as a token-based constrastive learning approach, specifically for ViT to extract self-supervised features from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The network pretrained with MoCo V3 outperformed the ImageNet-pretrained one, which demonstrated the effectiveness of token-based self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' In this paper, we follow the direction and propose a token-wise perturbation based self-supervised learning framework specifically for medical image classification with vision Transformer, namely Bootstrap Own Latent of Transformer (BOLT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Similar to the existing Bootstrap Your Own Latent (BYOL) [12], our BOLT consists of two networks, namely online and target branches, for self-supervised representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Instead of image-wise transformation adopted by BYOL, the online network of our BOLT is trained to predict the target network representa- tion of the same patch embedding tokens with a different perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Moreover, to encour- age the vision Transformer to deeply exploit useful information from limited medical data, we propose an auxiliary difficulty ranking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The difference between the original patch embedding tokens and the perturbed ones is measured as the difficulty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' the larger dif- STUDENT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' PROF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' COLLABORATOR: BMVC AUTHOR GUIDELINES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Linear Projection of Flattened Patches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Linear Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Split ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Sliding Window ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Token Permutation Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Vision Transformer �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Vision Transformer �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Perturbation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Perturbation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='��(��) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� (��) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Difficulty-awareness Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Exponential Moving Average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Similarity Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Online ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Embedded Token �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Content perturbed Token �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Long Token �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Figure 1: The architecture of our BOLT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Compared to the original BYOL, our BOLT consists of two main revisions: 1) The proposed BOLT generates two views of em- bedding tokens for self-supervised learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 2) A novel difficulty-awareness loss is proposed to encourage the ViT to deeply exploit useful information from raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' sg(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=') means stop- gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' ference means more difficult for the vision Transformer to process), which is then adopted as the supervision signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' In other words, the vision Transformer is required to identify which branch (online/target) is processing the more difficult perturbed tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Under the co-supervision of the two tasks, the vision Transformer is encouraged to endeavour itself to distill the transformation-invariantfeatures from the perturbed tokens, which should be capa- ble for simultaneous difficulty measurement and maintain the consistency of self-supervised representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' In summary, the main contributions of our work can be concluded into four-fold: A token perturbation based self-supervised learning approach, namely BOLT, specif- ically designed for vision Transformer is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' A token perturbation module is integrated to the existing BYOL framework for the more effective ViT pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' An auxiliary self-supervised task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', difficulty ranking, is proposed to encourage ViTs to deeply exploit useful information from limited medical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The self-supervised signal of this auxiliary task also derives from the perturbed tokens generated by our perturbation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' To our best knowledge, this is the first SSL framework based on the difficulty-awareness paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The proposed BOLT is evaluated on three medical image processing tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', skin lesion classification, knee fatigue fracture grading and diabetic retinopathy grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The experimental results demonstrate the superiority of our BOLT, compared to the widely-used ImageNet pretrained weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Last but not least, we pretrain the ViT using different self-supervised learning ap- proaches on a large-scale private fundus image dataset captured from a collaborating hospital for diabetic retinopathy grading task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The dataset consists of 350,000 fundus images of normal cohort and patients with various diseases, which may be the largest fundus image dataset in the worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The pretraining on our private large-scale dataset is verified to benefit the related downstream target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' To advance the de- velopment of automated fundus image processing, we will release the ViT pretrained models to the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 4 STUDENT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' PROF,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' COLLABORATOR: BMVC AUTHOR GUIDELINES ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Linear Projection of Flattened Patches ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='6 1 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 3 9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8 2 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Linear Projection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Split ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Sliding Window ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Token Permutation Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Vision Transformer �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Vision Transformer �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Patch ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Permutation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Module ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Difficulty awareness Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Exponential Moving Average ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Similarity Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Online ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Embedded Token �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Content-perturbed Token �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Long Token �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='Figure 2: The architecture of the proposed token perturbation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The module consists of three operations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', permutation, linear projection and split) to perturb the order and content of embedded tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Note that nine embedding tokens in this figure are taken as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The exact number (N) of embedding tokens is decided by HW P2 , where H and W are the height and width of the original image, respectively, and (P, P) is the size of each image patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 2 Method In this section, we introduce the proposed BOLT framework in details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The pipeline of our Bootstrap Own Latent of Transformer (BOLT) is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Similar to BYOL, the proposed BOLT adopts two branches to extract useful information from raw data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', the online and target branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The online branch consists of a set of weights θ, including a vision Transformer fθ, a projector gθ and a predictor qθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The target branch is of the same architecture with a different set of weights ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The target branch generates the regression targets for the online branch to learn, and its parameters ξ are an exponential moving average of the online branch parameters θ, which can be defined as: ξ ← τξ + (1 − τ)θ (1) where τ ∈ [0,1] is the decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Compared to the existing BYOL [12], the proposed BOLT has two differences: First, instead of image-based perturbation, we implement a token-based perturbation module for the constrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The underlying reason for the token-based perturbation is that the vision Transformer is insensitive to the order of input embedded tokens due to the mechanism of self-attention, which neutralizes the effectiveness of typical image-based transformation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', Jigsaw puzzle permutation [24]) made to the self-supervised learning of ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Inspired by recent studies [8, 36], our token perturbation module involves permutation, fusion and split operations to simultaneously disarrange the order and content of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Second, since the recent study [29] demonstrated the difficulty-awareness can boost the performance of CNNs, a difficulty-awareness auxiliary task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', requiring the ViT to identify which branch (online/target) is processing the more difficult perturbed tokens, is integrated to the existing BYOL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 Token Perturbation Module Instead of permuting the image content, we propose a token perturbation module to per- turb the order and content of embedded tokens for the self-supervised learning of a vision Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The architecture of our token perturbation module is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 2, which involves three operations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', permutation, linear projection and split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Similar to the typical vision Transformer, the input image x ∈ RH×W×C is cropped into a sequence of flattened 2D patches xp ∈ RN×(P2C), where H and W are the height and width of the original image, respectively, C is the number of channels, (P, P) is the size of each image patch, and N = HW P2 is the resulting number of patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Therefore, the embedded tokens zo can be written as: zo = [x1 pE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='x2 pE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='··· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='xN p E], (2) where E ∈ R(P2C)×D is a trainable linear projection (D is the latent vector size of the vision Transformer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Then, the permuted tokens zp are obtained using a permutation operation (Perm(·)), which randomly disarranges the order of zo: zp = Perm(zo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 2 shows an example, the order of zo is disarranged to [z6 o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='z1 o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='z5 o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='z7 o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='z3 o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='z9 o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='z8 o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='z2 o;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='z4 o].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Linear Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' After the permutation, we concatenate M adjacent tokens using a sliding window with a stride S = W P , which results in K = N S long tokens (z′ p) with the length of M × D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The obtained tokens are then fed to a linear projection layer (Efuse ∈ RMD×SD) for information fusion, which yields K content-perturbed long tokens (zl): zl = z′ pEfuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' (3) Split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' As previously mentioned, the typical vision Transformer uses the constant latent vec- tor size D through all of its layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' hence, the fused tokens with the length of S×D need to be reshaped back to the length of D to fulfill the input requirement of ViT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' To achieve that, the proposed token perturbation module adopts a split operation to separate each long token into S D-length tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The splitted tokens (zs) is then fed to ViT for self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 Loss Function As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 1, our BOLT is jointly supervised by two loss functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', similarity loss and difficulty-awareness loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The similarity loss is consistent to the existing BYOL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Concretely, for a set of embedded tokens zo, our BOLT produces two augmented perturbed tokens zt and z′ t for online and target branches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The perturbed tokens zt are then fed to a ViT fθ, which yields a representation yθ = fθ(zt) and a projection zθ = gθ(yθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' For the perturbed tokens for the target branch, a representation yξ = fξ(z′ t) and a projection zξ = gξ(yξ) are accordingly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Consistent to BYOL, a prediction network qθ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=') is adopted to yield the prediction of zξ and l2-norm is calculated for network training: Lθ = ��qθ(zθ)− zξ ��2 2 (4) where θ denotes the network weights of the online branch including fθ, gθ and qθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The loss LBOLT θ = Lθ + ˜Lθ only optimizes the weights of online branch θ, where ˜Lθ is the symmetric loss of Lθ by feeding z′ t and zt to online and target branches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 6 STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES Difficulty-awareness Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Apart from the similarity loss, inspired by the curriculum learn- ing [17], we propose an auxiliary task—identifying which branch is processing the tokens with a larger level of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Such an auxiliary task can drive ViTs to self-adaptively pay more attention on the hard case and accordingly better exploit the semantic information from the embedded tokens, since they are required to understand the content of tokens for the accurate difficulty ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' To formulate the auxiliary task, the self-supervised signal needs to be first generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' As- suming the perturbed tokens feeding to online and target branches as zt and z′ t, respectively, the self-supervised signal ysel f can be defined as: ysel f = � 0, MSE(Perm−1 zt (zt)−zo) < MSE(Perm−1 z′t (z′ t)−zo) 1, MSE(Perm−1 zt (zt)−zo) ⩾ MSE(Perm−1 z′t (z′ t)−zo) (5) where MSE(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=') is the mean squared error function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Perm−1(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=') is the inverse permutation operation rearranging the perturbed tokens back to the original order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' After the self-supervision is obtained, the features extracted by the online and target ViTs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', yθ and yξ) are concatenated (Cat(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=')) and sent to a fully-connected layer (FC(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=')) for difficulty classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Specifically, the process can be written as: LDif f fθ = −ysel f ∗ log(p)− (1 − ysel f)∗ log(1 − p)) (6) where p = FC(Cat(yθ,yξ))) is the probability of ysel f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Similar to LBOLT θ,ξ , the difficulty- awareness loss only optimizes the online branch (fθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' We notice that the recent study [29] has already proposed a difficulty-awareness loss for scleral spur localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Hence, it is worthwhile to emphasize the difference between it and our loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Concretely, Tao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' [29] explicitly enforced networks to predict the Dice score of input images using segmentation ground truth to achieve difficulty-awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Due to the lack of manual annotations, few study introduces the idea of difficulty-awareness for self-supervised learning (SSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' In this study, we obtain the difficulty-related information in a self-supervised manner using the token perturbation module, and implicitly formulate the difficulty-ranking proxy task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' To our best knowledge, this is the first SSL framework based on the difficulty-awareness paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Overall Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Combining the aforementioned loss functions (LBOLT and LDif f ), the full objective L for the optimization of the online branch can be written as: L = LBOLT θ + αLDif f fθ (7) where α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 is the loss weight of LDif f fθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' (1), the weights of target branch ξ are updated via exponential moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 3 Experiments We evaluate the proposed BOLT on three target tasks, including skin lesion classification, knee fatigue grading and diabetic retinopathy grading, using publicly available and private datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Conventional self-supervised learning approaches often pretrain the models on a large-scale unlabeled dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', proxy set), and then finetune them on the relatively smaller target set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' In this paper, three different medical image processing tasks are involved STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES 7 for performance evaluation and the corresponding proxy and target datasets (example images are shown in Supplementary Material) for each task are introduced in the followings: Skin Lesion Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The publicly available ISIC 2019 dataset1 is used to validate the effectiveness of the proposed BOLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Specifically, the dataset [30] is provided by the ISIC 2019 challenge, which encourages researchers to develop the automated systems pre- dicting eight skin disease categories with dermoscopic images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', squamous cell carci- noma, melanocytic nevus, benign keratosis, actinic keratosis, dermatofibroma, basal cell carcinoma, vascular lesion, and melanoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The whole ISIC 2019 dataset, consisting of over 20,000 dermoscopic images, is adopted as the proxy set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Due to the class imbalance prob- lem of original ISIC dataset, consistent to [21], 628 images are randomly sampled from each class to establish a balanced target set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' It is worthwhile to mention that the images from the two classes consisting of fewer than 628 images are all taken into the target set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' After that, the balanced target set with 4,260 images is randomly separated into training, validation and test sets based on the ratio of 70:10:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Note that the ViT is first pretrained on the proxy set and finetuned on the training and validation sets, and then evaluated on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Knee Fatigue Grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The publicly available MURA dataset2 (musculoskeletal radio- graphs) [28], which is a large dataset of bone X-rays (over 40,000 images), is adopted as the proxy set to pretrain ViTs for the subsequent target task (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', knee fatigue grading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' For the knee fatigue grading, 2,725 X-ray images are collected from a collaborating hospital as the target set [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The positions of fatigue fracture are different, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', navicular bone, tibia and fibula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Each X-ray image is labeled by three physicians, and the final grade is decided via majority-voting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' In particular, the target set has 1,785 normal, 190 grade-1, 452 grade-2, 196 grade-3 and 102 grade-4 cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' For the evaluation on our private knee fatigue grading dataset, the target set is divided to training, validation and test sets according to the ratio of 70:10:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Similar to [20], due to the imbalance problem (normal vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' fatigue fracture and grade-2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' other fracture grades), an equal number (20) of test images from each cate- gory are randomly sampled to form an uniform-distribution set for performance evaluation, instead of using the whole test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Diabetic Retinopathy Grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' For the diabetic retinopathy grading task, we pretrain the ViT on a large-scale private dataset captured from a collaborating hospital (proxy set), with approval obtained from the institutional review board of the hospital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The dataset consists of 350,000 fundus images of normal cohort and patients with various diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Then, the pre- trained ViT is finetuned on the publicly available APTOS 2019 blindness detection dataset (target set) for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 In particular, there are 3,662 fundus images con- tained in the target set and the severity of diabetic retinopathy (DR) can be classified to four grades, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', normal (1,805), mild DR (370), moderate DR (999), severe DR (193) and proliferative DR (295).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Consistent to [22], a five fold cross-validation is conducted on this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 Baselines & Evaluation Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' To demonstrate the effectiveness of our BOLT pretrain- ing, we finetune ViTs with ImageNet pretrained weights on the target tasks and evaluate 1https://challenge2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='isic-archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='com/ 2https://stanfordmlgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='io/competitions/mura/ 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='com/c/aptos2019-blindness-detection 4The ViT pretrained on our private large-scale dataset may benefit the related downstream target tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' To advance the development of automated fundus image processing, we will release the ViT pretrained models to the community soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 8 STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES Table 1: The classification accuracy (ACC) presented in percentage (%) of ViTs using dif- ferent training strategies with different amounts of training data on the ISIC 2019 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 100% 50% 10% Train-from-scratch 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 ImageNet Pretrained 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 SimSam [5] 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 BYOL [12] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 MoCo V3 [7] 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 BOLT w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' LDi f f 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 BOLT (ours) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 ImageNet Pretrained ResNet-50 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='5 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 their performances on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Consistent to MoCo V3 [7], the basic ViT-B/16 is adopted as backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The original BYOL [12], state-of-the-art self-supervised learning approach SimSam [5] and token-based self-supervised learning approach MoCo V3 [7] are assessed for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' It is worthwhile to mention that the backbones of representation networks of BYOL and SimSam implemented in this study are ViT-B/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The average classification accuracy (ACC) is adopted as metric for the performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 Performance Evaluation In this section, we evaluate the effectiveness of different training strategies on different datasets and present the experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The widely-used ImageNet pretrained ResNet- 50 is also adopted as a baseline for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Some detailed discussions are presented in Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Skin Lesion Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' First, the different training strategies are evaluated on the pub- licly available ISIC 2019 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The evaluation results of models finetuned with all train- ing data (100%) on the test set are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The ImageNet pretrained ViT is ob- served to surpass the ImageNet pretrained ResNet-50 by a large margin (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8%), which demonstrates the superiority of ViT for medical image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Compared to the state- of-the-art self-supervised learning approaches (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', SimSam, BYOL and MoCo V3), our token-based BOLT achieves a higher ACC (80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' By using the difficulty-awareness loss (LDif f ), the ACC of BOLT can be further improved to 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='5%, which outperforms the runner- up (MoCo V3) by a margin of +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The goal of self-supervised learning approach primarily is to deal with the insufficient training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Hence, to better verify the superiority of our BOLT approach, we conduct an experiment to assess the performance of BOLT pretrained ViTs with different numbers of labeled samples used for finetuning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', 10% and 50% in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' It can be observed that our BOLT can effectively tackle the situation with few labeled training samples—the proposed BOLT with difficulty-awareness loss achieves the best ACC under both 50% and 10% settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Knee Fatigue Grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Consistent to the previous study [20], apart from classification ac- curacy, the F1 score is also adopted for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The experimental results on the uniform test set are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' As shown, the ViT pretrained with the proposed BOLT outperforms the ones using existing self-supervised learning approaches and the Ima- geNet pretrained weights, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', an ACC of 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0% is achieved (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0% higher than the runner- STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES 9 Table 2: The accuracy (ACC and F1 score) presented in percentage (%) of different training strategies on knee fatigue grading and diabetic retinopathy grading tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Knee Fatigue Grading Diabetic Retinopathy Grading ACC F1 ACC F1 Train-from-scratch 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 ImageNet Pretrained 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 SimSam [5] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='5 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 BYOL [12] 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 MoCo V3 [7] 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 BOLT w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' LDif f 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='2 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='3 BOLT (ours) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8 ImageNet Pretrained ResNet-50 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='0 up).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Similar trend to ISIC 2019 is observed—the ACC of ImageNet pretrained ViT (51%) is significantly higher than that of ImageNet pretrained ResNet-50 (36%), demonstrating the effectiveness of ViT backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' We notice that the improvements to train-from-scratch yielded by pretraining are more obvious on our knee fatigue grading dataset (over +20%), compared to the skin lesion classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The reason may be that the target set of knee fatigue grading contains less training samples (around 1,000 X-ray images);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' thus, it is more difficult to well train the model from scratch, compared to the skin lesion classification task with a target set of 4,260 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Diabetic Retinopathy Grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Consistent to [22], we split the APTOS 2019 dataset into five folds for cross-validation and adopt the F1 score for performance evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The grad- ing accuracy of models using different training strategies is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The proposed BOLT pretrained ViT achieves the best ACC (85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='9%) and F1 score (85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='8%) among the listed approaches, which are +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1% and +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='1% higher than the original BYOL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' 4 Conclusion In this paper, a self-supervised learning approach, termed Boostrap Own Latent of Trans- former (BOLT), was proposed specifically for medical image classification with the vision Transformer backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The proposed BOLT involved online and target branches, which ex- tracted the self-supervised representation from raw data via contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Concretely, the online network was trained to predict the target network representation of the same patch embedding tokens with a different perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Furthermore, we proposed an auxiliary dif- ficulty ranking task to enable the vision Transformer to exploit diverse information from the limited medical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The difference between the original patch embedding tokens and the perturbed ones was calculated as the difficulty measurement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', the larger difference means more difficult for the vision Transformer to process), which was then adopted as the supervi- sion signal for self-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The vision Transformer was trained to identify the branch (online/target) processing for the more difficult perturbed tokens, which enabled it to distill the transformation-invariant features from the perturbed tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The experimental results on three medical image classification tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', skin lesion classification, knee fa- tigue fracture grading and dabetic retinopathy grading) demonstrated the effectiveness of the proposed BOLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' We notice several limitations of this study and plan to address them in the future works: Extension to Medical Image Segmentation Task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The proposed BOLT can be easily ex- 10 STUDENT, PROF, COLLABORATOR: BMVC AUTHOR GUIDELINES tended to medical image segmentation in a similar way like [40], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=', pretraining the encoder and using a random initialization for the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Yet, the randomly initialized decoder may neutralize the performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Therefore, we plan to explore a more effective way extending our pretrained ViTs for medical image segmentation task in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Pretrained Weights for ViT Variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' Recently, many powerful ViT-based backbones, such as Swin Transformer [23], have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' The weights of these ViT variants pretrained on our large-scale fundus image dataset will be continuously provided in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' References [1] Sara Atito, Muhammad Awais, and Josef 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' De- formable DETR: Deformable Transformers for end-to-end object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} +page_content='04159, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tAzT4oBgHgl3EQfEPpQ/content/2301.00989v1.pdf'} diff --git a/5dE2T4oBgHgl3EQfOgbi/vector_store/index.pkl b/5dE2T4oBgHgl3EQfOgbi/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..076590e4a5a13e6e63d3a006de9dfc30614b60fa --- /dev/null +++ b/5dE2T4oBgHgl3EQfOgbi/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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a/8tE4T4oBgHgl3EQfdQys/content/tmp_files/2301.05090v1.pdf.txt b/8tE4T4oBgHgl3EQfdQys/content/tmp_files/2301.05090v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb7fde93fd1b396e3024a917582d78909d66635f --- /dev/null +++ b/8tE4T4oBgHgl3EQfdQys/content/tmp_files/2301.05090v1.pdf.txt @@ -0,0 +1,4510 @@ +arXiv:2301.05090v1 [math.MG] 12 Jan 2023 +Divide and Conquer: A Distributed Approach +to Five Point Energy Minimization +Richard Evan Schwartz +January 13, 2023 +1 +Introduction +The purpose of this work is to rigorously verify the phase-transition for 5 +point energy minimization first observed in [MKS], in 1977, by T. W. Mel- +nyk, O, Knop, and W. R. Smith. +Our results contain, as special cases, +solutions to Thomson’s 5-electron problem and Polya’s 5-point problem. +This work is an updated version of my monograph from 6 years ago. I +simplified the proof significantly and also I wrote this version in an experi- +mental style designed to facilitate the verification process. This work is just +over half as long as the original. +I wrote the proof in a tree-like form. +Thus, the Main Theorem is an +immediate consequence of Lemma A, Lemma B, and Lemma C. These three +Lemmas are independent from each other. Lemma A is an immediate conse- +quence of Lemma A1 and Lemma A2. And so on. All the “ends” of the tree, +such as Lemma B21121, either have short and straightforward proofs or are +computer calculations which I will describe in enough detail that a compe- +tent programmer could reproduce them. At the same time, all my computer +programs are available to download and use. Figures 0 and 01 below map +out the complete logical structure of the proof of the Main Theorem. +The rest of this introduction states the results and explains how to divide +the verification of the proof into small pieces. Following this, §2 contains a +discussion of the history and context of the results, a high-level discussion of +the ideas in the proof, a discussion of the computer experiments I did, and a +guide to the relevant software I wrote. Following this we get to the proof. +1 + +Results: Let S2 be the unit sphere in R3. Given a configuration {pi} ⊂ S2 +of N distinct points and a function F : (0, 2] → R, define the F-potential of +the configuration: +EF(P) = +� +1≤i 0. +(2) +This is also called the s-potential and the power law potential. +The Triangular Bi-Pyramid (TBP) is the 5 point configuration having one +point at the north pole, one point at the south pole, and 3 points arranged +in an equilateral triangle on the equator. A Four Pyramid (FP) is a 5-point +configuration having one point at the north pole and 4 points arranged in a +square equidistant from the north pole. +Define +15+ = 15 + 24 +512, +15++ = 15 + 25 +512. +(3) +Theorem 1.1 (Main) There existש∈(15+,15++) such that: +1. For s ∈ (0,ש) the TPB is the unique minimizer for Rs. +2. For s =ש the TBP and some FP are the two minimizers for Rs. +3. For each s ∈ (ש,15++) some FP is the unique minimizer for Rs. +In Statement 3, the FP presumably depends on s. The numberש is a new +constant of nature. Its decimal expansion startsש=15.0480773927797... +In §3.5 I will also explain the main details the following theorem. +Theorem 1.2 (Auxiliary) The TBP is the unique minimizer for the Fejes- +Toth potential F(r) = −r−s for all s ∈ (−2, 0). +The original monograph bundled these results together. Here I separate +the results and concentrate on the Main Theorem. +2 + +Lemma Trees: Figure 0 shows most of the lemmas proved in this mono- +graph and how they contribute to the proof of the main theorem. The only +thing missing are the trees of implication for Lemmas B and C1, which we +show in Figure 01 below. The color coding indicates different independent +parts of the monograph. There are 7 colors. Each color (so to speak) may +be read independently from the others. I have indicated the nature of each +colored part, as well as the page numbers containing it, with tags. In the +interest of space I have not given the full name of every lemma. Thus, the +rightmost branch of the tree ends at Lemma C313. + + + + +E +5 +1 +3 +2 +1 +error estimate +41-52 +interpolation +53-61 +local analysis +36-40 +outline +16-21 +A135 +3 +main calculation +22-35 +Figure 0: The tree of implications. +3 + +VThe starred lemmas are the divide-and-conquer computer calculations. +The big one, for A13, is done with interval arithmetic. The others are done +with exact integer arithmetic, though I use floating point operations, in a way +that does not interfere with the logic of the proof, to guide the algorithm. The +general logic is that I’ll compute something with floating point calculations +to see if it is a good candidate for formal verification and then, if so, I’ll +formally verify it with exact calculations. The square nodes indicate sizeable +calculations done with exact evaluations of polynomials in Mathematica. +B3 +1 +1 +Figure 01: The trees of implication for Lemma B and Lemma C1. +Figure 01 shows the trees of implication for Lemma B and for Lemma +C1. This part of the monograph is 20 pages long. I have indicated how it +may be divided up into 4 independent parts, each of which could be verified +separately from the others. The various parts all require the material on +pp 65-66, which explains the computational methods. The proofs of Lemma +B and C1 also involve computer-assisted calculations, but these are done +exactly, in Mathematica, by analyzing the properties of the coefficients of in- +teger polynomials. The square nodes of the tree indicate the lemmas which +rely on such calculations. +4 + +DDtria70厂 +53PDVerification: As Figure 0 indicates, a team of 7 readers could check the +mathematical part of the proof, with the team-leader reading the 6-page +outline and the rest of the people reading self-contained sections at most +20 pages long. (Some readers might also want to consult the discussion on +pp 6-15 to get insight into where the ideas come from.) Figure 01 indicates +how the 20-page portion on symmetrization could be further divided into 4 +independent pieces, each no more than 8 pages. +At the same time, I wrote the computer code in such a way that the pro- +grams for each part are independent from the programs for each other part. +It is probably easier in each case to reproduce the code rather than check +that mine is correct. Each reader could team up with a strong computer +programmer who could reproduce the relevant code. This would be a serious +job only for the calculation in Lemma A13. The rest of the code could be +reproduced, in each case, in a day. I’d say that the code for Lemma A13 +could be reproduced in a few days by one person. +Acknowledgements: I thank Henry Cohn, Doug Hardin, John Hughes, +Abhinav Kumar, Curtis McMullen, Stephen D. Miller, Jill Pipher, Ed Saff, +Sergei Tabachnikov, and Alexander Tumanov for discussions related to this +monograph. I thank I.C.E.R.M. for facilitating this project. My interest +in this problem was rekindled during discussions about point configuration +problems around the time of the I.C.E.R.M. Science Advisory Board meeting +in Nov. 2015. After writing the original version of the monograph around +2016, I let the thing languish on my website for some years. After giving the +Lewis Lectures at Rutgers University in Nov, 2022, I got inspired to re-write +my monograph and try again to make it publishable. Finally, I thank the +National Science Foundation for their continued support, currently in the +form of NSF grant DMS-2102802. +5 + +2 +Discussion +This chapter discusses the history and context for the result, and the high +level ideas in the proof. Following this, I explain some of my experimental +methods and discuss the computer parts of the proof. None of this is logically +needed for the proof, but it will shed light on why the proof looks like it does. +2.1 +History and Context +We take up the question discussed in the introduction: Which configurations +of points on the sphere minimize a given potential function F : (0, 2] → R. +The classic choice for this question is F = Rs, the Riesz potential, defined +in Equation 2. Again, Rs(d) = d−s. The Riesz potential is defined when +s > 0. When s < 0 the corresponding function Rs(d) = −d−s is called the +Fejes-Toth potential. The main difference is the minus sign out in front. +The case s = 1 is specially called the Coulomb potential or the electro- +static potential. This case of the energy minimization problem is known as +Thomson’s problem. See [Th]. The case of s = −1, in which one tries to +maximize the sum of the distances, is known as Polya’s problem. +There is a large literature on the energy minimization problem. See [F¨o] +and [C] for some early local results. See [MKS] for a definitive numerical +study on the minimizers of the Riesz potential for n relatively small. The +website [CCD] has a compilation of experimental results which stretches all +the way up to about n = 1000. The paper [SK] gives a nice survey of results, +with an emphasis on the case when n is large. See also [RSZ]. The paper +[BBCGKS] gives a survey of results, both theoretical and experimental, +about highly symmetric configurations in higher dimensions. +When n = 2, 3 the problem is fairly trivial. In [KY] it is shown that when +n = 4, 6, 12, the most symmetric configurations – i.e. vertices of the relevant +Platonic solids – are the unique minimizers for all Rs with s ∈ (−2, ∞)−{0}. +See [A] for just the case n = 12 and see [Y] for an earler, partial result +in the case n = 4, 6. The result in [KY] is contained in the much more +general and powerful result [CK, Theorem 1.2] concerning the so-called sharp +configurations. +The case n = 5 has been notoriously intractable. There is a general feeling +that for a wide range of energy choices, and in particular for the power law +potentials (when s > −2) the global minimizer is either the TBP or an FP. +Here is a run-down on what is known so far: +6 + +• The paper [HS] has a rigorous computer-assisted proof that the TBP is +the unique minimizer for the potential F(r) = −r. (Polya’s problem). +• My paper [S1] has a rigorous computer-assisted proof that the TBP +is the unique minimizer for R1 (Thomson’s problem) and R2. Again +Rs(d) = d−s. +• The paper [DLT] gives a traditional proof that the TBP is the unique +minimizer for the logarithmic potential. +• In [BHS, Theorem 7] it is shown that, as s → ∞, any sequence of +5-point minimizers w.r.t. Rs must converge (up to rotations) to the +FP having one point at the north pole and the other 4 points on the +equator. In particular, the TBP is not a minimizer w.r.t Rs when s is +sufficiently large. +• In 1977, T. W. Melnyk, O. Knop, and W. R. Smith, [MKS] conjectured +the existence of the phase transition constant, around s = 15.04808, at +which point the TBP ceases to be the minimizer w.r.t. Rs. This is the +phase transition which our Main Theorem estabishes. +• Define +Gk(r) = (4 − r2)k, +k = 1, 2, 3, ... +(4) +In [T], A. Tumanov proves that the TBP is the unique minimizer for +G2. The minimizers for G1 are those configurations whose center of +mass is the origin. The TBP is included amongst these. +Tumanov points out that the G2 potential does not have an obvious ge- +ometric interpretation, but it is amenable to a traditional analysis. He also +mentions that his result might be a step towards proving that the TBP min- +imizes a range of power law potentials. Inspired by similar material in [CK], +he observes that if the TBP is the unique minimizer for G2, G3 and G5, then +the TBP is the unique minimizer for Rs provided that s ∈ (0, 2]. +We will establish implications like this during the course of our proof of +the Main Theorem. The family of potentials {Gk} behaves somewhat like the +Riesz potentials. The TBP is the unique minimizer for G3, G4, G5, G6 (as a +consequence of our work here) but not a minimizer for any of G7, G8, G9, G10. +I am sure the pattern continues, but I did not formally check or prove it. +7 + +2.2 +Ideas in the Proof +Here are the three ingredients in the proof of the Main Theorem. +• The divide-and-conquer approach taken in [S1]. +• Elaboration of Tumanov’s observation. +• A symmetrization trick that works on a small domain. +Divide and Conquer: For certain choices of F, we are interested in search- +ing through the moduli space of all 5-point configurations and eliminating +those which have higher F-potential than the TBP. We win if we eliminate +everything but the TBP. For the functions we consider, most of the configu- +rations have much higher energy than the TBP and we can eliminate most +of the configuration space just by crude calculations. What is left is just a +small neighborhood Ω0 of the TBP. The TBP is a critical point for EF, and +(it turns out) that the function EF is convex in Ω0. In this case, we can say +that the TBP must be the unique global minimizer. +To implement this, we normalize so that (0, 0, 1) is a point of the con- +figuration, and then we map the other 4 points into R2 using stereographic +projection: +Σ(x, y, z) = +� +x +1 − z, +y +1 − z +� +. +(5) +We call the 4-point planar configuration the avatar. We use crude a priori +estimates to produce a subset Ω of a 7-dimensional rectangular solid that (up +to symmetry) contains all avatars that could have lower potential than the +TBP for all the relevant functions. Inside Ω the divide-and-conquer algorithm +is easy to manage. Our basic object is a block, a rectangular solid subset of Ω. +The main mathematical feature of the paper is a result which gives a lower +bound on the energy of any configuration in a block based on the potentials +of the configurations corresponding to the vertices, and an error term. +Having an efficient error term makes the difference between a feasible +calculation and one which would outlast the universe. +Our error term is +fairly sharp, and also the error term is a rational function of the vertices of +the block. For the potentials we end up using, we could run all our com- +puter programs using exact integer arithmetic. Such integer calculations are +too slow (in this century). I implemented the big calculations using interval +arithmetic. Since everything in sight is rational, our calculations only involve +the operations plus, minus, times, divide, min, and max. +8 + +Elaborations of Tumanov’s Observation: So far we have discussed one +function at a time, but we are interested in a 1-parameter family of power +laws and we can only run our program finitely many times. Using the divide +and conquer approach we show that the TBP is the unique global minimizer +for Gk when k = 3, 4, 5, 6 and also for the wierd energy hybrids G5 − 25G1 +and G♯♯ +10 = G10+28G5+102G2. Converting these results to statements about +the power law potentials comes down to variants of Tumanov’s observation. +After a lot of experimenting I found variants which cover large ranges of +exponents. The results for the potentials above combine to prove that the +TBP is the unique minimizer for Rs as long as s ∈ (−2, 0) ∪ (0, 13]. +Symmetrization: The methods above cannot be sharp enough to arrive +at the exact statement of our Main Theorem, because of the phase transi- +tion. We get around this problem as follows. First, we use the divide and +conquer approach to identify a small subset Υ of the configuration space such +that every configuration not in Υ, and not the TBP, has higher G♯ +10-energy, +where G♯ +10 = G10 + 13G5 + 68G2. This combines with the previous calcula- +tions to show that every configuration not in Υ, and not the TBP, has higher +s potential than the TBP whenever s ∈ [13, 15++]. The configurations in Υ +very nearly have 4-fold symmetry. +To analyze configurations in Υ we use a symmetrization operation which +maps Υ to the subset Υ4 ⊂ Υ consisting of configurations having 4-fold +symmetry. This retraction turns out to reduce the s-potential for exponent +values s ∈ [13, 15++]. Finally (and slightly simplifying) we produce a re- +traction from Υ4 to a subset Υ8 ⊂ Υ4 consisting entirely of FPs. This new +retraction reduces the s-potential when s ∈ [15, 15++]. Now we are left with +an analysis of the s-potential on a 1-dimensional set. +Extending the Range: My techniques run out just past the valueש. The +obvious conjecture, already observed by Melnyk, Knop, and Smith, is some +FP is the minimizer for any Riesz potential with exponent larger thanש. The +original version of my monograph contained a proof that some FP beat the +TBP for all exponents up to 100. I omitted this material because it didn’t +seem like such a strong result and mostly it was just a tedious calculation. +I would say that the main bottleneck to proving that some FP is the mini- +mizer for Rs for all s >ש is the delicacy of the symmetrization process which +I discuss above and also below. +9 + +2.3 +Experimentation +The proof of the Main Theorem is mostly just a verification of the things I +discovered experimentally using the software I created. I will follow 3 main +lines of experimental investigation in this discussion. +Experiments with Interpolation: This discussion has to do with what +I called “Tumanov’s observation” in the preceding section. These kinds of +methods go under the name of interpolation. +For the purpose of giving results about the Riesz potentials, the functions +Gk lose their usefulness at k = 7 because the TBP is not a minimizer for +G7, G8, ... At the same time, the general method requires Gk for k large in +order to extend all the way to the phase transition, a phenomenon that occurs +atש=15.04... +I built a graphical user interface which allows me to explore combinations +of the form � ckGk and see whether various lists of these energy hybrids +produce the desired results. The computer program takes a quadruple of +hybrids, Γ1, Γ2, Γ3, Γ4, and then solves a linear algebra problem to find a +linear combination +Λs = a0 + +4 +� +i=1 +ai(s)Γi +(6) +which matches the values of Rs at the values +√ +2, +√ +3, +√ +4, the distances in- +volved in the TBP. (I will usually write 2 as +√ +4 because then the distances +involved in the TBP are easier to remember.) +As an aside, let me say how I knew to use quadruples rather than, say, +triples or quintuples. If you match the values at +√ +2, +√ +3, +√ +4 and the deriva- +tives, at +√ +2, +√ +3 you get a 5 variable problem with 5 unknowns. (You don’t +have to worry about matching the derivative at +√ +4 because this is an end- +point to the domain of distances between points on the unit sphere.) +Concerning Equation 6, what we need for the quadruple to “work” on the +interval (s0, s1) is that the functions a1(s), a2(s), a3(s), a4(s) are nonnegative +for s ∈ (s0, s1) and that simultaneously the comparison function +1 − Λs +Rs +is positive on (0, 2) − { +√ +2, +√ +3, +√ +4}. +So, my computer program lets you +manipulate the coefficients defining the energy hybrids and then see plots of +the functions just mentioned. +10 + +At the same time as this, my program computes the energy hybrid eval- +uated on the space of FPs to see how it compares to the value on the TBP. +I call this the TBP/FP competition. On intervals (s0, s1) ⊂ (0,ש) we want +the TBP to win the competition, as judged by the given energy hybrids. +Repeatedly running these competitions and looking at the plots of the co- +efficients and the comparison function, I eventually arrived at the energy +hybrids mentioned in the previous section. +You can use this program too. If you actually get my Java program to +run on your computer, you can get the same intuition I eventually got about +what works and what doesn’t. If you don’t play around with the software, +then choices like +G♭ +5 = G5 − 25G1, +G♯♯ +10 = G10 + 28G5 + 102G2 +will just seem like random lucky guesses. In fact they are practically the +unique (at most 3 term) energy hybrids which do the job! +To extend all the way toש, I had to accept an energy hybrid for which +the TBP would lose the TBP/FP competition. At the same time, the TBP +would still do well in the overall competition, beating most of the other +configurations. Eventually I hit upon the energy hybrid G♯ +10 and the small +neighborhood Υ mentioned above and defined precisely in the next chapter. +The quadruple (G1, G2, G♭ +5, G♯ +10) extends a bit pastש, up to 15++, and G♯ +10 +is a pretty kind judge: With respect to this judge, the TBP wins against all +configurations outside the tiny Υ. +The intuition I came away with is that you need to use some Gk for fairly +large k, to get enough extension, and then you need to tune it by sharp- +ening and flattening. To sharpen means to add in more of the lower Gks. +To flatten means to do the opposite. When you sharpen, you get an energy +hybrid which is a kinder but less extensive judge: It works better but on a +smaller range of exponents. When you flatten, you get a harsher but more +extensive judge. The miraculous quadruple (G1, G2, G♭ +5, G♯ +10) extends to the +neighborhood [13, 15++]. The TBP is the minimizer for the first 3 potentials, +and for G♯ +10 the TBP wins outside of Υ. +Experiments with Symmetrization: Most successful energy minimiza- +tion results are about symmetry. The work culminating in that of Cohn- +Kumar [CK] shows how to exploit the extreme symmetry of some special +configurations, like the Leech cell, to show that they are the energy mini- +mizers with respect to a wide range of potentials. These methods only work +11 + +for very special numbers of points. The number N = 5 is not special in this +way, because there are no Platonic solids with 5 points. +For N = 5 the TBP and the FPs are competitors for the most symmetric +configurations. +They have different symmetries. +However, they do have +one thing in common: 4 fold dihedral symmetry. One dream for proving the +Main Theorem is to use a kind of symmetrization operation which replaces an +arbitrary configuration with one having 4-fold dihedral symmetry and lower +potential energy. This would reduce the overall problem to an exploration of +a 2-dimensional moduli space and would possibly bring the result within the +range of rather ordinary calculus. Symmetrization operations are extremely +prevalent in some areas of mathematics – e.g., in proofs of isoperimetric type +results. +Such a symmetrization operation in general will surely fail due to the vast +range of possible configurations. However, certain operations might work well +in very specific parts of the configuration space and for very specific func- +tions. Fortunately, the divide-and-conquer-plus-interpolation method rules +out everything of interest except the magical domain Υ and the exponent +range [13, 15++]. What I did is test various symmetrizations and various +choices of Υ until I found a pair that worked. +Let me say a word about the relation between symmetrization and Υ. +The smaller the choice of Υ, the more likely symmetrization is to work. On +the other hand, the smaller Υ is, the more computation we have to use to +eliminate all the competitors outside Υ. The domain I finally settled on was +large enough to make this elimination process a feasible computation but +small enough for the symmetrization to work. +Once I found a symmetrization operation which worked, the question +became: How to prove it? Proving that symmetrization lowers the energy +seems to involve studying what happens on the tiny but still 7-dimensional +moduli space Υ. The secret to the proof is that, within Υ, the symmetrization +operation is so good that it reduces the energy in pieces. What I mean is +that the energy of a configuration is a 10 term sum e1 + ... + e10. What I +show is that one can write +e1 + .... + e10 = (e1 + e2) + (e3 + e4) + (e5 + e6 + e7) + (e8 + e9 + e10) +so that the symmetrization operation decreases each bracketed sum sepa- +rately. This replaces one big verification by a bunch of smaller ones, con- +ducted over lower dimensional configuration spaces. +12 + +Not only did I have to experiment to find the symmetrization operation +itself, I had to experiment with how to break up the expression for the energy +to make for a provable result. +As it is, showing that an expression like +(e5 + e6 + e7) decreases amounts to showing that a polynomial in 5-variables, +with thousands of terms, is positive on the unit cube. I have a positivity +certificate I use, which I call positive dominance, which works like magic on +the relevant polynomials. Positive dominance kills these big monsters with +one thrust of the sword for each monster – but only so because I tuned the +definition of Υ so that the verification process produced killable monsters. I +got lucky in that such a useful domain Υ exists at all. +I should also mention that I use a second symmetrization which improves +a configuration with 4-fold symmetry to one with 8-fold symmetry. +This +symmetrization, though rather simple, is extremely delicate. It works on a +tiny domain �Ψ4 ⊂ Υ and only for power laws with potential greater than +about 13.53. At the same time, this symmetrization has what I would call +miraculous algebraic properties when restricted to the tiny domain where I +use it. I found this operation, once again, by experimentation, and then the +algebraic properties took me by surprise. In the first version of the this work, +the 180 page monograph, I hadn’t noticed the good algebra. +Experiments with Local Analysis: Another part of the monograph deals +with configurations that are very near the TBP. Here we are fortunate be- +cause the functions EF are convex near the TBP. Put another way, their +Hessians are positive definite near the TBP. That means that the TBP is +the unique minimizer in a small neighborhood around the TBP. I want to +emphasize that what we need is not just a calculation at the TBP. In order to +use this information effectively in a computational proof, we need an explicit +neighborhood of convexity. Proving this sets up a recursive problem. +Consider the simpler situation where we would like to show that some +function f is positive on some interval I = [0, ǫ]. Let’s say that we have +free access to the values f(0), f ′(0), f ′′(0), ... and we can also look at the +explicit expressions for f and its derivatives. If we had some information +about maxI |f ′| we could combine it with information about f(0) to perhaps +complete the job. But how do we get information about maxI |f ′|? Well, if +we had information about maxI |f ′′| we could combine it with information +about f ′(0) to perhaps complete the job. And so on. +This is the situation we find ourselves in. We can compute all the partial +derivatives of EF at the TBP, though we have a function of 7 variables, and +13 + +so eventually it gets expensive to compute them all. However, no matter +how many derivatives we compute, it seems that we need to compute more +of them to get the bounds we need. +There is something that saves us: The error multiplier in Taylor’s The- +orem with Remainder. This multiplier is essentially ǫN/N!, a number that +becomes tiny as N increases. If we can get any kind of reasonable bounds on +high derivatives of our function, then we get pretty good bounds when we +multiply through by the tiny number. +I eventually found a combinatorial trick for bounding the derivatives of +EF, at least for the relevant choices of F, without having to evaluate them +anywhere. The magic formula is Equation 47. After a lot of experimentation +I found that I could get a reasonably sized neighborhood of convexity for +EF by explicitly evaluating the kth partial derivatives up to k = 6 and then +using Equation 47 for a global bound on the 7th partials. +2.4 +Guide to the Software +The software for my proof can be dowloaded from +http//www.math.brown.edu/ ∼ res/Java/TBP.tar +Once you untar this program, you get a directory with a suite of smaller +programs. There are 5 subdirectories, corresponding to 5 of the 6 parts of +the monograph. The part having to do with the error estimate does not rely +on any computer assists. The reader who is interested in verifying any part +of the monograph need only look at the programs for that part. +Main: This does the interval arithmetic calculation for the main divide- +and-conquer result, Lemma A135. +This program is quite extensive, and +spread out in about 20 Java files, but almost all the length comes from the +visual/experimental part. I show the calculations in action, allow the user +to experiment with fairly arbitrary energy hybrids, and also give detailed +written instructions on the operation of the program. +The reason for the extensive program is debugging. The big infrastructure +is designed to prevent errors in the actual computation. I have also included +a stripped down version that runs without all the bells and whistles. I try to +explain the main computation in §5 in enough detail that a reasonably good +programmer would be able to reproduce it. +14 + +Interpolation: The main program in this section, contained in the direc- +tory JavaMain, does all the experimentation with the energy hybrids dis- +cussed above, and also formally proves that the given energy hybrids extend +to their advertised ranges. However, the code I actually use in this version +of the proof is different. The subdirectory Proof has this shorter method. +Why both? I only discovered the method in Proof recently, and the method +in JavaMain is more robust. It doesn’t require the kind of ad hoc argument +I give in §11.2 and also (a very small part of) it is still needed logically for +the Auxiliary Theorem. I include a PDF file which explains the method in +JavaMain. +Independent from all this, I also include 5 Mathematica files, LemmaA221.m +and LemmaA222.m and LemmaA231.m and Lemma A232.m and LemmaA233.m, +which generate all the plots for the corresponding lemmas. One can compare +the Mathematica and Java plots and see that they are the same. +Local Analysis: This directory has 3 Mathematica files, LemmaL21.m and +LemmaL22.m and Lemma23.m, which perform the straightforward and exact +calculations needed for these lemmas. The calculations involve manipulating +rational polynomials and evaluating them at special points. These files can +easily be reproduced by anyone who knows how to manipulate polynomials in +Mathematica. Of course, one could also reproduce the programs in e.g. Sage. +Symmetrization: This directory has 6 Mathematica files, with names like +Lemma B3522.m. +These do the calculations for the corresponding lemmas +in this part of the monograph. These short files essentially just manipulate +rational polynomials using standard operations in Mathematica. They all +could be reproduced by anyone who knows how to manipulate polynomials +in Mathematica. +Endgame: This directly contains a program which is a baby version of +the main divide-and-conquer program. This program does the calculation +for Lemma C2. It is not a very complicated program, and I explain it in +detail in §18. The calculation is done with exact integer arithmetic over a 3- +dimensional space. It only takes about 2 minutes to run. Without the exact +integer arithmetic – meaning with floating point calculations – the program +is about 1000 times faster and and nearly instantaneous. +15 + +3 +Main Theorem: Proof Outline +3.1 +Preliminaries +Stereographic Projection: Let S2 ⊂ R3 be the unit 2-sphere. +Stere- +ographic projection is the map Σ : S2 → R2 ∪ ∞ given by the following +formula. +Σ(x, y, z) = +� +x +1 − z, +y +1 − z +� +. +(7) +Here is the inverse map: +Σ−1(x, y) = +� +2x +1 + x2 + y2, +2y +1 + x2 + y2, 1 − +2 +1 + x2 + y2 +� +. +(8) +Σ−1 maps circles in R2 to circles in S2 and Σ−1(∞) = (0, 0, 1). +Avatars: Stereographic projection gives us a correspondence between 5- +point configurations on S2 having (0, 0, 1) as the last point and planar con- +figurations: +V0, V1, V2, V3, (0, 0, 1) ∈ S2 +⇐⇒ p0, p1, p2, p3 ∈ R2, +pk = Σ(Vk). +(9) +We call the planar configuration the avatar of the corresponding configura- +tion in S2. By a slight abuse of notation we write EF(p1, p2, p3, p4) when we +mean the F-potential of the corresponding 5-point configuration. +Figure 3.1 shows the two possible avatars (up to rotations) of the trian- +gular bi-pyramid, first separately and then superimposed. We call the one +on the left the even avatar, and the one in the middle the odd avatar. The +points for the even avatar are (±1, 0) and (0, ± +√ +3/3). When we superimpose +the two avatars we see some extra geometric structure that is not relevant +for our proof but worth mentioning. The two circles respectively have radii +1/2 and 1 and the 6 segments shown are tangent to the inner one. +0 +1 +2 +3 +0 +2 +3 +1 +0 +2 +even +odd +both +Figure 3.1: Even and odd avatars of the TBP. +16 + +The Special Domain: We let Υ ⊂ (R2)4 denote those planar configurations +p0, p1, p2, p3 such that +1. ∥p0∥ ≥ ∥pk∥ for k = 1, 2, 3. +2. 512p0 ∈ [433, 498] × [0, 0]. (That is, p0 ∈ [433/512, 498/512] × {0}.) +3. 512p1 ∈ [−16, 16] × [−464, −349]. +4. 512p2 ∈ [−498, −400] × [0, 24]. +5. 512p3 ∈ [−16, 16] × [349, 464]. +We discuss the significance of Υ extensively in §2.3. The set Υ contains the +avatars that compete with the TBP near the exponentש. +p0 +p1 +p2 +p3 +Figure 3.2: The sets defining Υ compared with two TBP avatars. +Symmetrization: Let (p0, p1, p2, p3) be a planar configuration with p0 ̸= p2. +We define +d02 = 2∥p0 − p2∥, +d13 = 2∥π02(p1 − p3)∥. +(10) +Here π02 is the projection onto the subspace perpendicular to the vector +p0 − p2. Finally, we define +p∗ +0 = (d02, 0), +p∗ +1 = (0, −d13), +p∗ +2 = (−d02, 0), +p∗ +3 = (0, d13). +(11) +The avatar p∗ +1, p∗ +2, p∗ +3, p∗ +4 is invariant under reflections in the coordinate axes. +17 + +3.2 +Reduction to Three Lemmas +We now reduce the Main Theorem to Lemmas A, B, C. Let +15+ = 15 + 24 +512, +15++ = 15 + 25 +512 +as in the Main Theorem. Let Υ be as in §3.1. Recall that Rs is the Riesz +potential. +Lemma 3.1 (A) For s ∈ (0, 13] the TBP uniquely minimizes the Rs-potential. +For s ∈ (13, 15++], any Rs-potential minimizer is either the TBP or else iso- +metric to a configuration whose avatar lies in Υ. +Lemma A focuses our attention on the small domain Υ and the parameter +range [13, 15++]. Now we bring in the symmetrization operation from §3.1. +Lemma 3.2 (B) Let s ∈ [12, 15++] and (p0, p1, p2, p3) ∈ Υ. Then +ERs(p∗ +0, p∗ +1, p∗ +2, p∗ +3) ≤ ERs(p0, p1, p2, p3) +with equality if and only if the two configurations are equal. +Let Υ4 denote the subset of Υ consisting of configurations which are +invariant under reflections in the coordinate axes. +Lemma B (which re- +dundantly works for s ∈ [12, 13]) focuses our attention on the same small +parameter range [13, 15++] and on the symmetric configurations living in +Υ4. +Lemma 3.3 (C) Let ξ0 denote a planar avatar of the TPB. There exist +ש∈(15+,15++) such that the following is true. +1. For s ∈ (13,ש) we have Es(ξ0) < Es(ξ) for all ξ ∈ Υ4. +2. For s ∈ (ש,15++) we have Es(ξ0) > Es(ξ) for some ξ ∈ Υ4. +Also, for s ∈ [15+, 15++] the restriction of Es to Υ4 has a unique minimum, +and this minimum represents an FP. +The Main Theorem is an obvious consequence of Lemma A (§3.3), Lemma +B (§12), and Lemma C (§3.4.) As a matter of convention we will point the +reader to where the proof of the given lemma starts. +Thus, the proof of +Lemma A starts in §3.3. +18 + +3.3 +Proof of Lemma A +Define +Gk(r) = (4 − r2)k. +(12) +Also define +G♭ +5 = G5 − 25G1, +G♯♯ +10 = G10 + 28G5 + 102G2, +G♯ +10 = G10 + 13G5 + 68G2 +(13) +Lemma 3.4 (A1) The following is true. +1. The TBP is the unique minimizer for G4, G♭ +5, G6. +2. The TBP is the unique minimizer for G♯ +10 among configurations which +are not isometric to ones which have avatars in Υ. +3. The TBP is the unique minimizer for G♯♯ +10 among configurations which +have avatars in Υ. +We note two implications of Lemma A1: +• Since G5 is a positive combination of G♭ +5 and G1, Lemma A1 immedi- +ately implies that the TBP is the unique minimizer for G5. +• Since G♯♯ +10 is a positive combination of G♯ +10 and G5 and G2, Lemma A1 +immediately implies that the TBP is the unique minimizer for G♯♯ +10. +Forcing: Let T0 be the TBP. We say that a pair (Γ3, Γ4) of functions forces +the interval I if the following is true: If T is another configuration such that +Γk(T0) < Γk(T) for k = 3, 4 then Es(T0) < Es(T) for all s ∈ I. +Lemma 3.5 (A2) The following is true. +1. The pair (G4, G6) forces (0, 6]. +2. The pair (G5, G♯♯ +10) forces [6, 13]. +3. The pair (G♭ +5, G♯ +10) forces [13, 15++]. +Lemma A is an immediate consequence of Lemma A1 (§4) and Lemma +A2 (§10). +19 + +3.4 +Proof of Lemma C +Let Ψ4 denote the set of planar configurations of the form +(x, 0), +(0, −y), +(−x, 0), +(0, y), +64(x, y) ∈ [43, 64]. +(14) +This is a 2-dimensional domain consisting of avatars having 4-fold dihedral +symmetry. We have Υ4 ⊂ Ψ4. We work with Ψ4 because it is more symmetric +than Υ4. We identify Ψ4 with the square [43/64, 1]2 and we think of ERs as a +function on this square. We usually write Es = ERs. Again, the point (a, b) +corresponds to the planar configuration with points −p2 = p0 = (a, 0) and +−p1 = p3 = (0, b). Though the TBP does not lie in Ψ4, it corresponds in the +same way to the point (1, +√ +3/3). +The FP configurations in Ψ4 lie along the main diagonal. We call this +diagonal Ψ8. We define an even smaller square +�Ψ4 = +�55 +56, 55 +56 +�2 +⊂ Ψ4. +(15) +We think of �Ψ4 as the sweet spot, the place where all the action happens. +We now define another symmetrization. +σ(x, y) = (z, z), +z = x + y + (x − y)2 +2 +. +(16) +We have σ : �Ψ4 → Ψ8. Here is the key result. +Lemma 3.6 (C1) If s ∈ [14, 16] and p ∈ �Ψ4 Then Es(σ(p)) ≤ Es(p) with +equality if and only if σ(p) = p. +Remarks: +(1) The operation σ is extremely delicate. If we take the exponent s = 13, +the operation actually seems to increase the energy for all points of �Υ4 − �Υ8. +The magic only kicks in around exponent 13.53. +(2) Lemma C1 has an algebraic proof, using the Positive Dominance cer- +tificate. See §12.2. It turns out that we get miraculously good algebraic +behavior for s ∈ [14, 16] and for configurations in �Ψ4. +(3) Lemma C1 is false if we use Υ4 or Ψ4 in place of �Ψ4. This is why we +restrict our attention to a very small region. The same proof works on the +somewhat larger domain [27/32, 29/32]2 but that is about as far as we can go. +20 + +Our next result eliminates all those configurations and exponents not +in �Ψ4 × [15+, 15++]. This result is an explicit calculation similar in spirit +to Lemma A2 but easier. Here we are just dealing with functions on a 2 +dimensional configuration space. Let �Ψ8 be the main diagonal of �Ψ4. +Lemma 3.7 (C2) Let ξ0 be the point in the plane representing the TBP. +1. If s ∈ [13, 15+] and p ∈ Ψ4 then Es(ξ0) < Es(ξ). +2. If s ∈ [15+, 15++] and p ∈ Ψ4 − �Ψ4 then Es(ξ0) < Es(ξ). +3. If s ∈ [15+, 15++] the restriction of Es to �Ψ8 has a unique minimum. +Statements 1 and 2 of Lemma C2 imply that for any s ∈ [13, 15++], any +minimizer ξ of Es, not equal to the TBP avatar, lies in �Ψ4. Furthermore, +such a ξ can only exist when s ∈ [15+, 15++]. Lemma C1 now says that ξ in +fact lies in �Ψ8. Statement 3 of Lemma C2 adds the information that ξ is the +unique minimizer in �Ψ8. The final lemma finishes the proof. +Lemma 3.8 (C3) There existש∈(15+,15++) such that: +1. For s ∈ (15+,ש) we have Es(ξ0) < Es(ξ) for all ξ ∈ �Ψ8. +2. For s ∈ (ש,15++) we have Es(ξ0) > Es(ξ) for some ξ ∈ �Ψ8. +Lemma C is a consequence of Lemma C1 (§12, §17), Lemma C2 (§18), +and Lemma C3 (§20). +3.5 +The Polya Case +Here I explain the main details of the proof of the Auxiliary Theorem: the +TBP is the unique minimizer for Fs(r) = −r−s for any s ∈ (−2, 0). I call +this the Polya case because the well-known Polya problem concerns the case +s = 1. All we need here is Lemma A for the interval (−2, 0). +Lemma A1 also applies to G3. +The only differences in the proof are +discussed in the remarks at the end of §4.5 and §6.4. Once these details are +in place, our software gives a computational proof of Lemma A1 for G3 in +the same way it does for the other potentials. +A variant of Lemma A2, with the same kind of proof, shows that the +pair (G3, G5) forces (−2, 0) with respect to Fs. The main extra detail needed +here is the matrix of power combos given in §10.5 . The reader can also +see pictures of the Polya case using my software, and my software gives a +rigorous positivity case in the Polya case just as for the other cases. +21 + +4 +Main Theorem: Proof of Lemma A1 +4.1 +Odd and Even Planar Configurations +We call a pair of points �p, �q ∈ S2 far if ∥�p − �q∥ ≥ 4/ +√ +5. Note that (�p, �q) is +a far pair if and only if (�q, �p) is a far pair. Our rather strange definition has +a more natural interpretation in terms of the planar avatars. If we rotate S2 +so that �p = (0, 0, 1) then q = Σ(�q) lies in the disk of radius 1/2 centered at +the origin if and only if (�p, �q) is a far pair. +We say that a point in a 5-point configuration is odd or even according +to the parity of the number of far pairs it makes with the other points in +the configuration. +Correspondingly, define the parity of the avatar to be +the parity of the number of points which are contained in the closed disk of +radius 1/2 about the origin. This definition extends our definition for the +TBP avatars. Here we repeat Figure 3.1 for convenience. +0 +1 +2 +3 +0 +2 +3 +1 +0 +2 +even +odd +both +Figure 3.1: Even and odd avatars of the TBP. +We call 2 planar configurations isomorphic if they are the avatars of +isometric configurations on the sphere. Thus, for instance, the odd and even +avatars of the TBP are isomorphic. Every planar configuration is isomorphic +to an even configuration. To see this, we form a subgraph of the complete +graph by joining two points in a 5-point configuration by an edge if and only +if they make a far pair. As for any graph, the sum of the degrees is even. +Hence there is some vertex having even degree. When we rotate so that this +vertex is (0, 0, 1), the avatar is even. +The definition of even planar configurations (or something similar which +would play the same role) is an important part of our computational proof +of Lemma A1. By focusing on the even planar configurations, and further +using symmetry, we arrive at a configuration space where there is just one +avatar of the TBP. +22 + +4.2 +The Domains +Now we will use the concept of an even planar configuration to define the +main domains we will use in our calculation. +The Relevant Domains: Given a planar configuration ξ = (p0, p1, p2, p3), +we write pk = (pk1, pk2). We define a domain Ω ⊂ R7 to be the set of planar +configurations ξ satisfying the following conditions. +1. ξ is an even configuration. +2. ∥p0∥ ≥ max(∥p1∥, ∥p2∥, ∥p3∥). +3. p12 ≤ p22 ≤ p32 and p22 ≥ 0. +4. p01 ∈ [0, 2] and p01 = 0. +5. pj ∈ [−3/2, 3/2]2 for j = 1, 2, 3. +6. min(p1k, p2k, p3k) ≤ 0 for k = 1, 2. +We define Ω♭ (to be used specially with G♭ +5) by the same conditions except +that we leave off Condition 6. +Closed Versus Open Conditions: If we want to check for inclusion in +the interior of Ω or Ω♭, all the inequalities above must be strict. We find it +useful to work with the interior of Ω and Ω♭ because we won’t need to spe- +cially treat some boundary cases. This will make for a cleaner calculation. +A Tiny Cube: We cannot make a finite calculation if we wish to treat +configurations arbitrarily close to the TBP avatar ξ0 ∈ Ω. We have to make +a cutoff and deal separately with points very near ξ0. When we string out +the points of ξ0, the coordinates we get are +1, 0 +0, − +√ +3/3, +−1, 0, +0, +√ +3/3. +(17) +Here p0 = (1, 0) and p1 = (0, − +√ +3/3), etc. Again, see Figure 3.1. We let Ω0 +denote the cube of side-length 2−17 centered at ξ0. This is our cutoff. +23 + +4.3 +Reduction to Simpler Lemmas +Recall that we mean EF(ξ) to be the F-potential of the 5-point configuration +on the sphere corresponding to a planar avatar ξ. Lemma A1 makes 3 claims: +1. When F is any of G4, G♭ +5, G6, the TBP avatars are the unique minimizer +for EF. +2. When F = G♯ +10, the TBP avatars are the unique minimizers for EF +among configurations which are not isomorphic to ones in Υ. +3. When F = G♯♯ +10, the TBP avatars have smaller EF value than all con- +figurations in Υ. +Lemma 4.1 (A11) Let F be any of G4, G♭ +5, G6, G♯ +10. Then ξ0 is the unique +minimizer for EF inside Ω0. +Lemma 4.2 (A12) The following is true: +1. Let F = G4, G6, G♯ +10. If ξ is not equivalent to any planar configuration +in Ω then then ξ does not minimize EF. +2. Let F = G♭ +5. If ξ is not equivalent to any planar configuration in Ω♭ +then then ξ does not minimize EF. +Let [F] be the EF value of the TBP avatars. +Lemma 4.3 (A13) The following is true. +1. The infimum of EG4 on interior(Ω) − Ω0 is at least [G4] + 2−50. +2. The infimum of EG6 on interior(Ω) − Ω0 at at least [G6] + 2−50. +3. The infimum of EG♭ +5 on interior(Ω) − Ω0 is at least [G♭ +5] + 2−50. +4. The infimum of EG♯ +10 on interior(Ω) − Υ − Ω0 is at least [G♯ +10] + 2−50. +5. The infimum of EG♯♯ +10 on Υ is at least [G♯♯ +10] + 2−50. +Lemma A13 is the main calculation. +It follows from continuity that +Lemma A13 remains true if we replace the interior of Ω by Ω itself. But +then Lemma A1 follows immediately from Lemma A11 (§4.4), Lemma A12 +(§4.5), and Lemma A13 (§5). Our choice of 2−50 is somewhat arbitrary. +24 + +4.4 +Proof of Lemma A11 +Recall that Ω0 is the cube of side length 2−17 centered at ξ0. For all our +choices of F, the function EF is a smooth function on R7. +Lemma 4.4 (A111) The gradient of EF vanishes at ξ0. +Proof: We make a direct calculation in all cases. ♠ +Lemma 4.5 (A112) The Hessian of EF, meaning the matrix of second par- +tial derivatives, is positive definite at every point of Ω0. +Let ξ ∈ Ω0 be other than ξ0. Lemmas A111 and A112 (§6) imply that +the restriction of EF to the line segment γ joining ξ0 to ξ is convex and has +0 derivative at ξ0. Hence EF(ξ) > EF(ξ0). This proves Lemma A11 +4.5 +Proof of Lemma A12 +Recall that ξ0 is the avatar of the TBP. Let [F] = EF(ξ0). Since the TBP +has 6 bonds of length +√ +2, and 3 of length +√ +3, and 1 of length +√ +4, we have +[Gk] = 6 × 2k + 3. Using this result, and Equation 13, we compute +[G4] = 99, +[G6] = 387, +[G♭ +5] = −180, +[G♯ +10] = 10518. +(18) +Let ξ = p0, p1, p2, p3 some other planar configuration. +Lemma 4.6 (A121) Let F = G4, G6, G♯ +10. +If ∥p0∥ > 2 then ξ does not +minimize EF. Also, if ∥p0∥ > 3/2 then ξ does not mininizer EG♭ +5. +Proof: Let rj = ∥Σ−1(pj), (0, 0, 1)∥. If ∥p0∥ > 2 then r0 < d = 2/ +√ +5. We +compute that +G4(d) > 104 > [G4], +G6(d) > 1073 > [G6], +G♯ +10(d) > 117642 > [G♯ +10]. +Since F is monotone decreasing and non-negative, just the one term in EF(ξ) +is larger than [F]. +We have [G♭ +5] = −180. Using 3/2 in place of 2 we get r0 > d′ = 4/ +√ +13 in +place of d. We check that G♭ +5(r) > 93 for r ∈ [0, d′] and G♭ +5 > −30 in [0, 2]. +Hence EG♭ +5(P) > 93 − 270 > [G♭ +5]. ♠ +25 + +Lemma 4.7 (A122) Let F = G4, G♭ +5, G6, G♯ +10. +Suppose ∥p0∥ ≥ 3/2 and +∥pj∥ > 3/2 for some j = 1, 2, 3. Then ξ does not minimize EF. +Proof: +We use the same notation from the previous proof. +We already +proved a stronger result for G♭ +5, so we let F be one of the other functions +listed. If ∥p0∥, ∥pj∥ > 3/2 then we have r0, rj < d′ = 4/ +√ +13. We compute +2G4(d′) > 117 > [G4], +2G6(d′) > 901 > [G6], +2G♯ +10(d′) > 58319 > [G♯ +10]. +Since F is monotone decreasing and non-negative, just these two terms in +EF(ξ) together are larger than [F]. ♠ +Lemma 4.8 (A123) Let F be any strictly monotone decreasing potential. +If min(p1k, p2k, p3k) > 0 for one of k = 1, 2 then ξ does not minimize EF. +Proof: The conditions imply that the 5-point configuration in S2 is con- +tained in a hemisphere H, and at least 3 of the points are in the interior of +H. If we take one of these interior points and reflect it across ∂H then we +increase at least 2 of the distances in the configuration and keep the rest the +same. ♠ +Assume ξ is a minimizer for EF. As we discussed in §4.1, we can normal- +ize so that ξ is an even configuration. Reordering p0, p1, p2, p3 and rotating, +about the origin, we make ∥p0∥ ≥ ∥pi∥ for i = 1, 2, 3 and we move p0 into the +positive x-axis. Reflecting in the x-axis if necessary and reordering the points +p1, p2, p3 if necessary, we arrange that p12 ≤ p22 ≤ p32 and p22 ≥ 0. Lemmas +A121 and A122 tell us that, in all cases, p01 ∈ [0, 2] and pj ∈ [−3/2, 3/2]2 +for j = 1, 2, 3. +We have also arranged that p02 = 0. +For F = G♭ +5 we +have nothing left to check. Otherwise, Lemma A123 shows that ξ satisfies +min(p1k, p2k, p3k) ≤ 0 for k = 1, 2, 3. +Remark: (The Polya Case) For G3, we need to take a number a bit larger +than 2 in Lemma A121. The constant 4 works easily. We specially sepa- +rate out the case of G3 in our code so that we are sure to use the larger +domain. Lemma A122 also works for G3 but we need to work harder: We +have [G5] = 51 but 2G3(d′) ∈ [42, 43]. However, the G3-potential of the 4 +point configuration on S2 not involving (0, 0, 1) is at least that of the regular +tetrahedron, 14+ 2 +9. Since 14+42 > 51 we get the same conclusion as Lemma +A122 for G3. +26 + +5 +Main Calculation: Lemma A13 +5.1 +Blocks +We first list the ingredients in our main calculation and then explain the +calculation itself. +Dyadic Subdivision: The dyadic subdivision of a D-dimensional cube is +the list of 2D cubes obtained by cutting the cube in half in all directions. We +sometimes blur this notation and say that any one of these 2D smaller cubes +is a dyadic subdivision of the big cube. +Blocks: We define a block to be a product of the form +B = Q0 × Q1 × Q2 × Q3 ⊂ □ := [0, 2] × [−2, 2]2 × [−2, 2]2 × [−2, 2]2. (19) +where Q0 is a segment and Q1, Q2, Q3 are squares, each obtained by iterated +dyadic subdivision of [0, 2] or [−2, 2]2. +We call B acceptable if Q0 has length at most 1 and Q1, Q2, Q3 have +sidelength at most 2. If B is not acceptable we let the offending index be +the lowest index where the condition fails. +The kth subdivision of a block amounts to performing dyadic subdivi- +sion to the kth factor and leaving the others alone. We call these operations +S0, S1, S2, S3. Thus S0 cuts B into two pieces and each other Sk cuts B into 4 +pieces. We let Sk(B) denote the list of the blocks obtained by performing Sk +on B. All the blocks our algorithm produces come from iterated subdivision +of □. +Rational Block Calculations: We say that a rational block computation +is a finite calculation, only involving the arithmetic operations and min and +max. The output of a rational block computation will be one of two things: +yes, or an integer. A return of an integer is a statement that the computa- +tion does not definitively answer to the question asked of it. If the integer +is −1 then there is no more information to be learned. If the integer lies +in {0, 1, 2, 3} we use this integer as a guide in our algorithm. For example, +we might ask if the block is acceptable. If not, then we would return the of- +fending index, and our algorithm would subdivide the block along this index. +27 + +5.2 +The Main Calculation +Recall that +ξ0 = (1, 0, − +√ +3/3, −1, 0, 0, +√ +3/3) ∈ Ω +is the avatar of the TBP and Ω0 is the cube of side length 2−17 around ξ0. +Recall also that Υ is the special domain defined in §3.1. +Lemma 5.1 (A131) There exists a rational block computation C1 such that +an output of yes for a block B implies that B ⊂ Ω0. +Lemma 5.2 (A132) There exists a rational block computation C3 such that +an output of yes for an acceptable block B implies that B is disjoint from +the interior of Ω. The same goes for Ω♭. +Lemma 5.3 (A133) There exists a rational block computation C♯ +3 such that +an output of yes for a block B implies that B ⊂ Υ. Likewise, there exists +a rational block computation C♯♯ +3 such that an output of yes for a block B +implies that B is disjoint from Υ. +The proofs of the Lemmas A131, A132, A133, given below, just amount to +checking the conditions in a fairly straightforward way. The final ingredient +is the main ingredient. It is much more involved. All the energy potentials +we consider are what we call energy hybrids. They have the form +F = +m +� +k=1 +ckGk, +Gk(r) = (4 − r2)k, +c1 ∈ Q, +c2, ..., ck ∈ Q+. +(20) +With some modification of Lemma E below we could also handle the case +when some of c2, ..., ck are negative. See Remark (1) after the statement of +Lemma E. +Lemma 5.4 (A134) For any function F given by Equation 20, there exists +a rational block computation C3,F such that an output of yes for an acceptable +block B implies that the minimum of EF on B is at least EF(ξ0) + 2−50. +Otherwise C3,F(B) is an integer in {0, 1, 2, 3}. +Here is the main calculation. +1. We start with the list L = {□}. +28 + +2. If L = ∅ then HALT. Otherwise let B = Q0 × Q1 × Q2 × Q3 be the +last block of L. +3. If B is not acceptable we delete B from L and append to L the subdi- +vision of B along the offending index. We then return to Step 2. Any +blocks considered beyond this step are acceptable. +4. If C1(B) = yes or C2(B) = yes we remove B from L and go to Step +2. Here we are eliminating blocks disjoint from the interior of Ω or else +contained in Ω0. +5. If F = G♯ +10 and C♯ +3(B) = yes we remove B from L and go to Step 2. If +F = G♯♯ +10 and C♯♯ +3 (B) = yes we remove B from L and go to Step 2. +6. If C4,F(B) = yes then we remove B from L and go to Step 2. Here we +have verified that the F-energy of any avatar in B exceeds [F] + 2−50. +7. If C4,F(B) = k ∈ {0, 1, 2, 3} then we delete B from L and append to L +the blocks of the subdivision Sk(B) and return to step 2. +If the algorithm reaches the HALT state for a given choice of F, this +constitutes a proof that the corresponding statement of Lemma A13 is true. +Lemma 5.5 (A135) The Main Computation reaches the HALT state for +each choice of F listed in Lemma A13. +Lemma A13 follows from Lemma A131 (§5.3), Lemma A132 (§5.4), Lemma +A133 (§5.5), Lemma A134 (§5.6) and Lemma A135 (§5.8). +5.3 +Proof of Lemma A131 +Define intervals I0, I1, I√ +3/3 such that +I0 = [−2−17, 2−17], +I1 = [1 − 2−17, 1 + 2−17] +230I√ +3/3 = [619916940, 619933323] (21) +I√ +3/3 is a rational interval that is just barely contained inside the interval +of length 2−17 centered at +√ +3/3. Define +Ω00 = (I1 × {0}) × (I0 × −I√ +3/3) × (−I1 × I0) × (I0 × I√ +3/3). +(22) +We have Ω00 ⊂ Ω0, though just barely. There are 128 vertices of B. We +simply check whether each of these vertices is contained in Ω00. If so then +we return yes. In practice our program scales up all the coordinates by 230 +so that this test just involves integer comparisons. +29 + +5.4 +Proof of Lemma A132 +Let B = Q0×Q1×Q2×Q3 be an acceptable block. These blocks are such that +the squares Q1, Q2, Q3 do not cross the coordinate axes. For such squares, +the minimum and maximum norm of a point in the square is realized at a +vertex. Thus, we check that a square lies inside (respectively outside) a disk +of radius r centered at the origin by checking that the square norms of each +vertex is at most (respectively at least) r2. +We check whether there is an index j ∈ {1, 2, 3} such that all vertices of +Qj have norm at least max Q0. We return yes if this happens, because then +all configurations in the interior of B will have some pj with ∥pj∥ > ∥p0∥. +We check whether there is an index j ∈ {1, 2, 3} such that all vertices +of Qj have norm at least 3/2. If so, we return yes. If this happens then +∥p0∥, ∥pj∥ > 3/2 for all configurations in the interior of B. +We count the number a of indices j such that the vertices of Qj all have +norm at most 1/2. We then count the number b of indices j such that all +vertices of Qj have norm at least 1/2. We return yes if a is odd and a+b = 4. +In this case, every configuration in the interior of B is even. +We write I ≤ J to indicate that all values in an interval I are less or +equal to all values in an interval J. We also allow I and J to be single points +in this notation. For each j = 0, 1, 2, 3 we let Qjk be the projection of Qj +onto the kth factor. Thus Qj1 and Qj2 are both line segments in R. +We return yes for any of the following reasons. +• If Qjk ≤ −3/2 or Qjk ≥ 3/2 for any j = 1, 2, 3 and k = 1, 2. +• Q12 ≥ Q22 or Q12 ≥ Q32 or Q22 ≥ Q32 or Q22 ≤ 0. +• Qj1 ≥ 0 for j = 1, 2, 3 or Qj2 ≥ 0 for j = 1, 2, 3. +If any of these things happens, all configurations in Q violate some condition +for membership in the interior of Ω. We don’t check the last item for Ω♭. ♠ +5.5 +Proof of Lemma A133 +For C♯ +3 we return yes if all the vertices of B lie in Υ. For C♯♯ +3 we return +yes if one of the factors of B is disjoint from the corresponding factor of Υ. +This amounts to checking whether a pair of rational squares in the plane are +disjoint. We do this using the projections defined for Lemma A132. +30 + +5.6 +Proof of Lemma A134 +We say that an acceptable block B = Q0 × Q1 × Q2 × Q3 is good if we +have Qj ∈ [−3/2, 3/2]2 for all j = 1, 2, 3. We first test whether B is a good +block. If not, we return the lowest index i such that Qi has a vertex outside +[−3/2, 3/2]2. Otherwise we proceed with the main calculation. +We let Q denote the set of components of good blocks – either segments +or squares. We also let {∞} be a member of Q. We first define some mea- +surements we take of members in Q. +0. The Flat Approximation: Let Σ−1 be inverse stereographic projection, +as in Equation 8. Given Q ∈ Q we define +Q• = Convex Hull(Σ−1(v(Q)). +(23) +The set Q• is either the point (0, 0, 1), a chord of S2 or else a convex planar +quadrilateral with vertices in S2 that is inscribed in a circle. We let d• be +the diameter of Q•. The quantity d2 +• is a rational function of the vertices of Q. +1. The Hull Approximation Constant: We think of Q• as the linear +approximation to +�Q = Σ−1(Q). +(24) +The constant we define here turns out to measure the distance between �Q +and Q•. When Q = {∞} we define δ(Q) = 0. Otherwise, let +χ(D, d) = d2 +4D + (d2)2 +4D3 . +(25) +This wierd function turns out to be an upper bound to a more geometrically +meaningful non-rational function that computes the distance between an +chord of length d of a circle of radius D and the arc of the circle it subtends. +When Q is a dyadic segment we define +δ(Q) = χ(2, ∥�q1 − �q2∥). +(26) +Here q1, q2 are the endpoints of Q. When Q is a dyadic square we define +δ(Q) = max(s0, s2) + max(s1, s3), +sj = χ(1, ∥qj − qj+1∥). +(27) +Here q1, q2, q3, q4 are the vertices of Q and the indices are taken cyclically. +These are rational computations because χ(2, d) is a polynomial in d2. +31 + +2. +The Dot Product Estimator: By way of motivation, we point out +that if V1, V2 ∈ S2 then Gk(∥V1 − V2∥) = (2 + 2V1 · V2)k. +Now suppose that Q1 and Q2 are two dyadic squares. We set δj = δ(Qj). +Given any p ∈ R2 ∪ ∞ let �p = Σ−1(p). Define +Q1 · Q2 = max +i,j (�q1i · �q2j) + (τ) × (δ1 + δ2 + δ1δ2). +(28) +Here {q1i} and {q2j} respectively are the vertices of Q1 and Q2. The constant +τ is 0 if one of Q1 or Q2 is {∞} and otherwise τ = 1. Finally, we define +T(Q1, Q2) = 2 + 2(Q1 · Q2). +(29) +3. The Local Error Term: For Q1, Q2 ∈ Q and k ≥ 1 we define +ǫk(Q1, Q2) = 1 +2k(k − 1)T k−2d2 +1 + 2kT k−1δ1, +(30) +where +d1 = d•(Q1), +δ1 = δ(Q1), +T = T(Q1, Q2). +One of the terms in the error estimate comes from the analysis of the flat +approximation and the second term comes from the analysis of the difference +between the flat approximation and the actual subset of the sphere. The +quantity is not symmetric in the arguments and ǫk({∞}, Q2) = 0. +4. +The Global Error Estimate: Given a block Q0 × Q1 × Q2 × Q3 +we define +ERRk(B) = +N +� +i=0 +ERRk(B, i), +ERRk(B, i) = +� +j̸=i +ǫ(Qi, Qj). +(31) +More generally, when F = � ckGk is as in Equation 20, we define +ERRF(B) = +N +� +k=0 +ERRF(B, i), +ERRF(B, i) = +� +|ck| ERRk(B, i) +(32) +Now we state the main error estimate, proved in §7. For the most part +we only care about the (+) case of the lemma. We only need the (−) case +when we deal with the potential G5 − 25G1. +32 + +Lemma 5.6 (E) Let B be a good block. Let F = Gk for any k ≥ 1 or +F = −G1. Then +min +p∈B EF(v) ≥ min +p∈v(B) Ek(v) − ERRk(B) +Proof of Lemma A134: Now we can prove lemma A134. Let B be an +acceptable block. Once again, we mention that we immediately return an +integer if our block B is not a good block. So, assume B is a good block. +Let F be an energy hybrid. Let [F] denote the F-potential of the TBP. If +min +p∈v(B) EF(v) − ERRk(B) ≥ [F] + 2−50 +(33) +we return yes. Otherwise we return the index i such that ERRF(B, i) is the +largest. (In the case of a tie, which probably never happens, we would pick +the lowest such index.) ♠ +Remarks: (1) Lemma E is true more generally for F = ±Gk but we do not +need the general result and so we ignore it. Keeping track of the minus sign +all the time greatly increases the chances of making a sign error and we don’t +want to fool around with this. +(2) The integer we return is designed to be a recommendation for the subdivi- +sion that is most likely to speed the computation along. We try to subdivide +in such a way as to decrease the error term as fast as possible. +5.7 +Discussion of the Implementation +Representing Blocks: We represent the coordinates of blocks by longs, +which have 31 digits of accuracy. What we list are 230 times the coordinates. +Our algorithm never does so many subdivisions that it defeats this method +of representation. In all but the main step (Lemma A134) in the algorithm +below we compute with exact integers. When the calculation (such as squar- +ing a long) could cause an overflow error, we first recast the longs as a +BigIntegers in Java and then do the calculations. +Interval Arithmetic: For the main step of the algorithm we use inter- +val arithmetic. We use the same implementation as we did in [S1], where we +explain it in detail. Here is how it works in brief. If we have a calculation +involving numbers r1, ..., rn, and we produce intervals I1, ..., In with dyadic +33 + +rational numbers represented exactly by the computer such that ri ∈ Ii for +i = 1, ..., n. We then perform the usual arithmetic operations on the inter- +vals, rounding outward at each step. The final output of the calculation, in +interval, contains the result of the actual calculation. +In our situation here, the numbers r1, ..., rn are, with one exception, +dyadic rationals. (The exception is that the coordinates of the point rep- +resenting the TBP are quadratic irrationals.) In principle we could do the +entire computation, save for this one small exception, with expicit integer +arithmetic. However, the complexity of the rationals involved, meaning the +sizes of their numerators and denominators, qets quite large this way and the +calculation is too slow. +One way to think about the difference between our explicitly defined ex- +act integer arithmetic and interval arithmetic is that the integer arithmetic +interrupts the calculation at each step and rounds outward so as to keep the +complexity of the rational numbers from growing too large. +Guess and Check: Here is how we speed up the calculation. When we +do Steps 6-7, we first do the calculation C4,F using floating point operations. +If the floating version returns an integer, we use this integer to subdivide the +box and return to step 2. If C4,F says yes then we retest the box using the +interval arithmetic. In this way, we only pass a box for which the interval +version says yes. This way of doing things keeps the calculation rigorous but +speeds it up by using the interval arithmetic as sparingly as possible. +Parallelization: We also make our calculation more flexible using some +parallelization. +We classify each block B = Q0 × Q1 × Q2 × Q3 with a +number in {0, ..., 7} according to the formula +type(B) = σ(c01 − 1) + 2σ(c11) + 4σ(c31) ∈ {0, ..., 7}. +Here cj1 is the first coordinate of the center of Bj and σ(x) is 0 if x < 0 and +1 if x > 0. Step 3 of our algorithm guarantees that σ(·) is always applied to +nonzero numbers. +We wrote our program so that we can select any subset S ⊂ {0, ..., 7} we +like and then (after Step 3) automatically pass any block whose type is not +in S. Running the algorithm in parallel over sets which partition {0, ..., 7} is +logically the same as running the basic algorithm without any parallelization. +To be able to do the big calculations in pieces, we run the program for various +subsets of {0, ..., j}, sometimes in parallel. +34 + +5.8 +Proof of Lemma A135 +Here I give an account of one time I ran the computations to completion +during January 2023. I used a 2017 iMac Pro with a 3.2 GHz Intel Zeon W +processor, running the Mojave operating system. I ran the programs using +Java 8 Update 201. (The Java version I use is not the latest one. The graph- +ical parts of my program use some methods in the Applet class in a very +minor but somehow essential way that I find hard to eliminate.) In listing +the calculations I will give the approximate time and the exact number of +blocks passed. Since we use floating point calculations to guide the algo- +rithm, the sizes of the partitions can vary slightly with each run. +For G4 : 2 hrs 14 min, 10848537 blocks. +For G6: 5 hr 11 min, 25159337 blocks. +For G♭ +5 types 1&2: 2 hr 31 min, 6668864 blocks. +For G♭ +5 types 3&4: 1 hr 55 min, 4787489 blocks. +For G♭ +5 types 5&6: 5 hr 33 min, 14160332 blocks. +For G♭ +5 types 7&8: 3 hr 49 min, 9219550 blocks. +For G♯ +10 type 1: 4 hr 23 min, 6885912 blocks. +For G♯ +10 type 2: 9 hr 47 min, 15982122 blocks. +For G♯ +10 type 3: 3 hr 47 min, 5872029 blocks. +For G♯ +10 type 4: 7 hr 59 min, 13475260 blocks. +For G♯ +10 type 5: 8 hr 30 min, 13313492 blocks. +For G♯ +10 type 6: 15 hr 16 min, 24110457 blocks. +For G♯ +10 type 7: 5 hr 19 min, 7862780 blocks. +For G♯ +10 type 8: 8 hr 33 min, 13478467 blocks. +For G♯♯ +10 (on the domain Υ): 28 minutes, 805242 blocks. +35 + +6 +Local Analysis: Proof of Lemma A112 +6.1 +Reduction to Simpler Statements +We set L=A122, so that we are trying to prove Lemma L. We consider F to +be any of the 4 functions +G4, +G6, +G♭ +5 = G5 − 25G1, +2−5G♯ +10 = 2−5(G10 + 13G5 + 68G2). +Scaling the last function by 2−5 makes our estimates more uniform. +Recall that Ω0 is the cube of side length 2−17 centered at the point +ξ0 = +� +1, 0, −1 +√ +3 +, −1, 0, 0, 1 +√ +3 +� +∈ R7 +(34) +In general, the point (x1, ..., x7) represents the planar avatar +p0 = (x1, 0), p1 = (x2, x3), p2 = (x4, x5), p3 = (x6, x7). +(35) +The quantity EF(x1, ..., x7) is the F-potential of the 5-point configuration +associated to the planar avatar under inverse stereographic projection Σ−1. +EF(x1, ..., x7) = +� +i 0 for all +points ξ ∈ Ω0. +Lemma 6.2 (L2) M3(EF) < 212λ(HEF(ξ0))) in all cases. +36 + +6.2 +Proof of Lemma L1 +Let +H0 = HEF(ξ0), +H = HEF(ξ), +∆ = H − H0. +(38) +For any real symmetric matrix X define the L2 matrix norm: +∥X∥2 = +�� +ij +X2 +ij = sup +∥v∥=1 +∥Xv∥. +(39) +Given a unit vector v ∈ R7 we have H0v · v ≥ λ. Hence +Hv · v = (H0v + ∆v) · v ≥ H0v · v − |∆v · v| ≥ λ − ∥∆v∥ ≥ λ − ∥∆∥2 > 0. +So, to prove Lemma L1 we just need to establish the implication +M3 < 212λ(H0) +=⇒ +∥∆∥2 < λ(H0). +Let t → γ(t) be the unit speed parametrized line segment connecting p0 to +p in Ω0. Note that γ has length L ≤ +√ +7×2−18. We write γ = (γ1, ..., γ7). Let +Ht denote the Hessian of EF evaluated at γ(t). Let Dt denote the directional +derivative along γ. +Now ∥Dt(Ht)∥2 is the speed of the path t → Ht in R49, and ∥∆∥2 is the +Euclidean distance between the endpoints of this path. Therefore +∥∆∥2 ≤ +� L +0 +∥Dt(Ht)∥2 dt. +(40) +Let (Ht)ij denote the ijth entry of Ht. From the definition of directional +derivatives, and from the Cauchy-Schwarz inequality, we have +(DtHt)2 +ij = +� +7 +� +k=1 +dγk +dt +∂Hij +∂k +�2 +≤ 7M2 +3. +∥Dt(Ht)∥2 ≤ 73/2M3. +(41) +The second inequality follows from summing the first one over all 72 pairs +(i, j) and taking the square root. Equation 40 now gives +∥∆∥2 ≤ L × 73/2M3 = 49 × 2−18M3 < 2−12M3 < λ(H0). +(42) +This completes the proof of Lemma L1. +37 + +6.3 +Proof of Lemma L2 +Let F be any of our functions. Let H0 = HEF(ξ0). +Lemma 6.3 (L21) λ(H0) > 39. +Proof: Let χ be the characteristic polynomial of H0. This turns out to be +a rational polynomial. We check in Mathematica that the signs of the coef- +ficients of χ(t + 39) alternate. Hence χ(t + 39) has no negative roots. The +file we use is LemmaL21.m. ♠ +Recalling that ξ0 ∈ R7 is the point representing the TBP, we define +µN(EF) = sup +|I|=N +|∂IEF(ξ0)|. +(43) +Lemma 6.4 (L22) For any of our functions we have the bound +µ3 < 45893, +(7 × 2−18)j +j! +µj+3 < 38, +j = 1, 2, 3. +(44) +Proof: We compute this in Mathematica. The file we use is LemmaL22.m. ♠ +Lemma 6.5 (L23) For any of our functions we have the bound +(7 × 2−18)4 +4! +M7 < 2354. +Lemma 6.6 (L24) We have +M3 ≤ µ3 + +3 +� +j=1 +(7 × 2−18)j +j! +µj+3 + (7 × 2−18)4 +4! +M7 +(45) +Proof: Choose any multi-index J with |J| = 3. Let γ be the line segment +connecting ξ0 to any ξ ∈ Ω. We parametrize γ by unit speed and furthermore +set γ(0) = ξ0. Let +f(t) = ∂JEF ◦ γ(t). +38 + +The bound for |MJ| follows from Taylor’s Theorem with remainder once we +notice that +0 ≤ t ≤ +√ +7 × 2−18, +���∂nf(0) +∂tn +��� ≤ ( +√ +7)nµn +���∂nf +∂tn +��� ≤ ( +√ +7)nMn. +Since this works for all J with |J| = 3 we get the same bound for M3. ♠ +Lemmas L21 - L23 and Equation 44 imply +M3 < 45893 + 3 × 38 + 2354 ≤ 65536 = 216 ≤ 212λ(H0). +This completes the proof of Lemma L2. +6.4 +Proof of Lemma L23 +Now we come to the interesting part of the proof, the one place where we +need to go beyond specific evaluations of our functions. When r, s ≥ 0 and +r + s ≤ 2d we have +sup +(x,y)∈R +2 +xrys +(1 + x2 + y2)d ≤ (1/2)min(r,s). +(46) +One can prove Equation 46 by factoring the expression into pieces with +quadratic denominators. Here is a more general version. Say that a function +φ : R4 → R is nice if it has the form +� +i +Ciaαibβicγidδi +(1 + a2 + b2)ui(1 + c2 + d2)vi , αi, βi, γi, δi ≥ 0, +αi+βi ≤ 2ui, +γi+δi ≤ 2vi. +It follows from Equation 46 that +sup +R +4 |φ| ≤ ⟨φ⟩, +⟨φ⟩ = +� +i +|Ci|(1/2)min(αi,βi)+min(γi,δi). +(47) +Equation 47 is useful to us because it allows us to bound certain kinds of +functions without having to evaluate then anywhere. We also note that if +φ is nice, then so is any iterated partial derivative of φ. Indeed, the nice +functions form a ring that is invariant under partial differentiation. This fact +makes it easy to identify nice functions. +39 + +For any φ : Rn → R we define +M 7(ψ) = sup +|J|=7 +M J(ψ), +M J(ψ) = sup +ξ∈R +n |∂J(φ)|. +(48) +We obviously have +M7(EF) ≤ M 7(EF). +(49) +Recall that �p = Σ−1(p), the inverse stereographic image of p. Define +f(a, b) = 4 − ∥� +(a, b) − (0, 0, 1)∥2 = 4(a2 + b2) +1 + a2 + b2. +(50) +g(a, b, c, d) = 4 − ∥� +(a, b) − � +(c, d)∥2 = 4(1 + 2ac + 2bd + (a2 + b2)(c2 + d2)) +(1 + a2 + b2)(1 + c2 + d2) +. (51) +Notice that g is nice. Hence gk is nice and ∂Igk is nice for any multi-index. +That means we can apply Equation 47 to ∂Igk. +EGk is a 10-term expression involving 4 instances of f k and 6 of gk. How- +ever, each variable appears in at most 4 terms. So, as soon as we take a +partial derivative, at least 6 of the terms vanish. Moreover, ∂If is a limiting +case of ∂Ig for any multi-index I. From these considerations, we see that +M7(EGk) ≤ 4 × M 7(gk). +(52) +The function ∂I(gk) is nice in the sense of Equation 47. Therefore +4 × M 7(gk) ≤ 4 × max +|I|=7 ⟨∂Igk⟩. +(53) +Using this estimate, and the Mathematica file LemmaL23.m, we get +max +k∈{1,2,3,4,5,6} +(7 × 2−18)4 +4! +× 4 × M 7(gk) ≤ +1 +1000. +2−5 × (7 × 2−18)4 +4! +× 4 × M 7(g10) ≤ 2353. +(54) +The bounds in Lemma L23 follow directly from Equation 52 and from the +definitions of our functions. +Remark: The Polya Case. +The analysis above works easily for G3. +In +this case, the minimum eigenvalue satisfies λ0 > 14. +The bounds satisfy +µ3 ≤ 316 and µj < 1 for j = 4, 5, 6. Again, M7 < 1/1000 in this case. +40 + +7 +Error Estimate: Proof of Lemma E +7.1 +Guide to the Proof +Lemma E is stated in §5.6. It is the main error estimate that feeds into +Lemma A134, which in turn feeds into Lemma A13, our main computation. +Our proof of Lemma E splits into two halves, an algebraic part and a +geometric part. +The algebraic part, which we do in this chapter, has no +geometric content. It simply promotes a “local” result to a “global result”. +The geometric part, done in the next chapter, explains the meaning of the +local error term ǫk(Q1, Q2) for Q1, Q2 ∈ Q. Here Q is the space of components +of good blocks, and also the point ∞. +The algebraic part promotes the +information about ǫk(Q1, Q2), which we use as a black box in this chapter, +to the information about the global sum given in Lemma E. +The secret to the algebraic part is what we call an averaging system. For +these purposes, we will treat every member of Q as a quadrilateral by the +trick of repeating vertices. Thus, if we have a dyadic segment with vertices +q1, q2 we will list them as q1, q1, q2, q2. For the point {∞} we will list the +single vertex q1 = ∞ as q1, q1, q1, q1. We do this so as to give a uniform +description without having to stop each time and say what we are doing in +each case. +We say that an averaging system for a member of Q is a collection of +maps λ1, λ2, λ3, λ4 : Q → [0, 1] such that +4 +� +i=1 +λi(z) = 1, +∀ z ∈ Q. +One might also call this a partition of unity. The functions need not vary +continously. In case Q is a segment, we would have λ1 = λ2 and λ3 = λ4. In +case Q = {∞} we would have λj = 1/4 for j = 1, 2, 3, 4. +We say that an averaging system for Q is a choice of averaging system +for each member Q of Q. The averaging systems for different members need +not have anything to do with each other. In this chapter we will posit some +additional properties of an averaging system and then prove Lemma E under +the assumption such such an averaging system exists. In the next chapter +we will use geometric considerations to prove the existence of the desired +averaging system. +41 + +7.2 +Reduction to a Local Result +We fix the function F = Gk for some k ≥ 1 or else F = −G1. Lemma E1 +is true more generally for F = ±Gk but we don’t need the result in this +generality. +We write E = EF. We let ǫ = ǫk, as in Equation 30. +Our algebraic +argument would work for any choice of F, but we need F = Gk to actually +get the averaging system we need. Let q1,1, q1,2, q1,3, q1,4 be the vertices of Q1. +Lemma 7.1 (E1) There exists an averaging system on Q with the following +property: Let Q1, Q2 be distinct members of Q. +Given any z1 ∈ Q1 and +z2 ∈ Q2 we have +4 +� +i=1 +λi(z1)F(∥�q1,i − �z2∥) − F(∥�z1 − �z2∥) ≤ ǫ(Q1, Q2). +(55) +See §8 for the proof. +We call the averaging system in Lemma E1 an +efficient averaging system. We now explore the consequences of having an +efficient averaging system. +We suppose that we have the good dyadic block B = Q0 × ... × QN. The +vertices of B are indexed by a multi-index +I = (i0, ..., in) ∈ {1, 2, 3, 4}N+1. +Given such a multi-index, which amounts to a choice of vertex of in each +component member of the block. We define the energy of the corresponding +vertex configuration: +E(I) = E(q0,i0, ..., qN,iN) +(56) +Here is one more piece of notation. Given z = (z0, ..., zn) ∈ B and a +multi-index I we define +λI(z) = +N +� +i=0 +λij(zj). +(57) +Here λij is defined relative to the averaging system on Qj. +Now we are ready to state our main global result. The global result uses +the existence of an efficient averaging system. That is, it relies on Lemma +E1. +42 + +Lemma 7.2 (E2) Let z = (z0, ..., zN) ∈ B. Then +� +I +λI(z)E(I) − E(z) ≤ +N +� +i=0 +N +� +j=0 +ǫ(Qi, Qj). +(58) +The sum is taken over all multi-indices with the convention that ǫ(Qi, Qi) = 0 +for all i. +Now let us deduce Lemma E from Lemma E2. +Lemma 7.3 (E3) Lemma E1 implies Lemma E. +Proof: Notice that +� +I +λI(z) = +N +� +j=0 +� +4 +� +a=1 +λa(zj) +� += 1. +(59) +The minimum is always less or equal to a convex average. Hence +min +p∈v(B) E(v) ≤ +� +I +λI(z)E(I). +(60) +Choose some (z1, ..., zN) ∈ B which minimizes E. We have +0 ≤ min +p∈v(B) E(v) − min +v∈B E(v) = min +p∈v(B) E(v) − E(z) ≤∗ +� +I +λI(z)E(I) − E(z) ≤ +N +� +i=0 +N +� +j=0 +ǫ(Qi, Qj). +(61) +The starred inequality is Equation 60. The last expression is ERR(B) when +N = 4 and Q4 = ∞. ♠ +43 + +7.3 +From Local to Global +Now we deduce the global Lemma E2 from the local Lemma E1. +Lemma 7.4 (E4) Lemma E2 holds when N = 1. +Proof: In this case, we have a block B = Q0 × Q1. Setting ǫij = ǫ(Qi, Qj), +Lemma E1 gives us +F(∥z0 − z1∥) ≥ +4 +� +α=1 +λα(z0)F(∥q0α − z1∥) − ǫ01. +(62) +Applying Lemma E1 to the pair of points (z1, q0α) ∈ Q1 × Q0 we have +F(∥z1 − q0α∥) ≥ +4 +� +β=1 +λβ(z1)F(∥q1β − q0α∥) − ǫ10. +(63) +Plugging the second equation into the first and using � λα(z0) = 1, we have +F(∥z0 − z1∥) ≥ +� +α,β +λα(z0)[λβ(z1)F(∥q1β − q0α∥) − ǫ10] − ǫ01 = +� +α,β +λα(z0)λβ(z1)F(∥q1β − q0α∥) − (ǫ10 + ǫ01). +(64) +Equation 64 is equivalent to Equation 58 when N = 1. ♠ +Now we do the general case. +Lemma 7.5 (E5) Lemma E2 holds when N ≥ 2. +Proof: We rewrite Equation 64 as follows: +F(∥z0 − z1∥) ≥ +� +A +λA0(z0)λA1(z1) F(∥q0A0 − q1A1∥) − (ǫ01 + ǫ10). +(65) +The sum is taken over multi-indices A of length 2. +We also observe that +� +I′ +λI′(z′) = 1, +z′ = (z2, ..., zN). +(66) +44 + +The sum is taken over all multi-indices I′ = (i2, ..., iN). Therefore, if we hold +A = (A0, A1) fixed, we have +λA0(z0)λA1(z1) = +� +I′′ +λI′′(z). +(67) +The sum is taken over all multi-indices of length N + 1 which have I0 = A0 +and I1 = A1. Combining these equations, we have +F(∥z0 − z1∥) ≥ +� +I +λI(z)F(∥q0I0 − q1I1∥) − (ǫ01 + ǫ10). +(68) +The same argument works for other pairs of indices, giving +F(∥zi − zj∥) ≥ +� +I +λI(z)F(∥qiIi − qjIj∥) − (ǫij + ǫji). +(69) +Let us restate this as +Xij − Yij ≥ Zij, +where +Xij = +� +I +λI(z)F(∥qiIi − qjIj∥), +Yij = F(∥zi − zj∥), +Zij = ǫij + ǫji. +When we sum Yij over all i < j we get the second term in Equation 58. +When we sum Zij over all i < j we get the third term in Equation 58. When +we sum Xij over all i < j we get +� +i A∗ +when d = 1/2. Hence A > A∗ on (0, 1). This implies the inequality. ♠ +51 + +9.3 +Proof of Lemma E112 +Let Q be a dyadic square and let z ∈ Q be a point. Let L be the vertical line +through x and let z01, z23 be the endpoints of the segment L ∩ Q. We label +the vertices of Q (in cyclic order) so that z01 lies on the edge joining q0 to q1 +and z23 lies on the edge joining q2 to q3. +Lemma 9.4 (E1121) If M is any horizontal or vertical line intersecting Q +then the circle Σ−1(M ∪ ∞) has diameter at least 1. +Proof: Since Q is a component of a good block, and M is horizontal or verti- +cal, M contains a point p with ∥p∥ ≤ 3/2. At the same time, M ∪∞ contains +∞. An easy calculation shows that ∥Σ−1(p)−(0, 0, 1)∥ ≥ 4/ +√ +13 > 1. Hence, +our circle contains two points which are greater than 1 unit apart. ♠ +Define +dj = ∥�pj − �pj+1∥. +The length of the segment σ joining the endpoints of Σ−1(L∩Q) varies mono- +tonically with the position of L. Hence, σ has length at most max(d1, d3). +At the same time, Σ−1(L ∩ Q) is contained in a circle of diameter at least 1. +The same argument as in the proof of Lemma E111 now shows that there is +a point z∗ ∈ σ which is within +t13 = χ(1, max(d1, d3)) = max(χ(1, d1), χ(1, d3)) +of �z. +The endpoints of σ respectively are on the spherical arcs obtained by +mapping the top and bottom edge of Q onto S2 via Σ−1. Hence, one endpoint +of σ is within χ(1, d0) of a point on the corresponding edge of ∂Q• and the +other endpoint of σ is within χ(1, d2) of a point on the opposite edge of ∂Q•. +But that means that either endpoint of σ is within +t02 = max(χ(1, d0), χ(1, d2)) +of a point in Q•. But then every point of σ is within t02 of some point of the +line segment joining these two points of Q•. In particular, there is a point +z• ∈ Q• which is within t of z∗. +We have shown that �z is within t13 of z∗ and z∗ is within t02 of z•. Hence +�z is within t02 + t13 = δ(Q) of �z. This completes the proof of Lemma E112. +52 + +10 +Interpolation: Proof of Lemma A2 +10.1 +Statement of the Result +The goal of this part of the monograph is to prove Lemma A2. Here we +repeat all the needed definitions. Rs(r) = r−s is the Riesz potential. The +other potentials we consider are: +Gk(d) = (4 − r2)k. +(90) +Also define +G♭ +5 = G5 − 25G1, +G♯♯ +10 = G10 + 28G5 + 102G2, +G♯ +10 = G10 + 13G5 + 68G2 +(91) +Given any function F, and a configuration p0, p1, p2, p3, p4 of 5 distinct +points on S2, we define +EF(P) = +� +i a0 + +4 +� +j=1 +ajEΓj(T0) = EΛ(T0) = Es(T0). +The central inequality is strict because a1, a2, a3, a4 ≥ 0 and at least one is +nonzero. ♠ +Lemma 10.3 (A22) The pair (G4, G6) specially forces (0, 6]. +Lemma 10.4 (A23) The following is true. +1. The pair (G5, G## +10 ) specially forces [6, 13]. +2. The pair (G♭ +5, G# +5 ) specially forces [13, 15++]. +54 + +10.3 +Proof of Lemma A22 +Before we start the proofs, we direct the reader’s attention to §2.4, which +explains how the reader can see plots of all relevant functions. Using the +software here is like eating a cooked meal rather than just reading a recipe. +Here is how we find the coefficients for the pair (Γ3, Γ4) = (G4, G6). The +same method works for the other cases. Let Γj = Gj for j = 1, 2. We define +Λs = a0(s) + +4 +� +j=1 +aj(s)Γj. +(94) +We then solve the equations +Λs(√m) = Rs(√m), +m = 2, 3, 4, +Λ′ +s(√m) = R′ +s(√m), +m = 2, 3. (95) +Here f ′ denotes the derivative of f, a function defined on (0, 2]. We don’t need +to constrain f ′(2). For each s this gives us a linear system with 5 variables +and 5 equations. In all cases, our solutions have the following structure +(a0, a1, a2, a3, a4) = M(2−s/2, 3−s/2, 4−s/2, s2−s/2, s3−s/2) +(96) +Solving this system of equations, we get +792M = + + +0 +0 +792 +0 +0 +792 +1152 +−1944 +−54 +−288 +−1254 +−96 +1350 +87 +376 +528 +−312 +−216 +−39 +−98 +−66 +48 +18 +6 +10 + + +(97) +Lemma 10.5 (A221) aj(s) > 0 for s ∈ (0, 6] and j = 1, 2, 3, 4. +We define the comparison function +Hs = 1 − Λs +Rs +. +(98) +We want to show that Hs > 0 on (0, 2) − { +√ +2, +√ +3}. We have +H′ +s(r) = rs−1(sΛs(r) + rΛ′ +s(r)). +(99) +55 + +The extrema of Hs occur at the same places as the roots of H′ +s. The roots +of H′ +s coincide with the roots of sΛs(r) + rΛ′ +s(r). Using the general equation +rG′ +k(r) = 2kGk(r) − 8kGk−1(r), +(100) +we see that sΛs(r) + rΛ′ +s(r) is a polynomial in 4 − r2. Hence, the roots of H′ +s +in (0, 2) are in bijection with the roots in (0, 4) of +ψs(t) = (sΛ(r) + rΛ′(r)) +��� +r=√4−t = t6 − +48 +12 + st5 + ... +(101) +Lemma 10.6 (A222) ψs(0) > 0 and ψs(4) > 0 for all s ∈ (0, 6]. +Lemma 10.7 (A223) The only minina for Hs occur at r = +√ +2 and r = +√ +3. +Proof: Since ψs has degree 6 we conclude that ψs has at most N = 6 roots, +counting multiplicity. By construction Hs(√m) = H′ +s(√m) = 0 for m = 2, 3 +and Hs( +√ +4) = 0. This means that Hs has extrema at r2 = +√ +2 and r3 = +√ +3 +and at points r23 ∈ ( +√ +2, +√ +3) and r34 ∈ ( +√ +3, +√ +4). Correspondingly ψs has +roots t1 = 1 and t2 = 2 and t01 ∈ (0, 1) and t12 ∈ (1, 2). The sum of all the +roots of ψs is 48/(12 + s) < 4. Since t1 + t2 + t01 + t12 > 4 we see that not all +roots can be positive. Hence N < 6. Since ψs is positive at t = 0, 4 we see +that N is even. Hence N = 4. This means that the only roots of ψs in (0, 4) +are the ones we already know about. Hence the only extrema of Hs in (0, 2) +are ones we already have mentioned. As s varies the nature of these extrema +cannot change. So, we check at s = 2 that +√ +2 and +√ +3 are minima and then +this fact persists for all a ∈ (0, 6]. ♠ +Lemma A22 is an immediate consequence of Lemmas A221, A222, A223 +10.4 +Proof of Lemma A23 +We find the coefficients for the case (G5, G♯♯ +10) just as we did for the case +(G4, G6). This time the matrix M is given below. To get an integer matrix, +we list 268536M, + + +0 +0 +268536 +0 +0 +88440 +503040 +−591480 +−4254 +−65728 +−77586 +−249648 +327234 +2361 +65896 +41808 +−19440 +−22368 +−2430 +−9076 +−402 +264 +138 +33 +68 + + +(102) +56 + +Lemma 10.8 (A231) With respect to the pair (G5, G♯♯ +10) the coefficients +a1(s), ..., a4(s) are positive for s ∈ [6, 13]. +Similarly, for the pair (G♭ +5, G♯ +10) we get the following matrix M, which we +list as 268536M + + +0 +0 +268536 +0 +0 +0 +982890 +116040 +−1098930 +−52629 +−267128 +0 +−91254 +−240672 +331926 +3483 +68208 +0 +35778 +−15480 +−20298 +−1935 +−8056 +0 +−402 +264 +138 +33 +68 +0 + + +(103) +Lemma 10.9 (A232) With respect to the pair (G♭ +5, G♯ +10) the coefficients +a1(s), ..., a4(s) are positive for s ∈ [13, 15++]. +Now we proceed as in the previous section. It turns out that Λs (and +hence Hs) is the same with respect to either pair (G5, G♯♯ +10) and (G♭ +5, G♯ +10). +This is not an accident. The reason is that for both pairs we are simply using +the four functions G1, G2, G5, G10 to under-approximate the power functions. +We derive the polynomial ψs exactly as we did for the pair (G4, G6). +Lemma 10.10 (A233) The function ψs has at most 4 roots in [0, 4]. +Lemma A233 shows that ψs only has the 4 roots we already know about +– the same ones enumerated for the pair (G4, G5). Lemma A233 then shows +that these are the only roots. As s varies, the nature of these roots cannot +change. When s = 6 we confirm that the roots 1 and 2 are the two roots +corresponding to minima. Hence, this is the case for all s ∈ [6, 16]. Hence, +√ +2 and +√ +3 are the only local minima for Hs for all s ∈ [6, 16]. This proves +Lemma A23. +10.5 +The Polya Case +For the pair (G3, G5), and with respect to the potentials −r−s, the matrix is: +1 +144 + + +0 +0 +−144 +0 +0 +−312 +−96 +408 +24 +80 +684 +−288 +−396 +−54 +−144 +−402 +264 +138 +33 +68 +30 +−24 +−6 +−3 +−4 + + +. +Rows 2,3,4,5 give positive power combos for s ∈ (−2, 0). +57 + +11 +Interpolation: Auxiliary Lemmas +11.1 +Proof of Lemmas A231 and A232 +All the expressions that arise in Lemma A231 and A232 (and Lemmas A221 +and A222) have the following form. +F(s) = +� +cistibs/2 +i +, +(104) +where bi, ci ∈ Q and ti ∈ Z and bi > 0. +For each summand we com- +pute a floating point value, xi. We then consider the floor and ceiling of +232xi and divide by 232. This gives us rational numbers xi0 and xi1 such +that xi0 ≤ xi ≤ xi1. Since we don’t want to trust floating point operations +without proof, we formally check these inequalities with what we call the +expanding out method. +Expanding Out Method: Suppose we want to establish an inequality like +( a +b) +p +q < c +d, where every number involved is a positive integer. This inequality +is true iff bpcq − apdq > 0. We check this using exact integer arithmetic. The +same idea works with (>) in place of (<). +To check the positivity of F on some interval [s0, s1] we produce, for each +term, the 4 rationals xi00, xi10, xi01, xi01. Where xijk is the approximation +computed with respect to sk. We then let yi be the minimum of these ex- +pressions. The sum � yi is a lower bound for Equation 104 for all s ∈ [s0, s1]. +On any interval exponent I where we want to show that Equation 104 is pos- +itive, we pick the smallest dyadic interval [0, 2k] that contains I and then run +the following subdivision algorithm. +1. Start with a list L of intervals. Initially L = {[0, 2k]}. +2. If L is empty, then HALT. Otherwise let Q be the last member of L. +3. If either Q ∩ I = ∅ or the method above shows that Equation 104 is +positive on Q we delete Q from L and go to Step 2. +4. Otherwise we delete Q from L and append to L the 2 intervals obtained +by cutting Q in half. Then we ago to to Step 2. +If this algorithm halts then it constitutes a proof that F(s) > 0 for all s ∈ I. +We check that the algorithm halts for all 8 quantities associated to Lemmas +A231 and A232, respectively for the intervals [6, 13] and [13, 15++]. +58 + +11.2 +Proof of Lemmas A221 and A222 +The 6 quantities associated to Lemma A221 and A222 all have the form given +in Equation 104. Using the same method as in the previous section we show +that all these quantities are positive on [1/4, 6]. To analyze what happens +on the interval (0, 1/4] we take Taylor series expansions. Up to constants +and factors of s the 6 expressions all have the form Y · V (s), where Y is an +integer vector and V (s) = (2−s/2, ..., s3−s/2). For Lemma A221, the various +choices of Y are the rows of the matrix in Equation 97. For Lemma A222: +11ψs(0) = + + +−88 +−128 ++216 ++6 ++32 ++11 + + +· + + +2−s/2 +3−s/2 +4−s/2 +s2−s/2 +s3−s/2 +s4−s/2 + + +, +11 +s ψs(4) = + + +−2112 ++1664 ++459 ++219 +288 +0 + + +· + + +2−s/2 +3−s/2 +4−s/2 +s2−s/2 +s3−s/2 +s4−s/2 + + +We work with the expressions on the right hand sides of these equations, so +that they look more like what we have in Lemma A221. Note that +sup +m=2,3,4 sup +s∈[0,1] +��� ∂6 +∂s6m−s/2��� < 1 +8. +(105) +Moreover the sum of the absolute values of the coefficients in each of the Y +vectors is at most 5000. This means that, when we take the 5th order Taylor +series expansion for Y · V (s), the error term is at most +5000 × 1 +8 × 1 +6! < 1. +We compute each Taylor series, set all non-leading positive terms to 0, and +crudely round down the other terms: +98s − 69s2 + 0s3 − 6s4 + 0s5 − 1s6 +14s − 3s2 − 2s3 + 0s4 − 1s5 − 1s6. +1s + 0s2 − 1s3 + 0s4 + 0s5 − 1s6. +.03s + 0s2 + 0s3 − .01s4 + 0s5 − 1s6. +.08s + 0s2 − .02s3 + 0s4 − .01s5 − 1s6. +11 + 0s + 0s2 − 1s3 − 1s4 + 0s5 − 1s6. +These under-approximations are all positive on (0, 1/4]. For example, if f(s) +is any one of these polynomials them g(u) = f(u/4) is weak positive dominant +in the sense of §12.2. My computer code does these calculations rigorously +with interval arithmetic, but it hardly seems necessary. +59 + +11.3 +Proof of Lemma A233 +I will describe a proof which took me quite a lot of experimentation to find. +We want to show that the degree 10 polynomial ψs(t) has at most 4 roots in +[0, 4] for all s ∈ [6, 16]. The same analysis shows that ψs has roots at 1, 2, +and in (0, 1) and in (1, 2). We just want to see that there are no other roots. +We can factor ψs as (t − 1)(t − 2)βs where βs is a degree 8 polynomial. +Taking derivatives with respect to t, we notice that +Lemma 11.1 (A2331) The following is true: +1. γs = 268536 × 12s/2 × (β′′ +s − β′ +s) is positive for s × t ∈ [6, 16] × [0, 4]. +2. −β′ +s(0) > 0 for all s ∈ [6, 16]. +3. β′ +s(4) > 0 for all s ∈ [6, 16]. +Lemma A2331 Statement 1 shows in particular that β′ +s never has a double +root. This combines with Statements 2 and 3 to show that the number of +roots of β′ +s in [0, 4] is independent of s ∈ [6, 16]. We check explicitly that +β′ +6 has only one root in [0, 4]. Hence β′ +s always has just one root. But this +means that βs has at most 2 roots in [0, 4]. This, in turn, means that ψs has +at most 4 roots in [0, 4]. This completes the proof. +11.4 +Proof of Lemma A2331 +We first give a formula for γs. Define matrices M3, M4, M6 respectively as: + + +−546840 +−1800480 +99720 +−397440 +−234600 +−33120 +173880 +−22080 +18366 +17112 +80766 +24288 +18630 +11592 +4830 +−1104 +0 +0 +0 +0 +0 +0 +0 +0 + + + + +−345600 +−1576320 +−509760 +−760320 +−448800 +−63360 +332640 +−42240 +−199296 +−698784 +75216 +−149376 +−79960 +5856 +94920 +−12992 +7104 +8432 +33960 +11968 +9180 +5712 +2380 +−544 + + + + +892440 +3376800 +410040 +1157760 +683400 +96480 +−506520 +64320 +−73350 +−246888 +−228942 +−165792 +−110370 +−41688 +27510 +−2064 +1473 +4092 +10557 +5808 +4455 +2772 +1155 +−264 + + +60 + +Define 3 polynomials P3, P4, P6 by the formula: +Pk(s, t) = (1, s, s2) · Mk · (1, ..., t7) = +2 +� +i=0 +7 +� +j=0 +(Mk)ijsitj, +k = 3, 4, 6. (106) +We have +γ = P33s/2 + P44s/2 + P66s/2. +(107) +To check the positivity of γs we check that each of the 16 functions +γs(v/4 + 1/4) = av,0 + av,1t + ...av,7t7 +(108) +is positive dominant in the sense of §12.2 for each v = 0, ..., 15. This amounts +to showing that each sum Av,k = av,0 + ... + av,k is positive for all s ∈ [6, 16]. +For each v = 0, ..., 15 and each k = 0, ...., 7 we have a 3 × 3 integer matrix +µv,k such that +Av,k = (1, s, s2) · µv,t · (3s/2, 4s/2, 6s,2). +(109) +This gives 128 matrices to check. We get two more such matrices from the +conditions −β′ +s(0) > 0 and β′ +s(4) > 0. All in all, we have to check that 130 +expressions of the form in Equation 109 are positive for s ∈ [6, 16]. These +expressions are all special cases of Equation 104, and we use the method +discussed above to show positivity in all 130 cases. The program runs in +several hours. +Remark: I found the 130 matrices by manipulating ψs in Mathematica +and then exporting the results to a file which our Java code reads. Our Java +code has an auxiliary debugger which uses the 130 matrices to reconstruct +the values of γs at randomly chosen points. The reader can check the print- +out against the output of our Mathematica file LemmaA233.m, which has γs +defined in a more direct way. In other words, I didn’t make mistakes when +computing these 130 matrices. +61 + +12 +Symmetrization: Preliminaries +In this part of the monograph we prove Lemma B and Lemma C1. In this +preliminary chapter we discuss a few useful lemmas. The reader can read +the proof of Lemma C1 in §17 directly after reading this chapter. +12.1 +Exponential Sums +We begin with two easy and well-known lemmas about exponential sums. +The first is an exercise with Lagrange multipliers. +Lemma 12.1 (Convexity) Suppose that α, β, γ ≥ 0 have the property that +α + β ≥ 2γ. Then αs + βs ≥ 2γs for all s > 1, with equality iff α = β = γ. +Lemma 12.2 (Descartes) Let 0 < r1 ≤ r1... ≤ rn < 1 be a sequence of +positive numbers. Let c1, ..., cn be a sequence of nonzero numbers. Define +E(s) = +n +� +i=1 +ci rs +i . +(110) +Let K denote the number of sign changes in the sequence c1, ..., cn. Then E +changes sign at most K times on R. +Proof: Suppose we have a counterexample. By continuity, perturbation, +and taking mth roots, it suffices to consider a counterexample of the form +� citei where t = rs and r ∈ (0, 1) and e1 > ... > en ∈ N. As s ranges in r, +the variable t ranges in (0, ∞). But P(t) changes sign at most K times on +(0, ∞) by Descartes’ Rule of Signs. This gives us a contradiction. ♠ +12.2 +Positive Dominance +See [S2] and [S3] for more details about the material here. Let G ∈ R[x1, ..., xn] +be a multivariable polynomial: +G = +� +I +cIXI, +XI = +n +� +i=1 +xIi +i . +(111) +62 + +Given two multi-indices I and J, we write I ⪯ J if Ii ≤ Ji for all i. Define +GJ = +� +I⪯J +cI, +G∞ = +� +I +cI. +(112) +We call G weak positive dominant (WPD) if GJ ≥ 0 for all J and G∞ > 0. +We call G positive dominant if GJ > 0 for all J. +Lemma 12.3 (Weak Positive Dominance) If G is weak positive domi- +nant then G > 0 on (0, 1]n. If G is positive dominant then G > 0 on [0, 1]n. +Proof: We prove the first statement. The second one has almost the same +proof. Suppose n = 1. Let P(x) = a0 + a1x + .... Let Ai = a0 + ... + ai. The +proof goes by induction on the degree of P. The case deg(P) = 0 is obvious. +Let x ∈ (0, 1]. We have +P(x) = a0 + a1x + x2x2 + · · · + anxn ≥ +x(A1 + a2x + a3x2 + · · · anxn−1) = xQ(x) > 0 +Here Q(x) is WPD and has degree n − 1. +Now we consider the general case. We write +P = f0 + f1xk + ... + fmxm +k , +fj ∈ R[x1, ..., xn−1]. +(113) +Since P is WBP so are the functions Pj = f0 + ... + fj. By induction on the +number of variables, Pj > 0 on (0, 1]n−1. But then, when we arbitrarily set +the first n − 1 variables to values in (0, 1), the resulting polynomial in xn is +WPD. By the n = 1 case, this polynomial is positive for all xn ∈ (0, 1]. ♠ +Polynomial Subdivision: Let P ∈ R[x1, ..., xn] as above. For any xj and +k ∈ {0, 1} we define +Sxj,k(P)(x1, ..., xn) = P(x1, ..., xj−1, x∗ +j, xj+1, ..., xn), +x∗ +j = k +2 + xj +2 . +(114) +If Sxj,k(P) > 0 on (0, 1]n for k = 0, 1 then we also have P > 0 on (0, 1]n. +Positive Numerator Selection: If f = f1/f2 is a bounded rational func- +tion on [0, 1]n, written in so that f1, f2 have no common factors, we always +choose f2 so that f2(1, ..., 1) > 0. If we then show, one way or another, that +f1 > 0 on (0, 1]n we can conclude that f2 > 0 on (0, 1]n as well. The point +is that f2 cannot change sign because then f blows up. But then we can +conclude that f > 0 on (0, 1]n. We write num+(f) = f1. +63 + +13 +Symmetrization: Proof of Lemma B +13.1 +The Domains +Now we define the domains involved in our proof. Recall that the domain Υ +is defined in §3.1 and shown in Figure 3.1. In this section we describe the +domains that play a role in the proof of Lemma B. The various transforma- +tions we make start in Υ but then move us into slightly different but related +domains. +Let Υ′ denote the domain of planar configurations p′ +0, p′ +1, p′ +2, p′ +3 such that +1. ∥p′ +0∥ ≥ ∥p′ +k∥ for k = 1, 2, 3. +2. 512p′ +0 ∈ [432, 498] × [0, 16]. (Compare [433, 498] × [0, 0].) +3. 512p′ +1 ∈ [−16, 32] × [−465, −348]. (Compare [−16, 16] × [−464, −349].) +4. 512p′ +2 ∈ [−498, −400] × [0, 16]. (Compare [−498, −400] × [0, 24].) +5. 512p′ +3 ∈ [−32, 16] × [348, 465]. (Compare [−16, 16] × [349, 464].) +6. p′ +02 = p′ +22. (Compare p02 = 0.) +The comparison conditions are what we had for Υ. +Up to a very small +perturbation Υ and Υ′ are the same domain. Condition 6 says that p′ +0 and +p′ +2 are on the same horizontal line. +Let Υ′′ denote the domain of configurations p′′ +0, p′′ +1, p′′ +2, p′′ +3 such that +1. 512p′′ +01 ∈ [416, 498] +2. 512p′′ +02 ∈ [0, 16]. +3. 512p′′ +12 ∈ [−465, −348]. +4. 512p′′ +32 ∈ [348, 465]. +5. (p′′ +21, p′′ +22) = (p′′ +01, −p′′ +21). +6. p′′ +11 = p′′ +31 = 0. +Conditions 5,6 say that the configuration is invariant under reflection in the +y-axis. +64 + +13.2 +Reduction to Smaller Steps +Lemma B concerns a map from Υ into the subset K4 of configurations having +4-fold symmetry. Here we describe this map as a 3 step process. We start +with the configuration X having points (p1, p2, p3, p4) ∈ Υ. +Balanced Rotation: We let (p′ +1, p′ +2, p′ +3, p′ +4) be the planar configuration which +is obtained by rotating X about the origin so that p′ +0 and p′ +2 lie on the same +horizontal line, with p′ +0 lying on the right. We call this operation balanced +rotation. Balanced rotation does not change the energy of the configuration. +Horizontal Symmetrization: Given a configuration X′ = (p′ +0, p′ +1, p′ +2, p′ +3), +there is a unique configuration X′′ = (p′′ +0, p′′ +1, p′′ +2, p′′ +3), invariant under under +reflection in the y-axis, such that p′ +j and p′′ +j lie on the same horizontal line for +j = 0, 1, 2, 3 and ∥p′′ +0 − p′′ +2∥ = ∥p′ +0 − p′ +2∥. We call this horizontal symmetriza- +tion. +Vertical Symmetrization: Let Υ′′ denote the union of all configurations +which are obtained from configurations in Υ′ by horizontal symmetrization. +Given a configuration X′′ = (p′′ +0, p′′ +1, p′′ +2, p′′ +3) ∈ Υ′′ there is a unique configura- +tion X′′′ = (p′′′ +0 , p′′′ +1 , p′′′ +2 , p′′′ +3 ) ∈ K4 such that p′′ +j and p′′′ +j lie on the same vertical +line for j = 0, 1, 2, 3. The configuration X′′′ coincides with the configuration +X∗ defined in Lemma B. We call this operation vertical symmetrization. +Lemma 13.1 (B1) Let P ∈ Υ and let P ′ be the balanced rotation of P ′. +Then P ′ ∈ Υ′. +Lemma 13.2 (B2) Let P ′ ∈ Υ′ and let P ′′ be the horizontal symmetrization +of P ′. Then P ′′ ∈ Υ′′ and Es(P ′′) ≤ Es(P ′) for all X′ ∈ Υ′ and all parameters +s ≥ 2. We have equality if and only if P ′ = P ′′. +Lemma 13.3 (B3) Let P ′′ ∈ Υ′ and let P ′′′ be the vertical symmetrization +of P ′′. Then P ′′′ ∈ Ψ4 and Es(P ′′′) ≤ Es(P ′′) for all s ∈ [12, 15++]. We have +equality if and only if P ′′ = P ′′′. +Lemma B follows immediately from Lemma B1 (§13.3), Lemma B2 (§13.4) +and Lemms B3 (§13.5) +Remark: Our proof of Lemma B2 is robust while our proof of Lemma B3 +is delicate. That accounts for the differing range of exponents. +65 + +13.3 +Proof of Lemma B1 +Let P ∈ Υ and let P ′ be the balanced rotation of P. Rotation about the +origin does not change the norms, so P ′ satisfies Condition 1. Moreover, +Condition 6 holds by construction. Now we verify the other properties. Let +ρθ denote the counterclockwise rotation through the angle θ. +Since p0 lies on the x axis and p2 lies on or above it, we have to rotate +by a small amount counterclockwise to get p′ +0 and p′ +2 on the same horizontal +line. That is, the rotation moves the right point up and the left one down. +Hence θ ≥ 0. This angle is maximized when p0 is an endpoint of its segment +of constraint and p2 is one of the two upper vertices of rectangle of constaint. +Not thinking too hard which of the 4 possibilities actually realizes the max, +we check for all 4 pairs (p0, p2) that the second coordinate of ρ1/34(p0) is +larger than the second coordinate of ρ1/34(p0). From this we conclude that +θ < 1/34. This yields +512 cos(θ) ∈ [0, 1], +512 sin(θ) ∈ [0, 16]. +(115) +From Equation 115, the map 512p0 → 512p′ +0 changes the first coordinate +by 512δ01 ∈ [0, 16] and 512δ02 ∈ [−1, 0]. This gives Condition 2 for Υ′. At +the same time, we have p′ +21 = p′ +01 and the change 512p2 → 512p′ +2 changes the +second coordinate by 512δ21 ∈ [0, 1]. This gives Condition 4 for Υ′ once we +observe that |p′ +21| ≤ |p′ +01|. +For Condition 3 we just have to check (using the same notation) that +512δ11 ∈ [0, 16] and 512δ12 ∈ [−1, 1]. The first bound comes from the inequal- +ity 512 sin(θ) < 16. For the second bound we note that the angle that p1 +makes with the y-axis is maximized when p1 is at the corners of its constraints +in Υ. That is, +p1 = +�±16 +512 , 349 +512 +� +. +Since tan(1/21) > 16/349 we conclude that this angle is at most 1/21. Hence +|512δ12| ≤ max +|x|≤1/21 +��� cos +� +x + 1 +34 +� +− cos(x) +��� < 1. +This gives Condition 3. The same argument gives Condition 5. +66 + +13.4 +Proof of Lemma B2 +Each planar configuration corresponds to a 5-point configuration on S2 via +stereographic projection. The energy of the 5 point configuration involves 10 +pairs of points. A typical term is: +Rs(pi, pj) = +1 +∥Σ−1(pi) − Σ−1(pj)∥s. +(116) +Given a list L of pairs of points in the set {0, 1, 2, 3, 4} we define Es(P, L) to +be the sum of the Rs-potentials just over the pairs in L. Thus, for instance +L = {(0, 2), (0, 4), (2, 4)} =⇒ Es(P, L) = Rs(p0, p2)+Rs(p0, p4)+Rs(p2, p4). +We call the subset L good for the parameter s if we have +Es(P ′′, L) ≤ Es(P ′, L). +(117) +Here P ′ ∈ Υ′ and P ′′ is obtained from P ′ by horizontal symmetrization. We +call L great if L is good and if we get equality in Equation 117 if and only +if the points involved on both sides of the equation coincide. For example, +if {(0, 2), (2, 4)} is great it means that we get equality if and only if p′′ +j = p′ +j +for j = 0, 2. +Lemma 13.4 (B21) {(0, 1), (1, 2)} and {(0, 3), (2, 3)} are good for s ≥ 2. +Lemma 13.5 (B22) {(0, 2)} and {(0, 4), (2, 4)} are great for all s ≥ 2. +Lemma 13.6 (B23) {(1, 3), (1, 4), (3, 4)} is great for all s ≥ 2. +Lemma B2 is an immediate consequence of Lemma B21 (§14), Lemma +B22 (§15), and Lemma B23 (§15.2). +Remark: We could probably derive greatness rather than goodness in Lemma +B21, but we don’t need it and we save some trouble by not worrying about +when precisely we get equality. +67 + +13.5 +Proof of Lemma B3 +We define the notation of goodness with respect to vertical symmetrization +just as we did for horizontal symmetrization. This time we are working with +points in the domain Υ′′ described in §13.1. For convenience we set qk = p′′ +k +and q′ +k = p′′′ +k . Let Q be the configuration q0, q1, q2, q3 and let Q′ be the image +under vertical symmetrization. +Lemma 13.7 (B31) The lists {(0, 4)} and {(2, 4)} are great for s ≥ 2. +Proof: Let Σ−1 denote inverse stereographic projection. Let us assume that +q0 ̸= q′ +0. Vertical symmetrization moves the point q0 = (q01, q21) to the point +q′ +0 = (q01, 0) which is closer to the origin. Therefore Σ−1(q′ +0) is farther from +(0, 0, 1) than is Σ−1(q0). This shows the result for {(0, 4)}. The result for +the list {(2, 4)} now follows from symmetry, namely reflection in the y-axis. ♠ +Lemma 13.8 (B32) {(0, 2)} is great for all s ≥ 2. +Proof: Since this inequality only involves 1 term, it suffices to prove the +result for s = 2. Setting x = q01 = −q21 and y = q02 = q22 we +L = {(0, 2)} +−→ +E2(L, Q) = +�1 + x2 + y2 +4x +�2 +(118) +Vertical symmetrization sets y = 0 and keeps x the same. Therefore, we have +that E(L, Q′) ≤ E2(L, Q) with equality if and only if y = 0. ♠ +Lemma 13.9 (B33) {(1, 3)} is great for all s ≥ 2. +Proof: Since our inequality only involves 1 term, it suffices to prove the +result for s = 2. We keep the notation from the previous lemma, except that +now we set y and h so that q1 = (0, −y + h) and q3 = (0, y + h). All we +need in our proof is that y ∈ (0, 1), which we have for configurations in Υ′′. +Vertical symmetrization sets h = 0 and keeps y the same. +We compute that +L = {(1, 3)} +−→ +E2(L, Q) = (1 + (y − h)2)(1 + (y + h)2) +(4y)2 +. +(119) +68 + +Call this function f(h). We suppress the dependence on y in the notation. +Given that our configuration Q lies Υ′′ we have y ∈ (0, 1). +Setting f ′(h) = 0 and solving for h, we find that the local extrema for +this function occur at h = 0 and at h = ± +� +y2 − 1. Since y2 < 1 the latter +roots are not real. Hence f has a unique local extremum, and this occurs at +h = 0. We compute +f ′′(0) = (1 − y2)/4y2 > 0. +We conclude that h = 0 is the unique global minimizer for f. ♠ +Lemma 13.10 (B34) {(1, 4), (3, 4)} is great for all s ≥ 2. +Proof: Consider first the case s = 2. Keeping the notation from the previous +section, we compute that +L = {(1, 4), (3, 4)} +E2(L, Q) = 1 + y2 + h2 +2 +. +(120) +Now we consider the case s > 2. Set +α = ∥Σ−1(q1), (0, 0, 1)∥−2 +β = ∥Σ−1(q3), (0, 0, 1)∥−2 +γ = ∥Σ−1(q′ +1), (0, 0, 1)∥−2 = ∥Σ−1(q′ +3), (0, 0, 1)∥−2. +We have just proved that α + β ≥ 2γ. The Convexity Lemma now says that +αt + βt ≥ 2γt with equality if and only if α = β = γ. ♠ +Lemma 13.11 (B35) {(0, 1), (0, 3)} is good for all s ≥ 12. +It follows from Lemma B35 and symmetry – namely reflection in the y- +axis – that {(2, 1), (2, 3)} is also good for all s ≥ 12. Finally, we observe that +vertical symmetrization maps Υ′′ into Ψ4 because we get a configurations +invariant under reflections in the coordinate axes which satisfy +p01 ∈ +�416 +512, 498 +512 +� +⊂ +�43 +64, 1 +� +, +p32 ∈ +�348 +512, 465 +512 +� +⊂ +�43 +64, 1 +� +. +Thus Lemma B3 follows from Lemmas B31, B32, B33, B34, and B35 (§16). +69 + +14 +Symmetrization: Proof of Lemma B21 +14.1 +Reduction to a Simpler Statement +Let D denote the set of triples of points (q0, q1, q2) ∈ (R2)3 such that +1. |q21| ≤ q01 and q22 = q02. +2. 512q0 ∈ [432, 498] × [−16, 16]. +3. 512q1 ∈ [−32, 32] × [348, 465]. +4. 512q2 ∈ [−498, −400] × [−16, 16]. +The domain D is a symmetrized version of Υ′ with the following properties. +If p0, p1, p2, p3 ∈ Υ′ then +• (q0, q1, q2) = (p0, p3, p2) ∈ D. +• (q0, q1, q2) = (r(p0), r(p1), r(p2)) ∈ D. +Here r is reflection in the x-axis. Thus, rather than consider the two lists +{(0, 1), (1, 2)} and {(0, 3), (3, 2)} separately, we just consider the symmetriza- +tion operation once, as it is applied to configurations in D. +The symmetrization operation is given by (q0, q1, q2) → (q′ +1, q′ +2, q′ +3), where +q′ +0 = +�q01 − q21 +2 +, q02 +� +, +q′ +1 = (0, q21), +q′ +2 = +�q21 − q01 +2 +, q22 +� +, +(121) +Define +λk = ∥Σ−1(qk) − Σ−1(qk+1)∥−2, +(122) +for k = 0, 1 and likewise define λ′ +k with respect to q′ +0, q′ +1, q′ +2. Note that λ′ +0 = λ′ +1 +by symmetry. To prove Lemma B21 it suffices to show +Lemma 14.1 (B11) With respect to all triples in D we have λ0 +λ1 ≥ 2λ′ +0. +The Convexity Lemma then shows that λs +0 + λs +1 ≥ (2λ′ +0)s for all s > 1, with +equality if and only if λ0 = λ1. This is exactly the statement that the lists +{(0, 1), (1, 2)} and {(0, 3), (3, 2)} are great for all s > 2. +70 + +14.2 +Proof of Lemma B211 +We define +[u, v]t = u(1 − t) + vt. +(123) +The map t → [u, v]t maps [0, 1] to [u, v]. +For all 4 choices of signs we define a map φ±,± : [0, 1]5 → (R2)3 by the +following formula +φ±,±(a, b, c, d, e) = q0(a, d, ±b), q1(±e, c), q2(a, d, ±b), +(124) +where +512q0(a, d, ±b) = ([416, 498]a + 49e, ±16b). +512q1(±d, c) = (±32d, [348 + 465]c) +512q2(a, d, ±b) = ([−416, −498]a + 49e, ±16b). +Lemma 14.2 (B2111) We have +D ⊂ φ+,+([0, 1]5) ∪ φ+,−([0, 1]5) ∪ φ−,+([0, 1]5) ∪ φ−.−([0, 1]5). +Proof: Let Dij denote the set of possible coordinates qij that can arise for +points in D. This, for instance D01 = [−16, 16]/512. Let D∗ +ij denote the set of +possible coordinates qij that can arise from the union of our parametrizations. +By construction Di2 ⊂ D∗ +i2 for i = 0, 1, 2 and D11 ⊂ D∗ +11. +Remembering that we have q01 ≥ |q21|, we see that the set of points pairs +(q01, q21) satisfying all the conditions for inclusion in D lies in the triangle X +with vertices +(498, −498), +(498, −400), +(432, −400) +At the same time, the set of pairs (512)(p∗ +01, p∗ +21) that we can reach with our +parametrization is the rectangle X∗ with vertices +(498, −498), +(416, −416), +(498, −498)+(49, 49), +(416, −416)+(49, 49). +We have X ⊂ X∗ because +(432, −400) = (416, −416)+(16, 16), +(498, −400) = (449, −449)+(49, 49). +This completes the proof. ♠ +71 + +In our coordinates, horizontal symmetrization corresponds to the map +(a, b, c, d, e) → (a, b, c, 0, 0). +(125) +We define F±,±(a, b, c, d, e) = λ0 + λ1, where these quantities are defined +relative to the triple φ±,±(a, bb, c, d, e). We define +Φ±,±(a, b, c, d, e) = num+(F±,±(a, b, c, d, e) − F±,±(a, b, c, 0, 0)). +(126) +Lemma 14.3 (B2112) For any sign choice Φ±,± > 0 on (0, 1)5. +As we discussed in §12.2, Lemma B2112 implies that +F±(a, b, c, d, e) ≥ F±(a, b, c, 0, 0) +. Lemma B211 thus follows from Lemma B2111 and Lemma B2112. +14.3 +Proof of Lemma B2112 +Our argument works the same for any of the 4 sign choices, so we set +Φ = Φ±,±, with the understanding that each statement about Φ is really +a statement about each of the 4 cases. Let Φa = ∂Φ/∂a, etc. +Lemma 14.4 (B21121) The following is true: +• F and Φd and Φe are zero when d = e = 0. +• Φd + 2Φe and Φdd and Φee are non-negative on [0, 1]5. +Proof: +The file LemmaB21121.m does these calculations. +For the second +item, we check that the 3 functions are weak positive dominant (§12.2). ♠ +Let Qd ⊂ [0, 1]5 be the sub-cube where d = 0. Let φ(d) be the restriction +of Φ to a line segment which starts at some point (a, b, c, 0, 0) and moves +parallel to (0, 0, 0, 1, 0). By Lemma B21121 we have φ(0) = φ′(0) and also +φ′′(d) ≥ 0. Hence φ(d) ≥ 0 for d ≥ 0. Hence Φ ≥ 0 on Qd. A similar +argument shows that likewise Φ ≥ 0 on Qe. +Any point in (0, 1)5 can be joined to a point in Qd ∪Qe by a line segment +L which is parallel to the vector (0, 0, 0, 1, 2). By Lemma B21121, Φ increases +along such a line segment as we move out of Qd ∪Qe. Hence Φ ≥ 0 on [0, 1]5. +72 + +15 +Symmetrization: Lemma B22 and B23 +15.1 +Proof of Lemma B22 +The relevant points for Lemma B22 are +p′ +0 = (x + d, y), +p′ +2 = (−x + d, y), +p′′ +0 = (x, y), +p′′ +2 = (−x, y), +(127) +All we will use in our proof is that x2+y2 < 1, which we have for points in the +domain Υ′. Let Σ−1 denote inverse stereographic projection. See Equation +8. +Lemma 15.1 (B221) The list L = {(0, 2)} is great for all s ≥ 2. +Proof: Write λ′ = ∥Σ−1(p′ +0) − Σ−1(p′ +2)∥−2 and likewise define λ′′. We have +λ′ − λ′′ = d2 +x2 × (2 + d2 − 2x2 + 2y2). +Since x2+y2 < 1 we have 2−2x2+2y2 > 0. Hence 0 < λ′′ ≤ λ′, with equality +if and only if d = 0. But then for t = s/2 > 1 we also have (λ′′)t < (λ′)t, +with equality iff d = 0. ♠ +Lemma 15.2 (B222) The list L = {(0, 4), (2, 4)} is great for s ≥ 2. +Proof: By the Convexity Lemma, it suffices to prove this for s = 2. We +have +Σ−1(p′ +4) = Σ−1(p′′ +4) = (0, 0, 1). +(128) +This time we have a beautiful formula: +� +∥Σ−1(p′ +0) − (0, 0, 1)∥−2 + ∥Σ−1(p′ +2) − (0, 0, 1)∥−2� +− +� +∥Σ−1(p′′ +0) − (0, 0, 1)∥−2 + ∥Σ−1(p′′ +2) − (0, 0, 1)∥−2� += d2 +2 , +(129) +This holds identically for all choices of x, y, d. ♠ +Lemma B22 follows immediately from Lemma B221, B222, B223. +73 + +15.2 +Proof of Lemma B23 +In this chapter we prove Lemma B23, The relevant points are q1 = (x1, y1) +and q3 = (x3, y3). Let �q = Σ−1(q) be the inverse stereographic image of q. +Horizontal symmetrization sets x1 = x3 = 0. Define +Fs(x1, x3, y1, y3) = As(x1, y1) + As(x3, y3) + Bs(x1, x3, y1, y3), +As(x, y) = ∥ � +(x, y) − (0, 0, 1)∥−s, +Bs(x1, x3, y1, y3) = ∥�q1 − �q3∥−s. +Lemma B23 says that +Fs(x1, x3, y1, y3) ≥ Fs(0, 0, y1, y3) +(130) +for all relevant choices of variables, with equality iff x1 = x3 = 0. All we need +for the proof (and we have it) is that s ≥ 2 and y1 < − +√ +3/3 and y3 > +√ +3/3. +The file LemmaB23.m has our calculations for this lemma. +Lemma 15.3 (B231) If Equation 130 is true when x1x3 ≤ 0 it is also true +when x1x3 > 0. +Proof: When x1, x3 > 0 the points �q1 and �q3 lie in the same hemisphere on +S2, namely the inverse stereographic image of the right halfplane. Replacing +(x1, y1) with (−x1, y1) increases the B-term and fixed the A-terms. Hence +Fs(x1, x3, y1, y3) > Fs(x1, −x3, y1, y3). The same goes when x1, x3 < 0. ♠ +Lemma 15.4 (B232) Equation 130 holds if x1x3 = 0. We have equality if +and only if x1 = x3 = 0. +Proof: +Without loss of generality we assume x1 = 0. +We analyze the +two terms of Fs separately and show that both decrease. +Just as in the +proof of Lemma B221 it suffices to take s = 2 for each of these terms. Set +y1 = − +√ +3/3 − t1 and y3 = +√ +3/3 + t3. We compute +A2(x3, y3) − A2(0, y3) = x2 +3 +4 > 0. +B2(0, x3, y1, y3) − B2(0, 0, y1, y3) = +3x2 +3(4 + 2 +√ +3t1 + 3t2 +1)(4 +√ +3t1 + 3t2 +1 + 2 +√ +3t3 + 6t1t3) +(4(2 +√ +3 + 3t1 + 3t3)2(4 + 3x2 +3 + 4 +√ +3t1 + 3t2 +1 + 4 +√ +3t3 + 6t1t3 + 3t2 +3) > 0. +When x3 ̸= 0 this has all positive terms because t1, t2 > 0. ♠ +74 + +Lemma 15.5 (B233) If Equation 130 has a counterexample with x1x3 < 0 +it also has a counterexample with x1x3 = 0. +Proof: +Without loss of generality we can assume x1 > 0 and x3 < 0. +The point q1 lies in the (+, −) quadrant and the point q3 lies in the (−, +) +quadrant. +Let ρ be the clockwise rotation about the origin, through the +smallest positive angle, such that one of the points q∗ +1 = ρ(q1) or q∗ +3 = ρ(q3) +lies on the y-axis. As our points move, their vertical distance from the origin +increases. Setting q∗ = (x∗, y∗), we have +y∗ +1 < y1 < − +√ +3/3, +y∗ +3 > y3 > +√ +3/3, +x∗ +1x∗ +3 = 0. +(131) +If (x1, x3, y1, y3) gives a counterexample to Equation 130 we have +Fs(0, 0, y1, y3) > Fs(x1, x3, y1, y3) = Fs(x∗ +1, x∗ +3, y∗ +1, y∗ +3) ≥∗ Fs(0, 0, y∗ +1, y∗ +3). +The middle equality is rotational symmetry. The starred inequality is Lemma +B232. We will get a contradiction by showing Fs(0, 0, y∗ +1, y∗ +3) > Fs(0, 0, y1, y3). +It suffices to prove this result when y1 = y∗ +1 because then we can reverse +the roles of the two points and apply the result twice to get the general case. +The key point in the proof (aside from calculation) is that Σ−1(0, y3) is closer +to (0, 0, 1) than it is to Σ−1(0, y1), and Σ−1(0, y∗ +3) is even closer to (0, 0, 1). +We set y1 = − +√ +3/3 − t1 and y3 = +√ +3/3 + t3. Here t1, t3 > 0. We com- +pute that ∂F2(0, 0, y1, y3)/∂t3 is a rational function of t1, t3 with all positive +coefficients and ∂F−2(0, 0, y1, y3)/∂t3 is a rational function of t1, t3 with all +negative coefficients. From this we conclude that the exponential sum +E(s) = Fs(0, 0, y1, y∗ +3) − Fs(0, 0, y1, y3) +satisfies E(2) > 0 and E(−2) < 0. We also have E(0) = 0. +The motion of the point (0, y3) → (0, y∗ +3) increases one of the terms in- +volved in Fs and decreases the other. From this we see that E(s) has at +most 2 sign changes in the sense of §12.1. So, by Descartes’ Lemma, E(s) +changes sign at most twice. +The first and last term in E(s) are positive +because the motion (0, y3) → (0, y∗ +3) decreases the shortest distance involved +and increases the longest. Hence E(s) > 0 when |s| is sufficiently large. +We now see that E(s) changes sign on (−∞, −2). If E′(0) ̸= 0 then E(s) +also changes sign at s = 0. But then E(s) does not change sign on (0, ∞) +and hence is positive there. If E′(0) = 0 and E(s) changes sign in (0, ∞) we +can perturb the points slightly, reduce to the case when E′(0) ̸= 0, and get +a contradictory situation with 3 sign changes. ♠ +75 + +16 +Symmetrization: Proof of Lemma B35 +For ease of notation set qk = p′′ +k. Let D be the set of configurations (q0, q1, q3) +such that +1. 512q01 ∈ [416, 498] +2. 512q02 ∈ [0, 16]. +3. 512q12 ∈ [−465, −348]. +4. 512q32 ∈ [348, 465]. +5. q11 = q31 = 0. +Lemma B35 does not involve the point p2, so we ignore it. +The subset +D ⊂ (R2)3 denotes the set of triples (q0, q1, q3) which satisfy the conditions +for inclusion in Υ′′. This set is not meant to be confused with the set from +the proof of Lemma B21, though it plays the same role in the proof here. We +let D± ⊂ D denote those configurations with +±(q12 + q32) ≥ 0. +(132) +Obviously D = D+ ∪ D−. +We adopt the convention in Equation 123, namely [u, v]t = u(1 − t) + vt. +We define map φ± : [0, 1]4 → (R2)3 as follows: +φ(a, b, c, d) = (q0(b, d), q1(a, c), q3(a, c)), +(133) +512q0(b, d) = ([416, 498]b, 16d). +512q1(a, c) = (0, −[348, 465]a ± 59c). +512q3(a, c) = (0, +[348, 465]a ± 59c). +In these coordinates, the symmetrization operation is (a, b, c, d) → (a, b, 0, 0). +Lemma 16.1 (B351) D± ⊂ φ±([0, 1]4). +Proof: This is just like the proof of Lemma B2111. The only non-obvious +point is why every pair (p12, p32) is reached by the map φ±. The essential +point is that for configurations in D± we have 512|p12 + p32| ≤ 2 × 59. ♠ +76 + +Following the same idea as in the proof of Lemma B21, we define +Fs,± = +� +∥Σ−1(q0)−Σ−1(q1)∥−s+∥Σ−1(q0)−Σ−1(q3)∥−s� +◦φ±(a, b, c, d) (134) +Here Σ−1 is the inverse of stereographic projection. Next, we define +Φs,±(a, b, c, d) = num+(Fs,±(a, b, c, d) − Fs,±(a, b, 0, 0)). +(135) +We can finish the proof by showing that φ2,+ ≥ 0 and φ12,− ≥ 0 on [0, 1]4. +The Convexity Lemma then takes care of all exponents greater than 2 on D+ +and all exponents greater than 12 on D−. +Lemma 16.2 (B352) Φ2,+ ≥ 0 on [0, 1]4. +Proof: Let Φ = Φ2,+. Let Φ|c=0 denote the polynomial we get by setting +c = 0. We define other such symbols similarly. Let Φc = ∂Φ/∂c, etc. The +Mathematica file LemmaB352.m computes that Φ|c=0 and Φ|d=0 and Φc + Φd +are weak positive dominant. Hence Φ ≥ 0 when c = 0 or d = 0 and the +directional derivative of Φ in the direction (0, 0, 1, 1) is non-negative. This +suffices to show that Φ ≥ 0 on [0, 1]4. ♠ +Lemma 16.3 (B353) Φ12,− ≥ 0 on [0, 1]4. +Proof: The file LemmaB353.m has the calculations for our argument. Let +Φ = Φ12,−. This polynomial is a monster. It has 102218 terms. The first +thing we notice is that most of the terms are much smaller than the biggest +terms. So, we first kill off these terms carefully. +Let M denote the maximum coefficient of Φ. We let Φ∗ be the polynomial +we get by taking each coefficient of c of Φ and replacing it with +floor +�1010c +M +� +. +This has the effect of killing off about half the terms of Φ, namely the posi- +tive terms that are less than 10−10M. The “small” negative coefficients are +changed to −1. The polynomial 1010Φ − MΦ∗ has all non-negative coeffi- +cients. Hence, if Φ∗ ≥ 0 on [0, 1]4 so is MΦ ≥ 0 and so is 1010Φ and finally +so is Φ. +77 + +We want to kill more terms. The polynomial Φ∗ has 37760 monomials in +which the coefficient is −1. We check that each such monomial is divisible +by one of c2 or d2 or cd. We therefore define +Ψ = Φ∗∗ − 37760(c2 + d2 + cd), +where Φ∗∗ is obtained from Φ∗ by setting all the (−1) monomials to 0. We +have Ψ ≤ Φ∗ on [0, 1]4. Hence, if Ψ is non-negative on [0, 1]4 then so is Φ. We +have reduced the problem to showing that Ψ ≥ 0 on [0, 1]4. The polynomial +Ψ has 5743 terms, which is more manageable. +Again we let Fa = ∂F/∂a, etc. +We check that Ψaaa is weak positive +dominant and hence non-negative on [0, 1]4. This massive calculation reduces +us to showing that the restrictions Ψ|a=0 and Ψa|a=0 and Ψaa|a=0 are all non- +negative on [0, 1]3. Letting F be any of these 3 functions, we consider +F|c=0, +F|d=0 +4Fc + Fd, +(136) +We show that all three functions are weak positive dominant for Ψa|a=0 and +Ψaa|a=0. +This shows that Ψa|a=0 and Ψaa|a=0 are non-negative on [0, 1]3. +Concerning the choice F = Ψ|a=0, all that remains is showing (in some other +way) that G = 4Fc + Fd ≥ 0. +We check that Gd is weak positive dominant and hence non-negative on +[0, 1]3. This reduces us to showing that H = G|d=0 is non-negative on [0, 1]2. +Here H is a 2-variable polynomial in b, c. We check that the two subdivi- +sions Sb,0(H) and Sb,1(H) are weak positive dominant. This proves that H +is non-negative on [0, 1]2. ♠ +Remark: The proof of Lemma B35 is pretty crazy. Initially we gave an +alternate, which works for s ≥ 13, based on the following fact: Suppose that +x1, y1, x2, y2 are positive numbers with +x2 = y2, +7x1+8y1 ≥ 7x2+8y2, +3x2 +1+3y2 +1−4x1y1 ≥ 3x2 +2+3y2 +2−4x2y2. +Then xt +1 + yt +1 ≥ xt +2 + yt +2 for all t ≥ 13/2. We leave this as an exercise to +the reader. This leaves us to check that vertical symmetrization does not +increase 7E1 +8E3 and 3E2 +1 +3E2 +3 −4E1E3 on D1. The polynomials involved +are much smaller and similar positive dominance methods work for them. +We have preserved the Mathematica files which do the relevant calculations. +78 + +17 +Symmetrization: Proof of Lemma C1 +17.1 +Reduction to Two Halves +Recall that �Ψ4 and Ψ8 respectively are defined by +64�Ψ4 = [55, 56]2, +64Ψ8 = {(t, t)| t ∈ [43, 64]}. +(137) +We have the symmetrization operation σ : �Ψ4 → Ψ8 given by +σ(x, y) = (z, z), +z = x + y + (x − y)2 +2 +. +(138) +Given (x, y) ∈ �Ψ4 the corresponding planar avatar p0, p1, p2, p4 is given +by −p2 = p0 = (x, 0) and −p1 = p2 = (0, y). The quantity Es(x, y) denotes +the s-potential of the 5-point configuration on S2 corresponding to the above +planar configuration under Σ−1, the inverse stereographic projection. Lemma +C1 says that Es ◦ σ ≤ Es on �Ψ4 for all s ∈ [14, 16], with equality iff x = y. +Define �pj = Σ−1(pj). We have (by symmetry): +Es(x, y) = As(x, y) + Bs(x, y), +As(x, y) = ∥�p0, �p2∥−s + ∥�p1, �p3∥−s, +Bs(x, y) = 2∥�p0, (0, 0, 1)∥−s + 2∥�p1, (0, 0, 1)∥−s + 4∥�p0, �p1∥−s. +(139) +Lemma 17.1 (C11) For (x, y) ∈ �Ψ4 and s ≥ 2 we have As(x, y) ≥ As(z, z). +When s > 2 we get equality if and only if x = y. +Proof: Let φ : [0, 1]2 → �Ψ4 be the affine isomorphism whose linear part is a +positive diagonal matrix. Define F2 : [0, 1]2 → R +F2 = num+(A2 ◦ φ − A2 ◦ σ ◦ φ). +(140) +We check that F2(a, b) = (a − b)2H2, where H2 is weak positive dominant. +Hence H2 > 0 on (0, 1)2. Hence A2(x, y) ≥ A2(z, z) on �Ψ4. Now we apply +the Convexity Lemma from §12.1. ♠ +Lemma 17.2 (C12) For (x, y) ∈ �Ψ4 and s ∈ [14, 16] we have the inequality +Bs(x, y) ≥ Bs(z, z). +Lemma C1 follows immediately from Lemmas C11 and C12. +79 + +17.2 +Proof of Lemma C12 +Let ∆ ⊂ (0, 1)2 be the open triangular subset above the main diagonal. +For integers k = 2, 14, 16 define +Gk = num+(Bk ◦ φ − Bk ◦ σ ◦ φ). +(141) +Lemma 17.3 (C121) The following is true: +1. B2(x, y) ≤ B2(z, z) for all (x, y) ∈ �Ψ4. +2. B14(x, y) ≤ B14(z, z) for all (x, y) ∈ �Ψ4. +3. B16(x, y) ≤ B16(z, z) for all (x, y) ∈ �Ψ4. +We get strict inequalities for points in the interior of �Ψ4 − Ψ8. +Proof: An algebraic miracle happens. We compute that: +1. −G2(a, b) = (a − b)2H2(a, b) and H2 is weak positive dominant. +2. G14(a, b) = (a − b)2H14(a, b) and H14 is weak positive dominant. +3. G16(a, b) = (a − b)2H16(a, b) and H16 is weak positive dominant. +This does it. ♠ +Now suppose there is some (x, y) ∈ �Ψ4 − Ψ8 and some s0 ∈ (14, 16) such +that Bs0(x, y) < Bs0(z, z). +Perturbing, we can assume that (x, y) lies in +the interior of �Ψ4 − Ψ8. Let p0, p1, p2, p3 and p′ +0, p′ +1, p′ +2, p′ +3 respectively be the +configurations corresponding to (x, y) and (z, z). Define +1. r01 = ∥Σ−1(p0) − Σ−1(p1)∥−1. +2. r0 = ∥Σ−1(p0) − (0, 0, 1)∥−1 and r1 = ∥Σ−1(p1) − (0, 0, 1)∥−1. +3. r′ +01 = ∥Σ−1(p′ +0) − Σ−1(p′ +1)∥−1. +4. r′ +0 = ∥Σ−1(p′ +0) − (0, 0, 1)∥−1 = ∥Σ−1(p′ +1) − (0, 0, 1)∥−1. +Replacing (x, y) by (y, x) if necessary, we arrange that r0 < r1. +Lemma 17.4 (C122) r0, r1, r′ +0 < 1/ +√ +2 < r01, r′ +01 +80 + +Proof: Let h = 1/2. We have x, y, z ∈ (0, 1). We compute +h − r2 +0 = 1 − x2 +4 +, +h − r2 +1 = 1 − y2 +4 +, +h − (r′ +0)2 = 1 − z2 +4 +, +(r01)2 − h = (1 − x2)(1 − y2) +4(x2 + y2) +> 0, +(r′ +01)2 − h = (1 − z2)2 +8z2 +> 0. +This does it. ♠ +Lemma 17.5 (C123) r01 < r′ +01. +Proof: We define +B∗ +2 = ∥�p0 − �p1∥−2 = r2 +01 +and then define G∗ +2 in terms of B2 just as we defined G2 in terms of B2 in +Equation 141. We check that G∗ +2(a, b) = −(a − b)2H∗(a, b) where H∗ is weak +positive dominant. Hence G∗ +2 > 0 on (0, 1)2. Hence B∗ +2(z, z) > B∗ +2(x, y). But +this implies that r01 < r′ +01. ♠ +We now deduce Lemma C12 from Lemmas C121, C122, C123. We fix +(x, y) and (z, z) = σ(x, y) and define +β(s) := Bs − Bs ◦ σ = +2rs +0 − 4(r′ +0)s + 2rs +1 + 4rs +01 − 4(r′ +01)s +(142) +Now we make the following observations. +• β(2) < 0 and β(14) > 0. Hence β changes sign in (2, 14). +• β(14) > 0 and β(s0) < 0. Hence β changes sign in (14, s0). +• β(s0) < 0 and β(16) > 0. Hence β changes sign in (s0, 16). +• β(s) < 0 for s sufficiently large because the term −4(r′ +01)s eventually +dominates. Hence β changes sign in (16, ∞). +Hence β vanishes at least 4 times. By Descartes’ Lemma, the sign sequence +must change signs at least 4 times. Combining Lemmas C121 and C122 we +see that the sign sequence must be one of +−, +, +, +, −, ++, −, +, +, −, ++, +, −, +, −. +In no case does it change sign at least 4 time. This is a contradiction. (In +fact, Equation 142 has the terms in the correct order.) +81 + +18 +Endgame: Proof of Lemma C2 +18.1 +The Goal +The goal of this chapter is to prove Lemma C2. We will reformulate the +result because we want to set up a rational calculation. Define squares Ψ4 +and �Ψ4. Here are their definitions: +64Ψ4 = [43, 64], +64�Ψ4 = [55, 56]. +(143) +Also, �Ψ8 is the main diagonal of �Ψ4. A point (x, y) in these domains defines +a planar configutation +p0 = (x, 0), +p1 = (0, −y), +p2(−x, 0), +p3(0, y) +(144) +We define Es(x, y) to be the s-potential of the corresponding avatar. +Recall that the point (1, +√ +3/3) represents the TBP avatar. We define +Θ(s, x, y) = Es(x, y) − E(1, +√ +3/3). +(145) +Lemma C2 is equivalent to the following three statements. +1. Θ > 0 on [13, 15+] × Ψ4. +2. Θ > 0 on [15+, 15++ × (Ψ4 − �Ψ4). +3. If s ∈ [15+, 15++] then Θ has a unique minimum in �Ψ8. +Let us deduce Statement 3 from Statements 1 and 2 and an auxiliary +lemma. Let Θx be the partial derivative of Θ with respect to x, etc. +Lemma 18.1 (C21) For all s ∈ [13, 15++] and (x, y) ∈ Ψ4 the quantities +Θxx, Θyy, Θxy are all positive. +Statement 3 is equivalent to the statement that the single variable func- +tion f(x) = Θ(s, t, t) has only one minimum for s ∈ I. Here 64I = [55, 56]. +By the Chain Rule, +ftt = Θxx + Θyy + 2Θxy > 0. +(146) +Hence f is a convex function on I. Hence f has a unique minimum in I. +This proves Statement 3 of Lemma C2. +82 + +18.2 +Integer Calculations +The calculation for Lemma C2 is relatively small. We work over a 3-parameter +space, and every coordinate we consider will be a dyadic rational. +The Expanding Out Method: This is a repeat of the definition in §11.1. +Suppose we want to establish an inequality like ( a +b) +p +q < c +d, where every num- +ber involved is a positive integer. This inequality is true iff bpcq − apdq > 0. +We check this using exact integer arithmetic. The same idea works with (>) +in place of (<). We call this the expanding out method. +More generally, we will want to verify inequalities like +10 +� +i=1 +b−s +i +− +10 +� +i=1 +a−s/2 +i +> C. +(147) +where all ai belong to the set {2, 3, 4}, and bi, c, s are all rational. +more +specifically s ∈ [13, 15++] will be a dyadic rational and c will be positive. +The expression on the left will be Es(p) − Es(p0) for various choices of p, and +the constant C will be a kind of cushion introduced to get around an error +term. +Here is how we handle expressions like this. For each index i ∈ {1, ..., 10} +we produce rational numbers Ai and Bi such that +As/2 +i +> ai +Bs +i < bi. +(148) +We use the expanding out method to check these inequalities. We then check +that +10 +� +i=1 +Bi − +10 +� +i=1 +Ai > C. +(149) +This last calculation is again done with integer arithmetic. Equations 148 +and 149 together imply Equation 147. +Logically speaking, the way that we produce the rational Ai and Bi does +not matter, but let us explain how we find them in practice. +For Ai we +compute 232a−s/2 +i +and round the result up to the nearest integer Ni. We then +set Ai = Ni/232. We produce Bi in a similar way. +When we have verified Equation 147 in this manner we say that we have +used the rational approximation method to verify Equation 147. We will only +need to make verifications like this on the order of 20000 times. +83 + +18.3 +Main Argument for Lemma C2 +We say that a block is a rectangular solid, having the following form: +X = I × Q ⊂ [0, 16] × [0, 1]2, +(150) +where I is a dyadic interval and Q is a dyadic square. In this context we +mean that I is obtained by repeatedly cutting [0, 16] in half and selecting +one of the halves. Similarly, Q is obtained by repeatedly cutting [0, 1]2 in +quarters and selecting one of the pieces. +The Energy Estimate: +We call the dyadic block X relevant if X ⊂ +[13, 16] × Ψ4. +We work with the larger domain just to have nice initial +conditions for our divide-and-conquer algorithm. +We define +|X|1 = |I|, +|X|2 = |Q|. +(151) +Here |I| is the length of I and |Q| is the side length of Q. +Lemma 18.2 (C22) The following is true for any relevant block X and any +good point: +min +X Θ ≥ min +v(X) Θ − (|X|2 +1/512 + |X|2 +2). +Here v(X) denotes the vertex set of X. Thus, to show that Θ|X > 0 we just +need to show that +Θv(X) > |X|2 +1 +512 + |X|2 +2. +Grading a Block: Given a block X = I × Q ⊂ [0, 16] × [0, 1]2 we perform +the following pass/fail evaluation. Let I = [s0, s1]. +1. If I ⊂ [0, 13] we pass X because X is irrelevant. +2. If I ⊂ [15 + 25/512, 16] we pass X because X is irrelevant. +3. If Q is disjoint from Ψ4 we pass X because X is irrelevant. We test +this by checking that either Q10 ≤ 43/64 or Q01 ≤ 43/64. +4. If s0 ≥ 15 + 24/512 and Q ⊂ �Ψ4 we pass X. +5. s0 < 13 and s1 > 13 we fail X because we don’t want to make any +computations which involve exponents less than 13. +84 + +6. If X has not been passed or failed, we try to use the rational approxi- +mation method to verify that Θ(v) > |X|2 +1/512 − |X|2 +2 for each vertex +v of X. If we succeed at this, then we pass X. Otherwise we fail X. +If we pass X it either means that X is irrelevant or that Lemma C2 holds +for all configurations in X. To prove Lemma C2 it suffices to find a partition +of [0, 16] × [0, 1]2 into blocks, all of which pass the grading step. +Subdivision: We will either divide a block X into two pieces or 4, de- +pending on the dimensions. We call X fat if 16|X|2 > |X|1, and otherwise +thin. If X is fat we subdivide X into 4 equal equal pieces by dyadically +subdividing Q. If X is thin we subdivide X into 2 pieces by dyadically sub- +dividing I. We found by trial and error that this scheme takes advantage of +the lopsided form of Lemma C22 and produces a small partition. +The Main Algorithm: We perform the following algorithm. +1. We start with a list L of blocks. Initially L has the single member +{0, 16} × {0, 1}2. +2. We let B be the last block on L. We grade B. If B passes, we delete +B from L. If L = ∅ then HALT. If B fails, we delete B from L and +append to L the subdivision of B. Then we go back to Step 1. +Proposition 18.3 (C23) When we run the algorithm, it halts with success +after 23213 steps and in about 2 minutes. The partition it produces has 15519 +blocks. +This proves Statements 1 and 2 of Lemma C2. We have already seen +that Statements 1 and 2, and Lemma C1, together imply Statement 3. This +completes the proof of Lemma C2. +85 + +19 +Endgame: Lemmas C21 and C22 +19.1 +Proof of Lemma C21 +We will show that Θxx > 0 and Θxy > 0 on Γ := [13, 15++] × Ψ4. The case +of Θyy follows from the case of Θxx and symmetry. +Setting u = s/2 we compute +Es(x, y) = A(s, x) + A(s, y) + 2B(s, x) + 2B(s, y) + 4C(s, x, y), +(152) +A(x) = a(x)u, +B(x) = b(x)u, +C(x) = c(x)u, +a(x) = (1 + x2)2 +16x2 +b(x) = 1 + x2 +4 +c(x, y) = (1 + x2)(1 + y2) +4(x2 + y2) +From this we compute +Θxx = Axx + 2Bxx + 4Cxx, +Θxy = 4Cxy. +(153) +For each choice of F = A, B, C, and correspondingly f = a, b, c, we have +Fxx = u(u − 1)f u−2f 2 +x + uf u−1fxx +(154) +We compute +axx = 3 + x4 +8x4 , +bxx = 1 +2, +cxx = (1 − y4)(3x2 − y2) +2(x2 + y2)3 +, +These are all positive on [43/64, 1). Hence Fxx ≥ 0 and on Γ and we have +strict inequality unless F is the C function. Hence Θxx > 0 in Γ. +We also have +Cxy = u(u − 1)cu−2cxcy + ucu−1cxy. +(155) +We compute +cx = x(y4 − 1) +2(x2 + y2)2 < 0, +cxy = 2xy(1 + x2y2) +(x2 + y2)3 +. +Likewise cy < 0. Equation 155 now shows that Cxy > 0 in Γ Hence Θxy > 0 +in Γ. +86 + +19.2 +Proof of Lemma C22 +Lemma C22 follows immediately from these three results: +Lemma 19.1 (C221) Lemma C22 is true provided that +1. |Θss(s, x, y)| ≤ 1/64 for all (s, x, y) ∈ Γ. +2. |Θxx(s, x, y)| < 4 and |Θyy(s, x, y)| < 4 for all (s, x, y) ∈ Γ. +Here Θss is the second partial derivative of Θ with respect to s, etc. +Lemma 19.2 (C222) |Θss(s, x, y)| ≤ 1/64 for all (s, x, y) ∈ Γ. +Lemma 19.3 (C223) Θxx(s, x, y), Θyy(s, x, y) ∈ (0, 4) for all (s, x, y) ∈ Γ. +19.3 +Proof of Lemma C221 +We begin with a familiar lemma about functions of one variable. +Lemma 19.4 (C2211) Suppose F : [0, 1] → R is a smooth function. Then +min +[0,1] F ≥ min(F(0), F(1)) − 1 +8 max +[0,1] |F ′′(x)|. +Proof: This is an application of Taylor’s Theorem with Remainder. ♠ +Lemma 19.5 (C2212) Let X ⊂ Rn be a rectangular solid with vertex set +v(X). Let d1, ..., dn denote the side lengths of X. Let G : X → R be a +smooth function. Let Gii = ∂2X/∂x2 +i . Then +min +X G ≥ min +v(X) G − 1 +8 +n +� +i=1 +max +X +|Gii|d2 +i . +Proof: The truth of the result does not change if we translate the domain +and/or range, and/or scale the coordinates separately. Thus, it suffices to +prove the result when X = [0, 1]n. In this case, the result follows in a straight- +forward way from Lemma C2211, the triangle inequality, and induction on +the dimension. ♠ +We apply Lemma C2212 using the bounds +max +X +|G11| ≤ 1/64, +max +X +|G22| ≤ 4 +max +X +|G33| ≤ 4. +This gives the desired bound. +87 + +19.4 +Proof of Lemma C222 +Lemma 19.6 (C2221) The 10 distances associated to a 5-point configura- +tion parametrized by a point in Ψ4 exceed 1.3 and at least 6 of them exceed +√ +2, and at least 2 of them exceed +√ +3. +Proof: Let x0 = 43/64. An exercise in calculus shows that A(−1, x), which +represents the distance between 2 of the pairs of points, exceed +√ +3 on [x0, 1]. +Similarly, B(−1, x) exceeds +√ +2 on [x0, 1]. Finally, C[−1, x, y], restricted to +[d, 1]2, takes its min at (x0, x0) and the value there exceeds 1.3. ♠ +The conclusion of this result also obviously holds the TBP. +Lemma 19.7 (C2222) Let ψ(s, b) = b−s. Let s ≥ 13. +• When b ≥ 1.3 we have ψss(s, b) ∈ (0, 1/440). +• When b ≥ +√ +2 we have ψss(s, b) ∈ (0, 1/753). +• When b ≥ +√ +3 we have ψss(s, b) ∈ (0, 1/4184). +Proof: As a function of s, and for b > 1 fixed, the second derivative +ψss(s, b) = b−s log(b)2 +(156) +is decreasing. Given the monotonicity, it suffices to prove our result when +s = 13. Choose b ≥ 1.3. The equation ψssb(13, b) = 0 has its unique solution +in [1, ∞) at the value b = exp(2/13) < 1.3. Moreover, the function ψss(13, b) +tends to 0 as b → ∞. Hence the restriction of the function b → ψss(13, b) +to [b, ∞) takes its max at b. We get the specific bounds by evaluating at +b = 1.3, +√ +2, +√ +3. ♠ +Now, 10 of the 20 terms comprising Θss(s, x, y) are positive and 10 are +negative. Also, for the terms of the same sign, all 10 of them are less than +1/440, and at least 6 of them are less than 1/753, and at least 2 of them are +less than 1/4184. Hence +|Θss| ≤ +4 +440 + +4 +753 + +2 +4184 < 1 +64. +88 + +19.5 +Proof of Lemma C223 +We already know Θxx > 0 on our domain Γ. We will show Θxx < 4 on Γ. The +case of Γyy follows from symmetry. Equations 153 and 154 give us formulas +for Θxx. We just need one more estimate. +Lemma 19.8 (C2231) We have the following bounds for x ∈ Ψ4. +a ∈ [.2, .3] +ax ∈ [−.33, 0] +axx ∈ [.5, 2] +b ∈ [.3, .5] +bx ∈ [+.33, .5] +bxx ∈ [.5, .5] +c ∈ [.5, .6] +cx ∈ [−.33, 0] +cxx = [0, .5] +Proof: Let g be one of the 9 functions listed. An exercise in calculus shows +that the gradient ∇g does not vanish on the interior of Ψ4. The only mildly +tricky case is cxx. In this case, the resultant of (x4 +y4)cxxx and (x4 +y4)cxxy +is +−7962624x11(1 − 2x2 − 3x4)2(−1 − 2x2 + 3x4)2, +and this has no roots in (43/64, 1). Hence, for all cases, the restriction of g +to Ψ4 achieves its extrema at the vertices of Ψ4. Computing at the vertices +and rounding outward, we get the advertised bounds on g. ♠ +From Lemma C2232 and Equation 154 we see that Θxx ≥ 0 on Γ. Com- +bining Lemmas C2231 and C2232 we get the following bounds. +Axx ≤ (u)(u − 1)(.3)u−2(.11) + (u)(.3)u−1(2) < 0.1 +(157) +Bxx ≤ (u)(u − 1)(.5)u−2(.25) + (u)(.5)u−1(.5) < 0.5 +(158) +Cxx ≤ (u)(u − 1)(.6)u−2(.11) + (u)(.6)u−1(.5) < 0.6 +(159) +For the final inequalities we note that the expressions are maximized when +u is as small as possible, namely u = 13/2. These calculations plug into +Equations 153 and 154 to show that Θxx < 4. This completes the proof of +Lemma C223. +89 + +20 +Endgame: Proof of Lemma C3 +20.1 +Reduction to a Simpler Statement +Let I = [55/64, 56/64]. Recall that �Ψ8 is the set of points of the form (x, x) +with x ∈ I, and that 15+ = 15 + +24 +512 and 15++ = 15 + +25 +512. +The point +p0 = (1, +√ +3/3) represents the TBP. As in the proof of Lemma C2, define +Θ(s, x, y) = Es(x, y) − E(1, +√ +3/3). +(160) +We write Θs = ∂Θ/∂s, etc. We prove the following result below. +Lemma 20.1 (C31) Θstt(15, t, t) < 0 on I. +Lemma 20.2 (C32) Θs < 0 in [15+, 15++] × �Ψ8. +Proof: Lemma C212 tells us that |Θss| ≤ 2−6. We also have 15++−15 < 2−4. +Therefore, to prove this result, it suffices to prove that +Θs(15, t, t) < −2−10 +(161) +for t ∈ I. Let t0 = 55/64, the left endpoint of I. We compute Θst(15, t0, t0) < +0. Lemma C31 now implies that Θst(15, t, t) < 0 for all t ∈ I. Next, we com- +pute Θs(15, t0, t0) < −2−7. Since Θst(15, t, t) < 0 we get Equation 161. ♠ +We already know Θ(15+, ∗) > 0 on Ψ4. Hence Θ(15+, ∗) > 0 on �Ψ8. +We compute that Θ(15++, x, x) < 0 at x = 445/512 ∈ I. Combining this +with Lemma C32, we see that there exists a smallest parameterש such that +Θ(ש, p∗) = 0 for some p∗ ∈ �Ψ8. +For s >ש, Lemma C31 now says that +Θ(s, p∗) < 0. These statements immediately imply Lemma C3. +20.2 +Proof of Lemma C31 +The file LemmaC31.m has the calculations for this lemma. We have +Es(t, t) = 2A(s, t) + 4B(s, t) + 4C(s, t), +(162) +where +A(s, t) = +�(1 + t2)2 +16t2 +�s/2 +, +B(s, t) = +�1 + t2 +4 +�s/2 +, +C(s, t) = +�(1 + t2)2 +8t2 +�s/2 +. (163) +90 + +Because the s-energy of the TBP does not depend on the t-variable, we have +Θstt(15, t, t) = 2Astt|s=15 + 4Bstt|s=15 + 4Cstt|s=15. +(164) +Call the three functions on the right α(t), β(t), and γ(t). To finish the proof, +we just need to see that each of these three terms is negative in I. We note +that I has length 2−6. We write f ∼ f ∗ if +f +f ∗ = 2utv(1 + t2)w(2 + t2 + t−2)x +for exponents u, v, w, x ∈ R. In this case, f and f ∗ have the same sign. +Lemma 20.3 (C311) β < 0 on I. +Proof: Taking (u, v, w, x, y) = (−14, 0, 11/2, 0) we have β ∼ −β∗, +β∗(t) = (−2 + 30 log(2)) + t2(−58 + 420 log(2)) − 15(1 + 14t2) log(1 + t2). +Noting that log(2) = 0.69... we eyeball β∗ and see that it is positive for t ∈ I. +The term +420 log(2)t2 dominates. Hence β < 0 on I. ♠ +Lemma 20.4 (C312) γ < 0 on I. +Proof: Taking (u, v, w, x, y) = (−41/2, −16, 12, 1/2) we have γ ∼ −γ∗, +γ∗(t) = (−31 + 360 log(2)) + t2(56 − 585 log(2)) + t4(−29 + 315 log(2))+ +15(−8 + 13t2 − 7t4) log(2 + t2 + t−2). +We have γ∗(55/64) > 24 and we estimate easily that γ∗ +t > −210 on I. Only +the underlined term has negative derivative in I. These results show that +γ∗ > 0 on I. Hence γ < 0 on I. ♠ +Lemma 20.5 (C313) α < 0 on I. +Proof: Taking (u, v, w, x, y) = (−29, −14, 10, 3/2) we have α ∼ −α∗, +α∗(t) = γ∗(t) + δ∗(t), +δ∗(t) = 15 log 2 × (8 − 13t2 + 7t4). +We see easily that δ∗ > 0 on I. So, from Lemma C312, we have α∗ > 0 on +I. 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Mat. 4 (1992), 115-121, translation in Discrete Math +Appl. 3 (1993) 75-81 +93 + diff --git a/8tE4T4oBgHgl3EQfdQys/content/tmp_files/load_file.txt b/8tE4T4oBgHgl3EQfdQys/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..409d22930ef5853a530b8eeb1d6cb74c8e5b9fa2 --- /dev/null +++ b/8tE4T4oBgHgl3EQfdQys/content/tmp_files/load_file.txt @@ -0,0 +1,2528 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf,len=2527 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content='05090v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content='MG] 12 Jan 2023 Divide and Conquer: A Distributed Approach to Five Point Energy Minimization Richard Evan Schwartz January 13, 2023 1 Introduction The purpose of this work is to rigorously verify the phase-transition for 5 point energy minimization first observed in [MKS], in 1977, by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Mel- nyk, O, Knop, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Our results contain, as special cases, solutions to Thomson’s 5-electron problem and Polya’s 5-point problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' This work is an updated version of my monograph from 6 years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' I simplified the proof significantly and also I wrote this version in an experi- mental style designed to facilitate the verification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' This work is just over half as long as the original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' I wrote the proof in a tree-like form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Thus, the Main Theorem is an immediate consequence of Lemma A, Lemma B, and Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' These three Lemmas are independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Lemma A is an immediate conse- quence of Lemma A1 and Lemma A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' And so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' All the “ends” of the tree, such as Lemma B21121, either have short and straightforward proofs or are computer calculations which I will describe in enough detail that a compe- tent programmer could reproduce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' At the same time, all my computer programs are available to download and use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Figures 0 and 01 below map out the complete logical structure of the proof of the Main Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' The rest of this introduction states the results and explains how to divide the verification of the proof into small pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Following this, §2 contains a discussion of the history and context of the results, a high-level discussion of the ideas in the proof, a discussion of the computer experiments I did, and a guide to the relevant software I wrote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Following this we get to the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' 1 Results: Let S2 be the unit sphere in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tE4T4oBgHgl3EQfdQys/content/2301.05090v1.pdf'} +page_content=' Given a configuration {pi} ⊂ S2 of N distinct points and a function F : (0, 2] → R, define the F-potential of the configuration: EF(P) = � 1≤i 104 production rules. +Keywords: Probabilistic Pushdown Automata · Probabilistic Model +Checking · Certified Algorithms · Probabilistic Recursive Programs. +1 +Introduction +Complex software is likely to contain bugs. This applies in particular to model +checking tools. This is a serious problem, as the possibility of such bugs com- +promises the trust one can put in the verification results, rendering the process +of formal modeling and analysis less useful. Ideally, the implementation of a +model checker should be formally verified itself [15]. However, due to the great +complexity of these tools, this is often out of reach in practice. Certifying algo- +rithms [31] mitigate this problem by providing an easy-to-check certificate along +with their regular output. This means that there exists a verifier that, given the +input problem, the output, and the certificate, constructs a formal proof that the +output is indeed correct. The idea is that the verifier is much simpler than the +algorithm, and thus likely to be bug-free or even amenable to formal verification. +This paper extends the recent line of research on probabilistic certifica- +tion [19,23,24,40] to probabilistic pushdown automata [13,30] (pPDA). pPDA and +related models have applications in, amongst others, pattern recognition [38], +computational biology [28], and speech recognition [25]. They are moreover a +natural operational model for programs with procedures, recursion, and (dis- +crete) probabilistic constructs such as the ability to flip coins. With the advent + +2 +Tobias Winkler and Joost-Pieter Katoen +X → a | XY Y +x = 1 +2(1 + xy2) +Y +→ b | X | Y Y +y = 1 +3(1 + x + y2) +0.4 +0.6 +0.8 +1 +0.4 +0.6 +0.8 +1 +≈ (.66, .7) +(1, 1) +x +y +Fig. 1: Left: A stochastic context-free grammar (SCFG) and the associated pos- +itive polynomial system (PPS) which encodes the termination probabilities of +each non-terminal, assuming production rules are taken uniformly at random. +Right: The curves defined by the two equations. The least fixpoint (lfp) is +≈ (0.66, 0.70). The thin colored area to the top right of the lfp is the set of +inductive, i.e., self-certifying upper bounds on the lfp. +of probabilistic programming [32] as a paradigm for model-based machine learn- +ing [6], such programs have received lots of attention recently. Moreover, several +efficient algorithms such as Hoare’s quicksort with randomized pivot selection +(e.g. [26]) are readily encoded as probabilistic recursive programs. +A pPDA can be seen as a purely probabilistic variant of a standard pushdown +automaton: Instead of reading an input word, it takes its transitions randomly +based on fixed probability distributions over successor states. Quantities of inter- +est in pPDA include reachability probabilities [13], expected runtimes [8], vari- +ances [14], satisfaction probabilities of temporal logic formulas [44,41], and others +(see [7] for an overview). pPDA are equivalent to recursive Markov chains [17]. +One of the difficulties of pPDA is that they induce infinite Markov chains. +Despite this fact, many interesting quantitative properties are decidable, albeit +with rather high complexity. Therefore, in the past two decades there have been +significant research efforts on efficient approximative algorithms for pPDA, espe- +cially a decomposed variant of Newton iteration [16,27,11,17,12,10,39] which pro- +vides guaranteed lower, and occasionally upper [10,12] bounds on key quantities. +However, even though implementations might be complex [43], these algorithms +do not produce certificates for their results. +Our technique for certificate generation is an adaption of Optimistic Value +Iteration [22] (OVI) to the pPDA setting. In a nutshell, OVI computes some +lower bound ⃗l on the solution—which can be done using an approximative iter- +ative algorithm—and then optimistically guesses an upper bound ⃗u = ⃗l + ⃗ε and +verifies that the guess was correct. Originally, OVI was formulated for Markov +Decision Processes (MDP) where it is used to compute lower and upper bounds +on minimal or maximal reachability probabilities and expected rewards. The up- +per bounds computed by OVI have a special property: They are self-certifying +(also called inductive in this paper). This means that, given the MDP and the +upper bounds, one can check that the bounds are correct without the need for +an additional certificate; and this check is conceptually and practically easier +than finding the bounds in the first place. + +Certificates for Probabilistic Pushdown Automata via OVI +3 +The analysis of pPDA, however, is more involved than that of MDP. In +MDP, many quantitative properties are characterized as least fixpoints (lfp) of +piece-wise linear equation systems and can be computed in PTIME via, e.g., +LP solving. In pPDA, on the other hand, the equation systems for the same +properties may contain non-linear polynomials, and the best known complexity +bounds are usually as high as PSPACE. An example of such a non-linear system +is illustrated in Figure 1 which shows the translation of a stochastic context-free +grammar (SCFG; special case of pPDA with a single state) to a polynomial +equation system encoding termination probabilities. An important observation +is that the polynomials arising in this context only have positive coefficients. +Such systems are called positive polynomial systems (PPS) in this paper. +Applications of PPS beyond the analysis of pPDA include the recent factor +graph grammars [9] as well as obtaining approximate counting formulas for many +classes of trees in the framework of analytic combinatorics [18]. +Contributions. In summary, this paper makes the following contributions: +– We present an optimistic algorithm for computing inductive, self-certifying +upper bounds of any desired precision ε > 0 on the lfp of a positive poly- +nomial system. Compared to OVI from [22], the key innovation of our algo- +rithm is to compute a certain direction ⃗v in which to guess, i.e., the guess is +⃗u = ⃗l + ε⃗v rather than ⃗u = ⃗l + ⃗ε. This is to ensure that we eventually hit an +inductive bound, even if the latter lie in a very “thin strip” as in Figure 1. +– We prove that our algorithm is sound and complete in the sense that if a +(non-trivial) inductive upper bound exists, then such a bound will be found. +– We show how inductive bounds on the lfp of PPS can be used to certify +various quantities of interest in pPDA and SCFG, such as non-termination +or bounds on expected rewards/costs. +– We implement our algorithm in the software tool pray and compare the new +technique to an out-of-the-box approach based on SMT solving. +Related Work. Certification of pPDA has not been addressed explicitly in the +literature, but some existing technical results go in this direction. We mention +[17, Prop. 8.7] which yields certificates for non almost-sure termination of SCFG. +However, checking such certificates is not straightforward as it requires an SCC +decomposition. The tool PReMo [43] implements iterative algorithms for lower +bounds, but it supports neither certificates nor upper bounds. +Beyond pPDA, OVI was recently generalized to stochastic games [1]. Farkas +certificates for MDP [19] are verified by checking a set of linear constraints, which +is in spirit similar to our certificates that requires checking a set of polynomial +constraints. A deductive approach for verifying probabilistic recursive programs +on the syntax level was studied in [35]. The same paper also includes inductive +proof rules for verifying upper bounds just like we do. Recently, a higher-order +generalization of pPDA called PHORS was introduced in [29], and an algorithm +for finding upper bounds inspired by the Finite Elements method was proposed. + +4 +Tobias Winkler and Joost-Pieter Katoen +Paper Outline. We review the relevant background information on PPS in Sec- +tion 2. Section 3 presents our theoretical results on inductive upper bounds in +PPS as well as the new Optimistic Value Iteration algorithm. In Section 4 we +explain how inductive bounds in PPS are used to certify quantitative properties +of pPPA. The experimental evaluation is in Section 5. We conclude in Section 6. +2 +Preliminaries +Notation for Vectors. All vectors in this paper are column vectors and are written +in boldface, e.g., ⃗u = (u1, . . . , un)T . For vectors ⃗u, ⃗u′, we write ⃗u ≤ ⃗u′ if ⃗u is +component-wise less than or equal to ⃗u′. Moreover, we write ⃗u < ⃗u′ if ⃗u ≤ ⃗u′ +and ⃗u ̸= ⃗u′, and ⃗u ≺ ⃗u′ if ⃗u is component-wise strictly smaller than ⃗u′. The zero +vector is denoted ⃗0. The max norm of a vector ⃗u is ||⃗u||∞ = max1≤i≤n |ui|. We +say that ⃗u is normalized if ||⃗u||∞ = 1. +Positive Polynomial Systems (PPS). Let n ≥ 1 and ⃗x = (x1, . . . , xn)T be a +vector of variables. An n-dimensional PPS is an equation system of the form +x1 = f1(x1, . . . , xn) +. . . +xn = fn(x1, . . . , xn) +where for all 1 ≤ i ≤ n, the function fi is a polynomial with non-negative real +coefficients. An example PPS is the system x = 1 +2(1+xy2), y = 1 +3(1+x+y2) from +Figure 1. We also use vector notation for PPS: ⃗x = ⃗f(⃗x) = (f1(⃗x), . . . , fn(⃗x))T . +We write R≥0 = R≥0 ∪ {∞} for the extended non-negative reals. By conven- +tion, for all a ∈ R≥0, a ≤ ∞, a + ∞ = ∞ + a = ∞, and a · ∞ = ∞ · a equals 0 if +a = 0 and ∞ otherwise. For n ≥ 1, the partial order (R +n +≥0, ≤) is a complete lat- +tice, i.e., all subsets of R +n +≥0 have an infimum and a supremum. In particular, there +exists a least element ⃗0 and a greatest element ⃗∞ = (∞, . . . , ∞)T . Every PPS +induces a monotone function ⃗f : R +n +≥0 → R +n +≥0, i.e., ⃗u ≤ ⃗v =⇒ ⃗f(⃗u) ≤ ⃗f(⃗v). By +the Knaster-Tarski fixpoint theorem, the set of fixpoints of ⃗f is also a complete +lattice, and thus there exists a least fixpoint (lfp) denoted by µ⃗f. +In general, the lfp µ⃗f is a vector which may contain ∞ as an entry. For +instance, this happens in the PPS x = x+1. A PPS ⃗f is called feasible if µ⃗f ≺ ⃗∞ +(or equivalently, µ⃗f ∈ Rn +≥0), i.e., the lfp is a vector of real numbers. Besides +existence of the lfp, the Knaster-Tarski theorem also implies the following: +Lemma 1 (Inductive upper bounds). For all ⃗u ∈ R +n +≥0 it holds that +⃗f(⃗u) ≤ ⃗u +implies +µ⃗f ≤ ⃗u . +Such a vector ⃗u with ⃗u ≺ ⃗∞ is called inductive upper bound. +Given a feasible PPS ⃗f, find an inductive upper bound ⃗u ≥ µ⃗f. +Problem statement of this paper + +Certificates for Probabilistic Pushdown Automata via OVI +5 +If ⃗f is feasible, then µ⃗f is obviously an inductive upper bound. In Section 3 +we show under which conditions there exist more useful inductive upper bounds. +A PPS is called clean if µ⃗f ≻ ⃗0. Every PPS can be cleaned in linear time by +identifying and removing the variables that are assigned 0 in the lfp [17,12]. +Given a PPS ⃗f and a point ⃗u ∈ Rn +≥0, we define the Jacobi matrix of ⃗f at ⃗u +as the n×n-matrix ∂ ⃗f(⃗u) with coefficients ∂ ⃗f(⃗u)1≤i,j≤n = +∂ +∂xj fi(⃗u). +Example 1. Consider the example PPS ⃗fex with variables ⃗x = (x, y)T : +x = f1(x, y) = y + 0.1 +y = f2(x, y) = 0.2x2 + 0.8xy + 0.1 . +The line and the hyperbola defined by these equations are depicted in Figure 2 +on Page 7. The fixpoints of ⃗fex are the intersections of these geometric objects; +in this case there are two. In particular, ⃗fex is feasible and its lfp is +µ⃗fex = +� +(27− +√ +229)/50 , (22− +√ +229)/50 +�T ≈ (0.237 , 0.137)T . +Therefore, ⃗fex is clean as µ⃗fex ≻ ⃗0. The Jacobi matrix of ⃗fex is +∂ ⃗fex(x, y) = +� +∂ +∂xf1 +∂ +∂yf1 +∂ +∂xf2 +∂ +∂yf2 +� += +� +0 +1 +0.4x + 0.8y 0.8x +� +. +Note that the lfp µ⃗fex contains irrational numbers. However, we can still give ex- +act expressions for these numbers (involving square roots) because the fixpoints +of ⃗fex are the zeros of a quadratic polynomial. However, there are PPS whose lfp +cannot be expressed using radicals, i.e., square roots, cubic roots, etc. [16]. This +means that in general, there is no easy way to compute least fixpoints exactly. +It is thus desirable to provide bounds, which we do in this paper. +△ +Matrices and Eigenvectors. Let M be a real n×n-matrix. We say that M is non- +negative (in symbols: M ≥ 0) if it has no negative entries. M is called irreducible +if for all 1 ≤ i, j ≤ n there exists 0 ≤ k < n such that (M k)i,j ̸= 0. It is easy +to show that M is irreducible iff the directed graph GM = ({1, . . . , n}, E) with +(i, j) ∈ E iff Mi,j ̸= 0 is strongly connected. A maximal irreducible submatrix +of M is a square submatrix induced by a strongly connected component of GM. +The period of a strongly connected M is the length of the shortest cycle in GM. +It is instructive to note that PPS ⃗x = ⃗f(⃗x) are generalizations of linear equation +systems of the form ⃗x = M⃗x + ⃗c, with M ≥ 0 and ⃗c ≥ ⃗0. Moreover, note that +for any PPS ⃗f it holds that ∂ ⃗f(⃗u) ≥ 0 for all ⃗u ≻ ⃗0. +An eigenvector of an n×n-matrix M with eigenvalue λ ∈ C is a (complex) +vector ⃗v ̸= ⃗0 satisfying M⃗v = λ⃗v. There are at most n different eigenvalues. The +spectral radius ρ(M) ∈ R≥0 is the largest absolute value of the eigenvalues of +M. The following is a fundamental theorem about non-negative matrices: +Theorem 1 (Perron-Frobenius). Let M ≥ 0 be irreducible. + +6 +Tobias Winkler and Joost-Pieter Katoen +(1) M has a strictly positive eigenvector ⃗v ≻ ⃗0 with eigenvalue ρ(M), the spectral +radius of M, and all other eigenvectors ⃗v′ ≻ ⃗0 are scalar multiples of ⃗v. +(2) The eigenvalues of M with absolute value ρ(M) are exactly the h numbers +ρ(M), ξρ(M), . . . , ξh−1ρ(M), where ξ is a primitive hth root of unity. +The unique eigenvector ⃗v ≻ ⃗0 with ||⃗v||∞ = 1 of an irreducible non-negative +matrix M is called the Perron-Frobenius eigenvector of M. +Strongly Connected Components. To each PPS ⃗f we associate a finite directed +graph G ⃗f = ({x1, . . . , xn}, E), which, intuitively speaking, captures the depen- +dency structure among the variables. Formally, (xi, xj) ∈ E if the polynomial fi +depends on xj, i.e., xj appears in at least one term of fi with a non-zero coef- +ficient. This is equivalent to saying that the partial derivative +∂ +∂xj fi is not the +zero polynomial. We say that ⃗f is strongly connected if G ⃗f is strongly connected, +i.e., for each pair (xi, xj) of variables, there exists a path from xi to xj in G ⃗f. +For instance, ⃗fex from Example 1 is strongly connected because the dependency +graph has the edges E = {(x, y), (y, x), (y, y)}. Strong connectivity of PPS is a +generalization of irreducibility of matrices; indeed, a matrix M is irreducible iff +the PPS ⃗x = M⃗x is strongly connected. We often use the fact that ∂ ⃗f(⃗u) for +⃗u ≻ ⃗0 is irreducible iff ⃗f is strongly connected. +PPS are usually analyzed in a decomposed fashion by considering the sub- +systems induced by the strongly connected components (SCCs) of G ⃗f in bottom- +up order [16]. Here we also follow this approach and therefore focus on strongly +connected PPS. The following was proved in [17, Lem. 6.5] and later generalized +in [12, Thm. 4.1] (also see remark below [12, Prop. 5.4] and [17, Lem. 8.2]): +Theorem 2 ([17,12]). If ⃗f is feasible, strongly connected and clean, then for +all ⃗u < µ⃗f, we have ρ(∂ ⃗f(⃗u)) < 1. As a consequence, ρ(∂ ⃗f(µ⃗f)) ≤ 1. +Theorem 2 partitions all PPS ⃗f which satisfy its precondition into two classes: +Either (1) ρ(∂ ⃗f(µ⃗f)) < 1, or (2) ρ(∂ ⃗f(µ⃗f)) = 1. In the next section we show +that ⃗f admits non-trivial inductive upper bounds iff it is in class (1). +Example 2. Reconsider the PPS ⃗fex from Example 1. It can be shown that +⃗v = (1, λ1)T where λ1 ≈ 0.557 is an eigenvector of ∂ ⃗fex(µ⃗fex) with eigenvalue λ1. +Thus by the Perron-Frobenius Theorem, ρ(∂ ⃗fex(µ⃗fex)) = λ1 < 1. As promised, +there exist inductive upper bounds as can be seen in Figure 2. +△ +3 +Finding Inductive Upper Bounds in PPS +In this section, we are concerned with the following problem: Given a feasible, +clean, and strongly connected PPS ⃗f, find a vector ⃗0 ≺ ⃗u ≺ ⃗∞ such that +⃗f(⃗u) ≤ ⃗u, i.e., an inductive upper bound on the lfp of ⃗f (see Lemma 1). + +Certificates for Probabilistic Pushdown Automata via OVI +7 +0.2 +0.4 +0.6 +0.8 +0.2 +0.4 +0.6 +0.8 +µ⃗fex +ε +⃗v +µ⃗˜fex +x +y +x = y + 0.1 +y = 0.2x2 + 0.8xy + 0.1 +y = 0.2x2 + 0.8xy + 0.1916 +Fig. 2: The PPS ⃗fex corresponds to the solid red line and the solid blue curve. Its +inductive upper bounds form the shaded area above the lfp µ⃗fex. Lemma 2(4) +ensures that one can fit the gray “cone” pointing in direction of the Perron- +Frobenius eigenvector ⃗v inside the inductive region. The PPS ⃗˜fex which com- +prises the dashed curve and the solid line does not have any non-trivial inductive +upper bounds. Note that the tangent lines at µ⃗˜fex are parallel to each other. +3.1 +Existence of Inductive Upper Bounds +An important first observation is that inductive upper bounds other than the +exact lfp do not necessarily exist. As a simple counter-example consider the 1- +dimensional PPS x = 1 +2x2 + 1 +2. If u is an inductive upper bound, then +1 +2u2 + 1 +2 ≤ u +=⇒ +u2 − 2u + 1 ≤ 0 +=⇒ +(u − 1)2 ≤ 0 +=⇒ +u = 1 , +and thus the only inductive upper bound is the exact lfp u = 1. Another example +is the PPS ⃗˜fex from Figure 2. What these examples have in common is the +following property: Their derivative evaluated at the lfp is not invertible. Indeed, +we have +∂ +∂x( 1 +2x2 + 1 +2 − x) = x − 1, and inserting the lfp x = 1 yields zero. The +higher dimensional generalization of this property to arbitrary PPS ⃗f is that the +Jacobi matrix of the function ⃗f − ⃗x evaluated at µ⃗f is singular; note that this +is precisely the matrix ∂ ⃗f(µ⃗f) − I. Geometrically, this means that the tangent +lines at µ⃗f are parallel, as can be seen in Figure 2 for the example PPS ⃗˜fex. It +should be intuitively clear from the figure that inductive upper bounds only exist +if the tangent lines are not parallel. The next lemma makes this more precise: +Lemma 2 (Existence of inductive upper bounds). +Let ⃗f be a feasible, +clean, and strongly connected PPS. Then the following are equivalent: +(1) The matrix I − ∂ ⃗f(µ⃗f) is non-singular. +(2) The spectral radius of ∂ ⃗f(µ⃗f) satisfies ρ(∂ ⃗f(µ⃗f)) < 1. +(3) There exists ⃗0 ≺ ⃗u ≺ ⃗∞ s.t. ⃗f(⃗u) < ⃗u (i.e. ⃗u is inductive but not a fixpoint). + +8 +Tobias Winkler and Joost-Pieter Katoen +(4) The matrix ∂ ⃗f(µ⃗f) has a unique (normalized) eigenvector ⃗v ≻ ⃗0 and there +exist numbers δmax > 0 and ε > 0 s.t. +⃗f( µ⃗f + δ · ⃗˜v ) +≺ +µ⃗f + δ · ⃗˜v +holds for all 0 < δ ≤ δmax and vectors ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε. +The proof of Lemma 2 (see appendix) relies on a linear approximation of +⃗f via Taylor’s familiar theorem as well as Theorems 1 and 2. Condition (4) of +Lemma 2 means that there exists a “truncated cone” +C(µ⃗f,⃗v, ε, δmax) = { µ⃗f + δ⃗˜v | 0 ≤ δ ≤ δmax,⃗˜v ≥ ⃗v, ||⃗˜v − ⃗v||∞ ≤ ε } +which is entirely contained in the inductive region. This cone is located at the +lfp µ⃗f and points in the direction of the Perron-Frobenius eigenvector ⃗v, as +illustrated in Figure 2 (assuming δmax = 1 for simplicity). The length δmax +and the radius ε of the cone depend quantitatively on ρ(∂ ⃗f(µ⃗f)), but for our +purposes it suffices that they are non-zero. The idea of our Optimistic Value +Iteration is to construct a sequence of guesses that eventually hits this cone. +3.2 +The Optimistic Value Iteration Algorithm +The basic idea of Optimistic Value Iteration (OVI) can be applied to monotone +functions of the form ⃗φ: Rn +≥0 → Rn +≥0 (in [22], ⃗φ is the Bellman operator of an +MDP). Kleene’s fixpoint theorem suggests a simple method for approximating +the lfp µ⃗φ from below: Simply iterate ⃗φ starting at ⃗0, i.e., compute the sequence +⃗l0 = ⃗0, ⃗l1 = ⃗φ(⃗l0), ⃗l2 = ⃗φ(⃗l1), etc.1 In the context of MDP, this iterative scheme +is known as Value Iteration (VI). VI is easy to implement, but it is difficult +to decide when to stop the iteration. In particular, standard stopping criteria +such as small absolute difference of consecutive approximations are formally un- +sound [20]. OVI and other algorithms [3,36] cope with this problem by computing +not only a lower but also an upper bound on µ⃗φ. In the case of OVI, an upper +bound with absolute error ≤ ε is obtained as follows (we omit some details): +(1) Compute ⃗lk ≤ µ⃗φ such that ||⃗lk −⃗lk−1||∞ ≤ τ, for some (small) τ > 0. +(2) Guess a candidate upper bound ⃗u = ⃗lk + ⃗ε. +(a) If ⃗φ(⃗u) ≤ ⃗u holds, i.e., ⃗u is inductive, then return ⃗u. +(b) If not, refine ⃗u (see [22] for details). If the refined ⃗u is still not inductive, +then go back to step (1) and try again with 0 < τ ′ < τ. +We present our variant of OVI for PPS as Algorithm 1. The main differences +to the above scheme are that (i) we do not insist on Kleene iteration for obtaining +the lower bounds ⃗l, and (ii) we approximate the eigenvector ⃗v from condition (4) +of Lemma 2 and compute the “more informed” guesses ⃗u = ⃗l + ε⃗v, for various ε. +Refining the guesses as original OVI does is not necessary (but see our remarks +in Section 3.3 regarding floating point computations). +1 In order for the Kleene seqence to converge to the lfp, i.e., limk→∞⃗lk = µφ, it suffices +that ⃗φ is ω-continuous. This already implies monotonicity. + +Certificates for Probabilistic Pushdown Automata via OVI +9 +Algorithm 1: Optimistic Value Iteration (OVI) for PPS +input +: strongly connected clean PPS ⃗f; maximum abs. error ε > 0 +output +: a pair (⃗l, ⃗u) of real vectors s.t. ⃗l ≤ µ⃗f, ⃗f(⃗u) ≤ ⃗u (hence +µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε +termination : guaranteed if ⃗f is feasible and I − ∂ ⃗f(µ⃗f) is non-singular +1 ⃗l ← ⃗0 ; N ← 0 ; +2 τ ← ε ; +/* τ is the current tolerance */ +3 while true do +4 +⃗l′ ← improveLowerBound(⃗f,⃗l) ; +/* e.g. Kleene or Newton update */ +/* guess and verify phase starts here +*/ +5 +if ||⃗l − ⃗l′||∞ ≤ τ then +6 +⃗v ← approxEigenvec(∂ ⃗f(⃗l), τ) ; +/* recall ⃗v is normalized */ +7 +for k from 0 to N do +8 +⃗u ← ⃗l + dkε · ⃗v ; +/* optimistic guess, d ∈ (0, 1) */ +9 +if ⃗f(⃗u) ≤ ⃗u then +10 +return (⃗l, ⃗u) ; +/* guess was successful */ +11 +N ← N + 1 ; +12 +τ ← c · τ ; +/* decrease tolerance for next guess, c ∈ (0, 1) */ +13 +⃗l ← ⃗l′ ; +The functions improveLowerBound and approxEigenvec used in Algorithm 1 +must satisfy the following contracts: +– The sequence ⃗l0 = ⃗0, ⃗li+1 = improveLowerBound(⃗f,⃗li) is a monotonically +increasing sequence converging to the lfp µ⃗f. +– approxEigenvec must satisfy the following: Let M ≥ 0 be an irreducible +matrix with (normalized) Perron-Frobenius eigenvector ⃗v ≻ ⃗0. Then for all +ε > 0, we require that there exists τ > 0 such that ||approxEigenvec(M, τ)− +⃗v||∞ ≤ ε. In words, approxEigenvec approximates ⃗v up to arbitrarily small +absolute error if the tolerance τ is chosen sufficiently small. +In practice, both the Kleene and the Newton [16,17,12] update operator can +be used to implement improveLowerBound. We outline a possible implementa- +tion of approxEigenvec further below in Section 3.3. +Example 3. Consider the following PPS ⃗f: x = 1 +4x2 + 1 +8, y = 1 +4xy + 1 +4y + 1 +4. The +table illustrates the execution of Algorithm 1 on ⃗f with ε = 0.1 and c = 0.5: +# N +τ +⃗l +⃗l′ +||⃗l −⃗l′||∞ +⃗v +⃗u +⃗f(⃗u) ≤ ⃗u +1 +0 +0.1 +(0, 0) +(0.4, 0.3) +0.4 +2 +0 +0.1 +(0.4, 0.3) +(0.5, 0.4) +0.1 +(1.0, 0.8) (0.5, 0.38) +✗ +3 +1 +0.05 (0.5, 0.4) (0.55, 0.41) +0.05 +(1.0, 0.9) (0.6, 0.49) +✓ + +10 +Tobias Winkler and Joost-Pieter Katoen +The algorithm has to improve the lower bound 3 times (corresponding to the +3 lines of the table). After the second improvement, the difference between the +current lower bound ⃗l2 and the new bound ⃗l′2 does not exceed the current tol- +erance τ2 = 0.1 and the algorithm enters the optimistic guessing stage. The first +guess ⃗u2 is not successful. The tolerance is then decreased to τ3 = c · τ2 = 0.05 +and the lower bound is improved to ⃗l′3. The next guess ⃗u3 is inductive. +△ +Theorem 3. Algorithm 1 is correct: when invoked with a strongly connected +clean PPS ⃗f and ε > 0, then (if it terminates) it outputs a pair (⃗l, ⃗u) s.t. ⃗l ≤ µ⃗f, +⃗f(⃗u) ≤ ⃗u (and thus µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε. Moreover, if ⃗f is feasible and +I − ∂ ⃗f(µ⃗f) is non-singular, then the algorithm terminates. +The proof of Theorem 3 (see appendix) crucially relies on condition (4) of +Lemma 2 that assures the existence of a “truncated cone” of inductive bounds +centered around the Perron-Frobenius eigenvector of ∂ ⃗f(µ⃗f) (see Figure 2 for +an illustration). Intuitively, since the lower bounds ⃗l computed by the algorithm +approach the lfp µ⃗f, the eigenvectors of ∂ ⃗f(⃗l) approach those of ∂ ⃗f(µ⃗f). As a +consequence, it is guaranteed that the algorithm eventually finds an eigenvector +that intersects the cone. The inner loop starting on line 7 is needed because the +“length” of the cone is a priori unknown; the purpose of the loop is to scale the +eigenvector down so that it is ultimately small enough to fit inside the cone. +3.3 +Considerations for Implementing OVI +As mentioned above, there are at least two options for improveLowerBound: +Kleene or Newton iteration. We now show that approxEigenvec can be effec- +tively implemented as well. Further below we make some remarks on floating +point arithmetic. +Approximating the Eigenvector. A possible implementation of approxEigenvec +relies on the power iteration method (e.g. [37, Thm. 4.1]). Given a square matrix +M and an initial vector ⃗v0 with M⃗v0 ̸= ⃗0, power iteration computes the sequence +(⃗vi)i≥0 such that for i > 0, ⃗vi = M⃗vi−1/||M⃗vi−1||∞. +Lemma 3. Let M ≥ 0 be irreducible. Then power iteration applied to M + I +and any ⃗v0 > ⃗0 converges to the Perron-Frobenius eigenvector ⃗v ≻ ⃗0 of M. +The convergence rate of power iteration is determined by the ratio |λ2|/|λ1| +where λ1 and λ2 are eigenvalues of largest and second largest absolute value, +respectively. Each time approxEigenvec is called in Algorithm 1, the result of +the previous call to approxEigenvec (if available) may be used as an initial +approximation ⃗v0. + +Certificates for Probabilistic Pushdown Automata via OVI +11 +Exact vs Floating Point Arithmetic. So far we have assumed exact arithmetic +for the computations in Algorithm 1, but an actual implementation should use +floating point arithmetic for efficiency. However, this may (and actually does) +lead to unsound results. More specifically, the condition ⃗f(⃗u) ≤ ⃗u may hold in +floating point arithmetic even though it is actually violated. As a remedy, we +propose to nevertheless run the algorithm with floats, but then verify its output ⃗u +with exact arbitrary-precision rational arithmetic. That is, we compute a rational +number approximation ⃗uQ of ⃗u and check ⃗f(⃗uQ) ≤ ⃗uQ with exact arithmetic. If +the check fails, we resort to the following refinement scheme which is an instance +of the general k-induction principle for complete lattices from [5]: We iteratively +check the conditions +⃗f(⃗uQ ⊓ ⃗f(⃗uQ)) ≤ ⃗uQ , +⃗f(⃗uQ ⊓ ⃗f(⃗uQ ⊓ ⃗f(⃗uQ))) ≤ ⃗uQ , +and so on, +where ⊓ denotes pointwise minimum. If one of the checks is satisfied, then µ⃗f ≤ +⃗uQ [5]. This scheme often works well in practice (see Section 5). The original +OVI from [22] uses a similar technique to refine its guesses. +4 +Certificates for Probabilistic Pushdown Automata +This section shows how the results from Section 3 can be applied to pPDA. +We introduce some additional notation. For finite sets A, D(A) denotes the +set of probability distributions on A. We often denote tuples or triples without +parentheses and separating commata when this causes no confusion, e.g., we may +write ab rather than (a, b). +Definition 1 (pPDA [13]). A probabilistic pushdown automaton (pPDA) is a +triple ∆ = (Q, Γ, P) where Q ̸= ∅ is a finite set of states, Γ ̸= ∅ is a finite stack +alphabet, and P : Q × Γ → D(Q × Γ ≤2) is a probabilistic transition function. +In the following, we often write qZ +p−→ rα instead of P(qZ)(rα) = p [13]. Intu- +itively, qZ +p−→ rα means that if the pPDA is in state q and Z is on top of the +stack, then with probability p, the pPDA moves to state r, pops Z and pushes α +on the stack. More formally, the semantics of a pPDA ∆ = (Q, Γ, P) is a count- +ably infinite Markov chain with state space Q × Γ ∗ and transition probability +matrix M such that for all q, r ∈ Q, Z ∈ Γ, α ∈ Γ ≤2, γ ∈ Γ ∗, we have +M(qZγ, rαγ) = P(qZ)(rα) , +M(qε, qε) = 1 , +and all other transition probabilities are zero. This Markov chain, where the +initial state is fixed to qZ, is denoted MqZ +∆ (see Figure 3 for an example). As +usual, one can formally define a probability measure PqZ +∆ on the infinite runs of +MqZ +∆ via the standard cylinder construction (e.g., [2, Sec. 10]). +Consider a triple qZr ∈ Q×Γ×Q. We define the return probability2 [qZr] as +the probability of reaching rε in the Markov chain MqZ +∆ , i.e., [qZr] = PqZ +∆ (♦{rε}), +where ♦{rε} is the set of infinite runs of MqZ +∆ that eventually hit state rε. +2 When modeling procedural programs with pPDA, [qZr] is the probability that a +given procedure returns a specific value to its calling context. These probabilities + +12 +Tobias Winkler and Joost-Pieter Katoen +q +r +(1/2, Z, ε) +(1/4, Z, ZZ) +(1/4, Z, ε) +(1, Z, ε) +qε +qZ +qZZ +· · · +rε +rZ +rZZ +· · · +1/4 +1/4 +1/2 +1/2 +1/2 +1/4 +1/4 +1/4 +1 +1 +1 +1 +1 +⟨qZq⟩ = +1/4 +� +⟨qZq⟩⟨qZq⟩ + ⟨qZr⟩⟨rZq⟩ +� ++ 1/2 +⟨rZq⟩ = 0 +⟨qZr⟩ = +1/4 +� +⟨qZq⟩⟨qZr⟩ + ⟨qZr⟩⟨rZr⟩ +� ++ 1/4 +⟨rZr⟩ = 1 +Fig. 3: Top left: The pPDA ∆ex = ({q, r}, {Z}, P) where P comprises the tran- +sitions qZ +1/4 +−−→ qZZ, qZ +1/2 +−−→ qε, qZ +1/4 +−−→ rε, rZ +1−→ rε. Top right: A fragment of +the infinite underlying Markov chain, assuming initial configuration qZ. Bottom: +The associated equation system from Theorem 4. +Theorem 4 (The PPS of return probabilities [13]). Let ∆ = (Q, Γ, P) be +a pPDA and (⟨qZr⟩)qZr ∈ Q×Γ ×Q be variables. For each ⟨qZr⟩, define +⟨qZr⟩ += +� +qZ +p−→sY X +p · +� +t∈Q +⟨sY t⟩ · ⟨tXr⟩ + +� +qZ +p−→sY +p · ⟨sY r⟩ + +� +qZ +p−→rε +p +and call the resulting PPS ⃗f∆. Then µ⃗f∆ = ([qZr])qZr ∈ Q×Γ ×Q. +We refer to [30, Sec. 3] for an intuitive explanation of the equations in ⃗f∆. +Example 4. Figure 3 shows a pPDA ∆ex and the associated PPS ⃗f∆ex. The +least non-negative solution is ⟨qZq⟩ = 2 − +√ +2 ≈ 0.586 and ⟨qZr⟩ = +√ +2 − 1 ≈ +0.414 (and, of course, ⟨rZq⟩ = 0, ⟨rZr⟩ = 1). Thus by Theorem 4, the return +probabilities are [qZq] = 2 − +√ +2 and [qZr] = +√ +2 − 1. +△ +The PPS ⃗f∆ is always feasible (because µ⃗f∆ ≤ ⃗1). ⃗f∆ is neither necessarily +strongly connected nor clean. Let ⃗ˆf∆ denote the cleaned up version of ⃗f∆. +Proposition 1 (Basic Certificates for pPDA). +A basic certificate for +∆ = (Q, Γ, P) is a rational inductive upper bound ⃗u ∈ QQ×Γ ×Q +≥0 +on the lfp of the +return probabilities system ⃗f∆ (see Thm. 4). They have the following properties: +– (Existence) ∀ε > 0 there exists a basic certificate ⃗u with ||µ⃗f∆ − ⃗u||∞ ≤ ε if +all maximal irreducible submatrices M of ∂ ⃗ˆf∆(µ⃗ˆf∆) satisfy ρ(M) < 1. +were called termination probabilities in previous works [12,7] but we believe this +term is more appropriate for the numbers [qZ↓] = � +r[qZr], i.e., the probability to +eventually reach the empty stack from initial configuration qZ. + +Certificates for Probabilistic Pushdown Automata via OVI +13 +– (Complexity) Let β be the maximum number of bits used to encode any of +the numerators and denominators of the fractions occurring in ⃗u ∈ QQ×Γ ×Q +≥0 +. +Then checking ⃗f∆(⃗u) ≤ ⃗u, i.e., whether ⃗u is basic certificate for ∆, can be +done in time polynomial in β and the size of ∆. +Existence of basic certificates follows from Lemma 2 applied to each SCC of +the cleaned-up version of ⃗f∆ individually. However, note that in order to merely +check the certificate, i.e., verify the inequality ⃗f(⃗u) ≤ ⃗u, neither do SCCs need +to be computed nor does the system has to be cleaned up. +Example 5. Reconsider the example pPDA and its associated (non-strongly con- +nected) system of return probabilities from Figure 3. We verify that ⃗uqZq = 3/5 +and ⃗uqZr = 1/2 (as well as ⃗urZq = 0, ⃗urZr = 1) is a basic certificate: +1 +4 +�3 +5 · 3 +5 + 1 +2 · 0 +� ++ 1 +2 = 59 +100 +✓ +≤ 3 +5 +, +1 +4 +�3 +5 · 1 +2 + 1 +2 · 1 +� ++ 1 +4 = 45 +100 +✓ +≤ 1 +2 . +Note that [qZq] ≈ 0.586 ≤ 3/5 = 0.6 and [qZr] ≈ 0.414 ≤ 1/2 = 0.5. +△ +In the following we outline how a variety of key quantities associated to pPDA +can be verified using basic certificates. More details are in the appendix. +Upper Bounds on Temporal Properties. We may use basic certificates to verify +that a bad state rbad is reached with low probability, e.g., at most p = 0.01. +To this end, we remove the outgoing transitions of rbad and add the transitions +rbadZ +1−→ rbadε for all Z ∈ Γ. Clearly, rbad is reached with probability at most p +from initial configuration qZ iff [qZrbad] ≤ p. The results of [13] imply that this +idea can be generalized to until-properties of the form C1 U C2, where C1 and C2 +are regular sets of configurations. (This requires a small extension of the basic +certificates, but the overall idea stays the same). +Certificates for the Output Distribution. Once a pPDA reaches the empty stack, +we say that it has terminated. When modeling procedural programs, this cor- +responds to returning from a program’s main procedure. Assuming initial con- +figuration qZ, the probability sub-distribution over the possible return values is +then given by the return probabilities {[qZr] | r ∈ Q}. Missing probability mass +models the probability of non-termination. A basic certificate can thus be used +immediately to verify a point-wise upper bound on the output distribution as +well as to certify that a program is not almost-surely terminating (AST). If a +pPDA ∆ is already known to be AST, then we can also certify a lower bound on +the output distribution: Suppose that ⃗u is a basic certificate for ∆ and assume +that ∆ is AST from initial configuration qZ. Define ε = � +r∈Q ⃗uqZr − 1. Then +for all r ∈ Q, we have ⃗uqZr − ε ≤ [qZr] ≤ ⃗uqZr. +Example 6. The pPDA ∆ex from Figure 3 is AST from initial configuration qZ, +as the transition qZ +1/4 +−−→ rε is eventually taken with probability 1, and the stack +is emptied certainly once r is reached. Using the basic certificate from Example 5 +we can thus (correctly) certify that 0.5 ≤ [qZq] ≤ 0.6 and 0.4 ≤ [qZr] ≤ 0.5. + +14 +Tobias Winkler and Joost-Pieter Katoen +Certificates for Expected Rewards or Costs. Suppose we have equipped a pPDA +with a state-based reward (or cost) function Q → R≥0. It was shown in [14] that +the expected total reward accumulated during the run of a pPDA is the solution +of a linear equation system where the return probabilities [qZr] appear as coef- +ficients. Given a basic certificate ⃗u, we can replace each coefficient [qZr] by ⃗uqZr +and thus obtain an equation system whose solution is an over-approximation of +the true expected reward. We may extend the basic certificate ⃗u by the solution +of this linear system to make verification straightforward. Note that a program’s +expected runtime [8,35] is a special case of total expected reward. +5 +Implementation and Experiments +Our Tool: pray. We implemented our algorithm in the prototypical Java-tool +pray (Probabilistic Recursion AnalYzer). It supports two input formats: (i) +Recursive probabilistic programs in a Java-like syntax (e.g. Figure 4); these +programs are automatically translated to pPDA. (ii) Explicit PPS in the same +syntax used by the tool PReMo [43]. The output of pray is a rational inductive +upper bound on the lfp of the return probability PPS of the input program’s +pPDA model (a basic certificate), or on the lfp of the explicitly given PPS. The +absolute precision ε is configurable. The implementation works as follows: +(1) It parses the input and, if the latter was a program, constructs a pPDA +model and the associated PPS of return probabilities. +(2) It computes an SCC decomposition of the PPS under consideration using +standard algorithms implemented in the jGraphT library [33]. +(3) It applies Algorithm 1 to the individual SCC in reverse topological order +using floating point arithmetic. Algorithm 1 is instantiated with Kleene it- +eration3, the power iteration for approximating eigenvectors as outlined in +Section 3.3, and constants c = 0.1, d = 0.5. We allow ≤ 10 guesses per SCC. +(4) If stage (3) is successful, the tool verifies the resulting floating point certifi- +cate using exact rational number arithmetic as described in Section 3.3. +Baselines. To the best of our knowledge, no alternative techniques for finding +inductive upper bounds in PPS have been described explicitly in the literature. +However, there is an (almost) out-of-the-box approach using an SMT solver: +Given a PPS ⃗x = ⃗f(⃗x), compute some lower bound ⃗l ≤ µ⃗f using an iterative +technique. Then query the SMT solver for a model (variable assignment) of the +quantifier-free first-order logic formula ϕ⃗f(⃗x) = �n +i=1 fi(⃗x) ≤ xi ∧⃗li ≤ xi ≤ ⃗li +ε +in the (decidable) theory of polynomial real arithmetic with inequality (aka +QF_NRA in the SMT community). If such a model ⃗u exists, then clearly µ⃗f ≤ ⃗u +and ||⃗l − ⃗u||∞ ≤ ε. If no model exists, then improve ⃗l and try again. We have +3 In fact, we use the slightly optimized Gauss-Seidel iteration (see [42, Sec. 5.2]) which +provides a good trade-off between ease of implementation and efficiency [42]. + +Certificates for Probabilistic Pushdown Automata via OVI +15 +bool and() { +prob { +1//2: return +(1//2: true | 1//2: false); +1//2: { +if(!or()) return false; +else return or(); } } } +bool or() { +prob { +1//2: return +(1//2: true | 1//2: false); +1//2: { +if(and()) return true; +else return and(); } } } +Fig. 4: Program evaluating a random and-or tree [8]. The prob-blocks execute +the contained statements with the respective probabilities (syntax inspired by +Java’s switch). Our tool automatically translates this program to a pPDA and +computes a basic certificate (Proposition 1) witnessing that calling and() returns +true and false with probability ≤ 382/657 ≈ 0.58 and 391/933 ≈ 0.42, resp. +implemented this approach using the state-of-the-art SMT solvers cvc5 [4] and +z3 [34], the winners of the 2022 SMT-COMP in the category QF_NRA4. +As yet another baseline, we have also implemented a variant of OVI for PPS +which is closer to the original MDP algorithm from [22]. In this variant, called +“standard OVI” from now on, we compute the candidate ⃗u based on the “relative” +update rule ⃗u = (1 + ε)⃗l, where ⃗l is the current lower bound [22]. +Research Questions. We aim to shed some light on the following questions: (A) +How well does our algorithm scale? (B) Is the algorithm suitable for PPS with dif- +ferent characteristics, e.g., dense or sparse? (C) Is the requirement ρ(∂ ⃗f(µ⃗f)) < 1 +restrictive in practice? (D) How does our OVI compare to the baselines? +Benchmarks. To answer the above questions we run our implementation on two +sets of benchmarks (Table 3 and Table 2, respectively). The first set consists of +various example programs from the literature as well as a few new programs, +which are automatically translated to pPDA. This translation is standard and +usually takes not more than a few seconds. The programs golden, and-or (see Fig- +ure 4), virus, gen-fun are adapted from [35,8,41] and [32, Program 5.6], respec- +tively. The source code of all considered programs is in the appendix. We have +selected only programs with possibly unbounded recursion depth which induce +infinite Markov chains. The second benchmark set comprises explicitly given +PPS5. The instances brown, lemonde, negra, swbd, tiger, tuebadz, and wsj all en- +code SCFG from the area of language processing (see [43] for details). random is +the return probability system of a randomly generated pPDA. +Summary of Experimental Results. We ran the experiments on a standard note- +book. The approach based on cvc5 turns out to be not competitive (see Ap- +pendix D). We thus focus on z3 in the following. Both pray and the z3 approach +could handle most of the programs from Table 3 within a 10 minute time limit. +The considered programs induce sparse PPS with 38 - 26,367 variables, and most +4 https://smt-comp.github.io/2022/results +5 These examples come with PReMo: https://cgi.csc.liv.ac.uk/~dominik/premo/ + +16 +Tobias Winkler and Joost-Pieter Katoen +Table 1: Experiments with PPS obtained from recursive probabilistic programs. +Columns vars and terms display the number of variables and terms in the PPS. +Columns sccs and sccmax indicate the number of non-trivial SCC and the size of +the largest SCC. G is total number of guesses made by OVI (at least one guess per +SCC). ttot is the total runtime excluding the time for model construction. tQ is +the percentage of ttot spent on exact rational arithmetic. D is the average number +of decimal digits of the rational numbers in the certificate. The timeout (TO) +was set to 10 minutes. Timings are in ms. The absolute precision is ε = 10−3. +benchmark +|Q| +|P| +|Γ| +vars terms sccs sccmax cert G D +tQ +ttot certz3 Dz3 +tz3 certstd Gstd Dstd +tstd +rw-0.499 +18 +29 +5 +38 +45 +1 +12 +✓ +5 5 17% +163 +✓ +2 +11 +✓ +4 +5 +59 +rw-0.500 +18 +29 +5 +38 +45 +1 +12 +✗ +10 +- +- +7327 +✓ +2 +10 +✗ +10 +- +8083 +rw-0.501 +18 +29 +5 +38 +45 +1 +12 +✓ +5 4 +6% +36 +✓ +13 +12 +✓ +4 +5 +23 +geom-offspring +24 +40 +5 +52 +80 +4 +24 +✓ +8 6 13% +15 +✓ +9 +16 +✓ +8 +6 +14 +golden +27 +49 +6 +81 +94 +1 +36 +✓ +1 5 30% +10 +✓ +7 +14 +✓ +2 +4 +12 +and-or +50 +90 +7 +149 +182 +1 +48 +✓ +2 4 26% +19 +✓ +12 +15260 +✓ +2 +4 +19 +gen-fun +85 +219 +7 +202 +327 +1 +16 +✓ +2 3 32% +22 +✓ +15 +141 +✓ +2 +3 +21 +virus +68 +149 +27 +341 +551 +1 +220 +✓ +1 5 38% +40 +✓ +7 +139 +✓ +1 +6 +59 +escape10 +109 +174 +23 +220 +263 +1 +122 +✓ +1 4 +5% +56 +✓ +7 +48 +✓ +1 +8 +71 +escape25 +258 +413 +53 +518 +621 +1 +300 +✓ +1 5 17% +245 +✓ +7 +15958 +✓ +1 +9 +172 +escape50 +508 +813 +103 +1018 +1221 +1 +600 +✓ +1 7 23% +653 +✓ +7 +410 +✗ +1 +- +400 +escape75 +760 1215 +153 +1522 +1825 +1 +904 +✓ +2 9 10% +3803 +✗ +- +TO +✗ +1 +- +635 +escape100 +1009 1614 +203 +2020 +2423 +1 +1202 +✗ +5 +- +- +29027 +✓ +6 +939 +✗ +1 +- +901 +escape200 +2008 3213 +403 +4018 +4821 +1 +2400 +✗ +6 +- +- +83781 +✗ +- +TO +✗ +1 +- +2206 +sequential5 +230 +490 +39 +1017 +1200 +10 +12 +✓ +15 +4 26% +103 +✓ +8 +1074 +✓ +15 +5 +204 +sequential7 +572 1354 +137 +3349 +3856 +14 +12 +✓ +21 +5 27% +1049 +✓ +8 +12822 +✓ +20 +5 +1042 +sequential10 +3341 8666 1036 26367 29616 +20 +12 +✓ +30 +5 +2% 100613 +✓ +8 453718 +✓ +30 +6 101554 +mod5 +44 +103 +10 +296 +425 +1 +86 +✓ +1 5 39% +28 +✓ +9 +34150 +✗ +2 +- +178 +mod7 +64 +159 +14 +680 +1017 +1 +222 +✓ +1 6 69% +172 +✓ +7 +443 +✗ +2 +- +624 +mod10 +95 +244 +20 +1574 +2403 +1 +557 +✗ +1 +- +- +675 +✓ +7 +1245 +✗ +2 +- +882 +of them have just a single SCC. Notably, the examples with greatest maximum +SCC size were only solved by z3. pray and z3 need at most 95 and 31 seconds, +respectively, for the instances where they succeed. In many cases (e.g., rw-5.01, +golden, virus, brown, swbd), the resulting certificates formally disprove AST. For +the explicit PPS in Table 2, pray solves all instances whereas z3 only solves +3/8 within the time limit, and only finds the trivial solution ⃗1. Most of these +benchmarks contain dense high-degree polynomials and our tool spends most +time on performing exact arithmetic. pray never needs more than 6 guesses per +SCC if it succeeds. +Evaluation of Research Questions. (A) Scalability: Our algorithm succeeded on +instances with maximum SCC size of up to 8,000 and number of terms over +50,000. pray could solve all instances with a maximum SCC size of ≤ 1,000 in +less than 2 minutes per instance. For the examples where our algorithm does +not succeed (e.g., escape100) it is mostly because it fails converting a floating +point to a rational certificate. (B) PPS with different flavors: The problems +in Table 3 (low degree and sparse, i.e., few terms per polynomials) and Table 2 +(higher degree and dense) are quite different. A comparison to the SMT approach +suggests that our technique might be especially well suited for dense problems +with higher degrees. (C) Non-singularity: The only instance where our algorithm +fails because of the non-singularity condition is the symmetric random walk rw- + +Certificates for Probabilistic Pushdown Automata via OVI +17 +Table 2: Experiments with explicitly given PPS (setup as in Table 3). +benchmark +vars terms sccs sccmax cert G D +tQ +ttot certz3 Dz3 +tz3 certstd Gstd Dstd +tstd +brown +37 22866 +1 +22 +✓ +2 +6 74% +3212 +✗ +- +TO +✓ +2 +8 +9065 +lemonde +121 32885 +1 +48 +✓ +2 +5 97% 40738 +✗ +- +TO +✓ +2 +5 38107 +negra +256 29297 +1 +149 +✓ +2 +7 89% 10174 +✓ +1 37248 +✓ +1 +7 +8873 +swbd +309 47578 +1 +243 +✓ +1 +7 93% 18989 +✗ +- +TO +✓ +1 +8 67314 +tiger +318 52184 +1 +214 +✓ +2 +8 98% 94490 +✓ +1 17454 +✓ +1 +8 90801 +tuebadz +196 +8932 +2 +168 +✓ +4 +9 85% +2666 +✓ +1 15323 +✓ +3 +9 +2700 +wsj +240 31170 +1 +194 +✓ +2 +9 96% 30275 +✗ +- +TO +✓ +2 +9 29038 +random +10000 20129 +1 +8072 +✓ +3 +7 +5% 17585 +✗ +- +TO +✓ +4 +8 16357 +0.500. We therefore conjecture that this condition is often satisfied in practice. +(D) Comparison to SMT: There is no clear winner. Some instances can only be +solved by one tool or the other (e.g. escape100 and brown). However, pray often +delivers more succinct certificates, i.e., the rational numbers have less digits. +Overall, z3 behaves less predictably than pray. +6 +Conclusion and Future Work +We have proposed using inductive bounds as certificates for various properties in +probabilistic recursive models. Moreoever, we have presented the first dedicated +algorithm for computing inductive upper bounds. While our algorithm already +scales to non-trivial problems, the main bottleneck is the generation of an exact +rational bound from a floating point approximation. This might be improved +using appropriate rounding modes as in [21]. Additional future work includes +further certificates for pPDA, especially for lower bounds and termination. +References +1. 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IEEE Computer Society (2005) + +20 +Tobias Winkler and Joost-Pieter Katoen +A +Full Proofs +A.1 +Proof of Lemma 2 +Lemma 2 (Existence of inductive upper bounds). +Let ⃗f be a feasible, +clean, and strongly connected PPS. Then the following are equivalent: +(1) The matrix I − ∂ ⃗f(µ⃗f) is non-singular. +(2) The spectral radius of ∂ ⃗f(µ⃗f) satisfies ρ(∂ ⃗f(µ⃗f)) < 1. +(3) There exists ⃗0 ≺ ⃗u ≺ ⃗∞ s.t. ⃗f(⃗u) < ⃗u (i.e. ⃗u is inductive but not a fixpoint). +(4) The matrix ∂ ⃗f(µ⃗f) has a unique (normalized) eigenvector ⃗v ≻ ⃗0 and there +exist numbers δmax > 0 and ε > 0 s.t. +⃗f( µ⃗f + δ · ⃗˜v ) +≺ +µ⃗f + δ · ⃗˜v +holds for all 0 < δ ≤ δmax and vectors ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε. +We now explain the proof of Lemma 2. The proof heavily relies on a linear +approximation of ⃗f around the lfp µ⃗f. Intuitively, this is where the Jacobi matrix +∂ ⃗f(µ⃗f) comes into play. This is formalized via Taylor’s familiar theorem. +Lemma 4 (Taylor’s Theorem; cf. [12, Lem. 2.3]). Let ⃗f be a feasible PPS. +Then for all vectors ⃗u ≥ ⃗0, we have +⃗f(µ⃗f + ⃗u) += +µ⃗f + ∂ ⃗f(µ⃗f)⃗u + R⃗u⃗u +where R⃗u is a matrix that depends on ⃗u such that lim⃗u→⃗0 R⃗u = 0. More specifi- +cally, it holds that ⃗0 ≤ R⃗u⃗u ≤ +� +∂ ⃗f(µ⃗f + ⃗u) − ∂ ⃗f(µ⃗f) +� +⃗u. +Proof (Proof of Lemma 2). “(1) =⇒ (2)”: By Theorem 2 we have ρ(∂ ⃗f(µ⃗f)) ≤ +1. Towards contradiction assume that ρ(∂ ⃗f(µ⃗f)) = 1. By the Perron-Frobenius +Theorem, 1 is an eigenvalue of ∂ ⃗f(µ⃗f), which means that there exists ⃗u ̸= ⃗0 such +that ∂ ⃗f(µ⃗f)⃗u = ⃗u. This ⃗u is in the kernel of I − ∂ ⃗f(µ⃗f), which contradicts the +assumption that I − ∂ ⃗f(µ⃗f) is non-singular. +“(2) =⇒ (1)”: It is a well-known result that for an arbitrary real matrix M +the series �∞ +k=0 M k converges iff ρ(M) < 1. The limit of the series is the inverse +of I − M because +(I − M) +∞ +� +k=0 +M = +∞ +� +k=0 +M k − +∞ +� +k=1 +M k = M 0 = I . +“(2) +=⇒ +(4)”: Let ρ(∂ ⃗f(µ⃗f)) =: λ < 1. By the Perron-Frobenius Theo- +rem, the Jacobi matrix ∂ ⃗f(µ⃗f) has a unique normalized eigenvector ⃗v ≻ ⃗0 wrt. +eigenvalue λ: +∂ ⃗f(µ⃗f)⃗v = λ⃗v ≺ ⃗v . +(1) + +Certificates for Probabilistic Pushdown Automata via OVI +21 +Our goal is to define the values ε and δmax whose existence we claimed in +Lemma 2(4). Let cmin > 0 be the smallest component of (1−λ)⃗v ≻ ⃗0. We define +ε := +cmin +3||∂ ⃗f(µ⃗f)||∞ +, +(2) +where ||∂ ⃗f(µ⃗f)||∞ = max||⃗y||∞=1 ||∂ ⃗f(µ⃗f)⃗y||∞ is the maximum row sum of +∂ ⃗f(µ⃗f). Note that || · ||∞ is the operator norm induced by the maximum norm. +Then it holds for all ⃗ε with ||⃗ε||∞ ≤ ε that +||∂ ⃗f(µ⃗f)⃗ε||∞ ≤ ||∂ ⃗f(µ⃗f)||∞||⃗ε||∞ ≤ ||∂ ⃗f(µ⃗f)||∞ +cmin +3||∂ ⃗f(µ⃗f)||∞ += 1 +3cmin . +(3) +The first inequality in (3) is a property of operator norms (which is straightfor- +ward in the case of the maximum norm). Since cmin was the smallest component +of (1 − λ)⃗v, (3) implies +∂ ⃗f(µ⃗f)⃗ε ≤ 1 +3(1 − λ)⃗v . +(4) +We now define δmax as follows: +δmax := sup {δ > 0 | ∀⃗ε ≥ ⃗0 s.t. ||⃗ε||∞ ≤ ε: Rδ(⃗v+⃗ε)(⃗v + ⃗ε) ≤ 1 +2(1 − λ)⃗v} , +(5) +where Rδ(⃗v+⃗ε) is the matrix from Lemma 4 which satisfies +⃗f(µ⃗f + δ(⃗v + ⃗ε)) = µ⃗f + δ∂ ⃗f(µ⃗f)(⃗v + ⃗ε) + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) . +We now argue that δmax > 0. This is not immediately obvious because of the +∀-quantification in (5). Let δ > 0 be arbitrary. Further, let ⃗ε ≥ ⃗0 be such that +||⃗ε||∞ ≤ ε. In the following, we write ⃗ε′ = (ε . . . ε). We have +Rδ(⃗v+⃗ε)(⃗v + ⃗ε) += 1 +δ Rδ(⃗v+⃗ε)δ(⃗v + ⃗ε) +≤ 1 +δ +� +∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε)) − ∂ ⃗f(µ⃗f) +� +δ(⃗v + ⃗ε) +(Lemma 4) += +� +∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε)) − ∂ ⃗f(µ⃗f) +� +(⃗v + ⃗ε) +≤ +� +∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε′)) − ∂ ⃗f(µ⃗f) +� +(⃗v + ⃗ε′) +(Jacobi matrix is monotonic) +=: Mδ(⃗v + ⃗ε′) +Note that Mδ does not depend on ⃗ε and limδ→0 Mδ = 0. We can therefore find a +specific δ∗ > 0 such that Mδ∗(⃗v+⃗ε′) ≤ 1 +2(1−λ)⃗v. On the other hand, we have just + +22 +Tobias Winkler and Joost-Pieter Katoen +shown for all ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε and all δ > 0 that Rδ(⃗v+⃗ε)(⃗v+⃗ε) ≤ Mδ(⃗v+⃗ε′). +So we have in particular for all ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε that +Rδ∗(⃗v+⃗ε)(⃗v + ⃗ε) ≤ Mδ∗(⃗v + ⃗ε′) ≤ 1 +2(1 − λ)⃗v . +Hence δmax ≥ δ∗ > 0. +Finally, let 0 < δ ≤ δmax and ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε, i.e., ⃗˜v = ⃗v + ⃗ε for +some ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε. Then +⃗f(µ⃗f + δ(⃗v + ⃗ε)) += µ⃗f + δ∂ ⃗f(µ⃗f)(⃗v + ⃗ε) + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) +(by Taylor’s Theorem (Lemma 4)) += µ⃗f + δλ⃗v + δ∂ ⃗f(µ⃗f)⃗ε + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) +(by (1)) +≤ µ⃗f + δλ⃗v + δ 1 +3(1 − λ)⃗v + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) +(by (4)) +≤ µ⃗f + δλ⃗v + δ 1 +3(1 − λ)⃗v + δ 1 +2(1 − λ)⃗v +(by (5)) +≺ µ⃗f + δλ⃗v + δ 1 +2(1 − λ)⃗v + δ 1 +2(1 − λ)⃗v +(because δ(1 − λ)⃗v ≻ ⃗0) += µ⃗f + δ⃗v +(simplification) +≤ µ⃗f + δ(⃗v + ⃗ε) +(because ⃗ε ≥ ⃗0) +“(4) =⇒ (3)”: Trivial. +“(3) =⇒ (2)”: By (3) there exists ⃗u such that ⃗f(⃗u) < ⃗u. By Lemma 1 this +implies that µ⃗f < ⃗u, so we can write ⃗u = µ⃗f + ⃗v for some ⃗v > ⃗0. +Using Taylor’s Theorem (Lemma 4), it follows that +⃗f(µ⃗f + ⃗v) = µ⃗f + ∂ ⃗f(µ⃗f)⃗v + R⃗v⃗v < µ⃗f + ⃗v . +(6) +Using that R⃗v⃗v ≥ ⃗0, (6) implies that +∂ ⃗f(µ⃗f)⃗v < ⃗v . +(7) +The claim now follows by applying the following lemma to the matrix ∂ ⃗f(µ⃗f) +and the vector ⃗v: +Lemma 5. Let M ≥ 0 be an irreducible n× n-matrix. If there exists ⃗u > ⃗0 such +that M⃗u < ⃗u, then ⃗u ≻ ⃗0, M n⃗u ≺ ⃗u and ρ(M) < 1. +Proof. First observe that since multiplication by M is monotone we have for all +0 ≤ k1 ≤ k2 that +⃗0 ≤ M k2⃗u ≤ M k1⃗u ≤ ⃗u . +We first show that ⃗u ≻ ⃗0, which is essentially [12, Lemma 5.3]. Since ⃗u > ⃗0, +there must be 1 ≤ i ≤ n such that ⃗ui > 0. Now let 1 ≤ j ≤ n be arbitrary. Since + +Certificates for Probabilistic Pushdown Automata via OVI +23 +M is irreducible there exists 0 ≤ k < n such that M k +j,i > 0. This implies that +(M k⃗u)j > 0. By monotonicty, ⃗u ≥ M k⃗u, and thus ⃗uj ≥ (M k⃗u)j > 0. Since j +was arbitrary, ⃗u ≻ ⃗0. +Next we show M n⃗u ≺ ⃗u. Since M⃗u < ⃗u holds by assumption, there exists +1 ≤ i ≤ n such that (M⃗u)i < ⃗ui. Let 1 ≤ j ≤ n be a arbitrary. Since M is +irreducible, there exists 0 ≤ k < n such that (M k)j,i > 0. We now show that +(M n⃗u)j < uj which implies that M n⃗u ≺ ⃗u as j was chosen arbitrarily: +(M n⃗u)j +≤ (M kM⃗u)j +(by monotonicity, and because k + 1 ≤ n) += (M k)j,i(M⃗u)i + +� +l̸=i +(M k)j,l(M⃗u)l +(Def. matrix-vector product) +< (M k)j,i⃗ui + +� +l̸=i +(M k)j,l(M⃗u)l +(because (M⃗u)i < ⃗ui and (M k)j,i > 0) +≤ (M k)j,i⃗ui + +� +l̸=i +(M k)j,l⃗ul +(because (M⃗u)l ≤ ⃗ul) += (M k⃗u)j ≤ ⃗uj +It remains to show that ρ(M) < 1. We do this by showing that the powers +of M (i.e., the sequence (M k)k≥0) converge to the zero matrix. Since M n⃗u ≺ ⃗u, +we can choose c < 1 such that M n⃗u ≤ c⃗u. Then for all m ≥ 1 it holds that +M nm⃗u ≤ cm⃗u, so we have +lim +k→∞ M k⃗u = ⃗0 . +Recall from above that we already know ⃗u ≻ ⃗0. Thus limk→∞ M k⃗u = ⃗0 means +that a positive linear combination of the entries of each individual row of M k +converges to zero, i.e., for all 1 ≤ i ≤ n we have limk→∞ +� +j M k +i,j⃗uj = 0, and +thus for all 1 ≤ j ≤ n, limk→∞ M k +i,j = 0. Thus limk→∞ M k = 0, which completes +the proof. +⊓⊔ +A.2 +Proof of Theorem 3 +Theorem 3. Algorithm 1 is correct: when invoked with a strongly connected +clean PPS ⃗f and ε > 0, then (if it terminates) it outputs a pair (⃗l, ⃗u) s.t. ⃗l ≤ µ⃗f, +⃗f(⃗u) ≤ ⃗u (and thus µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε. Moreover, if ⃗f is feasible and +I − ∂ ⃗f(µ⃗f) is non-singular, then the algorithm terminates. +Proof. Correctness is obvious, so we only show termination assuming that ⃗f is +feasible and I − ∂ ⃗f(µ⃗f) is non-singular. Clearly, the algorithm terminates iff it +eventually finds a ⃗u in line 8 which is inductive. +Assume towards contradiction that the algorithm never terminates, i.e., it +never finds an inductive ⃗u. For all i ≥ 1 let ⃗li, ⃗vi, τi be the values of the variables +⃗l, ⃗v and τ at the ith time the inner loop at line 7 is reached (note that we +then have N = i − 1). Clearly, limi→∞ τi = 0. By the contract satisfied by + +24 +Tobias Winkler and Joost-Pieter Katoen +improveLowerBound, we have limi→∞ ∂ ⃗f(⃗li) = ∂ ⃗f(µ⃗f). Since the eigenvectors +of ∂ ⃗f(µ⃗f) depend continuously on those of the matrices ∂ ⃗f(⃗li), and because of +the contract satisfied by approxEigenvec, the sequence ⃗v1,⃗v2, . . . converges to +the true unique normalized Perron-Frobenius eigenvector ⃗vtrue of ∂ ⃗f(µ⃗f). +We now apply condition (4) of Lemma 2. The condition ensures that the cone +C(µ⃗f,⃗vtrue, ε′, δmax) = { µ⃗f + δ⃗˜v | 0 ≤ δ ≤ δmax, ||⃗˜v − ⃗vtrue||∞ ≤ ε′ } +which is located at µ⃗f, points in direction ⃗vtrue and has radius ε′ and length +δmax contains only inductive points. For the sake of illustration suppose that the +algorithm already knows δmax and computes ⃗ui = ⃗li +δ⃗vi for some 0 < δ < δmax +instead of executing the loop starting at line 7. But then the sequence (⃗ui)i≥1 +converges to µ⃗f + δ⃗vtrue, which is a point that lies inside the interior of C, so +there must be some i ≥ 1 such that ⃗ui ∈ C, i.e., ⃗ui is inductive. +The remaining difficulty is that δmax is of course unknown in practice. We +handle this using the inner loop that starts at line 7. Eventually, the variable +N is sufficiently large such that dkε < δmax for some k ≤ N. Termination then +follows by applying the argument in the previous paragraph to δ = dkε. +⊓⊔ +A.3 +Proof of Lemma 3 +Lemma 3. Let M ≥ 0 be irreducible. Then power iteration applied to M + I +and any ⃗v0 > ⃗0 converges to the Perron-Frobenius eigenvector ⃗v ≻ ⃗0 of M. +Proof. Consider the following conditions for an irreducible matrix M ≥ 0 and a +vector M⃗v0 with M⃗v0 ̸= ⃗0: +1. M has a unique dominant eigenvalue |λ1| > |λ2| ≥ . . . ≥ |λn|. +2. λ1 is semisimple, i.e., its algebraic multiplicity6 equals its geometric multi- +plicity7. +3. ⃗v0 is not orthogonal to the eigenspace {⃗v | M⃗v = λ1⃗v}. +It is known that if all these conditions are satisfied, then the power iteration +sequence (⃗vi)i∈N converges to a (normalized) eigenvector ⃗v with eigenvalue λ1 +(e.g. [37, Theorem 4.1]). +We now show that these conditions are satisfied for the irreducible matrix +M + I ≥ 0 and every initial vector ⃗v0 > ⃗0. The eigenvectors of M and M + I +are exactly the same but the eigenvalues are all shifted by +1. Indeed, if ⃗v is +some eigenvector of M with eigenvalue λ, then (M + I)⃗v = λ⃗v + ⃗v = (λ + 1)⃗v. +However, unlike M, the matrix M +I always has period 1, and so it has a unique +dominant eigenvalue λ1 by Theorem 1(2). Therefore the first of the above three +conditions is satisfied by the matrix M + I. +6 The algebraic multiplicity is the multiplicity of a given eigenvalue as a root of the +characteristic polynomial. +7 The geometric multiplicity is the dimension of the eigenspace associated with a +particular eigenvalue. + +Certificates for Probabilistic Pushdown Automata via OVI +25 +Next, by Theorem 1(1) it holds that the geometric multiplicity of λ1 is 1. As +the algebraic multiplicity is bounded by the geometric multiplicity, it must also +be 1 and thus the matrix M + I satisfies the second condition as well. +Finally, the third condition is satisfied for any ⃗v0 > ⃗0 because the scalar +product ⃗v0 · ⃗v is non-zero (either strictly positive or strictly negative) for all +non-zero eigenvectors ⃗v of λ1 by Theorem 1(1). +⊓⊔ +A.4 +Proof of Proposition 1 +Proposition 1 (Basic Certificates for pPDA). A basic certificate for ∆ = +(Q, Γ, P) is a rational inductive upper bound ⃗u ∈ QQ×Γ ×Q +≥0 +on the lfp of the +return probabilities system ⃗f∆ (see Thm. 4). They have the following properties: +– (Existence) ∀ε > 0 there exists a basic certificate ⃗u with ||µ⃗f∆ − ⃗u||∞ ≤ ε if +all maximal irreducible submatrices M of ∂ ⃗ˆf∆(µ⃗ˆf∆) satisfy ρ(M) < 1. +– (Complexity) Let β be the maximum number of bits used to encode any of +the numerators and denominators of the fractions occurring in ⃗u ∈ QQ×Γ ×Q +≥0 +. +Then checking ⃗f∆(⃗u) ≤ ⃗u, i.e., whether ⃗u is basic certificate for ∆, can be +done in time polynomial in β and the size of ∆. +Proof. This proof closely follows the general idea of decomposed analysis of +PPS [16]. +We first address existence. Note that ⃗f∆ is guaranteed to be feasible, in fact +⃗0 ≤ µ⃗f∆ ≤ ⃗1. For all qZr with (µ⃗f∆)qZr = 0 we set ⃗uqZr = 0. By removing +these variables from ⃗f∆ we obtain the clean PPS ⃗ˆf∆ with ⃗0 ≺ µ⃗ˆf∆. +Now consider the decomposition of ⃗ˆf∆ into the subsystems induced by the +strongly connected components of the graph G ⃗ˆf∆: ⃗ˆf 1 +∆, . . . , ⃗ˆf m +∆ . Note that in these +subsystems, some variables might only appear on the right hand sides but not on +the left (e.g. x1 = 0.5x1+0.5x2, x2 = 0.5x1+0.5x3). Since µ⃗ˆf∆ ≻ ⃗0, there is a 1 - 1 +correspondence of these subsystems and the maximal irreducible submatrices Mi +of ∂ ⃗ˆf∆(µ⃗ˆf∆). More specifically, Mi = ∂ ⃗ˆf i +∆(µ⃗ˆf∆)8. By assumption, ρ(Mi) < 19. +Now assume w.l.o.g. that ⃗ˆf 1 +∆ is a bottom SCC (i.e., in the dependency graph +G ⃗ˆ +f∆ there is no path from the variables in ⃗ˆf 1 +∆ to any variable not in ⃗ˆf 1 +∆). Then +⃗ˆf 1 +∆ is a strongly connected PPS with ∂ ⃗ˆf 1 +∆(µ⃗ˆf∆) = ∂ ⃗ˆf 1 +∆(µ⃗ˆf 1 +∆) and we can apply +Lemma 2(4) to obtain a rational ⃗u1 with ⃗ˆf 1 +∆(⃗u1) ≤ ⃗u1 and ||µ⃗ˆf 1 +∆ − ⃗u1||∞ ≤ ε (in +fact, we can do this for any ε > 0). +Suppose we have done the above for all bottom SCCs and now start traversing +the DAG of SCCs bottom-up, i.e., in reverse topological order. Let ⃗u be the +8 The Jacobi matrix of a sub-PPS with n′ < n equations is an n′ × n′ matrix where +all variables that occur only on the right hand sides are considered constants. +9 The spectral radius of the zero matrix is zero. + +26 +Tobias Winkler and Joost-Pieter Katoen +bound we have constructed to far (i.e., ⃗u contains ⃗u1 and the bounds from +the other bottom SCC as subvectors and is zero elsewhere). Note that we can +always make ⃗u smaller while retaining the inductivity property. W.l.o.g. suppose +that subsystem ⃗ˆf 2 +∆ is one of the first non-bottom SCCs in the reverse topological +order. The idea is now to modify ⃗ˆf 2 +∆ to a strongly connected PPS ˜⃗f 2 +⃗u by replacing +all variables that occur only in right hand sides by their value in ⃗u. Clearly, +lim⃗u→µ⃗ˆ +f∆ ∂ ˜⃗f 2 +⃗u(µ ˜⃗f 2 +⃗u) = ∂ ⃗ˆf 2 +∆(µ⃗ˆf∆). This means we can choose ⃗u sufficiently close +to µ⃗ˆf∆ such that the spectral radius of ∂ ˜⃗f 2 +⃗u(µ ˜⃗f 2 +⃗u) is strictly smaller than 1. We +can then apply Lemma 2(4) to ˜⃗f 2 +⃗u to obtain a rational ⃗u2 with ˜⃗f 2 +⃗u(⃗u2) ≤ ⃗u2 to +enlarge our current ⃗u with. +We can repeat this scheme for all finitely many subsystems until we have +constructed a rational ⃗u with ˜⃗f i +⃗u(⃗u) ≤ ⃗u for all i. Clearly, this ⃗u also satisfies +⃗ˆf∆(⃗u) ≤ ⃗u. Finally, we may extend ⃗u by zero entries corresponding to the vari- +ables that are assigned zero in the lfp of the (not necessarily clean) ⃗f∆. This +yields an inductive upper bound for ⃗f∆. We stress that in order to verify this +bound, we neither have to clean ⃗f∆ nor do we have to compute the SCCs. +For complexity observe that ⃗f∆ is cubic in the size of ∆ and that all polyno- +mials in ⃗f∆ have degree at most 2. Since multiplication and addition of rational +numbers can be done in polynomial time in the number of their bits, evaluat- +ing a polynomial of fixed maximum degree can also be done in polynomial time +in the size of the polynomial and the number of bits representing the rationals +where the polynomial is to be evaluated. Note that this is not true for arbitrary +polynomials where exponents are encoded in binary: For instance, evaluating the +polynomial x2n (which can be represented with O(n) bits) at x = 2 yields 22n, +a number that needs O(2n) bits. This means that in order to verify certificates +efficiently with exact rational arithmetic, it is important that the polynomials in +the PPS do not have very high degrees. Fortunately, this is the case for pPDA. + +Certificates for Probabilistic Pushdown Automata via OVI +27 +B +Certificates for Expected Rewards +We can certify upper bounds on the expected value of rewards collected during +the run of a pPDA. To simplify the presentation, in this section we assume +w.l.o.g. that qZ +p−→ rα with p > 0 implies |α| ∈ {0, 2}, i.e., all transitions either +decrease or increase the stack height by 1. Let R: Q → R≥0 be a state-based +reward function. Consider the following PPS ⃗f∆,R with variables {⟨EqZr⟩ | qZr ∈ +Q × Γ × Q}: +⟨EqZr⟩ = +� +qZ +p−→sY X +p · +� +t∈Q +[sY t] · [tXr] · KqZ,sY X + +� +qZ +p−→rε +p · R(r) , +where KqZ,sY X = R(r) + ⟨EsY t⟩ + ⟨EtXr⟩. Note that ⃗f∆,R is linear but uses +the return probabilities which are themselves characterized as the lfp of the +non-linear system ⃗f R +∆ from Theorem 4 as coefficients. +Suppose that in the lfp µ⃗f∆,R, each variable EqZr is assigned the quantity +EqZr ∈ R≥0. It follows from the results of [14] that EqZr equals the expected +value of the following random variable V r +R under the probability measure PqZ +∆ : +V r +R(q0γ0, q1γ1, . . .) = +firstHit(rε) +� +i>0 +R(qi) +where firstHit(rε) is the minimum integer k such that qkγk = rε, or 0 if no such +k exists. In words, EqZr is the expected reward accumulated on the runs from +qZ to rε, where it is assumed that runs which never reach rε contribute zero +reward. Consequently, E(qZ) = � +r∈Q EqZr is the expected reward accumulated +on all terminating runs. +Example 7. Setting R = 1 we can characterize the expected runtime of pPDA. +Reconsider Example 4. The equation system for expected runtimes becomes +⟨EqZq⟩ =1 +4([qZq]2(1+2⟨EqZq⟩) + [qZr][rZq](1+⟨EqZr⟩+⟨ErZq⟩)) + 1 +2 +⟨EqZr⟩ =1 +4([qZq][qZr](1+⟨EqZq⟩+⟨EqZr⟩)+[qZr][rZr](1+⟨EqZr⟩+⟨ErZr⟩)) + 1 +4 +as well as ⟨ErZq⟩ = 0 and ⟨ErZr⟩ = 1. The solution is ⟨EqZq⟩ = 2063/2624 ≈ +0.786 and ⟨EqZr⟩ = 59/82 ≈ 0.712, so the total expected runtime is E(qZ) ≈ +1.506. +△ +C +Benchmark Programs + +28 +Tobias Winkler and Joost-Pieter Katoen +void f() { +if flip(p) { +f(); +f(); +} +} +# main block +{ +f(); +} +(a) rw-p +void f() { +if flip(1//2) { +f(); +f(); +f(); +} +} +# main block +{ +f(); +} +(b) golden +void offspring() { +while flip(2//5) { +offspring(); +while flip(3//5) { +offspring(); +} +} +} +# main block +{ +offspring(); +} +(c) geom-offspring +void gen_operator() { +uniform(4); +} +void gen_expression() { +prob { +4//10: uniform(10); +3//10: { } +3//10: { +gen_operator(); +gen_expression(); +gen_expression(); +} +} +} +void gen_function() { +gen_operator(); +gen_expression(); +gen_expression(); +} +# main block +{ +gen_function(); +} +(d) gun-fun +void young() { +int y = uniform(4); +while(y > 0) { +young(); +y = y-1; +} +int e = uniform(3); +while(e > 0) { +elder(); +e = e-1; +} +} +void elder() { +int y = uniform(2); +while(y > 0) { +young(); +y = y-1; +} +int e = uniform(5); +while(e > 0) { +elder(); +e = e-1; +} +} +# main block +{ +young(); +} +(e) virus +bool f() { +prob { +1//2: +return flip(1//2); +1//2: +if f() +{ +return f(); +} else { +return false; +} +} +} +# main blcok +{ +bool res1 = f(); +... +bool resN = f(); +} +(f) sequentialN + +Certificates for Probabilistic Pushdown Automata via OVI +29 +int f(int n, int m) { +prob { +(n+1)//(n+2) : { +f((n + 1) % m, m); +f((n + 1) % m, m); +return 0; +} +1//(n+2) : +return 0; +} +} +# main block +{ +f(0, N); +} +(a) escapeN +void f(int n) { +while(n > 0) { +prob { +2//3: f(n-1); +1//3: f((n+1) % N); +} +n = n-1; +} +} +# main block +{ +f(1); +} +(b) modN + +30 +Tobias Winkler and Joost-Pieter Katoen +D +Z3 vs CVC5 +Table 3: Comparison of the SMT-approach (see §Baselines in Section 5) using z3 +and cvc5 on SCFG given as explicit PPS (right), and on programs automatically +translated to pPDA (left). +benchmark +certz3 +tz3 +certcvc5 +tcvc5 +rw-0.499 +✓ +11 +✓ +92 +rw-0.500 +✓ +10 +✓ +87 +rw-0.501 +✓ +12 +✓ +104 +geom-offspring +✓ +16 +✓ +4687 +golden +✓ +14 +✓ +1097 +and-or +✓ +15260 +✗ +TO +gen-fun +✓ +141 +✗ +TO +virus +✓ +139 +✓ +163727 +escape10 +✓ +48 +✓ +12031 +escape25 +✓ +15958 +✗ +TO +escape50 +✓ +410 +✗ +TO +escape75 +✗ +TO +✗ +TO +escape100 +✓ +939 +✗ +TO +escape200 +✗ +TO +✗ +TO +sequential5 +✓ +1074 +✗ +TO +sequential7 +✓ +12822 +✗ +TO +sequential10 +✓ +453718 +✗ +TO +mod5 +✓ +34150 +✗ +TO +mod7 +✓ +443 +✗ +TO +mod10 +✓ +1245 +✗ +TO +benchmark +certz3 +tz3 +certcvc5 +tcvc5 +brown +✗ +TO +✗ +TO +lemonde +✗ +TO +✗ +TO +negra +✓ +37248 +✓ +10144 +swbd +✗ +TO +✗ +Error +tiger +✓ +17454 +✓ +16118 +tuebadz +✓ +15323 +✓ +5534 +wsj +✗ +TO +✗ +TO +random +✗ +TO +✗ +TO + diff --git a/BdFAT4oBgHgl3EQfsR4z/content/tmp_files/load_file.txt b/BdFAT4oBgHgl3EQfsR4z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..69fea13b9d1536c7b6cf156a1a37445816cb3f7f --- /dev/null +++ b/BdFAT4oBgHgl3EQfsR4z/content/tmp_files/load_file.txt @@ -0,0 +1,1823 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf,len=1822 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='08657v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='FL] 20 Jan 2023 Certificates for Probabilistic Pushdown Automata via Optimistic Value Iteration Tobias Winkler and Joost-Pieter Katoen RWTH Aachen University, Germany Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Probabilistic pushdown automata (pPDA) are a standard model for discrete probabilistic programs with procedures and recur- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In pPDA, many quantitative properties are characterized as least fixpoints of polynomial equation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In this paper, we study the problem of certifying that these quantities lie within certain bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' To this end, we first characterize the polynomial systems that admit easy-to-check certificates for validating bounds on their least fixpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Second, we present a sound and complete Optimistic Value Iteration al- gorithm for computing such certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Third, we show how certificates for polynomial systems can be transferred to certificates for various quan- titative pPDA properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Experiments demonstrate that our algorithm computes succinct certificates for several intricate example programs as well as stochastic context-free grammars with > 104 production rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Keywords: Probabilistic Pushdown Automata · Probabilistic Model Checking · Certified Algorithms · Probabilistic Recursive Programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 1 Introduction Complex software is likely to contain bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This applies in particular to model checking tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This is a serious problem, as the possibility of such bugs com- promises the trust one can put in the verification results, rendering the process of formal modeling and analysis less useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Ideally, the implementation of a model checker should be formally verified itself [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, due to the great complexity of these tools, this is often out of reach in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certifying algo- rithms [31] mitigate this problem by providing an easy-to-check certificate along with their regular output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This means that there exists a verifier that, given the input problem, the output, and the certificate, constructs a formal proof that the output is indeed correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The idea is that the verifier is much simpler than the algorithm, and thus likely to be bug-free or even amenable to formal verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This paper extends the recent line of research on probabilistic certifica- tion [19,23,24,40] to probabilistic pushdown automata [13,30] (pPDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' pPDA and related models have applications in, amongst others, pattern recognition [38], computational biology [28], and speech recognition [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' They are moreover a natural operational model for programs with procedures, recursion, and (dis- crete) probabilistic constructs such as the ability to flip coins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' With the advent 2 Tobias Winkler and Joost-Pieter Katoen X → a | XY Y x = 1 2(1 + xy2) Y → b | X | Y Y y = 1 3(1 + x + y2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8 1 ≈ (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='66, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='7) (1, 1) x y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 1: Left: A stochastic context-free grammar (SCFG) and the associated pos- itive polynomial system (PPS) which encodes the termination probabilities of each non-terminal, assuming production rules are taken uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Right: The curves defined by the two equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The least fixpoint (lfp) is ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='66, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The thin colored area to the top right of the lfp is the set of inductive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', self-certifying upper bounds on the lfp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' of probabilistic programming [32] as a paradigm for model-based machine learn- ing [6], such programs have received lots of attention recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Moreover, several efficient algorithms such as Hoare’s quicksort with randomized pivot selection (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' [26]) are readily encoded as probabilistic recursive programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A pPDA can be seen as a purely probabilistic variant of a standard pushdown automaton: Instead of reading an input word, it takes its transitions randomly based on fixed probability distributions over successor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Quantities of inter- est in pPDA include reachability probabilities [13], expected runtimes [8], vari- ances [14], satisfaction probabilities of temporal logic formulas [44,41], and others (see [7] for an overview).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' pPDA are equivalent to recursive Markov chains [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' One of the difficulties of pPDA is that they induce infinite Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Despite this fact, many interesting quantitative properties are decidable, albeit with rather high complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Therefore, in the past two decades there have been significant research efforts on efficient approximative algorithms for pPDA, espe- cially a decomposed variant of Newton iteration [16,27,11,17,12,10,39] which pro- vides guaranteed lower, and occasionally upper [10,12] bounds on key quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, even though implementations might be complex [43], these algorithms do not produce certificates for their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Our technique for certificate generation is an adaption of Optimistic Value Iteration [22] (OVI) to the pPDA setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In a nutshell, OVI computes some lower bound ⃗l on the solution—which can be done using an approximative iter- ative algorithm—and then optimistically guesses an upper bound ⃗u = ⃗l + ⃗ε and verifies that the guess was correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Originally, OVI was formulated for Markov Decision Processes (MDP) where it is used to compute lower and upper bounds on minimal or maximal reachability probabilities and expected rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The up- per bounds computed by OVI have a special property: They are self-certifying (also called inductive in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This means that, given the MDP and the upper bounds, one can check that the bounds are correct without the need for an additional certificate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' and this check is conceptually and practically easier than finding the bounds in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for Probabilistic Pushdown Automata via OVI 3 The analysis of pPDA, however, is more involved than that of MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In MDP, many quantitative properties are characterized as least fixpoints (lfp) of piece-wise linear equation systems and can be computed in PTIME via, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', LP solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In pPDA, on the other hand, the equation systems for the same properties may contain non-linear polynomials, and the best known complexity bounds are usually as high as PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' An example of such a non-linear system is illustrated in Figure 1 which shows the translation of a stochastic context-free grammar (SCFG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' special case of pPDA with a single state) to a polynomial equation system encoding termination probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' An important observation is that the polynomials arising in this context only have positive coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Such systems are called positive polynomial systems (PPS) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Applications of PPS beyond the analysis of pPDA include the recent factor graph grammars [9] as well as obtaining approximate counting formulas for many classes of trees in the framework of analytic combinatorics [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In summary, this paper makes the following contributions: – We present an optimistic algorithm for computing inductive, self-certifying upper bounds of any desired precision ε > 0 on the lfp of a positive poly- nomial system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Compared to OVI from [22], the key innovation of our algo- rithm is to compute a certain direction ⃗v in which to guess, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', the guess is ⃗u = ⃗l + ε⃗v rather than ⃗u = ⃗l + ⃗ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This is to ensure that we eventually hit an inductive bound, even if the latter lie in a very “thin strip” as in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' – We prove that our algorithm is sound and complete in the sense that if a (non-trivial) inductive upper bound exists, then such a bound will be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' – We show how inductive bounds on the lfp of PPS can be used to certify various quantities of interest in pPDA and SCFG, such as non-termination or bounds on expected rewards/costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' – We implement our algorithm in the software tool pray and compare the new technique to an out-of-the-box approach based on SMT solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certification of pPDA has not been addressed explicitly in the literature, but some existing technical results go in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We mention [17, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='7] which yields certificates for non almost-sure termination of SCFG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, checking such certificates is not straightforward as it requires an SCC decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The tool PReMo [43] implements iterative algorithms for lower bounds, but it supports neither certificates nor upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Beyond pPDA, OVI was recently generalized to stochastic games [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Farkas certificates for MDP [19] are verified by checking a set of linear constraints, which is in spirit similar to our certificates that requires checking a set of polynomial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A deductive approach for verifying probabilistic recursive programs on the syntax level was studied in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The same paper also includes inductive proof rules for verifying upper bounds just like we do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Recently, a higher-order generalization of pPDA called PHORS was introduced in [29], and an algorithm for finding upper bounds inspired by the Finite Elements method was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 4 Tobias Winkler and Joost-Pieter Katoen Paper Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We review the relevant background information on PPS in Sec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Section 3 presents our theoretical results on inductive upper bounds in PPS as well as the new Optimistic Value Iteration algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In Section 4 we explain how inductive bounds in PPS are used to certify quantitative properties of pPPA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The experimental evaluation is in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 2 Preliminaries Notation for Vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' All vectors in this paper are column vectors and are written in boldface, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', ⃗u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , un)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For vectors ⃗u, ⃗u′, we write ⃗u ≤ ⃗u′ if ⃗u is component-wise less than or equal to ⃗u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Moreover, we write ⃗u < ⃗u′ if ⃗u ≤ ⃗u′ and ⃗u ̸= ⃗u′, and ⃗u ≺ ⃗u′ if ⃗u is component-wise strictly smaller than ⃗u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The zero vector is denoted ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The max norm of a vector ⃗u is ||⃗u||∞ = max1≤i≤n |ui|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We say that ⃗u is normalized if ||⃗u||∞ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Positive Polynomial Systems (PPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let n ≥ 1 and ⃗x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , xn)T be a vector of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' An n-dimensional PPS is an equation system of the form x1 = f1(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , xn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' xn = fn(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , xn) where for all 1 ≤ i ≤ n, the function fi is a polynomial with non-negative real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' An example PPS is the system x = 1 2(1+xy2), y = 1 3(1+x+y2) from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We also use vector notation for PPS: ⃗x = ⃗f(⃗x) = (f1(⃗x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , fn(⃗x))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We write R≥0 = R≥0 ∪ {∞} for the extended non-negative reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By conven- tion, for all a ∈ R≥0, a ≤ ∞, a + ∞ = ∞ + a = ∞, and a · ∞ = ∞ · a equals 0 if a = 0 and ∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For n ≥ 1, the partial order (R n ≥0, ≤) is a complete lat- tice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', all subsets of R n ≥0 have an infimum and a supremum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In particular, there exists a least element ⃗0 and a greatest element ⃗∞ = (∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , ∞)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Every PPS induces a monotone function ⃗f : R n ≥0 → R n ≥0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', ⃗u ≤ ⃗v =⇒ ⃗f(⃗u) ≤ ⃗f(⃗v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By the Knaster-Tarski fixpoint theorem, the set of fixpoints of ⃗f is also a complete lattice, and thus there exists a least fixpoint (lfp) denoted by µ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In general, the lfp µ⃗f is a vector which may contain ∞ as an entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For instance, this happens in the PPS x = x+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A PPS ⃗f is called feasible if µ⃗f ≺ ⃗∞ (or equivalently, µ⃗f ∈ Rn ≥0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', the lfp is a vector of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Besides existence of the lfp, the Knaster-Tarski theorem also implies the following: Lemma 1 (Inductive upper bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For all ⃗u ∈ R n ≥0 it holds that ⃗f(⃗u) ≤ ⃗u implies µ⃗f ≤ ⃗u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Such a vector ⃗u with ⃗u ≺ ⃗∞ is called inductive upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Given a feasible PPS ⃗f, find an inductive upper bound ⃗u ≥ µ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Problem statement of this paper Certificates for Probabilistic Pushdown Automata via OVI 5 If ⃗f is feasible, then µ⃗f is obviously an inductive upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In Section 3 we show under which conditions there exist more useful inductive upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A PPS is called clean if µ⃗f ≻ ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Every PPS can be cleaned in linear time by identifying and removing the variables that are assigned 0 in the lfp [17,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Given a PPS ⃗f and a point ⃗u ∈ Rn ≥0, we define the Jacobi matrix of ⃗f at ⃗u as the n×n-matrix ∂ ⃗f(⃗u) with coefficients ∂ ⃗f(⃗u)1≤i,j≤n = ∂ ∂xj fi(⃗u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Consider the example PPS ⃗fex with variables ⃗x = (x, y)T : x = f1(x, y) = y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 y = f2(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8xy + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The line and the hyperbola defined by these equations are depicted in Figure 2 on Page 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The fixpoints of ⃗fex are the intersections of these geometric objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' in this case there are two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In particular, ⃗fex is feasible and its lfp is µ⃗fex = � (27− √ 229)/50 , (22− √ 229)/50 �T ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='237 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='137)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Therefore, ⃗fex is clean as µ⃗fex ≻ ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The Jacobi matrix of ⃗fex is ∂ ⃗fex(x, y) = � ∂ ∂xf1 ∂ ∂yf1 ∂ ∂xf2 ∂ ∂yf2 � = � 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4x + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that the lfp µ⃗fex contains irrational numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, we can still give ex- act expressions for these numbers (involving square roots) because the fixpoints of ⃗fex are the zeros of a quadratic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, there are PPS whose lfp cannot be expressed using radicals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', square roots, cubic roots, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This means that in general, there is no easy way to compute least fixpoints exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It is thus desirable to provide bounds, which we do in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' △ Matrices and Eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let M be a real n×n-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We say that M is non- negative (in symbols: M ≥ 0) if it has no negative entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' M is called irreducible if for all 1 ≤ i, j ≤ n there exists 0 ≤ k < n such that (M k)i,j ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It is easy to show that M is irreducible iff the directed graph GM = ({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , n}, E) with (i, j) ∈ E iff Mi,j ̸= 0 is strongly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A maximal irreducible submatrix of M is a square submatrix induced by a strongly connected component of GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The period of a strongly connected M is the length of the shortest cycle in GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It is instructive to note that PPS ⃗x = ⃗f(⃗x) are generalizations of linear equation systems of the form ⃗x = M⃗x + ⃗c, with M ≥ 0 and ⃗c ≥ ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Moreover, note that for any PPS ⃗f it holds that ∂ ⃗f(⃗u) ≥ 0 for all ⃗u ≻ ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' An eigenvector of an n×n-matrix M with eigenvalue λ ∈ C is a (complex) vector ⃗v ̸= ⃗0 satisfying M⃗v = λ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' There are at most n different eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The spectral radius ρ(M) ∈ R≥0 is the largest absolute value of the eigenvalues of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The following is a fundamental theorem about non-negative matrices: Theorem 1 (Perron-Frobenius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let M ≥ 0 be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 6 Tobias Winkler and Joost-Pieter Katoen (1) M has a strictly positive eigenvector ⃗v ≻ ⃗0 with eigenvalue ρ(M), the spectral radius of M, and all other eigenvectors ⃗v′ ≻ ⃗0 are scalar multiples of ⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (2) The eigenvalues of M with absolute value ρ(M) are exactly the h numbers ρ(M), ξρ(M), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , ξh−1ρ(M), where ξ is a primitive hth root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The unique eigenvector ⃗v ≻ ⃗0 with ||⃗v||∞ = 1 of an irreducible non-negative matrix M is called the Perron-Frobenius eigenvector of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Strongly Connected Components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' To each PPS ⃗f we associate a finite directed graph G ⃗f = ({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , xn}, E), which, intuitively speaking, captures the depen- dency structure among the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Formally, (xi, xj) ∈ E if the polynomial fi depends on xj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', xj appears in at least one term of fi with a non-zero coef- ficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This is equivalent to saying that the partial derivative ∂ ∂xj fi is not the zero polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We say that ⃗f is strongly connected if G ⃗f is strongly connected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', for each pair (xi, xj) of variables, there exists a path from xi to xj in G ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For instance, ⃗fex from Example 1 is strongly connected because the dependency graph has the edges E = {(x, y), (y, x), (y, y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Strong connectivity of PPS is a generalization of irreducibility of matrices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' indeed, a matrix M is irreducible iff the PPS ⃗x = M⃗x is strongly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We often use the fact that ∂ ⃗f(⃗u) for ⃗u ≻ ⃗0 is irreducible iff ⃗f is strongly connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' PPS are usually analyzed in a decomposed fashion by considering the sub- systems induced by the strongly connected components (SCCs) of G ⃗f in bottom- up order [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Here we also follow this approach and therefore focus on strongly connected PPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The following was proved in [17, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5] and later generalized in [12, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1] (also see remark below [12, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4] and [17, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2]): Theorem 2 ([17,12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If ⃗f is feasible, strongly connected and clean, then for all ⃗u < µ⃗f, we have ρ(∂ ⃗f(⃗u)) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' As a consequence, ρ(∂ ⃗f(µ⃗f)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Theorem 2 partitions all PPS ⃗f which satisfy its precondition into two classes: Either (1) ρ(∂ ⃗f(µ⃗f)) < 1, or (2) ρ(∂ ⃗f(µ⃗f)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In the next section we show that ⃗f admits non-trivial inductive upper bounds iff it is in class (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Reconsider the PPS ⃗fex from Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It can be shown that ⃗v = (1, λ1)T where λ1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='557 is an eigenvector of ∂ ⃗fex(µ⃗fex) with eigenvalue λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Thus by the Perron-Frobenius Theorem, ρ(∂ ⃗fex(µ⃗fex)) = λ1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' As promised, there exist inductive upper bounds as can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' △ 3 Finding Inductive Upper Bounds in PPS In this section, we are concerned with the following problem: Given a feasible, clean, and strongly connected PPS ⃗f, find a vector ⃗0 ≺ ⃗u ≺ ⃗∞ such that ⃗f(⃗u) ≤ ⃗u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', an inductive upper bound on the lfp of ⃗f (see Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for Probabilistic Pushdown Automata via OVI 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8 µ⃗fex ε ⃗v µ⃗˜fex x y x = y + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8xy + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2x2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8xy + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1916 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 2: The PPS ⃗fex corresponds to the solid red line and the solid blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Its inductive upper bounds form the shaded area above the lfp µ⃗fex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Lemma 2(4) ensures that one can fit the gray “cone” pointing in direction of the Perron- Frobenius eigenvector ⃗v inside the inductive region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The PPS ⃗˜fex which com- prises the dashed curve and the solid line does not have any non-trivial inductive upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that the tangent lines at µ⃗˜fex are parallel to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 Existence of Inductive Upper Bounds An important first observation is that inductive upper bounds other than the exact lfp do not necessarily exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' As a simple counter-example consider the 1- dimensional PPS x = 1 2x2 + 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If u is an inductive upper bound, then 1 2u2 + 1 2 ≤ u =⇒ u2 − 2u + 1 ≤ 0 =⇒ (u − 1)2 ≤ 0 =⇒ u = 1 , and thus the only inductive upper bound is the exact lfp u = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Another example is the PPS ⃗˜fex from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' What these examples have in common is the following property: Their derivative evaluated at the lfp is not invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Indeed, we have ∂ ∂x( 1 2x2 + 1 2 − x) = x − 1, and inserting the lfp x = 1 yields zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The higher dimensional generalization of this property to arbitrary PPS ⃗f is that the Jacobi matrix of the function ⃗f − ⃗x evaluated at µ⃗f is singular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' note that this is precisely the matrix ∂ ⃗f(µ⃗f) − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Geometrically, this means that the tangent lines at µ⃗f are parallel, as can be seen in Figure 2 for the example PPS ⃗˜fex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It should be intuitively clear from the figure that inductive upper bounds only exist if the tangent lines are not parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The next lemma makes this more precise: Lemma 2 (Existence of inductive upper bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let ⃗f be a feasible, clean, and strongly connected PPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then the following are equivalent: (1) The matrix I − ∂ ⃗f(µ⃗f) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (2) The spectral radius of ∂ ⃗f(µ⃗f) satisfies ρ(∂ ⃗f(µ⃗f)) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (3) There exists ⃗0 ≺ ⃗u ≺ ⃗∞ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗f(⃗u) < ⃗u (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗u is inductive but not a fixpoint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 8 Tobias Winkler and Joost-Pieter Katoen (4) The matrix ∂ ⃗f(µ⃗f) has a unique (normalized) eigenvector ⃗v ≻ ⃗0 and there exist numbers δmax > 0 and ε > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗f( µ⃗f + δ · ⃗˜v ) ≺ µ⃗f + δ · ⃗˜v holds for all 0 < δ ≤ δmax and vectors ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The proof of Lemma 2 (see appendix) relies on a linear approximation of ⃗f via Taylor’s familiar theorem as well as Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Condition (4) of Lemma 2 means that there exists a “truncated cone” C(µ⃗f,⃗v, ε, δmax) = { µ⃗f + δ⃗˜v | 0 ≤ δ ≤ δmax,⃗˜v ≥ ⃗v, ||⃗˜v − ⃗v||∞ ≤ ε } which is entirely contained in the inductive region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This cone is located at the lfp µ⃗f and points in the direction of the Perron-Frobenius eigenvector ⃗v, as illustrated in Figure 2 (assuming δmax = 1 for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The length δmax and the radius ε of the cone depend quantitatively on ρ(∂ ⃗f(µ⃗f)), but for our purposes it suffices that they are non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The idea of our Optimistic Value Iteration is to construct a sequence of guesses that eventually hits this cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2 The Optimistic Value Iteration Algorithm The basic idea of Optimistic Value Iteration (OVI) can be applied to monotone functions of the form ⃗φ: Rn ≥0 → Rn ≥0 (in [22], ⃗φ is the Bellman operator of an MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Kleene’s fixpoint theorem suggests a simple method for approximating the lfp µ⃗φ from below: Simply iterate ⃗φ starting at ⃗0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', compute the sequence ⃗l0 = ⃗0, ⃗l1 = ⃗φ(⃗l0), ⃗l2 = ⃗φ(⃗l1), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 In the context of MDP, this iterative scheme is known as Value Iteration (VI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' VI is easy to implement, but it is difficult to decide when to stop the iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In particular, standard stopping criteria such as small absolute difference of consecutive approximations are formally un- sound [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' OVI and other algorithms [3,36] cope with this problem by computing not only a lower but also an upper bound on µ⃗φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In the case of OVI, an upper bound with absolute error ≤ ε is obtained as follows (we omit some details): (1) Compute ⃗lk ≤ µ⃗φ such that ||⃗lk −⃗lk−1||∞ ≤ τ, for some (small) τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (2) Guess a candidate upper bound ⃗u = ⃗lk + ⃗ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (a) If ⃗φ(⃗u) ≤ ⃗u holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', ⃗u is inductive, then return ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (b) If not, refine ⃗u (see [22] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If the refined ⃗u is still not inductive, then go back to step (1) and try again with 0 < τ ′ < τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We present our variant of OVI for PPS as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The main differences to the above scheme are that (i) we do not insist on Kleene iteration for obtaining the lower bounds ⃗l, and (ii) we approximate the eigenvector ⃗v from condition (4) of Lemma 2 and compute the “more informed” guesses ⃗u = ⃗l + ε⃗v, for various ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Refining the guesses as original OVI does is not necessary (but see our remarks in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3 regarding floating point computations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 1 In order for the Kleene seqence to converge to the lfp, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', limk→∞⃗lk = µφ, it suffices that ⃗φ is ω-continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This already implies monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for Probabilistic Pushdown Automata via OVI 9 Algorithm 1: Optimistic Value Iteration (OVI) for PPS input : strongly connected clean PPS ⃗f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' maximum abs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' error ε > 0 output : a pair (⃗l, ⃗u) of real vectors s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗l ≤ µ⃗f, ⃗f(⃗u) ≤ ⃗u (hence µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε termination : guaranteed if ⃗f is feasible and I − ∂ ⃗f(µ⃗f) is non-singular 1 ⃗l ← ⃗0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' N ← 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 2 τ ← ε ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' /* τ is the current tolerance */ 3 while true do 4 ⃗l′ ← improveLowerBound(⃗f,⃗l) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' /* e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Kleene or Newton update */ /* guess and verify phase starts here / 5 if ||⃗l − ⃗l′||∞ ≤ τ then 6 ⃗v ← approxEigenvec(∂ ⃗f(⃗l), τ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' /* recall ⃗v is normalized */ 7 for k from 0 to N do 8 ⃗u ← ⃗l + dkε · ⃗v ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' /* optimistic guess, d ∈ (0, 1) */ 9 if ⃗f(⃗u) ≤ ⃗u then 10 return (⃗l, ⃗u) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' /* guess was successful */ 11 N ← N + 1 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 12 τ ← c · τ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' /* decrease tolerance for next guess, c ∈ (0, 1) */ 13 ⃗l ← ⃗l′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The functions improveLowerBound and approxEigenvec used in Algorithm 1 must satisfy the following contracts: – The sequence ⃗l0 = ⃗0, ⃗li+1 = improveLowerBound(⃗f,⃗li) is a monotonically increasing sequence converging to the lfp µ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' – approxEigenvec must satisfy the following: Let M ≥ 0 be an irreducible matrix with (normalized) Perron-Frobenius eigenvector ⃗v ≻ ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then for all ε > 0, we require that there exists τ > 0 such that ||approxEigenvec(M, τ)− ⃗v||∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In words, approxEigenvec approximates ⃗v up to arbitrarily small absolute error if the tolerance τ is chosen sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In practice, both the Kleene and the Newton [16,17,12] update operator can be used to implement improveLowerBound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We outline a possible implementa- tion of approxEigenvec further below in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Consider the following PPS ⃗f: x = 1 4x2 + 1 8, y = 1 4xy + 1 4y + 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The table illustrates the execution of Algorithm 1 on ⃗f with ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 and c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5: # N τ ⃗l ⃗l′ ||⃗l −⃗l′||∞ ⃗v ⃗u ⃗f(⃗u) ≤ ⃗u 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 (0, 0) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='8) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='38) ✗ 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='05 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='55, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='41) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='05 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='9) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='49) ✓ 10 Tobias Winkler and Joost-Pieter Katoen The algorithm has to improve the lower bound 3 times (corresponding to the 3 lines of the table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' After the second improvement, the difference between the current lower bound ⃗l2 and the new bound ⃗l′2 does not exceed the current tol- erance τ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 and the algorithm enters the optimistic guessing stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The first guess ⃗u2 is not successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The tolerance is then decreased to τ3 = c · τ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='05 and the lower bound is improved to ⃗l′3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The next guess ⃗u3 is inductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' △ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Algorithm 1 is correct: when invoked with a strongly connected clean PPS ⃗f and ε > 0, then (if it terminates) it outputs a pair (⃗l, ⃗u) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗l ≤ µ⃗f, ⃗f(⃗u) ≤ ⃗u (and thus µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Moreover, if ⃗f is feasible and I − ∂ ⃗f(µ⃗f) is non-singular, then the algorithm terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The proof of Theorem 3 (see appendix) crucially relies on condition (4) of Lemma 2 that assures the existence of a “truncated cone” of inductive bounds centered around the Perron-Frobenius eigenvector of ∂ ⃗f(µ⃗f) (see Figure 2 for an illustration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Intuitively, since the lower bounds ⃗l computed by the algorithm approach the lfp µ⃗f, the eigenvectors of ∂ ⃗f(⃗l) approach those of ∂ ⃗f(µ⃗f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' As a consequence, it is guaranteed that the algorithm eventually finds an eigenvector that intersects the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The inner loop starting on line 7 is needed because the “length” of the cone is a priori unknown;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' the purpose of the loop is to scale the eigenvector down so that it is ultimately small enough to fit inside the cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3 Considerations for Implementing OVI As mentioned above, there are at least two options for improveLowerBound: Kleene or Newton iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We now show that approxEigenvec can be effec- tively implemented as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Further below we make some remarks on floating point arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Approximating the Eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A possible implementation of approxEigenvec relies on the power iteration method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' [37, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Given a square matrix M and an initial vector ⃗v0 with M⃗v0 ̸= ⃗0, power iteration computes the sequence (⃗vi)i≥0 such that for i > 0, ⃗vi = M⃗vi−1/||M⃗vi−1||∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let M ≥ 0 be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then power iteration applied to M + I and any ⃗v0 > ⃗0 converges to the Perron-Frobenius eigenvector ⃗v ≻ ⃗0 of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The convergence rate of power iteration is determined by the ratio |λ2|/|λ1| where λ1 and λ2 are eigenvalues of largest and second largest absolute value, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Each time approxEigenvec is called in Algorithm 1, the result of the previous call to approxEigenvec (if available) may be used as an initial approximation ⃗v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for Probabilistic Pushdown Automata via OVI 11 Exact vs Floating Point Arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' So far we have assumed exact arithmetic for the computations in Algorithm 1, but an actual implementation should use floating point arithmetic for efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, this may (and actually does) lead to unsound results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' More specifically, the condition ⃗f(⃗u) ≤ ⃗u may hold in floating point arithmetic even though it is actually violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' As a remedy, we propose to nevertheless run the algorithm with floats, but then verify its output ⃗u with exact arbitrary-precision rational arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' That is, we compute a rational number approximation ⃗uQ of ⃗u and check ⃗f(⃗uQ) ≤ ⃗uQ with exact arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If the check fails, we resort to the following refinement scheme which is an instance of the general k-induction principle for complete lattices from [5]: We iteratively check the conditions ⃗f(⃗uQ ⊓ ⃗f(⃗uQ)) ≤ ⃗uQ , ⃗f(⃗uQ ⊓ ⃗f(⃗uQ ⊓ ⃗f(⃗uQ))) ≤ ⃗uQ , and so on, where ⊓ denotes pointwise minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If one of the checks is satisfied, then µ⃗f ≤ ⃗uQ [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This scheme often works well in practice (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The original OVI from [22] uses a similar technique to refine its guesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 4 Certificates for Probabilistic Pushdown Automata This section shows how the results from Section 3 can be applied to pPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We introduce some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For finite sets A, D(A) denotes the set of probability distributions on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We often denote tuples or triples without parentheses and separating commata when this causes no confusion, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', we may write ab rather than (a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Definition 1 (pPDA [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A probabilistic pushdown automaton (pPDA) is a triple ∆ = (Q, Γ, P) where Q ̸= ∅ is a finite set of states, Γ ̸= ∅ is a finite stack alphabet, and P : Q × Γ → D(Q × Γ ≤2) is a probabilistic transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In the following, we often write qZ p−→ rα instead of P(qZ)(rα) = p [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Intu- itively, qZ p−→ rα means that if the pPDA is in state q and Z is on top of the stack, then with probability p, the pPDA moves to state r, pops Z and pushes α on the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' More formally, the semantics of a pPDA ∆ = (Q, Γ, P) is a count- ably infinite Markov chain with state space Q × Γ ∗ and transition probability matrix M such that for all q, r ∈ Q, Z ∈ Γ, α ∈ Γ ≤2, γ ∈ Γ ∗, we have M(qZγ, rαγ) = P(qZ)(rα) , M(qε, qε) = 1 , and all other transition probabilities are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This Markov chain, where the initial state is fixed to qZ, is denoted MqZ ∆ (see Figure 3 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' As usual, one can formally define a probability measure PqZ ∆ on the infinite runs of MqZ ∆ via the standard cylinder construction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', [2, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Consider a triple qZr ∈ Q×Γ×Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We define the return probability2 [qZr] as the probability of reaching rε in the Markov chain MqZ ∆ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', [qZr] = PqZ ∆ (♦{rε}), where ♦{rε} is the set of infinite runs of MqZ ∆ that eventually hit state rε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 2 When modeling procedural programs with pPDA, [qZr] is the probability that a given procedure returns a specific value to its calling context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' These probabilities 12 Tobias Winkler and Joost-Pieter Katoen q r (1/2, Z, ε) (1/4, Z, ZZ) (1/4, Z, ε) (1, Z, ε) qε qZ qZZ · · rε rZ rZZ · · 1/4 1/4 1/2 1/2 1/2 1/4 1/4 1/4 1 1 1 1 1 ⟨qZq⟩ = 1/4 � ⟨qZq⟩⟨qZq⟩ + ⟨qZr⟩⟨rZq⟩ � + 1/2 ⟨rZq⟩ = 0 ⟨qZr⟩ = 1/4 � ⟨qZq⟩⟨qZr⟩ + ⟨qZr⟩⟨rZr⟩ � + 1/4 ⟨rZr⟩ = 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 3: Top left: The pPDA ∆ex = ({q, r}, {Z}, P) where P comprises the tran- sitions qZ 1/4 −−→ qZZ, qZ 1/2 −−→ qε, qZ 1/4 −−→ rε, rZ 1−→ rε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Top right: A fragment of the infinite underlying Markov chain, assuming initial configuration qZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Bottom: The associated equation system from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Theorem 4 (The PPS of return probabilities [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let ∆ = (Q, Γ, P) be a pPDA and (⟨qZr⟩)qZr ∈ Q×Γ ×Q be variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For each ⟨qZr⟩, define ⟨qZr⟩ = � qZ p−→sY X p · � t∈Q ⟨sY t⟩ · ⟨tXr⟩ + � qZ p−→sY p · ⟨sY r⟩ + � qZ p−→rε p and call the resulting PPS ⃗f∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then µ⃗f∆ = ([qZr])qZr ∈ Q×Γ ×Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We refer to [30, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 3] for an intuitive explanation of the equations in ⃗f∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Figure 3 shows a pPDA ∆ex and the associated PPS ⃗f∆ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The least non-negative solution is ⟨qZq⟩ = 2 − √ 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='586 and ⟨qZr⟩ = √ 2 − 1 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='414 (and, of course, ⟨rZq⟩ = 0, ⟨rZr⟩ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Thus by Theorem 4, the return probabilities are [qZq] = 2 − √ 2 and [qZr] = √ 2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' △ The PPS ⃗f∆ is always feasible (because µ⃗f∆ ≤ ⃗1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗f∆ is neither necessarily strongly connected nor clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let ⃗ˆf∆ denote the cleaned up version of ⃗f∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Proposition 1 (Basic Certificates for pPDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A basic certificate for ∆ = (Q, Γ, P) is a rational inductive upper bound ⃗u ∈ QQ×Γ ×Q ≥0 on the lfp of the return probabilities system ⃗f∆ (see Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' They have the following properties: – (Existence) ∀ε > 0 there exists a basic certificate ⃗u with ||µ⃗f∆ − ⃗u||∞ ≤ ε if all maximal irreducible submatrices M of ∂ ⃗ˆf∆(µ⃗ˆf∆) satisfy ρ(M) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' were called termination probabilities in previous works [12,7] but we believe this term is more appropriate for the numbers [qZ↓] = � r[qZr], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', the probability to eventually reach the empty stack from initial configuration qZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for Probabilistic Pushdown Automata via OVI 13 – (Complexity) Let β be the maximum number of bits used to encode any of the numerators and denominators of the fractions occurring in ⃗u ∈ QQ×Γ ×Q ≥0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then checking ⃗f∆(⃗u) ≤ ⃗u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', whether ⃗u is basic certificate for ∆, can be done in time polynomial in β and the size of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Existence of basic certificates follows from Lemma 2 applied to each SCC of the cleaned-up version of ⃗f∆ individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, note that in order to merely check the certificate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', verify the inequality ⃗f(⃗u) ≤ ⃗u, neither do SCCs need to be computed nor does the system has to be cleaned up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Reconsider the example pPDA and its associated (non-strongly con- nected) system of return probabilities from Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We verify that ⃗uqZq = 3/5 and ⃗uqZr = 1/2 (as well as ⃗urZq = 0, ⃗urZr = 1) is a basic certificate: 1 4 �3 5 · 3 5 + 1 2 · 0 � + 1 2 = 59 100 ✓ ≤ 3 5 , 1 4 �3 5 · 1 2 + 1 2 · 1 � + 1 4 = 45 100 ✓ ≤ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that [qZq] ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='586 ≤ 3/5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='6 and [qZr] ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='414 ≤ 1/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' △ In the following we outline how a variety of key quantities associated to pPDA can be verified using basic certificates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' More details are in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Upper Bounds on Temporal Properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We may use basic certificates to verify that a bad state rbad is reached with low probability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', at most p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' To this end, we remove the outgoing transitions of rbad and add the transitions rbadZ 1−→ rbadε for all Z ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Clearly, rbad is reached with probability at most p from initial configuration qZ iff [qZrbad] ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The results of [13] imply that this idea can be generalized to until-properties of the form C1 U C2, where C1 and C2 are regular sets of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (This requires a small extension of the basic certificates, but the overall idea stays the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for the Output Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Once a pPDA reaches the empty stack, we say that it has terminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' When modeling procedural programs, this cor- responds to returning from a program’s main procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Assuming initial con- figuration qZ, the probability sub-distribution over the possible return values is then given by the return probabilities {[qZr] | r ∈ Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Missing probability mass models the probability of non-termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A basic certificate can thus be used immediately to verify a point-wise upper bound on the output distribution as well as to certify that a program is not almost-surely terminating (AST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If a pPDA ∆ is already known to be AST, then we can also certify a lower bound on the output distribution: Suppose that ⃗u is a basic certificate for ∆ and assume that ∆ is AST from initial configuration qZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Define ε = � r∈Q ⃗uqZr − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then for all r ∈ Q, we have ⃗uqZr − ε ≤ [qZr] ≤ ⃗uqZr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The pPDA ∆ex from Figure 3 is AST from initial configuration qZ, as the transition qZ 1/4 −−→ rε is eventually taken with probability 1, and the stack is emptied certainly once r is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Using the basic certificate from Example 5 we can thus (correctly) certify that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5 ≤ [qZq] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4 ≤ [qZr] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 14 Tobias Winkler and Joost-Pieter Katoen Certificates for Expected Rewards or Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Suppose we have equipped a pPDA with a state-based reward (or cost) function Q → R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It was shown in [14] that the expected total reward accumulated during the run of a pPDA is the solution of a linear equation system where the return probabilities [qZr] appear as coef- ficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Given a basic certificate ⃗u, we can replace each coefficient [qZr] by ⃗uqZr and thus obtain an equation system whose solution is an over-approximation of the true expected reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We may extend the basic certificate ⃗u by the solution of this linear system to make verification straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that a program’s expected runtime [8,35] is a special case of total expected reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 5 Implementation and Experiments Our Tool: pray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We implemented our algorithm in the prototypical Java-tool pray (Probabilistic Recursion AnalYzer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It supports two input formats: (i) Recursive probabilistic programs in a Java-like syntax (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Figure 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' these programs are automatically translated to pPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (ii) Explicit PPS in the same syntax used by the tool PReMo [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The output of pray is a rational inductive upper bound on the lfp of the return probability PPS of the input program’s pPDA model (a basic certificate), or on the lfp of the explicitly given PPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The absolute precision ε is configurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The implementation works as follows: (1) It parses the input and, if the latter was a program, constructs a pPDA model and the associated PPS of return probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (2) It computes an SCC decomposition of the PPS under consideration using standard algorithms implemented in the jGraphT library [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (3) It applies Algorithm 1 to the individual SCC in reverse topological order using floating point arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Algorithm 1 is instantiated with Kleene it- eration3, the power iteration for approximating eigenvectors as outlined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3, and constants c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We allow ≤ 10 guesses per SCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (4) If stage (3) is successful, the tool verifies the resulting floating point certifi- cate using exact rational number arithmetic as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' To the best of our knowledge, no alternative techniques for finding inductive upper bounds in PPS have been described explicitly in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, there is an (almost) out-of-the-box approach using an SMT solver: Given a PPS ⃗x = ⃗f(⃗x), compute some lower bound ⃗l ≤ µ⃗f using an iterative technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then query the SMT solver for a model (variable assignment) of the quantifier-free first-order logic formula ϕ⃗f(⃗x) = �n i=1 fi(⃗x) ≤ xi ∧⃗li ≤ xi ≤ ⃗li +ε in the (decidable) theory of polynomial real arithmetic with inequality (aka QF_NRA in the SMT community).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If such a model ⃗u exists, then clearly µ⃗f ≤ ⃗u and ||⃗l − ⃗u||∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If no model exists, then improve ⃗l and try again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We have 3 In fact, we use the slightly optimized Gauss-Seidel iteration (see [42, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2]) which provides a good trade-off between ease of implementation and efficiency [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for Probabilistic Pushdown Automata via OVI 15 bool and() { prob { 1//2: return (1//2: true | 1//2: false);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 1//2: { if(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='or()) return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' else return or();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } } bool or() { prob { 1//2: return (1//2: true | 1//2: false);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 1//2: { if(and()) return true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' else return and();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } } Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 4: Program evaluating a random and-or tree [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The prob-blocks execute the contained statements with the respective probabilities (syntax inspired by Java’s switch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Our tool automatically translates this program to a pPDA and computes a basic certificate (Proposition 1) witnessing that calling and() returns true and false with probability ≤ 382/657 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='58 and 391/933 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='42, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' implemented this approach using the state-of-the-art SMT solvers cvc5 [4] and z3 [34], the winners of the 2022 SMT-COMP in the category QF_NRA4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' As yet another baseline, we have also implemented a variant of OVI for PPS which is closer to the original MDP algorithm from [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In this variant, called “standard OVI” from now on, we compute the candidate ⃗u based on the “relative” update rule ⃗u = (1 + ε)⃗l, where ⃗l is the current lower bound [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Research Questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We aim to shed some light on the following questions: (A) How well does our algorithm scale?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (B) Is the algorithm suitable for PPS with dif- ferent characteristics, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', dense or sparse?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (C) Is the requirement ρ(∂ ⃗f(µ⃗f)) < 1 restrictive in practice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (D) How does our OVI compare to the baselines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' To answer the above questions we run our implementation on two sets of benchmarks (Table 3 and Table 2, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The first set consists of various example programs from the literature as well as a few new programs, which are automatically translated to pPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This translation is standard and usually takes not more than a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The programs golden, and-or (see Fig- ure 4), virus, gen-fun are adapted from [35,8,41] and [32, Program 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='6], respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The source code of all considered programs is in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We have selected only programs with possibly unbounded recursion depth which induce infinite Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The second benchmark set comprises explicitly given PPS5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The instances brown, lemonde, negra, swbd, tiger, tuebadz, and wsj all en- code SCFG from the area of language processing (see [43] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' random is the return probability system of a randomly generated pPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Summary of Experimental Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We ran the experiments on a standard note- book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The approach based on cvc5 turns out to be not competitive (see Ap- pendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We thus focus on z3 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Both pray and the z3 approach could handle most of the programs from Table 3 within a 10 minute time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The considered programs induce sparse PPS with 38 - 26,367 variables, and most 4 https://smt-comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='io/2022/results 5 These examples come with PReMo: https://cgi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='csc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='uk/~dominik/premo/ 16 Tobias Winkler and Joost-Pieter Katoen Table 1: Experiments with PPS obtained from recursive probabilistic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Columns vars and terms display the number of variables and terms in the PPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Columns sccs and sccmax indicate the number of non-trivial SCC and the size of the largest SCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' G is total number of guesses made by OVI (at least one guess per SCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ttot is the total runtime excluding the time for model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' tQ is the percentage of ttot spent on exact rational arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' D is the average number of decimal digits of the rational numbers in the certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The timeout (TO) was set to 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Timings are in ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The absolute precision is ε = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' benchmark |Q| |P| |Γ| vars terms sccs sccmax cert G D tQ ttot certz3 Dz3 tz3 certstd Gstd Dstd tstd rw-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='499 18 29 5 38 45 1 12 ✓ 5 5 17% 163 ✓ 2 11 ✓ 4 5 59 rw-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='500 18 29 5 38 45 1 12 ✗ 10 7327 ✓ 2 10 ✗ 10 8083 rw-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='501 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} 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+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', rw-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='01, golden, virus, brown, swbd), the resulting certificates formally disprove AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For the explicit PPS in Table 2, pray solves all instances whereas z3 only solves 3/8 within the time limit, and only finds the trivial solution ⃗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Most of these benchmarks contain dense high-degree polynomials and our tool spends most time on performing exact arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' pray never needs more than 6 guesses per SCC if it succeeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Evaluation of Research Questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (A) Scalability: Our algorithm succeeded on instances with maximum SCC size of up to 8,000 and number of terms over 50,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' pray could solve all instances with a maximum SCC size of ≤ 1,000 in less than 2 minutes per instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For the examples where our algorithm does not succeed (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', escape100) it is mostly because it fails converting a floating point to a rational certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (B) PPS with different flavors: The problems in Table 3 (low degree and sparse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', few terms per polynomials) and Table 2 (higher degree and dense) are quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A comparison to the SMT approach suggests that our technique might be especially well suited for dense problems with higher degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (C) Non-singularity: The only instance where our algorithm fails because of the non-singularity condition is the symmetric random walk rw- Certificates for Probabilistic Pushdown Automata via OVI 17 Table 2: Experiments with explicitly given PPS (setup as in Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='benchmark ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We therefore conjecture that this condition is often satisfied in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (D) Comparison to SMT: There is no clear winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Some instances can only be solved by one tool or the other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' escape100 and brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, pray often delivers more succinct certificates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', the rational numbers have less digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Overall, z3 behaves less predictably than pray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 6 Conclusion and Future Work We have proposed using inductive bounds as certificates for various properties in probabilistic recursive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Moreoever, we have presented the first dedicated algorithm for computing inductive upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' While our algorithm already scales to non-trivial problems, the main bottleneck is the generation of an exact rational bound from a floating point approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This might be improved using appropriate rounding modes as in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Additional future work includes further certificates for pPDA, especially for lower bounds and termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' References 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Hartmanns, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Kaminski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' : Optimistic value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In: CAV (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Lecture Notes in Computer Science, vol.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Katsouros, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Carayannis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=': Recognition of online handwritten mathematical formulas using probabilistic SVMs and stochastic context free gram- mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Pattern Recognit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 53, 85–92 (2015) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Stewart, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Etessami, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Yannakakis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=': Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ACM 62(4), 30:1–30:33 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 425–443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Springer (2020) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Winkler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Gehnen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Katoen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=': Model Checking Temporal Properties of Re- cursive Probabilistic Programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In: FoSSaCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Lecture Notes in Computer Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 13242, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 449–469.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Springer (2022) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Wojtczak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=': Recursive probabilistic models : efficient analysis and implementa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' thesis, University of Edinburgh, UK (2009) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Wojtczak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Etessami, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=': PReMo : An Analyzer for Probabilistic Recursive Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In: TACAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Lecture Notes in Computer Science, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 4424, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 66–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Springer (2007) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Yannakakis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', Etessami, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=': Checking LTL properties of recursive markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In: QEST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 155–165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' IEEE Computer Society (2005) 20 Tobias Winkler and Joost-Pieter Katoen A Full Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1 Proof of Lemma 2 Lemma 2 (Existence of inductive upper bounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let ⃗f be a feasible, clean, and strongly connected PPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then the following are equivalent: (1) The matrix I − ∂ ⃗f(µ⃗f) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (2) The spectral radius of ∂ ⃗f(µ⃗f) satisfies ρ(∂ ⃗f(µ⃗f)) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (3) There exists ⃗0 ≺ ⃗u ≺ ⃗∞ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗f(⃗u) < ⃗u (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗u is inductive but not a fixpoint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (4) The matrix ∂ ⃗f(µ⃗f) has a unique (normalized) eigenvector ⃗v ≻ ⃗0 and there exist numbers δmax > 0 and ε > 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗f( µ⃗f + δ · ⃗˜v ) ≺ µ⃗f + δ · ⃗˜v holds for all 0 < δ ≤ δmax and vectors ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We now explain the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The proof heavily relies on a linear approximation of ⃗f around the lfp µ⃗f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Intuitively, this is where the Jacobi matrix ∂ ⃗f(µ⃗f) comes into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This is formalized via Taylor’s familiar theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Lemma 4 (Taylor’s Theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' [12, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let ⃗f be a feasible PPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then for all vectors ⃗u ≥ ⃗0, we have ⃗f(µ⃗f + ⃗u) = µ⃗f + ∂ ⃗f(µ⃗f)⃗u + R⃗u⃗u where R⃗u is a matrix that depends on ⃗u such that lim⃗u→⃗0 R⃗u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' More specifi- cally, it holds that ⃗0 ≤ R⃗u⃗u ≤ � ∂ ⃗f(µ⃗f + ⃗u) − ∂ ⃗f(µ⃗f) � ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Proof (Proof of Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' “(1) =⇒ (2)”: By Theorem 2 we have ρ(∂ ⃗f(µ⃗f)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Towards contradiction assume that ρ(∂ ⃗f(µ⃗f)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By the Perron-Frobenius Theorem, 1 is an eigenvalue of ∂ ⃗f(µ⃗f), which means that there exists ⃗u ̸= ⃗0 such that ∂ ⃗f(µ⃗f)⃗u = ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This ⃗u is in the kernel of I − ∂ ⃗f(µ⃗f), which contradicts the assumption that I − ∂ ⃗f(µ⃗f) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' “(2) =⇒ (1)”: It is a well-known result that for an arbitrary real matrix M the series �∞ k=0 M k converges iff ρ(M) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The limit of the series is the inverse of I − M because (I − M) ∞ � k=0 M = ∞ � k=0 M k − ∞ � k=1 M k = M 0 = I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' “(2) =⇒ (4)”: Let ρ(∂ ⃗f(µ⃗f)) =: λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By the Perron-Frobenius Theo- rem, the Jacobi matrix ∂ ⃗f(µ⃗f) has a unique normalized eigenvector ⃗v ≻ ⃗0 wrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' eigenvalue λ: ∂ ⃗f(µ⃗f)⃗v = λ⃗v ≺ ⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (1) Certificates for Probabilistic Pushdown Automata via OVI 21 Our goal is to define the values ε and δmax whose existence we claimed in Lemma 2(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let cmin > 0 be the smallest component of (1−λ)⃗v ≻ ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We define ε := cmin 3||∂ ⃗f(µ⃗f)||∞ , (2) where ||∂ ⃗f(µ⃗f)||∞ = max||⃗y||∞=1 ||∂ ⃗f(µ⃗f)⃗y||∞ is the maximum row sum of ∂ ⃗f(µ⃗f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that || · ||∞ is the operator norm induced by the maximum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then it holds for all ⃗ε with ||⃗ε||∞ ≤ ε that ||∂ ⃗f(µ⃗f)⃗ε||∞ ≤ ||∂ ⃗f(µ⃗f)||∞||⃗ε||∞ ≤ ||∂ ⃗f(µ⃗f)||∞ cmin 3||∂ ⃗f(µ⃗f)||∞ = 1 3cmin .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (3) The first inequality in (3) is a property of operator norms (which is straightfor- ward in the case of the maximum norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since cmin was the smallest component of (1 − λ)⃗v, (3) implies ∂ ⃗f(µ⃗f)⃗ε ≤ 1 3(1 − λ)⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (4) We now define δmax as follows: δmax := sup {δ > 0 | ∀⃗ε ≥ ⃗0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ||⃗ε||∞ ≤ ε: Rδ(⃗v+⃗ε)(⃗v + ⃗ε) ≤ 1 2(1 − λ)⃗v} , (5) where Rδ(⃗v+⃗ε) is the matrix from Lemma 4 which satisfies ⃗f(µ⃗f + δ(⃗v + ⃗ε)) = µ⃗f + δ∂ ⃗f(µ⃗f)(⃗v + ⃗ε) + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We now argue that δmax > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This is not immediately obvious because of the ∀-quantification in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let δ > 0 be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Further, let ⃗ε ≥ ⃗0 be such that ||⃗ε||∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In the following, we write ⃗ε′ = (ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We have Rδ(⃗v+⃗ε)(⃗v + ⃗ε) = 1 δ Rδ(⃗v+⃗ε)δ(⃗v + ⃗ε) ≤ 1 δ � ∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε)) − ∂ ⃗f(µ⃗f) � δ(⃗v + ⃗ε) (Lemma 4) = � ∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε)) − ∂ ⃗f(µ⃗f) � (⃗v + ⃗ε) ≤ � ∂ ⃗f(µ⃗f + δ(⃗v + ⃗ε′)) − ∂ ⃗f(µ⃗f) � (⃗v + ⃗ε′) (Jacobi matrix is monotonic) =: Mδ(⃗v + ⃗ε′) Note that Mδ does not depend on ⃗ε and limδ→0 Mδ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We can therefore find a specific δ∗ > 0 such that Mδ∗(⃗v+⃗ε′) ≤ 1 2(1−λ)⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' On the other hand, we have just 22 Tobias Winkler and Joost-Pieter Katoen shown for all ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε and all δ > 0 that Rδ(⃗v+⃗ε)(⃗v+⃗ε) ≤ Mδ(⃗v+⃗ε′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' So we have in particular for all ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε that Rδ∗(⃗v+⃗ε)(⃗v + ⃗ε) ≤ Mδ∗(⃗v + ⃗ε′) ≤ 1 2(1 − λ)⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Hence δmax ≥ δ∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Finally, let 0 < δ ≤ δmax and ⃗˜v ≥ ⃗v with ||⃗v − ⃗˜v||∞ ≤ ε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', ⃗˜v = ⃗v + ⃗ε for some ⃗ε ≥ ⃗0 with ||⃗ε||∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then ⃗f(µ⃗f + δ(⃗v + ⃗ε)) = µ⃗f + δ∂ ⃗f(µ⃗f)(⃗v + ⃗ε) + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) (by Taylor’s Theorem (Lemma 4)) = µ⃗f + δλ⃗v + δ∂ ⃗f(µ⃗f)⃗ε + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) (by (1)) ≤ µ⃗f + δλ⃗v + δ 1 3(1 − λ)⃗v + δRδ(⃗v+⃗ε)(⃗v + ⃗ε) (by (4)) ≤ µ⃗f + δλ⃗v + δ 1 3(1 − λ)⃗v + δ 1 2(1 − λ)⃗v (by (5)) ≺ µ⃗f + δλ⃗v + δ 1 2(1 − λ)⃗v + δ 1 2(1 − λ)⃗v (because δ(1 − λ)⃗v ≻ ⃗0) = µ⃗f + δ⃗v (simplification) ≤ µ⃗f + δ(⃗v + ⃗ε) (because ⃗ε ≥ ⃗0) “(4) =⇒ (3)”: Trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' “(3) =⇒ (2)”: By (3) there exists ⃗u such that ⃗f(⃗u) < ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By Lemma 1 this implies that µ⃗f < ⃗u, so we can write ⃗u = µ⃗f + ⃗v for some ⃗v > ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Using Taylor’s Theorem (Lemma 4), it follows that ⃗f(µ⃗f + ⃗v) = µ⃗f + ∂ ⃗f(µ⃗f)⃗v + R⃗v⃗v < µ⃗f + ⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (6) Using that R⃗v⃗v ≥ ⃗0, (6) implies that ∂ ⃗f(µ⃗f)⃗v < ⃗v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' (7) The claim now follows by applying the following lemma to the matrix ∂ ⃗f(µ⃗f) and the vector ⃗v: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let M ≥ 0 be an irreducible n× n-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' If there exists ⃗u > ⃗0 such that M⃗u < ⃗u, then ⃗u ≻ ⃗0, M n⃗u ≺ ⃗u and ρ(M) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' First observe that since multiplication by M is monotone we have for all 0 ≤ k1 ≤ k2 that ⃗0 ≤ M k2⃗u ≤ M k1⃗u ≤ ⃗u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We first show that ⃗u ≻ ⃗0, which is essentially [12, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since ⃗u > ⃗0, there must be 1 ≤ i ≤ n such that ⃗ui > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Now let 1 ≤ j ≤ n be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since Certificates for Probabilistic Pushdown Automata via OVI 23 M is irreducible there exists 0 ≤ k < n such that M k j,i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This implies that (M k⃗u)j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By monotonicty, ⃗u ≥ M k⃗u, and thus ⃗uj ≥ (M k⃗u)j > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since j was arbitrary, ⃗u ≻ ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Next we show M n⃗u ≺ ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since M⃗u < ⃗u holds by assumption, there exists 1 ≤ i ≤ n such that (M⃗u)i < ⃗ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let 1 ≤ j ≤ n be a arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since M is irreducible, there exists 0 ≤ k < n such that (M k)j,i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We now show that (M n⃗u)j < uj which implies that M n⃗u ≺ ⃗u as j was chosen arbitrarily: (M n⃗u)j ≤ (M kM⃗u)j (by monotonicity, and because k + 1 ≤ n) = (M k)j,i(M⃗u)i + � l̸=i (M k)j,l(M⃗u)l (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' matrix-vector product) < (M k)j,i⃗ui + � l̸=i (M k)j,l(M⃗u)l (because (M⃗u)i < ⃗ui and (M k)j,i > 0) ≤ (M k)j,i⃗ui + � l̸=i (M k)j,l⃗ul (because (M⃗u)l ≤ ⃗ul) = (M k⃗u)j ≤ ⃗uj It remains to show that ρ(M) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We do this by showing that the powers of M (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', the sequence (M k)k≥0) converge to the zero matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since M n⃗u ≺ ⃗u, we can choose c < 1 such that M n⃗u ≤ c⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then for all m ≥ 1 it holds that M nm⃗u ≤ cm⃗u, so we have lim k→∞ M k⃗u = ⃗0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Recall from above that we already know ⃗u ≻ ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Thus limk→∞ M k⃗u = ⃗0 means that a positive linear combination of the entries of each individual row of M k converges to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', for all 1 ≤ i ≤ n we have limk→∞ � j M k i,j⃗uj = 0, and thus for all 1 ≤ j ≤ n, limk→∞ M k i,j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Thus limk→∞ M k = 0, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⊓⊔ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='2 Proof of Theorem 3 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Algorithm 1 is correct: when invoked with a strongly connected clean PPS ⃗f and ε > 0, then (if it terminates) it outputs a pair (⃗l, ⃗u) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗l ≤ µ⃗f, ⃗f(⃗u) ≤ ⃗u (and thus µ⃗f ≤ ⃗u), and ||⃗l − ⃗u||∞ ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Moreover, if ⃗f is feasible and I − ∂ ⃗f(µ⃗f) is non-singular, then the algorithm terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Correctness is obvious, so we only show termination assuming that ⃗f is feasible and I − ∂ ⃗f(µ⃗f) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Clearly, the algorithm terminates iff it eventually finds a ⃗u in line 8 which is inductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Assume towards contradiction that the algorithm never terminates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', it never finds an inductive ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For all i ≥ 1 let ⃗li, ⃗vi, τi be the values of the variables ⃗l, ⃗v and τ at the ith time the inner loop at line 7 is reached (note that we then have N = i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Clearly, limi→∞ τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By the contract satisfied by 24 Tobias Winkler and Joost-Pieter Katoen improveLowerBound, we have limi→∞ ∂ ⃗f(⃗li) = ∂ ⃗f(µ⃗f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since the eigenvectors of ∂ ⃗f(µ⃗f) depend continuously on those of the matrices ∂ ⃗f(⃗li), and because of the contract satisfied by approxEigenvec, the sequence ⃗v1,⃗v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' converges to the true unique normalized Perron-Frobenius eigenvector ⃗vtrue of ∂ ⃗f(µ⃗f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We now apply condition (4) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The condition ensures that the cone C(µ⃗f,⃗vtrue, ε′, δmax) = { µ⃗f + δ⃗˜v | 0 ≤ δ ≤ δmax, ||⃗˜v − ⃗vtrue||∞ ≤ ε′ } which is located at µ⃗f, points in direction ⃗vtrue and has radius ε′ and length δmax contains only inductive points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For the sake of illustration suppose that the algorithm already knows δmax and computes ⃗ui = ⃗li +δ⃗vi for some 0 < δ < δmax instead of executing the loop starting at line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' But then the sequence (⃗ui)i≥1 converges to µ⃗f + δ⃗vtrue, which is a point that lies inside the interior of C, so there must be some i ≥ 1 such that ⃗ui ∈ C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', ⃗ui is inductive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The remaining difficulty is that δmax is of course unknown in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We handle this using the inner loop that starts at line 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Eventually, the variable N is sufficiently large such that dkε < δmax for some k ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Termination then follows by applying the argument in the previous paragraph to δ = dkε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⊓⊔ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='3 Proof of Lemma 3 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let M ≥ 0 be irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then power iteration applied to M + I and any ⃗v0 > ⃗0 converges to the Perron-Frobenius eigenvector ⃗v ≻ ⃗0 of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Consider the following conditions for an irreducible matrix M ≥ 0 and a vector M⃗v0 with M⃗v0 ̸= ⃗0: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' M has a unique dominant eigenvalue |λ1| > |λ2| ≥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ≥ |λn|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' λ1 is semisimple, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', its algebraic multiplicity6 equals its geometric multi- plicity7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⃗v0 is not orthogonal to the eigenspace {⃗v | M⃗v = λ1⃗v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It is known that if all these conditions are satisfied, then the power iteration sequence (⃗vi)i∈N converges to a (normalized) eigenvector ⃗v with eigenvalue λ1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' [37, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We now show that these conditions are satisfied for the irreducible matrix M + I ≥ 0 and every initial vector ⃗v0 > ⃗0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The eigenvectors of M and M + I are exactly the same but the eigenvalues are all shifted by +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Indeed, if ⃗v is some eigenvector of M with eigenvalue λ, then (M + I)⃗v = λ⃗v + ⃗v = (λ + 1)⃗v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' However, unlike M, the matrix M +I always has period 1, and so it has a unique dominant eigenvalue λ1 by Theorem 1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Therefore the first of the above three conditions is satisfied by the matrix M + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 6 The algebraic multiplicity is the multiplicity of a given eigenvalue as a root of the characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 7 The geometric multiplicity is the dimension of the eigenspace associated with a particular eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for Probabilistic Pushdown Automata via OVI 25 Next, by Theorem 1(1) it holds that the geometric multiplicity of λ1 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' As the algebraic multiplicity is bounded by the geometric multiplicity, it must also be 1 and thus the matrix M + I satisfies the second condition as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Finally, the third condition is satisfied for any ⃗v0 > ⃗0 because the scalar product ⃗v0 · ⃗v is non-zero (either strictly positive or strictly negative) for all non-zero eigenvectors ⃗v of λ1 by Theorem 1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' ⊓⊔ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='4 Proof of Proposition 1 Proposition 1 (Basic Certificates for pPDA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' A basic certificate for ∆ = (Q, Γ, P) is a rational inductive upper bound ⃗u ∈ QQ×Γ ×Q ≥0 on the lfp of the return probabilities system ⃗f∆ (see Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' They have the following properties: – (Existence) ∀ε > 0 there exists a basic certificate ⃗u with ||µ⃗f∆ − ⃗u||∞ ≤ ε if all maximal irreducible submatrices M of ∂ ⃗ˆf∆(µ⃗ˆf∆) satisfy ρ(M) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' – (Complexity) Let β be the maximum number of bits used to encode any of the numerators and denominators of the fractions occurring in ⃗u ∈ QQ×Γ ×Q ≥0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then checking ⃗f∆(⃗u) ≤ ⃗u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', whether ⃗u is basic certificate for ∆, can be done in time polynomial in β and the size of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This proof closely follows the general idea of decomposed analysis of PPS [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We first address existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that ⃗f∆ is guaranteed to be feasible, in fact ⃗0 ≤ µ⃗f∆ ≤ ⃗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For all qZr with (µ⃗f∆)qZr = 0 we set ⃗uqZr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By removing these variables from ⃗f∆ we obtain the clean PPS ⃗ˆf∆ with ⃗0 ≺ µ⃗ˆf∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Now consider the decomposition of ⃗ˆf∆ into the subsystems induced by the strongly connected components of the graph G ⃗ˆf∆: ⃗ˆf 1 ∆, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' , ⃗ˆf m ∆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that in these subsystems, some variables might only appear on the right hand sides but not on the left (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' x1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5x1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5x2, x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5x1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='5x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since µ⃗ˆf∆ ≻ ⃗0, there is a 1 - 1 correspondence of these subsystems and the maximal irreducible submatrices Mi of ∂ ⃗ˆf∆(µ⃗ˆf∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' More specifically, Mi = ∂ ⃗ˆf i ∆(µ⃗ˆf∆)8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' By assumption, ρ(Mi) < 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Now assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' that ⃗ˆf 1 ∆ is a bottom SCC (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', in the dependency graph G ⃗ˆ f∆ there is no path from the variables in ⃗ˆf 1 ∆ to any variable not in ⃗ˆf 1 ∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Then ⃗ˆf 1 ∆ is a strongly connected PPS with ∂ ⃗ˆf 1 ∆(µ⃗ˆf∆) = ∂ ⃗ˆf 1 ∆(µ⃗ˆf 1 ∆) and we can apply Lemma 2(4) to obtain a rational ⃗u1 with ⃗ˆf 1 ∆(⃗u1) ≤ ⃗u1 and ||µ⃗ˆf 1 ∆ − ⃗u1||∞ ≤ ε (in fact, we can do this for any ε > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Suppose we have done the above for all bottom SCCs and now start traversing the DAG of SCCs bottom-up, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', in reverse topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let ⃗u be the 8 The Jacobi matrix of a sub-PPS with n′ < n equations is an n′ × n′ matrix where all variables that occur only on the right hand sides are considered constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 9 The spectral radius of the zero matrix is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 26 Tobias Winkler and Joost-Pieter Katoen bound we have constructed to far (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', ⃗u contains ⃗u1 and the bounds from the other bottom SCC as subvectors and is zero elsewhere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that we can always make ⃗u smaller while retaining the inductivity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' suppose that subsystem ⃗ˆf 2 ∆ is one of the first non-bottom SCCs in the reverse topological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The idea is now to modify ⃗ˆf 2 ∆ to a strongly connected PPS ˜⃗f 2 ⃗u by replacing all variables that occur only in right hand sides by their value in ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Clearly, lim⃗u→µ⃗ˆ f∆ ∂ ˜⃗f 2 ⃗u(µ ˜⃗f 2 ⃗u) = ∂ ⃗ˆf 2 ∆(µ⃗ˆf∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This means we can choose ⃗u sufficiently close to µ⃗ˆf∆ such that the spectral radius of ∂ ˜⃗f 2 ⃗u(µ ˜⃗f 2 ⃗u) is strictly smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We can then apply Lemma 2(4) to ˜⃗f 2 ⃗u to obtain a rational ⃗u2 with ˜⃗f 2 ⃗u(⃗u2) ≤ ⃗u2 to enlarge our current ⃗u with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We can repeat this scheme for all finitely many subsystems until we have constructed a rational ⃗u with ˜⃗f i ⃗u(⃗u) ≤ ⃗u for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Clearly, this ⃗u also satisfies ⃗ˆf∆(⃗u) ≤ ⃗u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Finally, we may extend ⃗u by zero entries corresponding to the vari- ables that are assigned zero in the lfp of the (not necessarily clean) ⃗f∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This yields an inductive upper bound for ⃗f∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' We stress that in order to verify this bound, we neither have to clean ⃗f∆ nor do we have to compute the SCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' For complexity observe that ⃗f∆ is cubic in the size of ∆ and that all polyno- mials in ⃗f∆ have degree at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Since multiplication and addition of rational numbers can be done in polynomial time in the number of their bits, evaluat- ing a polynomial of fixed maximum degree can also be done in polynomial time in the size of the polynomial and the number of bits representing the rationals where the polynomial is to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that this is not true for arbitrary polynomials where exponents are encoded in binary: For instance, evaluating the polynomial x2n (which can be represented with O(n) bits) at x = 2 yields 22n, a number that needs O(2n) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' This means that in order to verify certificates efficiently with exact rational arithmetic, it is important that the polynomials in the PPS do not have very high degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Fortunately, this is the case for pPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Certificates for Probabilistic Pushdown Automata via OVI 27 B Certificates for Expected Rewards We can certify upper bounds on the expected value of rewards collected during the run of a pPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' To simplify the presentation, in this section we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' that qZ p−→ rα with p > 0 implies |α| ∈ {0, 2}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=', all transitions either decrease or increase the stack height by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Let R: Q → R≥0 be a state-based reward function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Consider the following PPS ⃗f∆,R with variables {⟨EqZr⟩ | qZr ∈ Q × Γ × Q}: ⟨EqZr⟩ = � qZ p−→sY X p · � t∈Q [sY t] · [tXr] · KqZ,sY X + � qZ p−→rε p · R(r) , where KqZ,sY X = R(r) + ⟨EsY t⟩ + ⟨EtXr⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Note that ⃗f∆,R is linear but uses the return probabilities which are themselves characterized as the lfp of the non-linear system ⃗f R ∆ from Theorem 4 as coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Suppose that in the lfp µ⃗f∆,R, each variable EqZr is assigned the quantity EqZr ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' It follows from the results of [14] that EqZr equals the expected value of the following random variable V r R under the probability measure PqZ ∆ : V r R(q0γ0, q1γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=') = firstHit(rε) � i>0 R(qi) where firstHit(rε) is the minimum integer k such that qkγk = rε, or 0 if no such k exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' In words, EqZr is the expected reward accumulated on the runs from qZ to rε, where it is assumed that runs which never reach rε contribute zero reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Consequently, E(qZ) = � r∈Q EqZr is the expected reward accumulated on all terminating runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Setting R = 1 we can characterize the expected runtime of pPDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' Reconsider Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The equation system for expected runtimes becomes ⟨EqZq⟩ =1 4([qZq]2(1+2⟨EqZq⟩) + [qZr][rZq](1+⟨EqZr⟩+⟨ErZq⟩)) + 1 2 ⟨EqZr⟩ =1 4([qZq][qZr](1+⟨EqZq⟩+⟨EqZr⟩)+[qZr][rZr](1+⟨EqZr⟩+⟨ErZr⟩)) + 1 4 as well as ⟨ErZq⟩ = 0 and ⟨ErZr⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' The solution is ⟨EqZq⟩ = 2063/2624 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='786 and ⟨EqZr⟩ = 59/82 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='712, so the total expected runtime is E(qZ) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' △ C Benchmark Programs 28 Tobias Winkler and Joost-Pieter Katoen void f() { if flip(p) { f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } # main block { f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } (a) rw-p void f() { if flip(1//2) { f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } # main block { f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } (b) golden void offspring() { while flip(2//5) { offspring();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' while flip(3//5) { offspring();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } } # main block { offspring();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } (c) geom-offspring void gen_operator() { uniform(4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } void gen_expression() { prob { 4//10: uniform(10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 3//10: { } 3//10: { gen_operator();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' gen_expression();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' gen_expression();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } } void gen_function() { gen_operator();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' gen_expression();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' gen_expression();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } # main block { gen_function();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } (d) gun-fun void young() { int y = uniform(4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' while(y > 0) { young();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' y = y-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } int e = uniform(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' while(e > 0) { elder();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' e = e-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } void elder() { int y = uniform(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' while(y > 0) { young();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' y = y-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } int e = uniform(5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' while(e > 0) { elder();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' e = e-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } # main block { young();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } (e) virus bool f() { prob { 1//2: return flip(1//2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 1//2: if f() { return f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } else { return false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } } # main blcok { bool res1 = f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' bool resN = f();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } (f) sequentialN Certificates for Probabilistic Pushdown Automata via OVI 29 int f(int n, int m) { prob { (n+1)//(n+2) : { f((n + 1) % m, m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' f((n + 1) % m, m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' return 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } 1//(n+2) : return 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } # main block { f(0, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } (a) escapeN void f(int n) { while(n > 0) { prob { 2//3: f(n-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' 1//3: f((n+1) % N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } n = n-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } } # main block { f(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' } (b) modN 30 Tobias Winkler and Joost-Pieter Katoen D Z3 vs CVC5 Table 3: Comparison of the SMT-approach (see §Baselines in Section 5) using z3 and cvc5 on SCFG given as explicit PPS (right), and on programs automatically translated to pPDA (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content=' benchmark certz3 tz3 certcvc5 tcvc5 rw-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='499 ✓ 11 ✓ 92 rw-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='500 ✓ 10 ✓ 87 rw-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='501 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='✓ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFAT4oBgHgl3EQfsR4z/content/2301.08657v1.pdf'} 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+RMSim: Controlled Respiratory Motion Simulation +on Static Patient Scans +Donghoon Lee, Ellen Yorke, Masoud Zarepisheh, +Saad Nadeem*, Yu-Chi Hu* +Department of Medical Physics, Memorial Sloan Kettering Cancer Center, New York, +NY, USA +E-mail: {leed10,yorkee,zarepism,nadeems,huj}@mskcc.org +*Corresponding Authors +Objective: +This work aims to generate realistic anatomical deformations from +static +patient +scans. +Specifically, +we +present +a +method +to +generate +these +deformations/augmentations via deep learning driven respiratory motion simulation +that provides the ground truth for validating deformable image registration (DIR) +algorithms and driving more accurate deep learning based DIR. +Approach: We present a novel 3D Seq2Seq deep learning respiratory motion simulator +(RMSim) that learns from 4D-CT images and predicts future breathing phases given +a static CT image. The predicted respiratory patterns, represented by time-varying +displacement vector fields (DVFs) at different breathing phases, are modulated through +auxiliary inputs of 1D breathing traces so that a larger amplitude in the trace results in +more significant predicted deformation. Stacked 3D-ConvLSTMs are used to capture +the spatial-temporal respiration patterns. +Training loss includes a smoothness loss +in the DVF and mean-squared error between the predicted and ground truth phase +images. +A spatial transformer deforms the static CT with the predicted DVF to +generate the predicted phase image. 10-phase 4D-CTs of 140 internal patients were +used to train and test RMSim. The trained RMSim was then used to augment a public +DIR challenge dataset for training VoxelMorph to show the effectiveness of RMSim- +generated deformation augmentation. +Main results: +We validated our RMSim output with both private and public +benchmark datasets (healthy and cancer patients). +The structure similarity index +measure (SSIM) for predicted breathing phases and ground truth 4D CT images was +0.92±0.04, demonstrating RMSim’s potential to generate realistic respiratory motion. +Moreover, the landmark registration error in a public DIR dataset was improved from +8.12±5.78mm to 6.58mm±6.38mm using RMSim-augmented training data. +Significance: The proposed approach can be used for validating DIR algorithms as +well as for patient-specific augmentations to improve deep learning DIR algorithms. +The code, pretrained models, and augmented DIR validation datasets will be released +at https://github.com/nadeemlab/SeqX2Y. The supplementary video can be found +at https://youtu.be/xIx8B_Q_R9o. +arXiv:2301.11422v1 [cs.CV] 26 Jan 2023 + +RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +2 +1. Introduction +Respiratory motion hampers accurate diagnosis as well as image-guided therapeutics. +For example, during radiotherapy, it may lead to poor local tumor control and increased +radiation toxicity to the normal organs [1]. It can also exhibit itself as motion artifacts +in the acquired images, making it difficult to differentiate nodule/tumor morphology +changes from those induced by respiratory motion. +This also makes the image +registration task across different breathing phases as well as across different time points +challenging. To validate the image registration accuracy/performance for commissioning +solutions available in clinical commercial systems, the American Association of +Physicists in Medicine(AAPM) TG-132 [2] recommended independent quality checks +using digital phantoms. Current commercial solutions such as ImSimQA allow creation +of synthetic deformation vector fields (DVFs) by user-defined transformations with only +a limited degree of freedom. +These monotonic transformations can not capture the +realistic respiratory motion. +For modeling respiration motion, an intuitive representation of motion is time- +varying displacement vector fields (DVFs) obtained by deformable image registrations +(DIR) in 4D images, acquired in a breathing cycle. Surrogate-driven approaches [3] +employ DVF as a function of the surrogate breathing signal. However, an exact and +direct solution in the high-dimensional space of DVFs is computationally intractable. +Still, motion surrogates have been widely studied in the field of radiotherapy for +building models establishing the relationship between surrogates and respiratory motion +estimated from the image data [3]. +For example, the 1D diaphragm displacement +has been reported as a reliable surrogate for tumor motion model [4] as well as for +PCA (principle component analysis) respiratory motion model to correct CT motion +artifacts [5]. +Recently, +Romaguera et al. [6] used a 2D sequence-to-sequence (Seq2Seq) +network [7] to predict 2D in-plane motion for a single future time point. +Krebs et +al. [8] applied a similar encoder-decoder network in a conditional variational autoencoder +(cVAE) framework in which network parameters were learned to approximate the +distribution of deformations in low-dimensional latent space with the encoder and decode +the latent features for 2D motion prediction with the decoder. Romaguera et al. [9] +integrated Voxelmorph [10] for assisting the VAE encoder to map deformations in latent +space conditioned on anatomical features from 3D images. Temporal information of 2D +surrogate cine images from a 2D Seq2Seq network was used to predict 3D DVF at a +single future time point. +In this paper, we present a novel deep learning respiratory motion simulator +(RMSim) that learns to generate realistic patient-specific respiratory motion represented +by time-varying DVFs at different breathing phases from a static 3D CT image. For the +first time, we also allow modulation of this simulated motion via arbitrary 1D breathing +traces as auxiliary input to create large variations. This in turn creates diverse patient- +specific data augmentations while also generating ground truth for DIR validation. + +RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +3 +Our work has several differences and advantages over the aforementioned deep learning +approaches: (1) we used 3D Seq2Seq architecture for the first time which has never been +attempted before for predicting deformations due to GPU memory limitations, (2) we +did not use VoxelMorph in its entirety but only the Spatial Transform module to train +our model end-to-end, and (3) as opposed to predicting just a single future time point, +we can predict 9 future time point breathing phases simultaneously (learnt from 4D-CT +images with 10 3D CT breathing phases) along with their 3D DVFs. We have thoroughly +validated our RMSim output with both private and public benchmark datasets (healthy +and cancer patients) and demonstrated that adding our patient-specific augmentations +to training data can improve performance/accuracy of state-of-the-art deep learning DIR +algorithms. We also showcase breathing trace-modulated respiratory motion simulations +for public static radiology scans (in the accompanying supplementary video). The +code, pretrained models, and augmented DIR validation datasets will be released at +https://github.com/nadeemlab/SeqX2Y. +Figure 1. The schematic image for the proposed deep learning model. The Seq2Seq +encoder-decoder framework was used as the backbone of the proposed model. The +model was built with 3D convolution layers for feature encoding and output decoding +and 3D convolutional Long Short-Term Memory (3D ConvLSTM) layers for spatial- +temporal correlation between time points. The last layer of the decoder was a spatial +transform layer to warp the initial phase image with the predicted Deformation Vector +Field (DVF). To modulate the respiratory motions the 1D breathing trace was given +as input along with the initial phase image. The dimension of image volume was 128 +× 128 × 128 and the input feature to 3D ConvLSTM is 64 × 64 × 64 × 96 (Depth × +Width × Height × Channel) + +1D Respiratory Signal +D +128×128×128x3 +Phase 1 +64×64×64×96 +3D Convolution +ConvLSTM3D +Multiplication +T +Spatial transform +Phase 1 +Phase 2 +Phase k +DVF +128×128×128RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +4 +2. Materials and Methods +2.1. Datasets +We used an internal lung 4D-CT dataset retrospectively collected and de-identified +from 140 non-small cell lung cancer (NSCLC) patients receiving radiotherapy in our +institution. The helical and cine mode 4D-CTs were acquired using Philips Brilliance +Big Bore or GE Advantage respectively and binned into 10 phases using the vendor’s +proprietary software with breathing signals from bellows or external fiducial markers. +The x-ray energy for the CT image was 120 kVp and tube current varies case by case +according to vendor-specific tube current modulations based on patient size. The mAs +range is [100, 400] for GE and [500, 800] for Philips. The image slice dimension was +512x512, while the number of image slices varied patient by patient. We used the 100:40 +split for training:testing. +We used 20 cases of the Lung Nodule Analysis (LUNA) challenge dataset [11] +containing 3D radiology CTs for lung tumor screening to show that our RMSim +model trained with the internal dataset can be effectively applied to an external +radiology/diagnostic dataset to generate realistic respiration motions (see accompanying +supplementary video). For quantitative evaluation of the model generality on an +external data set, we used POPI [12] dataset which contains 6 10-phase 4D-CTs +with segmented lung masks as well as annotated landmarks on the vessel and airway +bifurcations. +To validate the effectiveness of data augmentation using synthetic respiratory +motion images generated from our RMSim model in the deformable registration task, +we used the Learn2Reg 2020 challenge dataset [13]. The Learn2Reg dataset consists of +30 subjects (20 for the training / 10 for the testing) with 3D CT thorax images taken +in inhale and exhale phases. For each Learn2Reg 20 inhale/exhale pairs, we generated +other phases of images using our RMSim model which was trained with the internal +dataset, therefore increasing the sample size to 200 in total to augment the training of +a well-known unsupervised deep learning DIR method, VoxelMorph [10]. Unfortunately +the inhale-exhale landmarks are not publicly available in Learn2Reg dataset to assess +the registration accuracy. For the landmarks evaluation in registration task, we used the +POPI dataset. Brief description/purpose of all the datasets used in this study is given in +Table 1. All datasets used in this study were cropped to eliminate the background and +resampled to 128×128×128 with 2mm voxel size due to the GPU memory constrains. +2.2. Realistic Respiratory Motion Simulation +Sequence-to-Sequence (Seq2Seg) is a many-to-many network architecture originally +developed for natural language processing tasks such as language translation. +Inspired by Seq2Seq, the proposed RMSim, illustrated in Figure 1, is a novel deep +learning encoder-decoder architecture that comprises three main parts including 3D +convolution, ConvLSTM3D (3D Convolutional Long-Short Term Memory), and spatial + +RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +5 +Table 1. Datasets used in this study. +Dataset +Size +Description +Purpose +Evaluation +Internal 4D-CTs +140 (100 train- +ing, 40 testing) +10-phase +radiother- +apy 4D-CTs +Training and testing RM- +Sim +Image similar- +ity +LUNA +20 +Radiology CTs for +lung nodule detec- +tion +Testing model generality +Visualization +and qualitative +POPI 4D-CTs +6 +10-phase +4D-CTs +with landmarks +Testing +model +general- +ity (evaluating DVF accu- +racy) +Target +Regis- +tration +Error +(TRE) +of +landmarks +Learn2Reg +30 (20 training, +10 testing) +Inspiration- +expiration +thorax +CT pairs with lung +segmentations +Training +and +testing +RMSim-augmented deep +learning +deformable +image registration (Vox- +elmorph) +Lung +segmen- +tation +(Dice +score) and im- +age similarity +Figure 2. Respiration motion surrogate extraction using a diaphragm point that has +the maximum superior-inferior displacement across the phases. LDDMM was used to +register the phase 1 (fixed) image to other phases (moving) to get the DVFs. The +diaphragm point’s trajectory in z-axis (shown in red) across the phases was considered +as the breathing trace. Yellow line shows the diaphragm position at phase 1. +transformation layer (adapted from VoxelMorph [10]). +The 3D convolution in the +encoder is used to reduce the matrix dimension and extract salient features from images. +We used 3×3×3 kernel size and 2×2×2 stride size to reduce the matrix dimension +to 1/8. The number of channels for 3D convolution layer is 96. LSTM has a more +complex cell structure than a neuron in classical recurrent neural network (RNN). +Apart from the cell state, it contains gate units to decide when to keep or override +information in and out of memory cells to better handle the gradient vanishing problem +in recurrent neural network. This helps in learning long term dependencies. ConvLSTM +[14] replaces Hadamard product with convolution operators in the input as well as the +state transitions to capture the spatial pattern of the feature representations aggregated +from different time points. We implemented ConvLSTM in 3D for handling the 3D +phase images from the 4D-CT. We used two stacked ConvLSTM3D layers to make the +network deeper, adding levels of abstraction to input observations similar to the typical +deep neural network. The hidden state output from ConvLSTM3D was fed to both the + +Phase 1 +Phase 2 +Phase 3 +Phase 4 +Phase 5 +Phase 6 +Phase7 +Phase 8 +Phase9 +Phase 10 +Fixed +Moving1 +Moving3 +Moving4 +Moving5 +Moving6 +Moving7 +Moving8 +Moving2 +Moving9 +DVF1 +DVF2 +DVF3 +DVF4 +DVF5 +DVF6 +DVF7 +DVF8 +DVF9RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +6 +next layer in the same stack and the next timepoint ConvLSTM3D layer. The output +of ConvLSTM3D in the decoder at each predicted time point was up-sampled to the +original input resolution and output channels were reduced via 3D convolution, resulting +in the 3D DVF for the final output. The initial phase CT image was then deformed to +a predicted phase image at different breathing phase using spatial transformation layer +and the predicted 3D DVFs. +Moreover, to modulate the predicted motion with a patient-specific pattern, we +used an auxiliary input of 1D breathing trace. +In this paper, we considered the +amplitude of diaphragm apex motion as the surrogate of the respiratory signal [4]. +The 1D breathing trace for each training case was extracted using DVF obtained +from large deformation diffeomorphic metric mapping (LDDMM) DIR provided by +ANTs (Advanced Normalization Tools). Specifically, using the DVF, the apex point +in diaphragm was propagated from the phase at the end of inhalation to other phases +to generate the 1D displacement trace. The apex of the diaphragm was determined by +finding the lung surface voxel with the maximum superior-inferior (z-axis) displacement +among the DVFs. The z-axis displacement of the apex voxel at each phase resembles +the 1D breathing trace. Figure 2 describes the process of preparing the 1D respiratory +signal. Feature-wise transformations, e.g. addition or multiplication, are simple and +effective mechanisms to incorporate conditioning information from another data source +to the features learned in the network. In this paper, the hidden state of ConvLSTM at +each phase is modulated by a simple element-wise multiplication of the phase-amplitude +of the trace: +m(Ht, bt) = btHt, +(1) +where Ht is the hidden state encoded from the sequence of phase images up to phase t +and bt is the amplitude of the breathing trace at phase t, +The loss function for training includes the mean-squared error of ground truth +phase image and predicted phase image, and the regularization on the gradient of DVF +by promoting smoothness of DVF: +Loss = +� +t>0 +[(Yt − T(X0, φt))2 + ||∇φt||2], +(2) +where X0 is the initial phase image (phase 1 in this paper), T is the spatial transform +(adapted from VoxelMorph), φt is the predicted DVF for phase t and Yt is the ground +truth phase image at phase t. +We developed RMSim using the PyTorch library (version 1.2.0). We used Adam +for optimization and set learning rate to be 0.001 (as done in the original Seq2Seq +paper [14]). Due to the large data size of 4D image sequence (10 3D CT phase images +constituting a single 4D-CT), the batch size was limited to 1 and the number of feature +channels was 96, considering GPU memory and training time. The model was trained +and tested on an internal high performance computing cluster with 4 NVIDIA A40 +GPUs with 48GB memory each. Our model consumed 35.2 GB GPU memory and the + +RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +7 +training time was approximately 72 hours. The inference time for 9 phases and 40 total +test cases from the internal dataset was less than 3 minutes. +2.3. Data augmentation by RMSim +Since RMSim can generate a series of realistic respiratory motion-induced images from +a single 3D CT, one of its use cases is data augmentation for training DIR algorithms. +For each of the 20 training cases in the Learn2Reg Grand Challenge dataset [13], we +randomly selected a 1D breathing trace from our internal dataset to modulate the motion +on the Learn2Reg inhalation image to generate 9 additional phase images, increasing +the training size 10-fold. We chose a popular deep learning DIR method, VoxelMorph, +suitable for unsupervised training for the propose of validating effectiveness of data +augmentation. We first trained a VoxelMorph model with the original 20 inhalation- +to-exhalation image pairs in the Learn2Reg training set. +We then trained another +VoxelMorph model with the augmented data including 200 pairs of inhalation-to-phase +images. We compared the registrations from the two VoxelMorph models for validating +the effectiveness of data augmentation. +2.4. Evaluation Metrics +For image similarity, we used structure similarity index measure (SSIM) [15] which +measures the similarity of two given images based on the degradation of structural +information, including luminance, contrast and structure. The closer the SSIM value +is to 1, the more similarity between the two images. SSIM was used for comparing +RMSim-predicted phase images and ground truth phase images in the internal test cases. +SSIM was also used for comparing deformable registration results from VoxelMorph to +validate data augmentation effectiveness in Learn2Reg test cases, which additionally +were evaluated with the provided lung segmentation using Dice score to compare the +ground truth lung contours and propagated lung contours. +For landmark comparison in the POPI dataset, we used Target Registration +Error (TRE), defined as the Euclidean distance between a landmark position spatially +transformed and the target position. +3. Results +For each test case in the internal 4D-CT dataset, we generated 9 simulated phase images +from the ground truth phase 1 image by deforming the phase 1 image using the predicted +DVF at each phase. We calculated SSIM to measure the image similarity (SSIMsim) +between the simulated phase image and the ground truth phase image. For comparison, +we also calculated the SSIM (SSIMgnd) between the ground truth phase 1 image and the +rest of the ground truth phase images. The average SSIMsim was 0.92±0.04, compared +to 0.86±0.08 of SSIMgnd (p < 0.01.) + +RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +8 +We also measured the diaphragm displacement between the reference respiratory +signal and the predicted signal (see Figure 3). +As can be seen, the error increased +from inhale to exhale phases. This is because prediction accuracy decreases at later +time points. However, the overall displacement error was within 3 mm. Adding more +realistic respiratory data for training can further reduce this displacement error. +Figure 3. The error between reference respiratory signal (diaphragm displacement in +mm) and predicted signal. +To demonstrate the modulation flexibility of the 1D breathing traces, we applied +different breathing traces to the same 3D CT image to generate different motion +simulations, as shown in Figure 4. The plot on the top illustrates the two 1D breathing +traces used for modulation. +The breathing trace 1 (BT1), denoted by orange color +line, represents the original respiratory signal for the case. BT2 denoted by gray line +is a trace from another patient that was used to generate the simulated images. The +white horizontal line indicates the position of the apex of the diaphragm in the initial +phase (the first column). It is used as a reference to show the relative positions of the +diaphragm at different phases. The diaphragm in images on the upper row clearly shows +the more significant movement as BT2 has higher amplitudes in the trace. +The amplitude range in our internal dataset was 0.14 – 40 mm. To validate the +prediction performance on out-of-range displacement, we predicted additional sequences +using a 5 times larger respiratory amplitude. The prediction results using a 5 times +larger respiratory signal achieve a higher diaphragm level which means the predicted +respiratory has larger fluctuation than the original respiratory signal but it was not +proportional to the respiratory signal that was used for inference (see Figure 5). +The results of propagating anatomical structures using the predicted DVFs are also +shown in Figure 4. We propagated the lung, heart, esophagus, and tumor from the initial +phase image. The propagated contours are well-matched with the predicted image and +the motion of structures looks very realistic. We also provided the supplementary +video of the simulated 4D-CT along with the ground truth 4D-CT and the 3D + +9 +8 +7 +6 +Error +5 +4 +3 +X +X +X +2 +X +X +1 +0RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +9 +Figure 4. +Two different breathing traces, BT1 and BT2 shown in the plot, were +used to simulate the respiration motion of an internal case, resulting in 2 series of +modulated phase images according to the breathing traces. The diaphragm has larger +displacement in images simulated with BT2 (upper row) than the displacement in +images simulated with shallower BT1 (bottom row.) The white horizontal line indicates +the position of the apex of the left diaphragm at the initial phase (left-most column.) +We also overlay the propagated lung(in yellow), heart(in red), esophagus(in blue) and +tumor(in green) contours using predicted DVFs. +volume-rendered visualizations. Specifically, the 3D volume-rendered visualizations on +LUNA challenge datasets as well as internal lung radiotherapy datasets with structure +propagation are included in the accompanying supplementary video with chained +predictions for 60-phase predictions for LUNA challenge (radiology lung nodule) and +30-phase predictions for the lung radiotherapy datasets. +In POPI dataset, there is only one case which contains lung segmentations on all +the phases. For this case, we extracted 1D breathing trace from the lung segmentations +as we did for our internal dataset. RMSim trained with our internal dataset predicted +the remaining phases from the inhale phase with the modulation from the 1D breathing +trace. The average TRE (Target Registration Error) of landmarks propagated with our +predicted DVFs in this case was 0.92±0.64mm, showing that RMSim can accurately +predict the patient-specific motion from the patient’s 1D breathing trace. +Figure 6 +shows the TRE results for all predicted phases in this case. For the three other 4D-CT +cases in POPI there were no lung segmentation masks so we performed semi-automatic +lung segmentation for extracting the 1D breathing traces and the results are shown in +Figure 7. +Additionally, we used the RMSim for augmenting the Learn2Reg Challenge dataset. +The Dice score of lung segmentation of 10 Learn2Reg testing cases using the VoxelMorph +without augmentation was 0.96 ± 0.01 while the model trained with RMSim data + +15 +10 +mm +5 +0 +2 +3 +5 +6 +7 +8 +9 +BT1---BT2RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +10 +Figure 5. The predicted phase 5 images using different 1D respiratory signal. Blue +line is original respiratory signal, orange line is 3 times amplitude and green line is 5 +times amplitude. +Figure 6. TRE results of all 9 phases from the 4DCT case in POPI. RMSim trained +with the internal dataset were able to achieve sub-mm accuracy in this external case. +augmentation was 0.97 ± 0.01 (p < 0.001 using the paired t-test). The SSIM between +the warped images and the ground truth images was 0.88 ± 0.02 for the model without +augmentation and 0.89 ± 0.02 (p < 0.001) for the model with augmentation. +To validate the improvement of DIR using VoxelMorph with augmentation, we +propagated the landmark points from the inhale phase to the exhale phase for the 6 + +Phase +Respiratory signal +140 +Amplitude(mm) +120 +100 +016.00 +Without Prediction +14.00 +With Prediction +12.00 +TRE (mm) +10.00 +8.00 +6.00 +X +4.00 +X +2.00 +0.00 +P.2 +P.3 +P.4 +P.5 +P.6 +P.7 +P.8 +P.9 +P.10RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +11 +Figure 7. Three other 4D-CT POPI cases including 10 phases with landmarks on each +phase (TRE plots for the three cases given below). For each case, we show original and +predicted phase images overlaid with the difference with respect to original phase 1 +input. In original DIR Validation 03 phase difference image, the diaphragm in the left +lung (viewer’s right) did not move due to the large tumor but it does in our prediction +(shown in red bounding boxes). This case does not deflect from the goals of this paper, +i.e. data augmentation and DIR validation. The difference in Case #1 appears minor +because the breathing is shallower (less diaphragm movement) and Case #2 and Case +#3 have larger differences due to deeper breathing. +cases available in POPI dataset and computed the TRE. On average, pre-DIR TRE +was 8.05±5.61mm, VoxelMorph w/o augmentation was 8.12±5.78mm compared to +6.58±6.38mm for VoxelMorph with augmentation (p < 3e-48). The TRE comparison +of all 6 cases are shown in Figure 8. + +Phase2 +Phase3 +Phase4 +Phase5 +Phase6 +Phase7 +Phase8 +Phase9 +Phase10 + Original +DIR_Validation_01 +Prediction + Original +Validation_02 +Prediction +DIR +Original +DIR_Validation_03 +Prediction +WithoutPrediction +WithPrediction +12 +35 +20 +30 +10 +5 +(mm) +15 +20 +TRE +15 +10 +10 +5 +0 +P2 P3 P4 P5P6 P7 P8 P9 P10 +P2 P3 P4 P5 P6 P7 P8 P9 P10 +P2 P3 P4 P5P6 P7 P8 P9 P10 +DIR Validation01 +DIR Validation 02 +DIRValidation 03RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +12 +Figure 8. TRE results of POPI dataset. VoxelMorph with RMSim augmentation +outperformed the VoxelMorph w/o augmentation in all 6 cases. +4. Discussion +In this work, we presented a 3D Seq2Seq network, referred to as RMSim, to predict +patient-specific realistic motion induced/modulated with 1D breathing trace. +We +successfully validated our RMSim output with both private and public benchmark +datasets (healthy and cancer patients) and demonstrated that adding our patient- +specific augmentations to training data can improve performance/accuracy of state- +of-the-art deep learning DIR algorithms. We also showcased breathing trace-modulated +respiratory motion simulations for public static radiology scans. +In this work, we +predicted the motion in one breathing cycle. +In the future, we will fine-tune our +current model to predict multiple cycles in one-shot. Possible solutions include making +our model bi-directional and using cross-attention to improve temporal dynamics in a +long sequence. Further research is needed to investigate the impact of training data +augmentation on different image modalities such as 4D-MRI. +Another application of our work is in external radiotherapy treatment planning. +RMSim simulated 4D-CT can be used to delineate the internal target volume (ITV) +which is the union of the target volumes in all respiratory phases. The entire ITV is +irradiated in radiation therapy to ensure all regions of tumor receive enough radiation. +There is a more sophisticated alternative to ITV, referred to as robust treatment +planning, where the key idea is to model the motion and directly incorporate it into the +planning [16]. This typically can be done by assuming a probability density function +(PDF) for the position of the target and doing plan optimization based on that [17, 18]. +It is also possible to assume a set of possible motion PDFs to account for uncertainty in +breathing and plan accordingly [19, 20]. The simulated 4D-CT can be used to extract + +40 +Vanilla VoxelMorph +35 +Pre_DIR +30 +VoxelMorph + Augmentation +TRE (mm) +25 +20 +15 +10 +5 +0 +#1 +#2 +#3 +#4 +#5 +#6RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +13 +the motion PDF or a set of motion PDFs from varied breathing patterns exhibited by +the patient. +Additional interesting future direction is the extension of our earlier work in +exhaustively simulating physics-based artifacts in CT and CBCT images for more robust +cross-modal deep learning translation, segmentation, and motion-correction algorithms +[21, 22, 23], available via our Physics-ArX library (https://github.com/nadeemlab/ +Physics-ArX). +Specifically, in our previous work we presented a proof-of-concept +pipeline for physics-based motion artifact simulation in CT/CBCT images using 4D- +CT phases [22]. Using the method proposed in the current paper, we can generate and +modulate large/diverse 4D-CT phases from any static 3D CT scan using the 1D RPM +signal. These generated 4D-CT variations can then be used to produce large realistic +motion-artifact variations via our earlier pipeline[22]. +Limitations: For simplicity, we used the maximal displacement on the diaphragm +as the surrogate of clinical breathing trace to drive the modulation. +We assume +(1) the breathing pattern is regular since we extracted the diaphragm displacements +from amplitude-binned 4D-CT, and (2) regional DVFs are linearly scaled according to +diaphragm motion. Note 1D breathing trace might not represent the actual cardiac +motion. +Because of the GPU memory constraints, our input and output dimension +was limited to 128x128x128. +Nevertheless, the precise estimation of motion is not +required for providing realistic motion-induced ground truth DVFs for the validation +of the DIR algorithms and data augmentation for training DIR algorithms, as shown in +this work. To extend our work to tumor tracking during radiation treatment, we will +use the signals from the actual external real-time motion management (RPM) device +to drive the modulation more precisely. We will also explore incorporating 2D MV/kV +projections acquired during the treatment to infer more realistic cardiac/tumor motion. +Acknowledgements +This work was supported partially by NCI/NIH P30 CA008748. +Conflict of interest +We have no conflict of interest to declare. +Code Availability Statement +The code, pretrained models, and augmented DIR validation datasets will be released +at https://github.com/nadeemlab/SeqX2Y. +Data Availability Statement +The public datasets used in this study and their urls are as follows: (1) Learn2Reg +Challenge Lung CT dataset (Empire10 Challenge Dataset): https://drive.google. + +RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans +14 +com/drive/folders/1yHWLQEK9c1xzggkCC4VX0X4To7BBDqu5, +(2) +LUNA +challenge +dataset (subset0.zip): +https://zenodo.org/record/3723295, (3) DIR Validation +POPI Dataset (6 4D CT patients with landmarks): https://www.creatis.insa-lyon. +fr/rio/dir_validation_data, and (4) POPI model dataset (one 4D CT patient +dataset with landmarks on all phases as well as lung segmentation mask): https: +//www.creatis.insa-lyon.fr/rio/popi-model_original_page. +References +[1] Jason K. 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Medical Physics, 48(9):5130–5141, 2021. + diff --git a/C9FJT4oBgHgl3EQfAyyD/content/tmp_files/load_file.txt b/C9FJT4oBgHgl3EQfAyyD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c3ca3cc12abfdc4dc93439d59b133a1f3c722e0c --- /dev/null +++ b/C9FJT4oBgHgl3EQfAyyD/content/tmp_files/load_file.txt @@ -0,0 +1,459 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf,len=458 +page_content='RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans Donghoon Lee, Ellen Yorke, Masoud Zarepisheh, Saad Nadeem*, Yu-Chi Hu* Department of Medical Physics, Memorial Sloan Kettering Cancer Center, New York, NY, USA E-mail: {leed10,yorkee,zarepism,nadeems,huj}@mskcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='org Corresponding Authors Objective: This work aims to generate realistic anatomical deformations from static patient scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Specifically, we present a method to generate these deformations/augmentations via deep learning driven respiratory motion simulation that provides the ground truth for validating deformable image registration (DIR) algorithms and driving more accurate deep learning based DIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Approach: We present a novel 3D Seq2Seq deep learning respiratory motion simulator (RMSim) that learns from 4D-CT images and predicts future breathing phases given a static CT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The predicted respiratory patterns, represented by time-varying displacement vector fields (DVFs) at different breathing phases, are modulated through auxiliary inputs of 1D breathing traces so that a larger amplitude in the trace results in more significant predicted deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Stacked 3D-ConvLSTMs are used to capture the spatial-temporal respiration patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Training loss includes a smoothness loss in the DVF and mean-squared error between the predicted and ground truth phase images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' A spatial transformer deforms the static CT with the predicted DVF to generate the predicted phase image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' 10-phase 4D-CTs of 140 internal patients were used to train and test RMSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The trained RMSim was then used to augment a public DIR challenge dataset for training VoxelMorph to show the effectiveness of RMSim- generated deformation augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Main results: We validated our RMSim output with both private and public benchmark datasets (healthy and cancer patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The structure similarity index measure (SSIM) for predicted breathing phases and ground truth 4D CT images was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='04, demonstrating RMSim’s potential to generate realistic respiratory motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Moreover, the landmark registration error in a public DIR dataset was improved from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='12±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='78mm to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='58mm±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='38mm using RMSim-augmented training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Significance: The proposed approach can be used for validating DIR algorithms as well as for patient-specific augmentations to improve deep learning DIR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The code, pretrained models, and augmented DIR validation datasets will be released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='com/nadeemlab/SeqX2Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The supplementary video can be found at https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='be/xIx8B_Q_R9o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='11422v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='CV] 26 Jan 2023 RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Introduction Respiratory motion hampers accurate diagnosis as well as image-guided therapeutics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For example, during radiotherapy, it may lead to poor local tumor control and increased radiation toxicity to the normal organs [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' It can also exhibit itself as motion artifacts in the acquired images, making it difficult to differentiate nodule/tumor morphology changes from those induced by respiratory motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' This also makes the image registration task across different breathing phases as well as across different time points challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' To validate the image registration accuracy/performance for commissioning solutions available in clinical commercial systems, the American Association of Physicists in Medicine(AAPM) TG-132 [2] recommended independent quality checks using digital phantoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Current commercial solutions such as ImSimQA allow creation of synthetic deformation vector fields (DVFs) by user-defined transformations with only a limited degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' These monotonic transformations can not capture the realistic respiratory motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For modeling respiration motion, an intuitive representation of motion is time- varying displacement vector fields (DVFs) obtained by deformable image registrations (DIR) in 4D images, acquired in a breathing cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Surrogate-driven approaches [3] employ DVF as a function of the surrogate breathing signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' However, an exact and direct solution in the high-dimensional space of DVFs is computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Still, motion surrogates have been widely studied in the field of radiotherapy for building models establishing the relationship between surrogates and respiratory motion estimated from the image data [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For example, the 1D diaphragm displacement has been reported as a reliable surrogate for tumor motion model [4] as well as for PCA (principle component analysis) respiratory motion model to correct CT motion artifacts [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Recently, Romaguera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' [6] used a 2D sequence-to-sequence (Seq2Seq) network [7] to predict 2D in-plane motion for a single future time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Krebs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' [8] applied a similar encoder-decoder network in a conditional variational autoencoder (cVAE) framework in which network parameters were learned to approximate the distribution of deformations in low-dimensional latent space with the encoder and decode the latent features for 2D motion prediction with the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Romaguera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' [9] integrated Voxelmorph [10] for assisting the VAE encoder to map deformations in latent space conditioned on anatomical features from 3D images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Temporal information of 2D surrogate cine images from a 2D Seq2Seq network was used to predict 3D DVF at a single future time point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' In this paper, we present a novel deep learning respiratory motion simulator (RMSim) that learns to generate realistic patient-specific respiratory motion represented by time-varying DVFs at different breathing phases from a static 3D CT image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For the first time, we also allow modulation of this simulated motion via arbitrary 1D breathing traces as auxiliary input to create large variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' This in turn creates diverse patient- specific data augmentations while also generating ground truth for DIR validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 3 Our work has several differences and advantages over the aforementioned deep learning approaches: (1) we used 3D Seq2Seq architecture for the first time which has never been attempted before for predicting deformations due to GPU memory limitations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' (2) we did not use VoxelMorph in its entirety but only the Spatial Transform module to train our model end-to-end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' and (3) as opposed to predicting just a single future time point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' we can predict 9 future time point breathing phases simultaneously (learnt from 4D-CT images with 10 3D CT breathing phases) along with their 3D DVFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We have thoroughly validated our RMSim output with both private and public benchmark datasets (healthy and cancer patients) and demonstrated that adding our patient-specific augmentations to training data can improve performance/accuracy of state-of-the-art deep learning DIR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We also showcase breathing trace-modulated respiratory motion simulations for public static radiology scans (in the accompanying supplementary video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The code, pretrained models, and augmented DIR validation datasets will be released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='com/nadeemlab/SeqX2Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The schematic image for the proposed deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The Seq2Seq encoder-decoder framework was used as the backbone of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The model was built with 3D convolution layers for feature encoding and output decoding and 3D convolutional Long Short-Term Memory (3D ConvLSTM) layers for spatial- temporal correlation between time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The last layer of the decoder was a spatial transform layer to warp the initial phase image with the predicted Deformation Vector Field (DVF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' To modulate the respiratory motions the 1D breathing trace was given as input along with the initial phase image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The dimension of image volume was 128 × 128 × 128 and the input feature to 3D ConvLSTM is 64 × 64 × 64 × 96 (Depth × Width × Height × Channel) 1D Respiratory Signal D 128×128×128x3 Phase 1 64×64×64×96 3D Convolution ConvLSTM3D Multiplication T Spatial transform Phase 1 Phase 2 Phase k DVF 128×128×128RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Materials and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Datasets We used an internal lung 4D-CT dataset retrospectively collected and de-identified from 140 non-small cell lung cancer (NSCLC) patients receiving radiotherapy in our institution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The helical and cine mode 4D-CTs were acquired using Philips Brilliance Big Bore or GE Advantage respectively and binned into 10 phases using the vendor’s proprietary software with breathing signals from bellows or external fiducial markers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The x-ray energy for the CT image was 120 kVp and tube current varies case by case according to vendor-specific tube current modulations based on patient size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The mAs range is [100, 400] for GE and [500, 800] for Philips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The image slice dimension was 512x512, while the number of image slices varied patient by patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We used the 100:40 split for training:testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We used 20 cases of the Lung Nodule Analysis (LUNA) challenge dataset [11] containing 3D radiology CTs for lung tumor screening to show that our RMSim model trained with the internal dataset can be effectively applied to an external radiology/diagnostic dataset to generate realistic respiration motions (see accompanying supplementary video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For quantitative evaluation of the model generality on an external data set, we used POPI [12] dataset which contains 6 10-phase 4D-CTs with segmented lung masks as well as annotated landmarks on the vessel and airway bifurcations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' To validate the effectiveness of data augmentation using synthetic respiratory motion images generated from our RMSim model in the deformable registration task, we used the Learn2Reg 2020 challenge dataset [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The Learn2Reg dataset consists of 30 subjects (20 for the training / 10 for the testing) with 3D CT thorax images taken in inhale and exhale phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For each Learn2Reg 20 inhale/exhale pairs, we generated other phases of images using our RMSim model which was trained with the internal dataset, therefore increasing the sample size to 200 in total to augment the training of a well-known unsupervised deep learning DIR method, VoxelMorph [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Unfortunately the inhale-exhale landmarks are not publicly available in Learn2Reg dataset to assess the registration accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For the landmarks evaluation in registration task, we used the POPI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Brief description/purpose of all the datasets used in this study is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' All datasets used in this study were cropped to eliminate the background and resampled to 128×128×128 with 2mm voxel size due to the GPU memory constrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Realistic Respiratory Motion Simulation Sequence-to-Sequence (Seq2Seg) is a many-to-many network architecture originally developed for natural language processing tasks such as language translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Inspired by Seq2Seq, the proposed RMSim, illustrated in Figure 1, is a novel deep learning encoder-decoder architecture that comprises three main parts including 3D convolution, ConvLSTM3D (3D Convolutional Long-Short Term Memory), and spatial RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 5 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Datasets used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Dataset Size Description Purpose Evaluation Internal 4D-CTs 140 (100 train- ing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' 40 testing) 10-phase radiother- apy 4D-CTs Training and testing RM- Sim Image similar- ity LUNA 20 Radiology CTs for lung nodule detec- tion Testing model generality Visualization and qualitative POPI 4D-CTs 6 10-phase 4D-CTs with landmarks Testing model general- ity (evaluating DVF accu- racy) Target Regis- tration Error (TRE) of landmarks Learn2Reg 30 (20 training,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' 10 testing) Inspiration- expiration thorax CT pairs with lung segmentations Training and testing RMSim-augmented deep learning deformable image registration (Vox- elmorph) Lung segmen- tation (Dice score) and im- age similarity Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Respiration motion surrogate extraction using a diaphragm point that has the maximum superior-inferior displacement across the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' LDDMM was used to register the phase 1 (fixed) image to other phases (moving) to get the DVFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The diaphragm point’s trajectory in z-axis (shown in red) across the phases was considered as the breathing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Yellow line shows the diaphragm position at phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' transformation layer (adapted from VoxelMorph [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The 3D convolution in the encoder is used to reduce the matrix dimension and extract salient features from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We used 3×3×3 kernel size and 2×2×2 stride size to reduce the matrix dimension to 1/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The number of channels for 3D convolution layer is 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' LSTM has a more complex cell structure than a neuron in classical recurrent neural network (RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Apart from the cell state, it contains gate units to decide when to keep or override information in and out of memory cells to better handle the gradient vanishing problem in recurrent neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' This helps in learning long term dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' ConvLSTM [14] replaces Hadamard product with convolution operators in the input as well as the state transitions to capture the spatial pattern of the feature representations aggregated from different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We implemented ConvLSTM in 3D for handling the 3D phase images from the 4D-CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We used two stacked ConvLSTM3D layers to make the network deeper, adding levels of abstraction to input observations similar to the typical deep neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The hidden state output from ConvLSTM3D was fed to both the Phase 1 Phase 2 Phase 3 Phase 4 Phase 5 Phase 6 Phase7 Phase 8 Phase9 Phase 10 Fixed Moving1 Moving3 Moving4 Moving5 Moving6 Moving7 Moving8 Moving2 Moving9 DVF1 DVF2 DVF3 DVF4 DVF5 DVF6 DVF7 DVF8 DVF9RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 6 next layer in the same stack and the next timepoint ConvLSTM3D layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The output of ConvLSTM3D in the decoder at each predicted time point was up-sampled to the original input resolution and output channels were reduced via 3D convolution, resulting in the 3D DVF for the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The initial phase CT image was then deformed to a predicted phase image at different breathing phase using spatial transformation layer and the predicted 3D DVFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Moreover, to modulate the predicted motion with a patient-specific pattern, we used an auxiliary input of 1D breathing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' In this paper, we considered the amplitude of diaphragm apex motion as the surrogate of the respiratory signal [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The 1D breathing trace for each training case was extracted using DVF obtained from large deformation diffeomorphic metric mapping (LDDMM) DIR provided by ANTs (Advanced Normalization Tools).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Specifically, using the DVF, the apex point in diaphragm was propagated from the phase at the end of inhalation to other phases to generate the 1D displacement trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The apex of the diaphragm was determined by finding the lung surface voxel with the maximum superior-inferior (z-axis) displacement among the DVFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The z-axis displacement of the apex voxel at each phase resembles the 1D breathing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Figure 2 describes the process of preparing the 1D respiratory signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Feature-wise transformations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' addition or multiplication, are simple and effective mechanisms to incorporate conditioning information from another data source to the features learned in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' the hidden state of ConvLSTM at each phase is modulated by a simple element-wise multiplication of the phase-amplitude of the trace: m(Ht,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' bt) = btHt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' (1) where Ht is the hidden state encoded from the sequence of phase images up to phase t and bt is the amplitude of the breathing trace at phase t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The loss function for training includes the mean-squared error of ground truth phase image and predicted phase image,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' and the regularization on the gradient of DVF by promoting smoothness of DVF: Loss = � t>0 [(Yt − T(X0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' φt))2 + ||∇φt||2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' (2) where X0 is the initial phase image (phase 1 in this paper),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' T is the spatial transform (adapted from VoxelMorph),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' φt is the predicted DVF for phase t and Yt is the ground truth phase image at phase t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We developed RMSim using the PyTorch library (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We used Adam for optimization and set learning rate to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='001 (as done in the original Seq2Seq paper [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Due to the large data size of 4D image sequence (10 3D CT phase images constituting a single 4D-CT), the batch size was limited to 1 and the number of feature channels was 96, considering GPU memory and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The model was trained and tested on an internal high performance computing cluster with 4 NVIDIA A40 GPUs with 48GB memory each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Our model consumed 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='2 GB GPU memory and the RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 7 training time was approximately 72 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The inference time for 9 phases and 40 total test cases from the internal dataset was less than 3 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Data augmentation by RMSim Since RMSim can generate a series of realistic respiratory motion-induced images from a single 3D CT, one of its use cases is data augmentation for training DIR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For each of the 20 training cases in the Learn2Reg Grand Challenge dataset [13], we randomly selected a 1D breathing trace from our internal dataset to modulate the motion on the Learn2Reg inhalation image to generate 9 additional phase images, increasing the training size 10-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We chose a popular deep learning DIR method, VoxelMorph, suitable for unsupervised training for the propose of validating effectiveness of data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We first trained a VoxelMorph model with the original 20 inhalation- to-exhalation image pairs in the Learn2Reg training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We then trained another VoxelMorph model with the augmented data including 200 pairs of inhalation-to-phase images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We compared the registrations from the two VoxelMorph models for validating the effectiveness of data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Evaluation Metrics For image similarity, we used structure similarity index measure (SSIM) [15] which measures the similarity of two given images based on the degradation of structural information, including luminance, contrast and structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The closer the SSIM value is to 1, the more similarity between the two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' SSIM was used for comparing RMSim-predicted phase images and ground truth phase images in the internal test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' SSIM was also used for comparing deformable registration results from VoxelMorph to validate data augmentation effectiveness in Learn2Reg test cases, which additionally were evaluated with the provided lung segmentation using Dice score to compare the ground truth lung contours and propagated lung contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For landmark comparison in the POPI dataset, we used Target Registration Error (TRE), defined as the Euclidean distance between a landmark position spatially transformed and the target position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Results For each test case in the internal 4D-CT dataset, we generated 9 simulated phase images from the ground truth phase 1 image by deforming the phase 1 image using the predicted DVF at each phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We calculated SSIM to measure the image similarity (SSIMsim) between the simulated phase image and the ground truth phase image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For comparison, we also calculated the SSIM (SSIMgnd) between the ground truth phase 1 image and the rest of the ground truth phase images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The average SSIMsim was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='04, compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='86±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='08 of SSIMgnd (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=') RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 8 We also measured the diaphragm displacement between the reference respiratory signal and the predicted signal (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' As can be seen, the error increased from inhale to exhale phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' This is because prediction accuracy decreases at later time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' However, the overall displacement error was within 3 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Adding more realistic respiratory data for training can further reduce this displacement error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The error between reference respiratory signal (diaphragm displacement in mm) and predicted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' To demonstrate the modulation flexibility of the 1D breathing traces, we applied different breathing traces to the same 3D CT image to generate different motion simulations, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The plot on the top illustrates the two 1D breathing traces used for modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The breathing trace 1 (BT1), denoted by orange color line, represents the original respiratory signal for the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' BT2 denoted by gray line is a trace from another patient that was used to generate the simulated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The white horizontal line indicates the position of the apex of the diaphragm in the initial phase (the first column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' It is used as a reference to show the relative positions of the diaphragm at different phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The diaphragm in images on the upper row clearly shows the more significant movement as BT2 has higher amplitudes in the trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The amplitude range in our internal dataset was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='14 – 40 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' To validate the prediction performance on out-of-range displacement, we predicted additional sequences using a 5 times larger respiratory amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The prediction results using a 5 times larger respiratory signal achieve a higher diaphragm level which means the predicted respiratory has larger fluctuation than the original respiratory signal but it was not proportional to the respiratory signal that was used for inference (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The results of propagating anatomical structures using the predicted DVFs are also shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We propagated the lung, heart, esophagus, and tumor from the initial phase image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The propagated contours are well-matched with the predicted image and the motion of structures looks very realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We also provided the supplementary video of the simulated 4D-CT along with the ground truth 4D-CT and the 3D 9 8 7 6 Error 5 4 3 X X X 2 X X 1 0RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Two different breathing traces, BT1 and BT2 shown in the plot, were used to simulate the respiration motion of an internal case, resulting in 2 series of modulated phase images according to the breathing traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The diaphragm has larger displacement in images simulated with BT2 (upper row) than the displacement in images simulated with shallower BT1 (bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=') The white horizontal line indicates the position of the apex of the left diaphragm at the initial phase (left-most column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=') We also overlay the propagated lung(in yellow), heart(in red), esophagus(in blue) and tumor(in green) contours using predicted DVFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' volume-rendered visualizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Specifically, the 3D volume-rendered visualizations on LUNA challenge datasets as well as internal lung radiotherapy datasets with structure propagation are included in the accompanying supplementary video with chained predictions for 60-phase predictions for LUNA challenge (radiology lung nodule) and 30-phase predictions for the lung radiotherapy datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' In POPI dataset, there is only one case which contains lung segmentations on all the phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For this case, we extracted 1D breathing trace from the lung segmentations as we did for our internal dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' RMSim trained with our internal dataset predicted the remaining phases from the inhale phase with the modulation from the 1D breathing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The average TRE (Target Registration Error) of landmarks propagated with our predicted DVFs in this case was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='64mm, showing that RMSim can accurately predict the patient-specific motion from the patient’s 1D breathing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Figure 6 shows the TRE results for all predicted phases in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For the three other 4D-CT cases in POPI there were no lung segmentation masks so we performed semi-automatic lung segmentation for extracting the 1D breathing traces and the results are shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Additionally, we used the RMSim for augmenting the Learn2Reg Challenge dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The Dice score of lung segmentation of 10 Learn2Reg testing cases using the VoxelMorph without augmentation was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='01 while the model trained with RMSim data 15 10 mm 5 0 2 3 5 6 7 8 9 BT1---BT2RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 10 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The predicted phase 5 images using different 1D respiratory signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Blue line is original respiratory signal, orange line is 3 times amplitude and green line is 5 times amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' TRE results of all 9 phases from the 4DCT case in POPI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' RMSim trained with the internal dataset were able to achieve sub-mm accuracy in this external case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' augmentation was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='01 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='001 using the paired t-test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The SSIM between the warped images and the ground truth images was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='02 for the model without augmentation and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='02 (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='001) for the model with augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' To validate the improvement of DIR using VoxelMorph with augmentation, we propagated the landmark points from the inhale phase to the exhale phase for the 6 Phase Respiratory signal 140 Amplitude(mm) 120 100 016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 Without Prediction 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 With Prediction 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 TRE (mm) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 X 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 X 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='00 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='3 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='5 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='7 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='9 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='10RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 11 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Three other 4D-CT POPI cases including 10 phases with landmarks on each phase (TRE plots for the three cases given below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' For each case, we show original and predicted phase images overlaid with the difference with respect to original phase 1 input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' In original DIR Validation 03 phase difference image, the diaphragm in the left lung (viewer’s right) did not move due to the large tumor but it does in our prediction (shown in red bounding boxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' This case does not deflect from the goals of this paper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' data augmentation and DIR validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The difference in Case #1 appears minor because the breathing is shallower (less diaphragm movement) and Case #2 and Case #3 have larger differences due to deeper breathing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' cases available in POPI dataset and computed the TRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' On average, pre-DIR TRE was 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='05±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='61mm, VoxelMorph w/o augmentation was 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='12±5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='78mm compared to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='58±6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='38mm for VoxelMorph with augmentation (p < 3e-48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The TRE comparison of all 6 cases are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Phase10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='DIR_Validation_01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Validation_02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='DIR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Original ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='DIR_Validation_03 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='WithoutPrediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='WithPrediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='(mm) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='TRE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='P2 P3 P4 P5P6 P7 P8 P9 P10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='P2 P3 P4 P5 P6 P7 P8 P9 P10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='P2 P3 P4 P5P6 P7 P8 P9 P10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='DIR Validation01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='DIR Validation 02 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='DIRValidation 03RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' TRE results of POPI dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' VoxelMorph with RMSim augmentation outperformed the VoxelMorph w/o augmentation in all 6 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Discussion In this work, we presented a 3D Seq2Seq network, referred to as RMSim, to predict patient-specific realistic motion induced/modulated with 1D breathing trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We successfully validated our RMSim output with both private and public benchmark datasets (healthy and cancer patients) and demonstrated that adding our patient- specific augmentations to training data can improve performance/accuracy of state- of-the-art deep learning DIR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We also showcased breathing trace-modulated respiratory motion simulations for public static radiology scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' In this work, we predicted the motion in one breathing cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' In the future, we will fine-tune our current model to predict multiple cycles in one-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Possible solutions include making our model bi-directional and using cross-attention to improve temporal dynamics in a long sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Further research is needed to investigate the impact of training data augmentation on different image modalities such as 4D-MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Another application of our work is in external radiotherapy treatment planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' RMSim simulated 4D-CT can be used to delineate the internal target volume (ITV) which is the union of the target volumes in all respiratory phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The entire ITV is irradiated in radiation therapy to ensure all regions of tumor receive enough radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' There is a more sophisticated alternative to ITV, referred to as robust treatment planning, where the key idea is to model the motion and directly incorporate it into the planning [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' This typically can be done by assuming a probability density function (PDF) for the position of the target and doing plan optimization based on that [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' It is also possible to assume a set of possible motion PDFs to account for uncertainty in breathing and plan accordingly [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' The simulated 4D-CT can be used to extract 40 Vanilla VoxelMorph 35 Pre_DIR 30 VoxelMorph + Augmentation TRE (mm) 25 20 15 10 5 0 #1 #2 #3 #4 #5 #6RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 13 the motion PDF or a set of motion PDFs from varied breathing patterns exhibited by the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Additional interesting future direction is the extension of our earlier work in exhaustively simulating physics-based artifacts in CT and CBCT images for more robust cross-modal deep learning translation, segmentation, and motion-correction algorithms [21, 22, 23], available via our Physics-ArX library (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='com/nadeemlab/ Physics-ArX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Specifically, in our previous work we presented a proof-of-concept pipeline for physics-based motion artifact simulation in CT/CBCT images using 4D- CT phases [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Using the method proposed in the current paper, we can generate and modulate large/diverse 4D-CT phases from any static 3D CT scan using the 1D RPM signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' These generated 4D-CT variations can then be used to produce large realistic motion-artifact variations via our earlier pipeline[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Limitations: For simplicity, we used the maximal displacement on the diaphragm as the surrogate of clinical breathing trace to drive the modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We assume (1) the breathing pattern is regular since we extracted the diaphragm displacements from amplitude-binned 4D-CT, and (2) regional DVFs are linearly scaled according to diaphragm motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Note 1D breathing trace might not represent the actual cardiac motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Because of the GPU memory constraints, our input and output dimension was limited to 128x128x128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Nevertheless, the precise estimation of motion is not required for providing realistic motion-induced ground truth DVFs for the validation of the DIR algorithms and data augmentation for training DIR algorithms, as shown in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' To extend our work to tumor tracking during radiation treatment, we will use the signals from the actual external real-time motion management (RPM) device to drive the modulation more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' We will also explore incorporating 2D MV/kV projections acquired during the treatment to infer more realistic cardiac/tumor motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Acknowledgements This work was supported partially by NCI/NIH P30 CA008748.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Conflict of interest We have no conflict of interest to declare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Code Availability Statement The code, pretrained models, and augmented DIR validation datasets will be released at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='com/nadeemlab/SeqX2Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Data Availability Statement The public datasets used in this study and their urls are as follows: (1) Learn2Reg Challenge Lung CT dataset (Empire10 Challenge Dataset): https://drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' RMSim: Controlled Respiratory Motion Simulation on Static Patient Scans 14 com/drive/folders/1yHWLQEK9c1xzggkCC4VX0X4To7BBDqu5, (2) LUNA challenge dataset (subset0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='zip): https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='org/record/3723295, (3) DIR Validation POPI Dataset (6 4D CT patients with landmarks): https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='creatis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='insa-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' fr/rio/dir_validation_data, and (4) POPI model dataset (one 4D CT patient dataset with landmarks on all phases as well as lung segmentation mask): https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content='creatis.' metadata={'source': 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and organs-at-risk segmentation using physics-based data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} +page_content=' Medical Physics, 48(9):5130–5141, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9FJT4oBgHgl3EQfAyyD/content/2301.11422v1.pdf'} diff --git a/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf b/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..90f50d0f7c2638776ea9d20c93efe3ca4ece7290 --- /dev/null +++ b/C9FKT4oBgHgl3EQfYi5Z/content/2301.11799v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9e1cf4ff29ea2f8599b27e82683916e704e1e8524d78b1c1b653bf2126953380 +size 554045 diff --git a/C9FKT4oBgHgl3EQfYi5Z/vector_store/index.faiss b/C9FKT4oBgHgl3EQfYi5Z/vector_store/index.faiss new file mode 100644 index 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b/CtE2T4oBgHgl3EQfoAiM/content/tmp_files/2301.04014v1.pdf.txt @@ -0,0 +1,461 @@ +arXiv:2301.04014v1 [quant-ph] 9 Jan 2023 +Ozawa’s Intersubjectivity Theorem as +objection to QBism individual agent +perspective +Andrei Khrennikov +Linnaeus University, International Center for Mathematical Modeling +in Physics and Cognitive Sciences V¨axj¨o, SE-351 95, Sweden +January 11, 2023 +Abstract +QBism’s foundational statement that “the outcome of a measure- +ment of an observable is personal” is in the straight contraversion with +Ozawa’s Intersubjectivity Theorem (OIT). The latter (proven within +the quantum formalism) states that two observers, agents within the +QBism terminology, performing joint measurements of the same ob- +servable A on a system S in the state ψ should get the same outcome +A = x. In Ozawa’s terminology, this outcome is intersubjective and it +can’t be treated as personal. This is the strong objection to QBism +which can’t survive without updating its principles. +The essential +aspect in understanding of the OIT-impact on QBism’s foundations +takes the notion of quantum observable. This paper comprises the +complementary discussion highlighting the difference between the ac- +curate, von Neumann, and inaccurate, noisy, quantum observables +which are represented by PVMs and POVMs respectively. Moreover, +we discuss the OIT-impact on the Copenhagen interpretation of quan- +tum mechanics. +1 +Introduction +In this paper I move ahead my critical analysis of QBism’s founda- +tions (see, e.g., [1]–[4] for QBism basics). This paper, as well as my +two previous articles [5, 6], straightly critiques the individual agent +perspective on measurement’s outcomes [7]. My previous appraisal +1 + +convinced QBists to specify the level of agent’s individuality. In con- +trast to the general subjective probability theory, the class of agents +should be restricted, at least to agents who were educated in basics +of quantum theory. So, Ivan who lives in a Siberian village, a busy +hunter, can’t be treated as a QBism’s agent. +Now I have an intention to offense QBism by using Ozawa’s Inter- +subjectivity Theorem (OIT) [8]. Qbism’s statement that “the outcome +of a measurement of an observable is personal” is in the straight con- +traversion with OIT. This theorem is not so widely known and one +of the present paper’s intention is the theorem’s advertizement. OIT +states that two observers, agents within the QBism terminology, per- +forming joint measurements of the same observable A on a system S in +the state ψ should register the same outcome A = x with probability +one. Hence, the outcome is intersubjective [8], and it’s unnatural to +consider outcomes of quantum observations as agent’s personal expe- +riences. +OIT is proven within the quantum formalism, it is the rigorous +mathematical statement. But, as many theorems having the quan- +tum foundational impact, its interpretation is not straightforward. +The analysis of the OIT-impact onto QBism is coupled to the foun- +dations of quantum measurement theory and especially the notion of +quantum observable. Therefore, this paper comprises the complemen- +tary discussion, highlighting the difference between the accurate, von +Neuman, and inaccurate, noisy, quantum observables, mathematically +represented by projection valued measures (PVMs) and positive oper- +ator valued measures (POVMs), respectively. QIT is about the agents +who are able to perform the joint accurate measurements. For such +agents, measurement’s outcome loses its personalization, in favour of +intersubjectivity. +The conclusion of our analysis is that QBism should update its +ideology by taking in consideration OIT. But, how? See section 6. +Thus, I am in line with the criticism of QBism presented in article +[8]. However, I depart from its conclusion that OIT contradicts to the +Copenhagen interpretation; in contrast, OIT peacefully coexist with +this interpretation. It is relevant to recall here that QBism fundamen- +tally differs from the Copenhagen interpretation [2]. +Right away we initiate with the mathematical formulation of OIT +and its proof. We set out to make the presentation very shortly (see +[8] for details). The indirect measurement scheme is the heart of OIT. +We go ahead with the recollection of the notion of quantum observ- +able, namely, Hermitian operator or PVM, and generalized quantum +observable (POVM) and the indirect measurements scheme for their +generation. +2 + +2 +Quantum observables vs. +general- +ized quantum observables +In quantum mechanics’ axiomatics, von Neumann [9] introduced quan- +tum observables as Hermitian operators acting in complex Hilbert +space H, the state space of a system.1 The spectral decomposition is +the essential part in this framework. +We restrict considerations to observables represented by the oper- +ators with totally discrete spectra X ⊂ R. Here +A = +� +x +xEA(x), +(1) +where EA(x) is projection on the eigensubspace corresponding to the +eigenvalue x; these projectors form the resolution of unity: +I = +� +x +EA(x). +(2) +The Born rule determines the probabilities of the outcomes of mea- +surements for a system S in the state ψ, +P(A = x|ψ) = ⟨ψ|EA(x)|ψ⟩. +(3) +Later generalized quantum observables were invented. Such ob- +servables are represented by POVMs. We restrict considerations to +POVMs with a discrete domain of definition X. POVM is a map +x → Π(x) : for each x ∈ X, Π(x) is a positive contractive self-adjoint +operator (i.e., 0 ≤ Π(x) ≤ I) (called an effect), and effects form the +resolution of unity +� +x +Π(x) = I. +(4) +This map defines an operator valued measure on algebra of all subsets +of set X. For O ⊂ X, +Π(O) = +� +x∈O +Π(x). +The condition (4) is the operator-measure counterpart of the condition +normalization by 1 for usual probability measures. +1Why did he select the Hermitian operators for mathematical representation of observ- +ables in quantum theory? Moreover, he considered only such observables as the genuine +quantum observables. I guess that he followed Schr¨odinger’s quantization rule for the +position and momentum observables which are realized by Hermitian operators in L2- +space. This rule implies that each classical observable given by the real-valued function +A = A(q, p) on the phase space is represented as a Hermitian operator in L2-space. +3 + +POVM Π represents statistics of measurements for observable A +with the following generalization of the Born’s rule: +P(Π = x|ψ) = ⟨ψ|Π(x)|ψ⟩. +(5) +We remark that equality (4) implies that +� +x +P(A = x|ψ) = 1. +Any quantum observable A can also be represented as POVM of the +special type – PVM EA = (EA(x)). +Quantum observables given by PVMs were interpreted by von Neu- +mann [9] as describing accurate measurements. And generalized ob- +servables given by POVMs which are not PVMs are interpreted as +representing inaccurate measurements. In von Neumann’s [9], the no- +tion of measurement’s precision was not completely formalized. Only +recently the consistent formalization of this notion was presented in +[11]. +We shall keep firmly the expression “quantum observable” for ob- +servable axiomatically introduced by von Neumann [9] and represented +by PVMs and the expression “generalized quantum observable” for +POVMs. +3 +Generalized quantum observables from +the indirect measurement scheme +The indirect measurement scheme involves the following components +• the states spaces H and K of the systems S and the apparatus +M for measurement of some observable A; +• the evolution operator U = U(t) representing the interaction- +dynamics for the system S + M; +• the meter observable M giving outputs of the pointer of the +apparatus M. +Here the quantum observables A and M can be represented as PVMs, +EA = (EA(x)), EM = (EM(x)), where EA(x), EM(x) are projections +in Hilbert spaces H and K respectively. It is assumed that the com- +pound system’s evolution is driven by the Schr¨odinger equation, so +the evolution operator is unitary. +Formally, an indirect measurement model for an observable A, in- +troduced in [10] as a “measuring process”, is a quadruple +(K, |ξ⟩, U, M) +4 + +where |ξ⟩ ∈ K represents the apparatus state. +We explore the Heisenberg picture. To describe meter’s evolution, +we represent it in the state space of the compound system, i.e., as +I ⊗ M. The meter observable evolves as +M(t) = U ⋆(t)(I ⊗ M)U(t). +(6) +By the Born rule +P(M(t) = x|ψξ) = ⟨ψξ|EM(t)(x)|ψξ⟩. +(7) +This is the probability distribution for the outputs of measure- +ments done by the apparatus and given by the meter. In principle, +one can ignore the representation of the measurement process as the +system-apparatus interaction and operate solely with system’s states. +In this picture one proceeds with generalized observables given by +POVMs. The meter observable generates the POVM Π = (Π(x)) +Π(x) = ⟨ξ|EM(T)(x)|ξ⟩, +(8) +where T is the time needed to complete the experiment. +The probability distribution of the generalized observable given by +a POVM is determined by (5). +Generally the probability distribution generated by a measurement +process does not coincide with the probability distribution of the quan- +tum observable A for which this process was constructed, i.e., generally +P(Π = x|ψ) = ⟨ψ|Π(x)|ψ⟩ ̸= P(A = x|ψ) = ⟨ψ|EA(x)|ψ⟩, +(9) +We remark that, as was proven by Ozawa [10], any generalized +observable (POVM) can be generated via the indirect measurement +scheme. +Typically one operates solely with generalized observables +by ignoring the indirect measurement scheme. This simplifies consid- +erations, but it can lead to misunderstanding of the foundations the +quantum measurement theory. +4 +Probability reproducibility condition +Definition. A measurement process (K, |ξ⟩, U, M) reproduces the prob- +ability distribution for quantum observable A (accurate von Neumann +observable) if +P(A = x|ψ) = P(M(T) = x|ψξ). +(10) +In this case +⟨ψξ|EM(T)(x)|ψξ⟩ = ⟨ψ|E(x)|ψ⟩. +(11) +5 + +or +⟨ψ|Π(x)|ψ⟩ = ⟨ψ|E(x)|ψ⟩, +(12) +and hence, +Π(x) = E(x), +Proposition. Probability reproducibility condition for a measure- +ment process is equivalent to the representation of the corresponding +generalized observable by the PVM EA of measured quantum observ- +able A. +5 +Intersubjectivity of outcomes of quan- +tum observables +Following [8], consider two remote observers O1 and O2 who perform +joint measurements on a system S, in mathematical terms it means +that the meter quantum observables of the corresponding measure- +ment processes commute, +[M1(t), M2(t)] = 0. +Here each apparatus has its own state space, i.e., K = K1 ⊗ K2. We +call such measurements local. In this situation the joint probability +distribution is well defined +P(M1(t) = x, M1(t) = y|ψξ1ξ2) = ⟨ψξ1ξ2|EM1(t)(x)EM1(t)(y)|ψξ1ξ2⟩ +(13) +Suppose that both observers perform the accurate measurements +of the quantum observable A given by PVM EA = (EA(x)). Then the +corresponding POVMs Πj, j = 1, 2, coincide with EA : +Π1(x) = Π2(x) = EA(x). +(14) +This equality implies: +Theorem. (OIT [8]) Two observers performing the joint local and +probability reproducible measurements of the same quantum observable +A on the system S should get the same outcome with probability 1: +P(M1(T) = x, M1(T) = y|ψξ1ξ2) = δ(x − y)P(E = x|ψ) = ∥E(x)ψ∥2. +(15) +6 + +6 +Intersubjectivity challenges QBism +We start with the following citation of Fuchs and Schack [2]: +“The fundamental primitive of QBism is the concept of experience. +According to QBism, quantum mechanics is a theory that any agent +can use to evaluate her expectations for the content of her personal +experience. ... +In QBism, a measurement is an action an agent takes to elicit an +experience. The measurement outcome is the experience so elicited. +The measurement outcome is thus personal to the agent who takes the +measurement action. In this sense, quantum mechanics, like probabil- +ity theory, is a single user theory. A measurement does not reveal a +pre-existing value. Rather, the measurement outcome is created in the +measurement action.” +However, OIT implies that, for accurate local observables, mea- +surement’s outcome is intersubjective which is the strong objection to +QBism. There is nothing concerning personal experiences and QBists +should response to this objection. My suggestion (see also [7]) is to fol- +low Brukner’s work [12] where he proceeds not with individual agents +and their personal experiences, but with a universal agent. I remark +that consideration of universal agents is common in general theory of +decision making. However, for QBists, such solution seems to be un- +acceptable, since it would destroy consistency of the QBism’s private +agency perspective. It would move QBism closer to Zeilinger-Brukner +information interpretation of quantum mechanics [13, 14, 15]. +This objection to QBism is foundationally interesting and gen- +erates the discussion on the notion of quantum observable. Due to +efforts Helstrom, Holevo, and Ozawa [16]–[19], [10], generalized quan- +tum observables which are mathematically represented by POVMs +became one of the basic tools of quantum information theory. Nowa- +days the special role of accurate observables represented by PVMs is +not emphasized. In particular, the notion of observables in QBism is +identified with generalized quantum observable given by POVM. How- +ever, the clash between QBism and OIT stimulates highlighting of the +accurate PVM- as the genuine quantum observables, and treating the +generalized quantum observables which are not accurate POVM as +imprecise and noisy ones. Of course, it is a well known fact, but the +clash between OIT and QBism is good occasion to emphasize this +difference. +What does this difference between accurate PVM and noisy POVM +observables mean for QBism? +I have the following picture of the situation. OIT holds only for the +accurate PVM-observables; for generalized quantum observables, it +7 + +can be violated and generally it is impossible to assign the same value +for measurements’ outcomes for observers O1 and O2. Thus, QBism +ideology of the personal experiences of observers (agents) can still be +kept for such generalizad observables. But, where does individuality +come from? The personal experiences come from noise! So, different +observers performing inaccurate measurements are coupled to different +noisy environments. This is just my personal view on consequences of +IOT for QBism. +In conclusion, QBism might response to the OIT-challenge by con- +sidering the universal agent who is able to perform accurate measure- +ments; individuality of agents’ experience is reduced to individuality +of noise generated in the process of measurement. +7 +Intersubjectivity and Copenhagen in- +terpretation +By the Copenhagen interpretation (at least by its Bohr’s version2) +measurements’ outcomes cannot be treated as the objective properties +of a system S. They are results of the complex process of interaction +of a system and an apparatus, see Bohr [21]: +“This crucial point ... implies the impossibility of any sharp sep- +aration between the behaviour of atomic objects and the interaction +with the measuring instruments which serve to define the conditions +under which the phenomena appear. In fact, the individuality of the +typical quantum effects finds its proper expression in the circumstance +that any attempt of subdividing the phenomena will demand a change +in the experimental arrangement introducing new possibilities of inter- +action between objects and measuring instruments which in principle +cannot be controlled. Consequently, evidence obtained under different +experimental conditions cannot be comprehended within a single pic- +ture, but must be regarded as complementary in the sense that only the +totality of the phenomena exhausts the possible information about the +objects.” +The indirect measurement scheme matches perfectly with the Copen- +hagen interpretation. Therefore it is surprising that OIT contradicts +to it. The clash between OIT and the the Copenhagen interpretation +was highlighted in the conclusion section of OIT-article [8]: +2As was stressed by Plotnitsky [20], one should recognize the diversity of views on the +Copenhagen interpretation. He suggested to speak about interpretations in the spirit of +Copenhagen. Even Bohr changed the views a few times during his life [20]. +8 + +“Schr¨odinger [22] argued that a measurement does not ascertain +the pre-existing value of the observable and is only required to be re- +peatable. Since the inception of quantum mechanics, this view has long +been supported as one of the fundamental tenets of quantum mechan- +ics. In contrast, we have shown that any probability reproducible mea- +surement indeed ascertains the value that the observable has, whether +the repeatability is satisfied or not.” +I disagree with the author of [8]. The seed of this misunderstand- +ing is in ignoring the two level structure of physical theories, ontic +and epistemic [23, 24, 25]. The former is about reality as it is and +the latter is about knowledge about reality. +Bohr and Schr¨odinger +wrote about the ontic reality, about impossibility to assign to quan- +tum systems preexisting values and here “preexisting” is the synonym +for “objective”, “ontic”. But OIT is not about such values, it is about +epistemic reality, reality of knowledge about the possible outcome of +measurement. +Hence, in my opinion OIT can peacefully coexist with the Copen- +hagen interpretation. +But, as was stressed, OIT is a challenge for QBism which operates +at the epistemic level of scientific description of quantum phenom- +ena. This is the good place to recall that QBism should be sharply +separated from the Copenhagen interpretation, see again Fuchs and +Schack [2]: +“According to QBism, quantum mechanics can be applied to any +physical system. QBism treats all physical systems in the same way, +including atoms, beam splitters, Stern-Gerlach magnets, preparation +devices, measurement apparatuses, all the way to living beings and +other agents. In this, QBism differs crucially from various versions +of the Copenhagen interpretation.” +Acknowledgments +This paper was written on the basis of the long discussions with +Masanao Ozawa and I would like to thank him; Arkady Plotnitsky +told me a lot about the Copenhagen interpretation and Bohr’s views +and I would like to thank him; Christopher Fuchs ignited my inter- +est to QBism at the second V¨axj¨o conference (in 2001) and I am +sorry if this paper would disturb QBists; I am also thankful to Harald +Atmanspacher who introduced me into ontic-epistemic approach to +scientific representation of natural phenomena. +9 + +References +[1] Fuchs, C. A. and Schack, R. (2011). A Quantum-Bayesian Route +to Quantum-State Space, Found. Phys. 41, p. 345. +[2] Fuchs, C. A. and Schack, R. (2014). QBism and the Greeks: +why a quantum state does not represent an element of physical +reality, Phys. Scr., 90, 015104. +[3] Fuchs, C. A., Mermin, N. D. and Schack, R. (2014). An In- +troduction to QBism with an Application to the Locality of +Quantum Mechanics, Am. J. Phys. 82, p. 749. +[4] DeBrota, J. B., Fuchs, C. A., Pienaar, J. L., and Stacey, B. C. +(2021). Born’s rule as a quantum extension of Bayesian coher- +ence. Physical Review A, 104(2), 022207. +[5] Khrennikov, +A. (2018). +External Observer Reflections on +QBism, Its Possible Modifications, and Novel Applications, In: +Quantum Foundations, STEAM-H: Science, Technology, Engi- +neering, Agriculture, Mathematics & Health; Khrennikov A. and +Toni B. Eds.; Springer, Cham, pp. 93–118. +[6] Khrennikov, A. (2018). Towards better understanding QBism, +Found. Sc., 23 (1), 181–195. +[7] Khrennikov, A. Reflections on Zeilinger–Brukner Information +Interpretation of Quantum Mechanics. Found Phys 46, 836–844 +(2016). +[8] Ozawa, M. (2019). Intersubjectivity of outcomes of quantum +measurements. https://arxiv.org/abs/1911.10893 +[9] von Neuman, J. (1955). Mathematical foundations of quan- +tum mechanics (Princeton Univ. Press, Princenton) [Originally +published: Mathematische Grundlagen der Quanten-mechanik, +Springer, Berlin, 1932]. +[10] Ozawa, M. (1984). Quantum measuring processes for continuous +observables. J. Math. Phys. 25, 79–87 +[11] Ozawa, M. Soundness and completeness of quantum root-mean- +square errors. Quant. Inf. 5, Article number: 1 (2019) +[12] Brukner, +C. +On +the +quantum +measurement +problem. +https://arxiv.org/abs/1507.05255. +[13] Zeilinger, A.: A foundational principle for quantum mechanics. +Found. Phys. 29, 31–641 (1999) +[14] Brukner, C., Zeilinger, A.: Malus’ law and quantum informa- +tion. Acta Phys. Slovava 49, 647–652 (1999) +10 + +[15] Brukner, C., Zeilinger, A.: Information invariance and quantum +probabilities. Found. Phys. 39, 677 (2009) +[16] C. W. Helstrom, Quantum Detection and Estimation Theory. +Academic, New York, 1976. +[17] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum +Theory. North-Holland, Amsterdam, 1982. +[18] M. Ozawa, Optimal measurements for general quantum systems. +Rep. Math. Phys. 18, 1980, 11–28 +[19] M. Ozawa, +Realization of measurement and the standard +quantum limit. Squeezed and Nonclassical Light, P. Tombesi +and E. R. Pike, Plenum, New York, 1989, +pp. 263–286, +arXiv:1505.01083 [quant-ph]. +[20] Plotnitsky, A. (2012). Niels Bohr and complementarity: An in- +troduction. Berlin and New York: Springer. +[21] Bohr, N.: (1987). The Philosophical Writings of Niels Bohr, 3 +vols. (Ox Bow Press, Woodbridge, CT). +[22] Schr¨odinger, E. The present situation in quantum mechanics: +A translation of Schr¨odinger’s “Cat Paradox” paper (by: J. D. +Trimmer). Proc. Am. Philos. Soc. 124, 323–338 (1980). [Orig- +inally published: Die gegenw¨artige Situation in der Quanten- +mechanik, Naturwissenschaften 23, 807–812, 823–828, 844– 849 +(1935)]. +[23] Atmanspacher, H. (1994). Is the ontic/epistemic-distinction suf- +ficient to describe quantum systems exhaustively. In: Sympo- +sium on the Foundations of Modern Physics (pp. 15-32). +[24] Atmanspacher, H. and Primas, H. (2003). Epistemic and ontic +quantum realities. In: Time, quantum and information (pp. 301- +321). Springer, Berlin, Heidelberg. +[25] Khrennikov, A. (2017). Quantum epistemology from subquan- +tum ontology: Quantum mechanics from theory of classical ran- +dom fields. Annals of Physics, 377, 147-163. +11 + diff --git a/CtE2T4oBgHgl3EQfoAiM/content/tmp_files/load_file.txt b/CtE2T4oBgHgl3EQfoAiM/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ff41a5abb3417f891676a9ef18e16d89d4f97cee --- /dev/null +++ b/CtE2T4oBgHgl3EQfoAiM/content/tmp_files/load_file.txt @@ -0,0 +1,306 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf,len=305 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='04014v1 [quant-ph] 9 Jan 2023 Ozawa’s Intersubjectivity Theorem as objection to QBism individual agent perspective Andrei Khrennikov Linnaeus University, International Center for Mathematical Modeling in Physics and Cognitive Sciences V¨axj¨o, SE-351 95, Sweden January 11, 2023 Abstract QBism’s foundational statement that “the outcome of a measure- ment of an observable is personal” is in the straight contraversion with Ozawa’s Intersubjectivity Theorem (OIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The latter (proven within the quantum formalism) states that two observers, agents within the QBism terminology, performing joint measurements of the same ob- servable A on a system S in the state ψ should get the same outcome A = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In Ozawa’s terminology, this outcome is intersubjective and it can’t be treated as personal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This is the strong objection to QBism which can’t survive without updating its principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The essential aspect in understanding of the OIT-impact on QBism’s foundations takes the notion of quantum observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This paper comprises the complementary discussion highlighting the difference between the ac- curate, von Neumann, and inaccurate, noisy, quantum observables which are represented by PVMs and POVMs respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Moreover, we discuss the OIT-impact on the Copenhagen interpretation of quan- tum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 1 Introduction In this paper I move ahead my critical analysis of QBism’s founda- tions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=', [1]–[4] for QBism basics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This paper, as well as my two previous articles [5, 6], straightly critiques the individual agent perspective on measurement’s outcomes [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' My previous appraisal 1 convinced QBists to specify the level of agent’s individuality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In con- trast to the general subjective probability theory, the class of agents should be restricted, at least to agents who were educated in basics of quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' So, Ivan who lives in a Siberian village, a busy hunter, can’t be treated as a QBism’s agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Now I have an intention to offense QBism by using Ozawa’s Inter- subjectivity Theorem (OIT) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Qbism’s statement that “the outcome of a measurement of an observable is personal” is in the straight con- traversion with OIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This theorem is not so widely known and one of the present paper’s intention is the theorem’s advertizement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' OIT states that two observers, agents within the QBism terminology, per- forming joint measurements of the same observable A on a system S in the state ψ should register the same outcome A = x with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Hence, the outcome is intersubjective [8], and it’s unnatural to consider outcomes of quantum observations as agent’s personal expe- riences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' OIT is proven within the quantum formalism, it is the rigorous mathematical statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' But, as many theorems having the quan- tum foundational impact, its interpretation is not straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The analysis of the OIT-impact onto QBism is coupled to the foun- dations of quantum measurement theory and especially the notion of quantum observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Therefore, this paper comprises the complemen- tary discussion, highlighting the difference between the accurate, von Neuman, and inaccurate, noisy, quantum observables, mathematically represented by projection valued measures (PVMs) and positive oper- ator valued measures (POVMs), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' QIT is about the agents who are able to perform the joint accurate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' For such agents, measurement’s outcome loses its personalization, in favour of intersubjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The conclusion of our analysis is that QBism should update its ideology by taking in consideration OIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' But, how?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' See section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Thus, I am in line with the criticism of QBism presented in article [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' However, I depart from its conclusion that OIT contradicts to the Copenhagen interpretation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' in contrast, OIT peacefully coexist with this interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' It is relevant to recall here that QBism fundamen- tally differs from the Copenhagen interpretation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Right away we initiate with the mathematical formulation of OIT and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' We set out to make the presentation very shortly (see [8] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The indirect measurement scheme is the heart of OIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' We go ahead with the recollection of the notion of quantum observ- able, namely, Hermitian operator or PVM, and generalized quantum observable (POVM) and the indirect measurements scheme for their generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 2 2 Quantum observables vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' general- ized quantum observables In quantum mechanics’ axiomatics, von Neumann [9] introduced quan- tum observables as Hermitian operators acting in complex Hilbert space H, the state space of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='1 The spectral decomposition is the essential part in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' We restrict considerations to observables represented by the oper- ators with totally discrete spectra X ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Here A = � x xEA(x), (1) where EA(x) is projection on the eigensubspace corresponding to the eigenvalue x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' these projectors form the resolution of unity: I = � x EA(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (2) The Born rule determines the probabilities of the outcomes of mea- surements for a system S in the state ψ, P(A = x|ψ) = ⟨ψ|EA(x)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (3) Later generalized quantum observables were invented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Such ob- servables are represented by POVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' We restrict considerations to POVMs with a discrete domain of definition X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' POVM is a map x → Π(x) : for each x ∈ X, Π(x) is a positive contractive self-adjoint operator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=', 0 ≤ Π(x) ≤ I) (called an effect), and effects form the resolution of unity � x Π(x) = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (4) This map defines an operator valued measure on algebra of all subsets of set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' For O ⊂ X, Π(O) = � x∈O Π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The condition (4) is the operator-measure counterpart of the condition normalization by 1 for usual probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 1Why did he select the Hermitian operators for mathematical representation of observ- ables in quantum theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Moreover, he considered only such observables as the genuine quantum observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' I guess that he followed Schr¨odinger’s quantization rule for the position and momentum observables which are realized by Hermitian operators in L2- space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This rule implies that each classical observable given by the real-valued function A = A(q, p) on the phase space is represented as a Hermitian operator in L2-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 3 POVM Π represents statistics of measurements for observable A with the following generalization of the Born’s rule: P(Π = x|ψ) = ⟨ψ|Π(x)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (5) We remark that equality (4) implies that � x P(A = x|ψ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Any quantum observable A can also be represented as POVM of the special type – PVM EA = (EA(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Quantum observables given by PVMs were interpreted by von Neu- mann [9] as describing accurate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' And generalized ob- servables given by POVMs which are not PVMs are interpreted as representing inaccurate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In von Neumann’s [9], the no- tion of measurement’s precision was not completely formalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Only recently the consistent formalization of this notion was presented in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' We shall keep firmly the expression “quantum observable” for ob- servable axiomatically introduced by von Neumann [9] and represented by PVMs and the expression “generalized quantum observable” for POVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 3 Generalized quantum observables from the indirect measurement scheme The indirect measurement scheme involves the following components the states spaces H and K of the systems S and the apparatus M for measurement of some observable A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' the evolution operator U = U(t) representing the interaction- dynamics for the system S + M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' the meter observable M giving outputs of the pointer of the apparatus M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Here the quantum observables A and M can be represented as PVMs, EA = (EA(x)), EM = (EM(x)), where EA(x), EM(x) are projections in Hilbert spaces H and K respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' It is assumed that the com- pound system’s evolution is driven by the Schr¨odinger equation, so the evolution operator is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Formally, an indirect measurement model for an observable A, in- troduced in [10] as a “measuring process”, is a quadruple (K, |ξ⟩, U, M) 4 where |ξ⟩ ∈ K represents the apparatus state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' We explore the Heisenberg picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' To describe meter’s evolution, we represent it in the state space of the compound system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=', as I ⊗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The meter observable evolves as M(t) = U ⋆(t)(I ⊗ M)U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (6) By the Born rule P(M(t) = x|ψξ) = ⟨ψξ|EM(t)(x)|ψξ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (7) This is the probability distribution for the outputs of measure- ments done by the apparatus and given by the meter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In principle, one can ignore the representation of the measurement process as the system-apparatus interaction and operate solely with system’s states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In this picture one proceeds with generalized observables given by POVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The meter observable generates the POVM Π = (Π(x)) Π(x) = ⟨ξ|EM(T)(x)|ξ⟩, (8) where T is the time needed to complete the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The probability distribution of the generalized observable given by a POVM is determined by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Generally the probability distribution generated by a measurement process does not coincide with the probability distribution of the quan- tum observable A for which this process was constructed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=', generally P(Π = x|ψ) = ⟨ψ|Π(x)|ψ⟩ ̸= P(A = x|ψ) = ⟨ψ|EA(x)|ψ⟩, (9) We remark that, as was proven by Ozawa [10], any generalized observable (POVM) can be generated via the indirect measurement scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Typically one operates solely with generalized observables by ignoring the indirect measurement scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This simplifies consid- erations, but it can lead to misunderstanding of the foundations the quantum measurement theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 4 Probability reproducibility condition Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' A measurement process (K, |ξ⟩, U, M) reproduces the prob- ability distribution for quantum observable A (accurate von Neumann observable) if P(A = x|ψ) = P(M(T) = x|ψξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (10) In this case ⟨ψξ|EM(T)(x)|ψξ⟩ = ⟨ψ|E(x)|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (11) 5 or ⟨ψ|Π(x)|ψ⟩ = ⟨ψ|E(x)|ψ⟩, (12) and hence, Π(x) = E(x), Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Probability reproducibility condition for a measure- ment process is equivalent to the representation of the corresponding generalized observable by the PVM EA of measured quantum observ- able A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 5 Intersubjectivity of outcomes of quan- tum observables Following [8], consider two remote observers O1 and O2 who perform joint measurements on a system S, in mathematical terms it means that the meter quantum observables of the corresponding measure- ment processes commute, [M1(t), M2(t)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Here each apparatus has its own state space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=', K = K1 ⊗ K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' We call such measurements local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In this situation the joint probability distribution is well defined P(M1(t) = x, M1(t) = y|ψξ1ξ2) = ⟨ψξ1ξ2|EM1(t)(x)EM1(t)(y)|ψξ1ξ2⟩ (13) Suppose that both observers perform the accurate measurements of the quantum observable A given by PVM EA = (EA(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Then the corresponding POVMs Πj, j = 1, 2, coincide with EA : Π1(x) = Π2(x) = EA(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (14) This equality implies: Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (OIT [8]) Two observers performing the joint local and probability reproducible measurements of the same quantum observable A on the system S should get the same outcome with probability 1: P(M1(T) = x, M1(T) = y|ψξ1ξ2) = δ(x − y)P(E = x|ψ) = ∥E(x)ψ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (15) 6 6 Intersubjectivity challenges QBism We start with the following citation of Fuchs and Schack [2]: “The fundamental primitive of QBism is the concept of experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' According to QBism, quantum mechanics is a theory that any agent can use to evaluate her expectations for the content of her personal experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In QBism, a measurement is an action an agent takes to elicit an experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The measurement outcome is the experience so elicited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The measurement outcome is thus personal to the agent who takes the measurement action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In this sense, quantum mechanics, like probabil- ity theory, is a single user theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' A measurement does not reveal a pre-existing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Rather, the measurement outcome is created in the measurement action.” However, OIT implies that, for accurate local observables, mea- surement’s outcome is intersubjective which is the strong objection to QBism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' There is nothing concerning personal experiences and QBists should response to this objection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' My suggestion (see also [7]) is to fol- low Brukner’s work [12] where he proceeds not with individual agents and their personal experiences, but with a universal agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' I remark that consideration of universal agents is common in general theory of decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' However, for QBists, such solution seems to be un- acceptable, since it would destroy consistency of the QBism’s private agency perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' It would move QBism closer to Zeilinger-Brukner information interpretation of quantum mechanics [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This objection to QBism is foundationally interesting and gen- erates the discussion on the notion of quantum observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Due to efforts Helstrom, Holevo, and Ozawa [16]–[19], [10], generalized quan- tum observables which are mathematically represented by POVMs became one of the basic tools of quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Nowa- days the special role of accurate observables represented by PVMs is not emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In particular, the notion of observables in QBism is identified with generalized quantum observable given by POVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' How- ever, the clash between QBism and OIT stimulates highlighting of the accurate PVM- as the genuine quantum observables, and treating the generalized quantum observables which are not accurate POVM as imprecise and noisy ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Of course, it is a well known fact, but the clash between OIT and QBism is good occasion to emphasize this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' What does this difference between accurate PVM and noisy POVM observables mean for QBism?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' I have the following picture of the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' OIT holds only for the accurate PVM-observables;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' for generalized quantum observables, it 7 can be violated and generally it is impossible to assign the same value for measurements’ outcomes for observers O1 and O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Thus, QBism ideology of the personal experiences of observers (agents) can still be kept for such generalizad observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' But, where does individuality come from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The personal experiences come from noise!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' So, different observers performing inaccurate measurements are coupled to different noisy environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This is just my personal view on consequences of IOT for QBism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In conclusion, QBism might response to the OIT-challenge by con- sidering the universal agent who is able to perform accurate measure- ments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' individuality of agents’ experience is reduced to individuality of noise generated in the process of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 7 Intersubjectivity and Copenhagen in- terpretation By the Copenhagen interpretation (at least by its Bohr’s version2) measurements’ outcomes cannot be treated as the objective properties of a system S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' They are results of the complex process of interaction of a system and an apparatus, see Bohr [21]: “This crucial point .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' implies the impossibility of any sharp sep- aration between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In fact, the individuality of the typical quantum effects finds its proper expression in the circumstance that any attempt of subdividing the phenomena will demand a change in the experimental arrangement introducing new possibilities of inter- action between objects and measuring instruments which in principle cannot be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Consequently, evidence obtained under different experimental conditions cannot be comprehended within a single pic- ture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects.” The indirect measurement scheme matches perfectly with the Copen- hagen interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Therefore it is surprising that OIT contradicts to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The clash between OIT and the the Copenhagen interpretation was highlighted in the conclusion section of OIT-article [8]: 2As was stressed by Plotnitsky [20], one should recognize the diversity of views on the Copenhagen interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' He suggested to speak about interpretations in the spirit of Copenhagen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Even Bohr changed the views a few times during his life [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 8 “Schr¨odinger [22] argued that a measurement does not ascertain the pre-existing value of the observable and is only required to be re- peatable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Since the inception of quantum mechanics, this view has long been supported as one of the fundamental tenets of quantum mechan- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In contrast, we have shown that any probability reproducible mea- surement indeed ascertains the value that the observable has, whether the repeatability is satisfied or not.” I disagree with the author of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The seed of this misunderstand- ing is in ignoring the two level structure of physical theories, ontic and epistemic [23, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' The former is about reality as it is and the latter is about knowledge about reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Bohr and Schr¨odinger wrote about the ontic reality, about impossibility to assign to quan- tum systems preexisting values and here “preexisting” is the synonym for “objective”, “ontic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' But OIT is not about such values, it is about epistemic reality, reality of knowledge about the possible outcome of measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Hence, in my opinion OIT can peacefully coexist with the Copen- hagen interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' But, as was stressed, OIT is a challenge for QBism which operates at the epistemic level of scientific description of quantum phenom- ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' This is the good place to recall that QBism should be sharply separated from the Copenhagen interpretation, see again Fuchs and Schack [2]: “According to QBism, quantum mechanics can be applied to any physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' QBism treats all physical systems in the same way, including atoms, beam splitters, Stern-Gerlach magnets, preparation devices, measurement apparatuses, all the way to living beings and other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' In this, QBism differs crucially from various versions of the Copenhagen interpretation.” Acknowledgments This paper was written on the basis of the long discussions with Masanao Ozawa and I would like to thank him;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Arkady Plotnitsky told me a lot about the Copenhagen interpretation and Bohr’s views and I would like to thank him;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Christopher Fuchs ignited my inter- est to QBism at the second V¨axj¨o conference (in 2001) and I am sorry if this paper would disturb QBists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' I am also thankful to Harald Atmanspacher who introduced me into ontic-epistemic approach to scientific representation of natural phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 9 References [1] Fuchs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Khrennikov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' and Toni B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Springer, Cham, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' 93–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' [6] Khrennikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Towards better understanding QBism, Found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=', 23 (1), 181–195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' [7] Khrennikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Reflections on Zeilinger–Brukner Information Interpretation of Quantum Mechanics.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content='10893 [9] von Neuman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Mathematical foundations of quan- tum mechanics (Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' Press, Princenton) [Originally published: Mathematische Grundlagen der Quanten-mechanik, Springer, Berlin, 1932].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} +page_content=' [10] Ozawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtE2T4oBgHgl3EQfoAiM/content/2301.04014v1.pdf'} 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sha256:05c57a8365e35acfe7f917794a0d1a5f4d0786844bb60171e39f86bd160b9045 +size 109271 diff --git a/F9E0T4oBgHgl3EQfzQJ7/vector_store/index.faiss b/F9E0T4oBgHgl3EQfzQJ7/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e526c97c9dfd8d6b2d088e44e5db3572c26a2f05 --- /dev/null +++ b/F9E0T4oBgHgl3EQfzQJ7/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c5534bf7488af4ad2427fcfe02099abe2adbb3bc2d2a701fe5fb55f8a928c5a5 +size 4456493 diff --git a/FNAzT4oBgHgl3EQfiv3U/content/tmp_files/2301.01506v1.pdf.txt b/FNAzT4oBgHgl3EQfiv3U/content/tmp_files/2301.01506v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..51c00d4675624e70ce2ff12b53b32bc66def3e6d --- /dev/null +++ b/FNAzT4oBgHgl3EQfiv3U/content/tmp_files/2301.01506v1.pdf.txt @@ -0,0 +1,1176 @@ +arXiv:2301.01506v1 [math.OC] 4 Jan 2023 +Impulse control of conditional McKean-Vlasov +jump diffusions +Nacira Agram1, Giulia Pucci1 & Bernt Øksendal2 +January 5, 2023 +Abstract +This paper establishes a verification theorem for impulse control problems +involving conditional McKean-Vlasov jump diffusions. We obtain a Markovian +system by combining the state equation of the problem with the stochastic +Fokker-Planck equation for the conditional probability law of the state. We +derive sufficient variational inequalities for a function to be the value function +of the impulse control problem, and for an impulse control to be the optimal +control. We illustrate our results by applying them to the study of an optimal +stream of dividends under transaction costs. We obtain the solution explic- +itly by finding a function and an associated impulse control which satisfy the +verification theorem. +Keywords : Jump diffusion; impulse control; common noise; conditional McKean- +Vlasov differential equation; stochastic Fokker-Planck equation; quasi-variational in- +equalities. +1 +Introduction +Consider a filtered probability space (Ω, F, P, F = {F}t≥0) on which we are given a +d-dimensional Brownian motion B = (B1, B2, . . . , Bd), a k-dimensional compensated +Poisson random measure �N(dt, dζ) such that +�N(dt, dζ) = N(dt, dζ) − ν(dζ)dt, +1Department of Mathematics, KTH Royal Institute of Technology 100 44, Stockholm, Sweden. +Email: nacira@kth.se, pucci@kth.se. +Work supported by the Swedish Research Council grant +(2020-04697). +2Department of Mathematics, University of Oslo, Norway. Email: oksendal@math.uio.no. +1 + +where N(dt, dζ) is a Poisson random measure and ν(dζ) the L´evy measure of N, and +a random variable Z ∈ L2(P) that is independent of F. We denote by L2(P) the set +of all the d−dimensional F-measurable random variables X such that E[X2] < ∞, +where E denotes the expectation with respect to P. We consider the state process +X(t) ∈ Rd given as the solution of the following conditional McKean-Vlasov jump +equation +X(t) = Z + +� t +0 +α(s, X(s), µs)dt + β(s, X(s), µs)dB(s) ++ +� t +0 +� +Rd γ(s, X(s−), µs−, ζ) � +N(ds, dζ), +(1.1) +where we denote by µt = L(X(t)|F(1) +t +) the conditional probability distribution of X(t) +given the filtration F(1) +t +generated by the first component B1(u); u ≤ t of the Brownian +motion up to time t. +Loosely speaking, the equation above models a McKean-Vlasov +dynamics which is subject to what is called a ”common noise” coming from the Brownian +motion B1(t), which is observed and is influencing the dynamics of the system. +So defined, µt is a Borel probability measure on Rd for all t ∈ [0, T], ω ∈ Ω. In particular, +µt ∈ M0, with M0 the set of deterministic Radon measures i.e. Borel measures finite on +compact sets, outer regular on all Borel sets and inner regular on all open sets. Notice that +all Borel probability measures on Rd are Radon measures. From now on we will indicate +with M the set of random measures λ(dx, ω) which are Radon measures with respect to x +for each ω. We refer to [9] for more information. +We suppose that α(t, x, µ): [0, T] × Rd × M → Rd, +β(t, x, µ): [0, T] × Rd × M → Rd×m, +γ(t, x, µ, ζ): [0, T] × Rd × M × Rd → Rd×k are bounded processes and F-predictable for all +x, µ, ζ and they are also continuous with respect to t and x for all µ, ζ. +We can easily see that, under hypothesis of Lipschitz continuity and at most linear growth, +there exists a unique solution for (1.1) for all t in [0, T]. +The purpose of this paper is to study impulse control problems for conditional McKean- +Vlasov jump diffusions. +In particular, we will define a performance criterion and then +attempt to find a policy that maximizes performance within the admissible impulse strate- +gies. Using a verification theorem approach, we establish a general form of quasi-variational +inequalities and identify the sufficient conditions that lead to an optimal function. See pre- +cise formulation below. Standard impulse control problems can be solved by using the +Dynkin formula. We refer to e.g. Bensoussan & Lions [4] in the continuous case and to +Øksendal and Sulem [12] in the setting of jump diffusions. +Impulse control problems naturally arise in many concrete applications, in particular when +an operator, because of the intervention costs, decides to control the system by intervening +only at a discrete set of times with a chosen intervention size: a sequence of stopping times +(τ1, τ2, . . . , τk, . . .) is chosen to intervene and exercise the control. At each time τk of the +player’s kth intervention, the player chooses an intervention of size ζk. The impulse control +2 + +consists of the sequence {(τk, ζk)}k≥1. +Impulse control has sparked great interest in the financial field and beyond. See, for ex- +ample, [10] for portfolio theory applications, [2] for energy markets, and [6] for insurance. +All of these works are based on quasi-variational inequalities and employ a verification +approach. +Despite its adaptability to more realistic financial models, few papers have studied the case +of mean field problems with impulse control. We refer to [3] for a discussion of a more +special type of impulse, where the only type of impulse is to add something to the system. +This is a mean field game (MFG) where the mean-field (only the empirical mean) appears +as an approximation of the many-player game. They use the smooth fit principle (as used +in the present work) to solve a specific MFG explicitly. +We refer also to [7] for a MFG impulse control approach. Specifically, a problem of optimal +harvesting in natural resource management is addressed. +A maximum principle for regime switching control problem for mean-field jump diffusions +is studied by [11] but in that paper the problem considered is not really an impulse control +problem because the intervention times are fixed in advance. +In our setting, we will not consider a MFG setup, as in the above mentioned works, we +will only consider a decision maker who chooses the control to optimise a certain reward. +Moreover, the mean-field appears as a conditional probability distribution and to over- +come the lack of the Markov property, we introduce the equation of the measure which is +of stochastic Fokker-Planck type. +In [8], the authors can handle a non-Markovian dynamics. However, the impulse control +is given in a particular compact form and only a given number of impulses are allowed. +They use a Snell envelope approach and related reflected backward stochastic differential +equations. +In the next section, we introduce some notations and present some preliminary results. As +part of Section 3, we state the optimal control problem and prove the verification theorem. +In Section 4, we apply the previous results to solve an explicit problem of optimal dividend +streams under transaction costs. +2 +Preliminaries +The process X(t) given by (1.1) is not in itself Markovian, so to be able to use the Dynkin +formula, we extend the system to the process Y defined by +Y (t) = (s + t, X(t), µt); +t ≥ 0; +Y (0) = (s, Z, µ0) =: y, +for some arbitrary starting time s ≥ 0, with state dynamics given by X(t), conditional law +of the state given by µt and with X(0) = Z, µ0 = L(X(0)). This system is Markovian, in +virtue of the following Fokker-Planck equation for the conditional law µt, proved in [1]. +3 + +Theorem 2.1 (Conditional stochastic Fokker-Planck equation) +Let X(t) be as in (1.1) and let µt = µt(dx, ω) be the regular conditional distribution of X(t) +given F(1) +t +. Then µt satisfies the following SPIDE (in the sense of distributions): +dµt = A∗ +0µtdt + A∗ +1µtdB1(t); +µ0 = L(X(0)), +(2.1) +where A∗ +0 is the integro-differential operator +A∗ +0µ = − +d +� +j=1 +Dj[αjµ] + 1 +2 +d +� +n,j=1 +Dn,j[(ββ(T))n,jµ] ++ +k +� +ℓ=1 +� +R +� +µ(γ(ℓ)) − µ + +d +� +j=1 +Dj[γ(ℓ) +j (s, ·, ζ)µ] +� +νℓ (dζ) , +and A∗ +1 is the differential operator +A∗ +1µ = − +d +� +j=1 +Dj[β1,jµ], +where β(T) denotes the transposed of the d × m - matrix β = +� +βj,k +� +1≤j≤d,1≤k≤m and γ(ℓ) is +column number ℓ of the matrix γ. +For notational simplicity, we use Dj, Dn,j to denote +∂ +∂xj and +∂2 +∂xn∂xj in the sense of +distributions. +We have also used the following notation, taken from [1]. +For fixed t, µ, ζ and ℓ = 1, 2, ...k, we write for simplicity γ(ℓ) = γ(ℓ)(t, x, µ, ζ) for column +number ℓ of the d × k-matrix γ. Then νℓ represents the L´evy measure of Nℓ for all ℓ. Note +that for given µ ∈ M the map +g �→ +� +Rd g(x + γ(ℓ))µ(dx) +is a bounded linear map on C0(Rd), which is defined to be the uniform closure of the space +Cc(Rd) of continuous functions with compact support. Therefore, since M is the dual of +C0(Rd), there is a unique measure µ(γ(ℓ)) ∈ M such that +⟨µ(γ(ℓ)), g⟩ := +� +Rd g(x)µ(γ(ℓ))(dx) = +� +Rd g(x + γ(ℓ))µ(dx), for all g ∈ C0(Rd), +where ⟨µ(γ(ℓ)), g⟩ denotes the action of the measure µ(γ(ℓ)) on g. We call µ(γ(ℓ)) the γ(ℓ)-shift +of µ. Note that µ(γ(ℓ)) is positive and absolutely continuous with respect to µ. +4 + +3 +A General Formulation and a Verification The- +orem +As noted above, in virtue of the Fokker-Planck equation (2.1) we can extend the system +(1.1) into a Markovian system by defining the following [0, ∞)×L2(P)×M - valued process +Y (t) := (s + t, X(t), µt) as follows: +dY (t) = F(Y (t))dt + G(Y (t))dB(t) + +� +Rk H(Y (t−), z) � +N(dt, dz) +:= + + +dt +dX(t) +dµt + + = + + +1 +α(Y (t)) +A∗ +0µt + + dt + + + +01×m +β(Y (t)) +A∗ +1µt, 0, 0..., 0 + + dB(t) ++ +� +Rd + + +01×k +γ(Y (t−), ζ) +01×k + + � +N(dt, dζ), +s ≤ t ≤ T, +(3.1) +where X(t) and µt satisfy the equations (1.1) and (2.1), respectively. Moreover, we have +used the shorthand notation +α(Y (t)) = α(s + t, X(t), µ(t)) +β(Y (t)) = β(s + t, X(t), µ(t)) +γ(Y (t−), ζ) = γ(s + t, X(t−), µ(t−), ζ). +The process Y (t) starts at y = (s, Z, µ). +We shall denote by µ the initial probability +distribution L(X(0)) or the generic value of the conditional law µt := L(X(t)|F(1) +t +), when +there is no ambiguity. Similarly, we use the following notation: +Notation 3.1 We use +• x to denote a generic value of the point X(t, ω) ∈ Rd, and +• X to denote a generic value of the random variable X(t, ·) ∈ L2(P). +• When the meaning is clear from the context we use x in both situations. +The concept of impulse control is simple and intuitive: at any time the agent can +make an intervention ζ into the system. Due to the cost of each intervention the agent can +intervene only at discrete times τ1, τ2, . . .. The impulse problem is to find out at what times +it is optimal to intervene and what is the corresponding optimal intervention sizes. We now +proceed to formulate precisely our impulse control problem for conditional McKean-Vlasov +jump diffusions. +Suppose that – if there are no interventions – the [0, ∞) × L2(P) × M - valued process +Y (t) = (s + t, X(t), µt) is the conditional McKean-Vlasov jump diffusion given by (3.1). +5 + +Suppose that at any time t and any state y = (s, X, µ) we are free to intervene and +give the state X an impulse ζ ∈ Z ⊂ Rd, where Z is a given set (the set of admissible +impulse values). Suppose the result of giving the state X the impulse ζ is that the state +jumps immediately from X to Γ(X, ζ), where Γ(X, ζ) : L2(P) × Z → L2(P) is a given +function. In many applications, the process shifts as a result of a simple translation, i.e. +Γ(y, ζ) = y + ζ. +Simultaneously, the conditional law jumps from µt = L(X|F(1) +t +) to +µΓ(X,ζ) +t +:= L(Γ(X, ζ)|F(1) +t +). +(3.2) +An impulse control for this system is a double (possibly finite) sequence +v = (τ1, τ2, . . . , τj, . . . ; ζ1, ζ2, . . . , ζj, . . .)j≤M, +M ≤ ∞, +where 0 ≤ τ1 ≤ τ2 ≤ · · · are Ft-stopping times (the intervention times) and ζ1, ζ2, . . . are +the corresponding impulses at these times. Mathematically, we assume that τj is a stopping +time with respect to a suitable filtration {Ft}t≥0, with τj+1 ≥ τj and ζj is Fτj-measurable +for all j. We let V denote the set of all impulse controls. +If v = (τ1, τ2, . . . ; ζ1, ζ2, . . .) ∈ V, the corresponding state process Y (v)(t) is defined by +Y (v)(0−) = y +and +Y (v)(t) = Y (t); +0 < t ≤ τ1, +(3.3) +Y (v)(τj) = +� +τj, Γ[ ˇX(v)(τ − +j ), ζj], L(Γ[ ˇX(v)(τ − +j ), ζj]|F1 +t ) +� +, +j = 1, 2, . . . +(3.4) +dY (v)(t) = F(Y (v)(t))dt + G(Y (v)(t))dB(t) ++ +� +Rk H(Y (v)(t−), z) � +N(dt, dz) +for τj < t < τj+1 ∧ τ ∗, +(3.5) +where we have used the notation +ˇX(v)(τ − +j ) = X(v)(τ − +j ) + ∆NX(τj), +∆NX(v)(t) being the jump of X(v) stemming from the jump of the random measure N(t, ·) +Note that we distinguish between the (possible) jump of X(v)(τj) stemming from the ran- +dom measure N, denoted by ∆NX(v)(τj) and the jump caused by the intervention v, given +by +∆vX(v)(τj) := Γ( ˇX(v)(τ − +j ), ζ) − ˇX(v)(τ − +j ). +Accordingly, at the time t = τj, X(v)(t) jumps from ˇX(v)(τ − +j ) to Γ[ ˇX(v)(τ − +j ), ζj] +and µτ − +j jumps to +µτj = L(Γ[ ˇX(v)(τ − +j ), ζj]|F1 +τj). +6 + +Consider a fixed open set (called the solvency region) S ⊂ [0, ∞) × Rd × M. It represents +the set in which the game takes place since it will end once the controlled process leaves +S. In portfolio optimization problems, for instance, the game ends in case of bankruptcy, +which may be modelled by choosing S to be the set of states where the capital is above a +certain threshold. Define +τS = inf{t ∈ (0, ∞); Y (v)(t) ̸∈ S}, +and +T = {τ ; stopping time, 0 ≤ τ ≤ τS} . +Suppose we are given a continuous profit function f : S → R and a continuous bequest +function g : S → R. Moreover, suppose the profit/utility of making an intervention with +impulse ζ ∈ Z when the state is y is K(y, ζ), where K : S × Z → R is a given continuous +function. +We assume we are given a set V of admissible impulse controls which is included in +the set of v = (τ1, τ2, . . . ; ζ1, ζ2, . . .) such that a unique solution Y (v) of (3.3)–(3.5) exist, +for all v ∈ V, and the following additional properties hold, assuring that the performance +functional below is well-defined: +Ey� � τS +0 +f −(Y (v)(s))ds +� +< ∞, +for all y ∈ S, v ∈ V, +Ey � +g−(Y (v)(τS))1[τS<∞] +� +< ∞, +for all y ∈ S, v ∈ V, +and +Ey + + � +τj≤τS +K−( ˇY (v)(τ − +j ), ζj) + + < ∞, +for all y ∈ S, v ∈ V, +where Ey denotes expectation given that Y (0) = y. +We now define the performance criterion, which consists of three parts: a continuous time +running profit in [0, τS], a terminal bequest value if the game ends, and a discrete-time +intervention profit, namely +J(v)(y) = Ey +� � τS +0 +f(Y (v)(t))dt + g(Y (v)(τS))1[τS<∞] + +� +τj≤τS +K( ˇY (v)(τ − +j ), ζj) +� +. +We consider the following impulse control problem: +Problem 3.2 Find Φ(y) and v∗ ∈ V such that +Φ(y) = sup{J(v)(y); v ∈ V} = J(v∗)(y), +y ∈ S. +The function Φ(y) is called the value function and v∗ is called an optimal control. +7 + +The following concept is crucial for the solution of this problem. +Definition 3.3 Let H be the space of all measurable functions h : S → R. The intervention +operator M : H → H is defined by +Mh(s, X, µ) = sup +ζ∈Z +{h(s, Γ(X, ζ), µΓ(X,ζ)) + K(y, ζ), ζ ∈ Z and (s, Γ(X, ζ), µΓ(X,ζ)) ∈ S}, +(3.6) +where µΓ(X,ζ) is given by (3.2). +Let C(1,2,2)(S) denote the family of functions ϕ(s, x, µ) : S → R which are continuously +differentiable w.r.t. +s and twice continuously Fr´echet differentiable w.r.t. +x ∈ Rd and +µ ∈ M. We let ∇µϕ ∈ L(M, R) (the set of bounded linear functionals on M) denote the +Fr´echet derivative (gradient) of ϕ with respect to µ ∈ M. +Similarly, D2 +µϕ denotes the +double derivative of ϕ with respect to µ and it belongs to L(M × M, R) (see Appendix for +further details). +The infinitesimal generator G of the Markov jump diffusion process Y (t) is defined on +functions ϕ ∈ C(1,2,2)(S) by +Gϕ = ∂ϕ +∂s + +d +� +j=1 +αj +∂ϕ +∂xj ++ ⟨∇µϕ, A∗ +0µ⟩ + 1 +2 +d +� +j,n=1 +(ββT )j,n +∂2ϕ +∂xj∂xn ++ 1 +2 +d +� +j=1 +βj,1 +∂ +∂xj +⟨∇µϕ, A∗ +1µ⟩ + 1 +2⟨A∗ +1µ, ⟨D2 +µϕ, A∗ +1µ⟩⟩ ++ +k +� +ℓ=1 +� +R +{ϕ(s, X + γ(ℓ), µ)) − ϕ(s, X, µ) − +d +� +j=1 +γ(ℓ) +j +∂ +∂xj ϕ(s, X, µ)}νℓ(dζ), +where, as before, A∗ +0 is the integro-differential operator +A∗ +0µ = − +d +� +j=1 +Dj[αjµ] + 1 +2 +d +� +n,j=1 +Dn,j[(ββ(T))n,jµ] ++ +k +� +ℓ=1 +� +R +� +µ(γ(ℓ)) − µ + +d +� +j=1 +Dj[γ(ℓ) +j (s, ·, ζ)µ] +� +νℓ (dζ) , +and +A∗ +1µ = − +d +� +j=1 +Dj[β1,jµ]. +We can now state a verification theorem for conditional McKean-Vlasov impulse control +problems, providing sufficient conditions that a given function is the value function and +8 + +a given impulse control is optimal. +The verification theorem links the impulse control +problem to a suitable system of quasi-variational inequalities. +Since the process Y (t) is Markovian, we can, with appropriate modifications, use the +approach in Chapter 9 in [12]. +For simplicity of notation we will in the following write +Γ(y, ζ) = (s, Γ(x, ζ), µΓ(x,ζ)), when y = (s, x, µ) ∈ [0, ∞) × L2(P) × M. +Theorem 3.4 Variational inequalities for conditional McKean-Vlasov impulse control +(a) Suppose we can find φ : ¯S → R such that +(i) φ ∈ C1(S) ∩ C( ¯S). +(ii) φ ≥ Mφ on S. +Define +D = {y ∈ S; φ(y) > Mφ(y)} +(the continuation region). +Assume +(iii) +Ey +�� τS +0 +Y (v)(t)1∂Ddt +� += 0 +for all y ∈ S, v ∈ V. +(iv) ∂D is a Lipschitz surface. +(v) φ ∈ C(1,2,2)(S \ ∂D) with locally bounded derivatives near ∂D. +(vi) Gφ + f ≤ 0 on S \ ∂D. +(vii) φ(y) = g(y) for all y ̸∈ S. +(viii) {φ−(Y (v)(τ)); τ ∈ T } is uniformly integrable, for all y ∈ S, v ∈ V. +(ix) Ey +� +|φ(Y (v)(τ))| + +� τS +0 +|Gφ(Y (v)(t))|dt +� +< ∞ for all τ ∈ T , v ∈ V, y ∈ S. +Then +φ(y) ≥ Φ(y) +for all y ∈ S. +(b) Suppose in addition that +(x) Gφ + f = 0 in D. +(xi) ˆζ(y) ∈ Argmax{φ(Γ(y, ·))+K(y, ·)} ∈ Z exists for all y ∈ S and ˆζ(·) is a Borel +measurable selection. +Put ˆτ0 = 0 and define ˆv = (ˆτ1, ˆτ2, . . . ; ˆζ1, ˆζ2, . . .) inductively by +ˆτj+1 = inf{t > ˆτj; Y (ˆvj)(t) ̸∈ D} ∧ τS and ˆζj+1 = ˆζ(Y (ˆvj)(ˆτ − +j+1)) +if ˆτj+1 < τS, where Y (ˆvj) is the result of applying +ˆvj := (ˆτ1, . . . , ˆτj; ˆζ1, . . . , ˆζj) to Y . Suppose +9 + +(xii) ˆv ∈ V and {φ(Y (ˆv)(τ)); τ ∈ T } is uniformly integrable. +Then +φ(y) = Φ(y) +and ˆv is an optimal impulse control. +Remark 3.5 We give the intuitive idea behind intervention operator as in (3.6): +MΦ(y) = sup +ζ∈Z +{Φ(Γ(y, ζ)) + K(y, ζ), ζ ∈ Z and Γ(y, ζ) ∈ S}, +(3.7) +Assume that the value function Φ is known. If y = (s, x, µ) is the current state of the pro- +cess, and the agent intervenes with impulse of size ζ, the resulting value can be represented +as Φ(Γ(y, ζ))+K(y, ζ), consisting of the sum of the value of Φ in the new state Γ(y, ζ) and +the intervention cost K. Therefore, MΦ(y) represents the optimal new value if the agent +decides to make an intervention at y. +Note that by (ii) Φ ≥ MΦ on S, so it is not always optimal to intervene. At the time +ˆτj, the operator should intervene with impulse ˆζj when the controlled process leaves the +continuation region, that is when Φ(Y ˆvj) ≤ MΦ(Y ˆvj). +Proof. +(a) By an approximation argument (see e.g. Theorem 3.1 in [12]) and (iii)–(v), +we may assume that φ∈C2(S)∩C( ¯S). Choose v=(τ1, τ2, . . . ; ζ1, ζ2, . . .)∈V and set τ0 = 0. +By another approximation argument we may assume that we can apply the Dynkin formula +to the stopping times τj. Then for j = 0, 1, 2, . . ., with Y = Y (v) +Ey[φ(Y (τj))] − Ey[φ( ˇY (τ − +j+1))] = −Ey +�� τj+1 +τj +Gφ(Y (t))dt +� +, +where ˇY (τ − +j+1) = Y (τ − +j+1) + ∆NY (τj+1), as before. Summing this from j = 0 to j = m we +get +φ(y) + +m +� +j=1 +Ey[φ(Y (τj)) − φ( ˇY (τ − +j ))] − Ey[φ( ˇY (τ − +m+1))] += −Ey +�� τm+1 +0 +Gφ(Y (t))dt +� +≥ Ey +�� τm+1 +0 +f(Y (t))dt +� +. +(3.8) +Now +φ(Y (τj)) = φ(Γ( ˇY (τ − +j ), ζj)) +≤ Mφ( ˇY (τ − +j )) − K( ˇY (τ − +j ), ζj) +if τj < τS by (3.6) +and +φ(Y (τj)) = φ( ˇY (τ − +j )) +if τj = τS by (vii). +10 + +Therefore +Mφ( ˇY (τ − +j )) − φ( ˇY (τ − +j )) ≥ φ(Y (τj)) − φ( ˇY (τ − +j )) + K( ˇY (τ − +j ), ζj), +and +φ(y) + +m +� +j=1 +Ey[{Mφ( ˇY (τ − +j )) − φ( ˇY (τ − +j ))}1[τj<τS]] +≥ Ey + + +� τm+1 +0 +f(Y (t))dt + φ( ˇY (τ − +m+1)) + +m +� +j=1 +K( ˇY (τ − +j ), ζj) + + . +Letting m → M and using quasi-left continuity of Y (·), we get +φ(y) ≥ Ey + + +� τS +0 +f(Y (t))dt + g(Y (τS))1[τS<∞] + +M +� +j=1 +K( ˇY (τ − +j ), ζj) + +=J(v)(y). +(3.9) +Hence φ(y) ≥ Φ(y). +(b) Next assume (x)–(xii) also hold. Apply the above argument to ˆv = (ˆτ1, ˆτ2, . . . ; ˆζ1, ˆζ2, . . .). +Then by (x) we get equality in (3.8) and by our choice of ζj = ˆζj we have equality in (3.9). +Hence +φ(y) = J(ˆv)(y), +which combined with (a) completes the proof. +□ +4 +Example: Optimal stream of dividends under +transaction costs +In this Section, we solve explicitly an optimal stream of dividends under transaction costs. +To this end, for v = (τ1, τ2, . . . ; ζ1, ζ2, . . .) with ζi ∈ R+, we define +Y (v)(t) = (s + t, X(v)(t), µ(v) +t ) by +dX(t) = E +� +X(t) | F(1) +t +� � +α0dt + σ1dB1(t) + σ2dB2(t) + +� +R +γ0(ζ) � +N(dt, dζ) +� +, +(4.1) +µ(v) +t += L(X(v)(t)|F(1) +t +); +τi < t < τi+1, +X(v)(τi+1) = ˇX(v)(τ − +i+1) − (1 + λ)ζi+1 − c, +µ(v) +τi+1 = L(X(v)(τi+1)|F(1) +τi+1); +i = 0, 1, 2, . . . , +X(v)(0−) = x > 0; a.s., +11 + +where α0, σ1 ̸= 0, σ2 ̸= 0, λ ≥ 0, and c > 0 are constants with −1 ≤ γ0(z) a.s. ν. +Here X(t) represents the amount available at time t of a cash flow. We assume that it +satisfies the McKean-Vlasov equation in (4.1). Note that at any time τi, i = 0, 1, 2, . . . , +the system jumps from ˇX(v)(τ − +i ) to +X(v)(τi) = Γ[ ˇX(v)(τ − +i ), ζi] = ˇX(v)(τ − +i ) − (1 + λ)ζi − c, +where the quantity c + λζi represents the transaction cost with a fixed part c and a pro- +portional part λζi, while ζi is the amount we decide to take out at time τi. +At the same time µτ − +i jumps to +µτi = L( ˇX(v)(τ − +i )|F1 +τi). +Problem 4.1 We want to find Φ and v∗ ∈ V such that +Φ(s, x, µ) = sup +v J(v)(s, x, µ) = J(v∗)(s, x, µ), +where +J(v)(s, x, µ) = J(v)(y) = Ey +� � +τk<τS +e−ρ(s+τk)ζk +� +(ρ > 0 constant) +is the expected discounted total dividend up to time τS, where +τS = τS(ω) = inf{t > 0; P y[Ey[X(v)(t)|F(1) +t +] ≤ 0] > 0} +is the time of bankruptcy. +To put this problem into the context above, we define +Y (v)(t) = + + +s + t +X(v)(t) +µ(v) +t + + , +Y (v)(0−) = + + +s +x +µ + + = y, +Γ(y, ζ) = Γ(s, x, µ) = (s, x − c − (1 + λ)ζ, L(x − c − (1 + λ)ζ)|F(1)), +x ∈ L2(P), +K(y, ζ) = e−ρsζ, +f ≡ g ≡ 0, +S = {(s, x, µ) : x > 0 a.s. } . +Comparing with our Theorem, we see that in this case we have d = 1, m = 2, k = 1 and +α1 = α0 ⟨µ, q⟩ , β1 = σ1 ⟨µ, q⟩ , β2 = σ2 ⟨µ, q⟩ , γ(s, x, µ, ζ) = γ0(t, ζ) ⟨µ, q⟩ , +12 + +where we have put q(x) = x so that ⟨µt, q⟩ = E +� +X(t) | F(1) +t +� +. +Therefore the operator G takes the form +Gϕ(s, x, µ) += +∂ϕ +∂s + α0 ⟨µ, q⟩ ∂ϕ +∂x + ⟨∇µϕ, A∗ +0µ⟩ ++ 1 +2(σ2 +1 + σ2 +2) ⟨µ, q⟩2 ∂2ϕ +∂x2 + 1 +2σ1 ⟨µ, q⟩ ∂ +∂x ⟨∇µϕ, A∗ +1µ⟩ ++ 1 +2 +� +A∗ +1µ, +� +D2 +µϕ, A∗ +1µ +�� ++ +� +R +� +ϕ(s, x + γ0 ⟨µ, q⟩ , µ) − ϕ(s, x, µ) − γ0 ⟨µ, q⟩ ∂ +∂xϕ(s, x, µ) +� +ν(dζ), +where +A∗ +0µ = −D[α0 ⟨µ, q⟩ µ] + 1 +2D2[(σ2 +1 + σ2 +2) ⟨µ, q⟩2 µ], +and +A∗ +1µ = −D[σ1 ⟨µ, q⟩ µ]. +The adjoints of the last two operators are +A0µ = α0 ⟨µ, q⟩ Dµ + 1 +2(σ2 +1 + σ2 +2) ⟨µ, q⟩2 D2µ, +and +A1µ = σ1 ⟨µ, q⟩ Dµ. +In this case the intervention operator gets the form +Mh(s, x, µ) = sup +� +h(s, x − c − (1 + λ)ζ, µx−c−(1+λ)ζ) + e−ρtζ; +0 ≤ ζ ≤ x − c +1 + λ +� +. +Note that the condition on ζ is due to the fact that the impulse must be positive and +x − c − (1 + λ)ζ must belong to S. We distinguish between two cases: +1. α0 > ρ. In this case, suppose we wait until some time t1 and then take out +ζ1 = X(t1) − c +1 + λ +. +Noting that Ey|X(t)] = x exp(α0t) for t < t1, we see that the corresponding performance +is +J(v1)(s, x, µ) = Ey +� +e−ρ(t1+s) +1 + λ +(X(t1) − c) +� += Ex +� +1 +1 + λ +� +xe−ρse(α0−ρ)t1 − c e−ρ(s+t1)� +� +→ ∞ as t1 → ∞. +13 + +Therefore we obtain Φ(s, x, µ) = +∞ in this case. +2. α0 < ρ. We look for a solution by using the results of Theorem 3.4. +We guess that the continuation region is of the form +D = {(s, x, µ) : 0 < ⟨µ, q⟩ < ¯x} +for some ¯x > 0 (to be determined), and in D we try a value function of the form +ϕ(s, x, µ) = e−ρsψ(⟨µ, q⟩). +This gives +Gφ(s, x, µ) = e−ρsG0ψ(⟨µ, q⟩), where +G0ψ(x, µ) = −ρψ(⟨µ, q⟩) + ⟨∇µψ, A∗ +0µ⟩ + 1 +2σ1 ⟨µ, q⟩ ∂ +∂x ⟨∇µψ, A∗ +1µ⟩ ++ 1 +2 +� +A∗ +1µ, +� +D2 +µψ, A∗ +1µ +�� ++ +� +R +� +ψ(x + γ0 ⟨µ, q⟩ , µ) − ψ(x, µ) − γ0 ⟨µ, q⟩ ∂ +∂xψ(x, µ) +� +ν(dζ). +By the chain rule for Fr´echet derivatives (see Appendix), we have +∇µψ(h) = ψ′(⟨µ, q⟩)⟨h, q⟩ and D2 +µψ(h, k) = ψ′′(⟨µ, q⟩)⟨h, q⟩⟨k, q⟩. +Therefore, +⟨∇µψ, A∗ +0µ⟩ = ψ′(⟨µ, q⟩)⟨A∗ +0µ, q⟩ = ψ′(⟨µ, q⟩)⟨µ, A0q⟩ = ψ′(⟨µ, q⟩)α0⟨µ, q⟩, +and similarly +1 +2⟨A∗ +1µ, ⟨D2 +µψ, A∗ +1µ⟩⟩ = 1 +2ψ′′(⟨µ, q⟩⟨A∗ +1µ, q⟩⟨A∗ +1µ, q⟩ = 1 +2ψ′′(⟨µ, q⟩)⟨µ, A1q⟩⟨µ, A1q⟩ += 1 +2ψ′′(⟨µ, q⟩)σ2 +1⟨µ, q⟩2. +Moreover, since ψ does not depend on x we see that +� +R +� +φ(s, x + γ0 ⟨µ, q⟩ , µ) − φ(s, x, µ) − γ0 ⟨µ, q⟩ ∂φ +∂x(s, x, µ) +� +ν(dζ) = 0. +Substituting this into the expression for G0ψ we get, with u = ⟨µ, q⟩, +G0ψ(u) = −ρψ(u) + α0uψ′(u) + 1 +2σ2 +1u2ψ′′(u). +By condition (x) we are required to have G0ψ(u) = 0 for all u ∈ (0, ¯x), and this equation +has the general solution +ψ(u) = C1uγ1 + C2uγ2, +u ∈ (0, ¯x), +14 + +where γ1 > 1, γ2 < 0, and C1, C2 are constants. Since we expect φ to be bounded near 0, +we guess that C2 = 0. +We guess that it is optimal to wait till u = ⟨µt, q⟩ = Ey[X(t)|F(1) +t +] reaches or exceeds a +value u = ¯u > c and then take out as much as possible, i.e., reduce Ey[X(t)|F(1) +t +] to 0. +Taking the transaction costs into account this means that we should take out +ˆζ(u) = u − c +1 + λ for u ≥ ¯u. +We therefore propose that ψ(u) has the form +ψ(u) = + + + +C1uγ1for 0 < u < ¯u +u − c +1 + λ for u ≥ ¯u. +Continuity and differentiability of ψ(u) at u = ¯u give the equations +C1¯uγ1 = ¯u − c +1 + λ, +and +C1γ1¯uγ1−1 = +1 +1 + λ. +Combining these we get +¯u = +γ1c +γ1 − 1 +and +C1 = ¯u − c +1 + λ ¯u−γ1. +With these values of ¯u and C1, we have to verify that ψ satisfies all the requirements of +Theorem 3.4. We check some of them: +(ii) φ ≥ Mφ on S. +In our case we have Γ(s, X, µ) = (s, X − c − (1 + λ)ζ, µX−c−(1+λ)ζ) and hence we get +Mφ(s, X, µ) = sup +ζ +� +φ(s, X − c − (1 + λ)ζ), µX−c−(1+λ)ζ) + e−ρsζ; 0 ≤ ζ ≤ ¯u − c +1 + λ +� += e−ρs sup +ζ +� +C1⟨µX−c−(1+λ)ζ, q⟩γ1 + ζ; 0 ≤ ζ ≤ ¯u − c +1 + λ +� += e−ρs sup +ζ +� +C1(⟨µ, q(x) − c − (1 + λ)ζ⟩γ1 + ζ; 0 ≤ ζ ≤ ¯u − c +1 + λ +� += e−ρs sup +ζ +� +C1(⟨µ, q⟩ − c − (1 + λ)ζ)γ1 + ζ; 0 ≤ ζ ≤ ¯u − c +1 + λ +� +. +If u − c − (1 + λ)ζ ≥ ¯u, then +ψ(u − c − (1 + λ)ζ) + ζ = u − 2c +1 + λ < u − c +1 + λ = ψ(u) +15 + +and if u − c − (1 + λ)ζ < ¯u then +h(ζ) := ψ(u − c − (1 + λ)ζ) + ζ = C1(u − c − (1 + λ)ζ)γ1 + ζ. +Since +h′ +�u − c +1 + λ +� += 1 and h′′(ζ) > 0, +we see that the maximum value of h(ζ); 0 ≤ ζ ≤ u − c +1 + λ, is attained at ζ = ˆζ(u) = u − c +1 + λ. +Therefore +Mψ(u) = max +�x − 2c +1 + λ , u − c +1 + λ +� += u − c +1 + λ for all u > c. +Hence Mψ(u) = ψ(u) for u ≥ ¯u. +For 0 < u < ¯u consider +k(u) := C1uγ1 − u − c +1 + λ. +Since +k(¯u) = k′(¯u) = 0 +and +k′′(u) > 0 for all u, +we conclude that +k(u) > 0 +for 0 < u < ¯u. +Hence +ψ(u) > Mψ(u) +for 0 < u < ¯u. +(vi) A0ψ(u) ≤ 0 for u ∈ S\ ¯D i.e., for u > ¯u. For u > ¯u, we have +A0ψ(u) = −ρ u − c +1 + λ + α0u +1 +1 + λ ++ +� +u+γuz<¯u +� +C1(u + γuz)γ1 − u + γuz − c +1 + λ +� +ν(dz) +≤ (1 + λ)−1[(µ − ρ)u + (ρ + ∥ν∥)c]. +16 + +Therefore we see that +A0ψ(u) ≤ 0 for all u > ¯u +⇔ (α0 − ρ)u + (ρ + ∥ν∥)c ≤ 0 for all u > ¯u +⇔ (α0 − ρ)¯u + (ρ + ∥ν∥)c ≤ 0 +⇔ ¯u ≥ (ρ + ∥ν∥)c +ρ − α0 +⇔ +γ1c +γ1 − 1 ≥ (ρ + ∥ν∥)c +ρ − α0 +⇔ γ1 ≤ ρ + ∥ν∥ +α0 + ∥ν∥. +Since +F +� ρ +µ +� +≥ −ρ + α0 +ρ +α0 ++ 1 +2σ2 ρ +α0 +� ρ +α0 +− 1 +� +> 0, +and F(γ1) = 0, γ1 > 1 we conclude that γ1 < +ρ +α0 and hence (vi) holds if ∥ν∥ is small +enough, say ∥ν∥ ≤ K. +Therefore, we have the following. +Theorem 4.2 Suppose ∥ν∥ ≤ K. Then the value function for Problem 4.1 is +Φ(s, x, µ) = + + + +e−ρsC1uγ1for 0 < u < ¯u +e−ρs u − c +1 + λ for u ≥ ¯u. +where u = ⟨µ, q⟩ = E[X(t)|F(1) +t +] and +¯u = +γ1c +γ1 − 1 +and +C1 = ¯u − c +1 + λ ¯u−γ1. +and γ = γ1 > 1 is the positive solution of the equation +−ρ + α0γ + 1 +2σ2 +1γ(γ − 1) = 0. +The optimal impulse control is to do nothing while u = E[X(t)|F(1) +t +] < ¯u and take out +immediately +ˆζ(u) = u − c +1 + λ when u ≥ ¯u. +This brings E[X(t)|F(1) +t +] down to 0, and the system stops. +Hence the optimal impulse +consists of at most one intervention. +17 + +5 +Appendix: Double Fr´echet derivatives +In this section we recall some basic facts we are using about the Fr´echet derivatives of a +function f : V �→ W, where V, W are given Banach spaces. +Definition 5.1 We say that f has a Fr´echet derivative ∇xf = Df(x) at x ∈ V if there +exists a bounded linear map A : V �→ W such that +lim +h→0 +||f(x + h) − f(x) − A(h)||W +||h||V += 0. +Then we call A the Fr´echet derivative of f at x and we put Df(x) = A. +Note that Df(x) ∈ L(V, W) (the space of bounded linear functions from V to W), for each +x. +Definition 5.2 We say that f has a double Fr´echet derivative D2f(x) at x if there exists +a bounded bilinear map A(h, k) : V × V �→ W such that +lim +k→0 +||Df(x + k)(h) − Df(x)(h) − A(h, k)||W +||h||V += 0. +Example 5.3 +• Suppose f : M �→ R is given by +f(µ) = ⟨µ, q⟩2, where q(x) = x. +Then +f(µ + h) − f(µ) = ⟨µ + h, q⟩2 − ⟨µ, q⟩2 += 2⟨µ, q⟩⟨h, q⟩ + ⟨h, q⟩2, +so we see that +Df(µ)(h) = 2⟨µ, q⟩⟨h, q⟩. +To find the double derivative we consider +Df(µ + k)(h) − Df(µ)(h) += 2⟨µ + k, q⟩⟨h, q⟩ − 2⟨µ, q⟩⟨h, q⟩ += 2⟨k, q⟩⟨h, q⟩, +and we conclude that +D2f(µ)(h, k) = 2⟨k, q⟩⟨h, q⟩. +• Next assume that g : M �→ R is given by g(µ) = ⟨µ, q⟩. Then, proceeding as above we +find that +Dg(µ)(h) = ⟨h, q⟩ (independent of µ) +and +D2g(µ) = 0. +18 + +References +[1] Agram, N., & Øksendal, B. (2021). Stochastic Fokker-Planck PIDE for conditional +McKean-Vlasov jump diffusions and applications to optimal control. arXiv preprint +arXiv:2110.02193v3. +[2] Basei, M. (2019). Optimal price management in retail energy markets: an impulse +control problem with asymptotic estimates. Mathematical Methods of Operations Re- +search, 89(3), 355-383. +[3] Basei, M., Cao, H., & Guo, X. (2022). Nonzero-sum stochastic games and mean-field +games with impulse controls. Mathematics of Operations Research, 47(1), 341-366. +[4] Bensoussan, A. (1984). Impulse control and quasi-variational inequalities. +[5] Bertucci, C. (2020). Fokker-Planck equations of jumping particles and mean field +games of impulse control. Annales de l’Institut Henri Poincar´e C, 37(5), 1211-1244. +[6] Cadenillas, A., Choulli, T., Taksar, M., & Zhang, L. (2006). Classical and impulse +stochastic control for the optimization of the dividend and risk policies of an insurance +firm. Mathematical Finance: An International Journal of Mathematics, Statistics and +Financial Economics, 16(1), 181-202. Chicago +[7] Christensen, S., Neumann, B. A., & Sohr, T. (2021). Competition versus Cooperation: +A class of solvable mean field impulse control problems. SIAM Journal on Control and +Optimization, 59(5), 3946-3972. +[8] Djehiche, B., Hamadene, S., & Hdhiri, I. (2010). Stochastic impulse control of non- +Markovian processes. Applied mathematics and optimization, 61(1), 1-26. +[9] Folland, G.B. (1984). Real Analysis. Modern Techniques and Their Applications. Wi- +ley. +[10] Korn, R. (1999). Some applications of impulse control in mathematical finance. Math- +ematical Methods of Operations Research, 50(3), 493-518. +[11] Li, C., Liu, Z., Wu, J., & Huang, X. (2020). The stochastic maximum principle for a +jump-diffusion mean-field model involving impulse controls and applications in finance. +Journal of Systems Science and Complexity, 33(1), 26-42. Chicago +[12] Øksendal, B. & Sulem, A. (2019). Applied Stochastic Control of Jump diffusions. 3rd +edition. Springer. +19 + diff --git a/FNAzT4oBgHgl3EQfiv3U/content/tmp_files/load_file.txt b/FNAzT4oBgHgl3EQfiv3U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b159ea64db35dd44900edc4c52a509408566b48f --- /dev/null +++ b/FNAzT4oBgHgl3EQfiv3U/content/tmp_files/load_file.txt @@ -0,0 +1,488 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf,len=487 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='01506v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='OC] 4 Jan 2023 Impulse control of conditional McKean-Vlasov jump diffusions Nacira Agram1, Giulia Pucci1 & Bernt Øksendal2 January 5, 2023 Abstract This paper establishes a verification theorem for impulse control problems involving conditional McKean-Vlasov jump diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We obtain a Markovian system by combining the state equation of the problem with the stochastic Fokker-Planck equation for the conditional probability law of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We obtain the solution explic- itly by finding a function and an associated impulse control which satisfy the verification theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Keywords : Jump diffusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' impulse control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' common noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' conditional McKean- Vlasov differential equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' stochastic Fokker-Planck equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' quasi-variational in- equalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 1 Introduction Consider a filtered probability space (Ω, F, P, F = {F}t≥0) on which we are given a d-dimensional Brownian motion B = (B1, B2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' , Bd), a k-dimensional compensated Poisson random measure �N(dt, dζ) such that �N(dt, dζ) = N(dt, dζ) − ν(dζ)dt, 1Department of Mathematics, KTH Royal Institute of Technology 100 44, Stockholm, Sweden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Email: nacira@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='se, pucci@kth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Work supported by the Swedish Research Council grant (2020-04697).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 2Department of Mathematics, University of Oslo, Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Email: oksendal@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='uio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 1 where N(dt, dζ) is a Poisson random measure and ν(dζ) the L´evy measure of N, and a random variable Z ∈ L2(P) that is independent of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We denote by L2(P) the set of all the d−dimensional F-measurable random variables X such that E[X2] < ∞, where E denotes the expectation with respect to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We consider the state process X(t) ∈ Rd given as the solution of the following conditional McKean-Vlasov jump equation X(t) = Z + � t 0 α(s, X(s), µs)dt + β(s, X(s), µs)dB(s) + � t 0 � Rd γ(s, X(s−), µs−, ζ) � N(ds, dζ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) where we denote by µt = L(X(t)|F(1) t ) the conditional probability distribution of X(t) given the filtration F(1) t generated by the first component B1(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' u ≤ t of the Brownian motion up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Loosely speaking, the equation above models a McKean-Vlasov dynamics which is subject to what is called a ”common noise” coming from the Brownian motion B1(t), which is observed and is influencing the dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' So defined, µt is a Borel probability measure on Rd for all t ∈ [0, T], ω ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In particular, µt ∈ M0, with M0 the set of deterministic Radon measures i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Borel measures finite on compact sets, outer regular on all Borel sets and inner regular on all open sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Notice that all Borel probability measures on Rd are Radon measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' From now on we will indicate with M the set of random measures λ(dx, ω) which are Radon measures with respect to x for each ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We refer to [9] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We suppose that α(t, x, µ): [0, T] × Rd × M → Rd, β(t, x, µ): [0, T] × Rd × M → Rd×m, γ(t, x, µ, ζ): [0, T] × Rd × M × Rd → Rd×k are bounded processes and F-predictable for all x, µ, ζ and they are also continuous with respect to t and x for all µ, ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We can easily see that, under hypothesis of Lipschitz continuity and at most linear growth, there exists a unique solution for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) for all t in [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The purpose of this paper is to study impulse control problems for conditional McKean- Vlasov jump diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In particular, we will define a performance criterion and then attempt to find a policy that maximizes performance within the admissible impulse strate- gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Using a verification theorem approach, we establish a general form of quasi-variational inequalities and identify the sufficient conditions that lead to an optimal function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' See pre- cise formulation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Standard impulse control problems can be solved by using the Dynkin formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We refer to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Bensoussan & Lions [4] in the continuous case and to Øksendal and Sulem [12] in the setting of jump diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Impulse control problems naturally arise in many concrete applications, in particular when an operator, because of the intervention costs, decides to control the system by intervening only at a discrete set of times with a chosen intervention size: a sequence of stopping times (τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' , τk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=') is chosen to intervene and exercise the control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' At each time τk of the player’s kth intervention, the player chooses an intervention of size ζk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The impulse control 2 consists of the sequence {(τk, ζk)}k≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Impulse control has sparked great interest in the financial field and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' See, for ex- ample, [10] for portfolio theory applications, [2] for energy markets, and [6] for insurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' All of these works are based on quasi-variational inequalities and employ a verification approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Despite its adaptability to more realistic financial models, few papers have studied the case of mean field problems with impulse control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We refer to [3] for a discussion of a more special type of impulse, where the only type of impulse is to add something to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' This is a mean field game (MFG) where the mean-field (only the empirical mean) appears as an approximation of the many-player game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' They use the smooth fit principle (as used in the present work) to solve a specific MFG explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We refer also to [7] for a MFG impulse control approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Specifically, a problem of optimal harvesting in natural resource management is addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' A maximum principle for regime switching control problem for mean-field jump diffusions is studied by [11] but in that paper the problem considered is not really an impulse control problem because the intervention times are fixed in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In our setting, we will not consider a MFG setup, as in the above mentioned works, we will only consider a decision maker who chooses the control to optimise a certain reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Moreover, the mean-field appears as a conditional probability distribution and to over- come the lack of the Markov property, we introduce the equation of the measure which is of stochastic Fokker-Planck type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In [8], the authors can handle a non-Markovian dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' However, the impulse control is given in a particular compact form and only a given number of impulses are allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' They use a Snell envelope approach and related reflected backward stochastic differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In the next section, we introduce some notations and present some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' As part of Section 3, we state the optimal control problem and prove the verification theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In Section 4, we apply the previous results to solve an explicit problem of optimal dividend streams under transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 2 Preliminaries The process X(t) given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) is not in itself Markovian, so to be able to use the Dynkin formula, we extend the system to the process Y defined by Y (t) = (s + t, X(t), µt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' t ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Y (0) = (s, Z, µ0) =: y, for some arbitrary starting time s ≥ 0, with state dynamics given by X(t), conditional law of the state given by µt and with X(0) = Z, µ0 = L(X(0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' This system is Markovian, in virtue of the following Fokker-Planck equation for the conditional law µt, proved in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 3 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1 (Conditional stochastic Fokker-Planck equation) Let X(t) be as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) and let µt = µt(dx, ω) be the regular conditional distribution of X(t) given F(1) t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then µt satisfies the following SPIDE (in the sense of distributions): dµt = A∗ 0µtdt + A∗ 1µtdB1(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' µ0 = L(X(0)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) where A∗ 0 is the integro-differential operator A∗ 0µ = − d � j=1 Dj[αjµ] + 1 2 d � n,j=1 Dn,j[(ββ(T))n,jµ] + k � ℓ=1 � R � µ(γ(ℓ)) − µ + d � j=1 Dj[γ(ℓ) j (s, ·, ζ)µ] � νℓ (dζ) , and A∗ 1 is the differential operator A∗ 1µ = − d � j=1 Dj[β1,jµ], where β(T) denotes the transposed of the d × m - matrix β = � βj,k � 1≤j≤d,1≤k≤m and γ(ℓ) is column number ℓ of the matrix γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' For notational simplicity, we use Dj, Dn,j to denote ∂ ∂xj and ∂2 ∂xn∂xj in the sense of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We have also used the following notation, taken from [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' For fixed t, µ, ζ and ℓ = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='k, we write for simplicity γ(ℓ) = γ(ℓ)(t, x, µ, ζ) for column number ℓ of the d × k-matrix γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then νℓ represents the L´evy measure of Nℓ for all ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Note that for given µ ∈ M the map g �→ � Rd g(x + γ(ℓ))µ(dx) is a bounded linear map on C0(Rd), which is defined to be the uniform closure of the space Cc(Rd) of continuous functions with compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Therefore, since M is the dual of C0(Rd), there is a unique measure µ(γ(ℓ)) ∈ M such that ⟨µ(γ(ℓ)), g⟩ := � Rd g(x)µ(γ(ℓ))(dx) = � Rd g(x + γ(ℓ))µ(dx), for all g ∈ C0(Rd), where ⟨µ(γ(ℓ)), g⟩ denotes the action of the measure µ(γ(ℓ)) on g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We call µ(γ(ℓ)) the γ(ℓ)-shift of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Note that µ(γ(ℓ)) is positive and absolutely continuous with respect to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 4 3 A General Formulation and a Verification The- orem As noted above, in virtue of the Fokker-Planck equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) we can extend the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) into a Markovian system by defining the following [0, ∞)×L2(P)×M - valued process Y (t) := (s + t, X(t), µt) as follows: dY (t) = F(Y (t))dt + G(Y (t))dB(t) + � Rk H(Y (t−), z) � N(dt, dz) := \uf8ee \uf8f0 dt dX(t) dµt \uf8f9 \uf8fb = \uf8ee \uf8f0 1 α(Y (t)) A∗ 0µt \uf8f9 \uf8fb dt + \uf8ee \uf8f0 01×m β(Y (t)) A∗ 1µt, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', 0 \uf8f9 \uf8fb dB(t) + � Rd \uf8ee \uf8f0 01×k γ(Y (t−), ζ) 01×k \uf8f9 \uf8fb � N(dt, dζ), s ≤ t ≤ T, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) where X(t) and µt satisfy the equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Moreover, we have used the shorthand notation α(Y (t)) = α(s + t, X(t), µ(t)) β(Y (t)) = β(s + t, X(t), µ(t)) γ(Y (t−), ζ) = γ(s + t, X(t−), µ(t−), ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The process Y (t) starts at y = (s, Z, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We shall denote by µ the initial probability distribution L(X(0)) or the generic value of the conditional law µt := L(X(t)|F(1) t ), when there is no ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Similarly, we use the following notation: Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1 We use x to denote a generic value of the point X(t, ω) ∈ Rd, and X to denote a generic value of the random variable X(t, ·) ∈ L2(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' When the meaning is clear from the context we use x in both situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The concept of impulse control is simple and intuitive: at any time the agent can make an intervention ζ into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Due to the cost of each intervention the agent can intervene only at discrete times τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='. The impulse problem is to find out at what times it is optimal to intervene and what is the corresponding optimal intervention sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We now proceed to formulate precisely our impulse control problem for conditional McKean-Vlasov jump diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Suppose that – if there are no interventions – the [0, ∞) × L2(P) × M - valued process Y (t) = (s + t, X(t), µt) is the conditional McKean-Vlasov jump diffusion given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 5 Suppose that at any time t and any state y = (s, X, µ) we are free to intervene and give the state X an impulse ζ ∈ Z ⊂ Rd, where Z is a given set (the set of admissible impulse values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Suppose the result of giving the state X the impulse ζ is that the state jumps immediately from X to Γ(X, ζ), where Γ(X, ζ) : L2(P) × Z → L2(P) is a given function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In many applications, the process shifts as a result of a simple translation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Γ(y, ζ) = y + ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Simultaneously, the conditional law jumps from µt = L(X|F(1) t ) to µΓ(X,ζ) t := L(Γ(X, ζ)|F(1) t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='2) An impulse control for this system is a double (possibly finite) sequence v = (τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' , τj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' , ζj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' )j≤M, M ≤ ∞, where 0 ≤ τ1 ≤ τ2 ≤ · · · are Ft-stopping times (the intervention times) and ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' are the corresponding impulses at these times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Mathematically, we assume that τj is a stopping time with respect to a suitable filtration {Ft}t≥0, with τj+1 ≥ τj and ζj is Fτj-measurable for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We let V denote the set of all impulse controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' If v = (τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=') ∈ V, the corresponding state process Y (v)(t) is defined by Y (v)(0−) = y and Y (v)(t) = Y (t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 0 < t ≤ τ1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='3) Y (v)(τj) = � τj, Γ[ ˇX(v)(τ − j ), ζj], L(Γ[ ˇX(v)(τ − j ), ζj]|F1 t ) � , j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='4) dY (v)(t) = F(Y (v)(t))dt + G(Y (v)(t))dB(t) + � Rk H(Y (v)(t−), z) � N(dt, dz) for τj < t < τj+1 ∧ τ ∗, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='5) where we have used the notation ˇX(v)(τ − j ) = X(v)(τ − j ) + ∆NX(τj), ∆NX(v)(t) being the jump of X(v) stemming from the jump of the random measure N(t, ·) Note that we distinguish between the (possible) jump of X(v)(τj) stemming from the ran- dom measure N, denoted by ∆NX(v)(τj) and the jump caused by the intervention v, given by ∆vX(v)(τj) := Γ( ˇX(v)(τ − j ), ζ) − ˇX(v)(τ − j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Accordingly, at the time t = τj, X(v)(t) jumps from ˇX(v)(τ − j ) to Γ[ ˇX(v)(τ − j ), ζj] and µτ − j jumps to µτj = L(Γ[ ˇX(v)(τ − j ), ζj]|F1 τj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 6 Consider a fixed open set (called the solvency region) S ⊂ [0, ∞) × Rd × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' It represents the set in which the game takes place since it will end once the controlled process leaves S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In portfolio optimization problems, for instance, the game ends in case of bankruptcy, which may be modelled by choosing S to be the set of states where the capital is above a certain threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Define τS = inf{t ∈ (0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Y (v)(t) ̸∈ S}, and T = {τ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' stopping time, 0 ≤ τ ≤ τS} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Suppose we are given a continuous profit function f : S → R and a continuous bequest function g : S → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Moreover, suppose the profit/utility of making an intervention with impulse ζ ∈ Z when the state is y is K(y, ζ), where K : S × Z → R is a given continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We assume we are given a set V of admissible impulse controls which is included in the set of v = (τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=') such that a unique solution Y (v) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='3)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='5) exist, for all v ∈ V, and the following additional properties hold, assuring that the performance functional below is well-defined: Ey� � τS 0 f −(Y (v)(s))ds � < ∞, for all y ∈ S, v ∈ V, Ey � g−(Y (v)(τS))1[τS<∞] � < ∞, for all y ∈ S, v ∈ V, and Ey \uf8ee \uf8f0 � τj≤τS K−( ˇY (v)(τ − j ), ζj) \uf8f9 \uf8fb < ∞, for all y ∈ S, v ∈ V, where Ey denotes expectation given that Y (0) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We now define the performance criterion, which consists of three parts: a continuous time running profit in [0, τS], a terminal bequest value if the game ends, and a discrete-time intervention profit, namely J(v)(y) = Ey � � τS 0 f(Y (v)(t))dt + g(Y (v)(τS))1[τS<∞] + � τj≤τS K( ˇY (v)(τ − j ), ζj) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We consider the following impulse control problem: Problem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='2 Find Φ(y) and v∗ ∈ V such that Φ(y) = sup{J(v)(y);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' v ∈ V} = J(v∗)(y), y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The function Φ(y) is called the value function and v∗ is called an optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 7 The following concept is crucial for the solution of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='3 Let H be the space of all measurable functions h : S → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The intervention operator M : H → H is defined by Mh(s, X, µ) = sup ζ∈Z {h(s, Γ(X, ζ), µΓ(X,ζ)) + K(y, ζ), ζ ∈ Z and (s, Γ(X, ζ), µΓ(X,ζ)) ∈ S}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='6) where µΓ(X,ζ) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Let C(1,2,2)(S) denote the family of functions ϕ(s, x, µ) : S → R which are continuously differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' s and twice continuously Fr´echet differentiable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' x ∈ Rd and µ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We let ∇µϕ ∈ L(M, R) (the set of bounded linear functionals on M) denote the Fr´echet derivative (gradient) of ϕ with respect to µ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Similarly, D2 µϕ denotes the double derivative of ϕ with respect to µ and it belongs to L(M × M, R) (see Appendix for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The infinitesimal generator G of the Markov jump diffusion process Y (t) is defined on functions ϕ ∈ C(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='2)(S) by Gϕ = ∂ϕ ∂s + d � j=1 αj ∂ϕ ∂xj + ⟨∇µϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' A∗ 0µ⟩ + 1 2 d � j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='n=1 (ββT )j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='n ∂2ϕ ∂xj∂xn + 1 2 d � j=1 βj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1 ∂ ∂xj ⟨∇µϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' A∗ 1µ⟩ + 1 2⟨A∗ 1µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ⟨D2 µϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' A∗ 1µ⟩⟩ + k � ℓ=1 � R {ϕ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' X + γ(ℓ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' µ)) − ϕ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' µ) − d � j=1 γ(ℓ) j ∂ ∂xj ϕ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' µ)}νℓ(dζ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' as before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' A∗ 0 is the integro-differential operator A∗ 0µ = − d � j=1 Dj[αjµ] + 1 2 d � n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='j=1 Dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='j[(ββ(T))n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='jµ] + k � ℓ=1 � R � µ(γ(ℓ)) − µ + d � j=1 Dj[γ(ℓ) j (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ζ)µ] � νℓ (dζ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' and A∗ 1µ = − d � j=1 Dj[β1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='jµ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We can now state a verification theorem for conditional McKean-Vlasov impulse control problems, providing sufficient conditions that a given function is the value function and 8 a given impulse control is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The verification theorem links the impulse control problem to a suitable system of quasi-variational inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Since the process Y (t) is Markovian, we can, with appropriate modifications, use the approach in Chapter 9 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' For simplicity of notation we will in the following write Γ(y, ζ) = (s, Γ(x, ζ), µΓ(x,ζ)), when y = (s, x, µ) ∈ [0, ∞) × L2(P) × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='4 Variational inequalities for conditional McKean-Vlasov impulse control (a) Suppose we can find φ : ¯S → R such that (i) φ ∈ C1(S) ∩ C( ¯S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (ii) φ ≥ Mφ on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Define D = {y ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' φ(y) > Mφ(y)} (the continuation region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Assume (iii) Ey �� τS 0 Y (v)(t)1∂Ddt � = 0 for all y ∈ S, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (iv) ∂D is a Lipschitz surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (v) φ ∈ C(1,2,2)(S \\ ∂D) with locally bounded derivatives near ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (vi) Gφ + f ≤ 0 on S \\ ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (vii) φ(y) = g(y) for all y ̸∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (viii) {φ−(Y (v)(τ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' τ ∈ T } is uniformly integrable, for all y ∈ S, v ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (ix) Ey � |φ(Y (v)(τ))| + � τS 0 |Gφ(Y (v)(t))|dt � < ∞ for all τ ∈ T , v ∈ V, y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then φ(y) ≥ Φ(y) for all y ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (b) Suppose in addition that (x) Gφ + f = 0 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (xi) ˆζ(y) ∈ Argmax{φ(Γ(y, ·))+K(y, ·)} ∈ Z exists for all y ∈ S and ˆζ(·) is a Borel measurable selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Put ˆτ0 = 0 and define ˆv = (ˆτ1, ˆτ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ˆζ1, ˆζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=') inductively by ˆτj+1 = inf{t > ˆτj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Y (ˆvj)(t) ̸∈ D} ∧ τS and ˆζj+1 = ˆζ(Y (ˆvj)(ˆτ − j+1)) if ˆτj+1 < τS, where Y (ˆvj) is the result of applying ˆvj := (ˆτ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' , ˆτj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ˆζ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' , ˆζj) to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Suppose 9 (xii) ˆv ∈ V and {φ(Y (ˆv)(τ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' τ ∈ T } is uniformly integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then φ(y) = Φ(y) and ˆv is an optimal impulse control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='5 We give the intuitive idea behind intervention operator as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='6): MΦ(y) = sup ζ∈Z {Φ(Γ(y, ζ)) + K(y, ζ), ζ ∈ Z and Γ(y, ζ) ∈ S}, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='7) Assume that the value function Φ is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' If y = (s, x, µ) is the current state of the pro- cess, and the agent intervenes with impulse of size ζ, the resulting value can be represented as Φ(Γ(y, ζ))+K(y, ζ), consisting of the sum of the value of Φ in the new state Γ(y, ζ) and the intervention cost K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Therefore, MΦ(y) represents the optimal new value if the agent decides to make an intervention at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Note that by (ii) Φ ≥ MΦ on S, so it is not always optimal to intervene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' At the time ˆτj, the operator should intervene with impulse ˆζj when the controlled process leaves the continuation region, that is when Φ(Y ˆvj) ≤ MΦ(Y ˆvj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (a) By an approximation argument (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1 in [12]) and (iii)–(v), we may assume that φ∈C2(S)∩C( ¯S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Choose v=(τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' )∈V and set τ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' By another approximation argument we may assume that we can apply the Dynkin formula to the stopping times τj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then for j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', with Y = Y (v) Ey[φ(Y (τj))] − Ey[φ( ˇY (τ − j+1))] = −Ey �� τj+1 τj Gφ(Y (t))dt � , where ˇY (τ − j+1) = Y (τ − j+1) + ∆NY (τj+1), as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Summing this from j = 0 to j = m we get φ(y) + m � j=1 Ey[φ(Y (τj)) − φ( ˇY (τ − j ))] − Ey[φ( ˇY (τ − m+1))] = −Ey �� τm+1 0 Gφ(Y (t))dt � ≥ Ey �� τm+1 0 f(Y (t))dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='8) Now φ(Y (τj)) = φ(Γ( ˇY (τ − j ), ζj)) ≤ Mφ( ˇY (τ − j )) − K( ˇY (τ − j ), ζj) if τj < τS by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='6) and φ(Y (τj)) = φ( ˇY (τ − j )) if τj = τS by (vii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 10 Therefore Mφ( ˇY (τ − j )) − φ( ˇY (τ − j )) ≥ φ(Y (τj)) − φ( ˇY (τ − j )) + K( ˇY (τ − j ), ζj), and φ(y) + m � j=1 Ey[{Mφ( ˇY (τ − j )) − φ( ˇY (τ − j ))}1[τj<τS]] ≥ Ey \uf8ee \uf8f0 � τm+1 0 f(Y (t))dt + φ( ˇY (τ − m+1)) + m � j=1 K( ˇY (τ − j ), ζj) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Letting m → M and using quasi-left continuity of Y (·), we get φ(y) ≥ Ey \uf8ee \uf8f0 � τS 0 f(Y (t))dt + g(Y (τS))1[τS<∞] + M � j=1 K( ˇY (τ − j ), ζj) \uf8f9 \uf8fb=J(v)(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='9) Hence φ(y) ≥ Φ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (b) Next assume (x)–(xii) also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Apply the above argument to ˆv = (ˆτ1, ˆτ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ˆζ1, ˆζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then by (x) we get equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='8) and by our choice of ζj = ˆζj we have equality in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Hence φ(y) = J(ˆv)(y), which combined with (a) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' □ 4 Example: Optimal stream of dividends under transaction costs In this Section, we solve explicitly an optimal stream of dividends under transaction costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' To this end, for v = (τ1, τ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ζ1, ζ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=') with ζi ∈ R+, we define Y (v)(t) = (s + t, X(v)(t), µ(v) t ) by dX(t) = E � X(t) | F(1) t � � α0dt + σ1dB1(t) + σ2dB2(t) + � R γ0(ζ) � N(dt, dζ) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1) µ(v) t = L(X(v)(t)|F(1) t );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' τi < t < τi+1, X(v)(τi+1) = ˇX(v)(τ − i+1) − (1 + λ)ζi+1 − c, µ(v) τi+1 = L(X(v)(τi+1)|F(1) τi+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' , X(v)(0−) = x > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', 11 where α0, σ1 ̸= 0, σ2 ̸= 0, λ ≥ 0, and c > 0 are constants with −1 ≤ γ0(z) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Here X(t) represents the amount available at time t of a cash flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We assume that it satisfies the McKean-Vlasov equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Note that at any time τi, i = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' , the system jumps from ˇX(v)(τ − i ) to X(v)(τi) = Γ[ ˇX(v)(τ − i ), ζi] = ˇX(v)(τ − i ) − (1 + λ)ζi − c, where the quantity c + λζi represents the transaction cost with a fixed part c and a pro- portional part λζi, while ζi is the amount we decide to take out at time τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' At the same time µτ − i jumps to µτi = L( ˇX(v)(τ − i )|F1 τi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Problem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1 We want to find Φ and v∗ ∈ V such that Φ(s, x, µ) = sup v J(v)(s, x, µ) = J(v∗)(s, x, µ), where J(v)(s, x, µ) = J(v)(y) = Ey � � τk<τS e−ρ(s+τk)ζk � (ρ > 0 constant) is the expected discounted total dividend up to time τS, where τS = τS(ω) = inf{t > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' P y[Ey[X(v)(t)|F(1) t ] ≤ 0] > 0} is the time of bankruptcy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' To put this problem into the context above, we define Y (v)(t) = \uf8ee \uf8f0 s + t X(v)(t) µ(v) t \uf8f9 \uf8fb , Y (v)(0−) = \uf8ee \uf8f0 s x µ \uf8f9 \uf8fb = y, Γ(y, ζ) = Γ(s, x, µ) = (s, x − c − (1 + λ)ζ, L(x − c − (1 + λ)ζ)|F(1)), x ∈ L2(P), K(y, ζ) = e−ρsζ, f ≡ g ≡ 0, S = {(s, x, µ) : x > 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Comparing with our Theorem, we see that in this case we have d = 1, m = 2, k = 1 and α1 = α0 ⟨µ, q⟩ , β1 = σ1 ⟨µ, q⟩ , β2 = σ2 ⟨µ, q⟩ , γ(s, x, µ, ζ) = γ0(t, ζ) ⟨µ, q⟩ , 12 where we have put q(x) = x so that ⟨µt, q⟩ = E � X(t) | F(1) t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Therefore the operator G takes the form Gϕ(s, x, µ) = ∂ϕ ∂s + α0 ⟨µ, q⟩ ∂ϕ ∂x + ⟨∇µϕ, A∗ 0µ⟩ + 1 2(σ2 1 + σ2 2) ⟨µ, q⟩2 ∂2ϕ ∂x2 + 1 2σ1 ⟨µ, q⟩ ∂ ∂x ⟨∇µϕ, A∗ 1µ⟩ + 1 2 � A∗ 1µ, � D2 µϕ, A∗ 1µ �� + � R � ϕ(s, x + γ0 ⟨µ, q⟩ , µ) − ϕ(s, x, µ) − γ0 ⟨µ, q⟩ ∂ ∂xϕ(s, x, µ) � ν(dζ), where A∗ 0µ = −D[α0 ⟨µ, q⟩ µ] + 1 2D2[(σ2 1 + σ2 2) ⟨µ, q⟩2 µ], and A∗ 1µ = −D[σ1 ⟨µ, q⟩ µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The adjoints of the last two operators are A0µ = α0 ⟨µ, q⟩ Dµ + 1 2(σ2 1 + σ2 2) ⟨µ, q⟩2 D2µ, and A1µ = σ1 ⟨µ, q⟩ Dµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In this case the intervention operator gets the form Mh(s, x, µ) = sup � h(s, x − c − (1 + λ)ζ, µx−c−(1+λ)ζ) + e−ρtζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 0 ≤ ζ ≤ x − c 1 + λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Note that the condition on ζ is due to the fact that the impulse must be positive and x − c − (1 + λ)ζ must belong to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We distinguish between two cases: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' α0 > ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In this case, suppose we wait until some time t1 and then take out ζ1 = X(t1) − c 1 + λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Noting that Ey|X(t)] = x exp(α0t) for t < t1, we see that the corresponding performance is J(v1)(s, x, µ) = Ey � e−ρ(t1+s) 1 + λ (X(t1) − c) � = Ex � 1 1 + λ � xe−ρse(α0−ρ)t1 − c e−ρ(s+t1)� � → ∞ as t1 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 13 Therefore we obtain Φ(s, x, µ) = +∞ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' α0 < ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We look for a solution by using the results of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We guess that the continuation region is of the form D = {(s, x, µ) : 0 < ⟨µ, q⟩ < ¯x} for some ¯x > 0 (to be determined), and in D we try a value function of the form ϕ(s, x, µ) = e−ρsψ(⟨µ, q⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' This gives Gφ(s, x, µ) = e−ρsG0ψ(⟨µ, q⟩), where G0ψ(x, µ) = −ρψ(⟨µ, q⟩) + ⟨∇µψ, A∗ 0µ⟩ + 1 2σ1 ⟨µ, q⟩ ∂ ∂x ⟨∇µψ, A∗ 1µ⟩ + 1 2 � A∗ 1µ, � D2 µψ, A∗ 1µ �� + � R � ψ(x + γ0 ⟨µ, q⟩ , µ) − ψ(x, µ) − γ0 ⟨µ, q⟩ ∂ ∂xψ(x, µ) � ν(dζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' By the chain rule for Fr´echet derivatives (see Appendix), we have ∇µψ(h) = ψ′(⟨µ, q⟩)⟨h, q⟩ and D2 µψ(h, k) = ψ′′(⟨µ, q⟩)⟨h, q⟩⟨k, q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Therefore, ⟨∇µψ, A∗ 0µ⟩ = ψ′(⟨µ, q⟩)⟨A∗ 0µ, q⟩ = ψ′(⟨µ, q⟩)⟨µ, A0q⟩ = ψ′(⟨µ, q⟩)α0⟨µ, q⟩, and similarly 1 2⟨A∗ 1µ, ⟨D2 µψ, A∗ 1µ⟩⟩ = 1 2ψ′′(⟨µ, q⟩⟨A∗ 1µ, q⟩⟨A∗ 1µ, q⟩ = 1 2ψ′′(⟨µ, q⟩)⟨µ, A1q⟩⟨µ, A1q⟩ = 1 2ψ′′(⟨µ, q⟩)σ2 1⟨µ, q⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Moreover, since ψ does not depend on x we see that � R � φ(s, x + γ0 ⟨µ, q⟩ , µ) − φ(s, x, µ) − γ0 ⟨µ, q⟩ ∂φ ∂x(s, x, µ) � ν(dζ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Substituting this into the expression for G0ψ we get, with u = ⟨µ, q⟩, G0ψ(u) = −ρψ(u) + α0uψ′(u) + 1 2σ2 1u2ψ′′(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' By condition (x) we are required to have G0ψ(u) = 0 for all u ∈ (0, ¯x), and this equation has the general solution ψ(u) = C1uγ1 + C2uγ2, u ∈ (0, ¯x), 14 where γ1 > 1, γ2 < 0, and C1, C2 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Since we expect φ to be bounded near 0, we guess that C2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We guess that it is optimal to wait till u = ⟨µt, q⟩ = Ey[X(t)|F(1) t ] reaches or exceeds a value u = ¯u > c and then take out as much as possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', reduce Ey[X(t)|F(1) t ] to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Taking the transaction costs into account this means that we should take out ˆζ(u) = u − c 1 + λ for u ≥ ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We therefore propose that ψ(u) has the form ψ(u) = \uf8f1 \uf8f2 \uf8f3 C1uγ1for 0 < u < ¯u u − c 1 + λ for u ≥ ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Continuity and differentiability of ψ(u) at u = ¯u give the equations C1¯uγ1 = ¯u − c 1 + λ, and C1γ1¯uγ1−1 = 1 1 + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Combining these we get ¯u = γ1c γ1 − 1 and C1 = ¯u − c 1 + λ ¯u−γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' With these values of ¯u and C1, we have to verify that ψ satisfies all the requirements of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' We check some of them: (ii) φ ≥ Mφ on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' In our case we have Γ(s, X, µ) = (s, X − c − (1 + λ)ζ, µX−c−(1+λ)ζ) and hence we get Mφ(s, X, µ) = sup ζ � φ(s, X − c − (1 + λ)ζ), µX−c−(1+λ)ζ) + e−ρsζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 0 ≤ ζ ≤ ¯u − c 1 + λ � = e−ρs sup ζ � C1⟨µX−c−(1+λ)ζ, q⟩γ1 + ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 0 ≤ ζ ≤ ¯u − c 1 + λ � = e−ρs sup ζ � C1(⟨µ, q(x) − c − (1 + λ)ζ⟩γ1 + ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 0 ≤ ζ ≤ ¯u − c 1 + λ � = e−ρs sup ζ � C1(⟨µ, q⟩ − c − (1 + λ)ζ)γ1 + ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 0 ≤ ζ ≤ ¯u − c 1 + λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' If u − c − (1 + λ)ζ ≥ ¯u, then ψ(u − c − (1 + λ)ζ) + ζ = u − 2c 1 + λ < u − c 1 + λ = ψ(u) 15 and if u − c − (1 + λ)ζ < ¯u then h(ζ) := ψ(u − c − (1 + λ)ζ) + ζ = C1(u − c − (1 + λ)ζ)γ1 + ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Since h′ �u − c 1 + λ � = 1 and h′′(ζ) > 0, we see that the maximum value of h(ζ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 0 ≤ ζ ≤ u − c 1 + λ, is attained at ζ = ˆζ(u) = u − c 1 + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Therefore Mψ(u) = max �x − 2c 1 + λ , u − c 1 + λ � = u − c 1 + λ for all u > c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Hence Mψ(u) = ψ(u) for u ≥ ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' For 0 < u < ¯u consider k(u) := C1uγ1 − u − c 1 + λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Since k(¯u) = k′(¯u) = 0 and k′′(u) > 0 for all u, we conclude that k(u) > 0 for 0 < u < ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Hence ψ(u) > Mψ(u) for 0 < u < ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (vi) A0ψ(u) ≤ 0 for u ∈ S\\ ¯D i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', for u > ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' For u > ¯u, we have A0ψ(u) = −ρ u − c 1 + λ + α0u 1 1 + λ + � u+γuz<¯u � C1(u + γuz)γ1 − u + γuz − c 1 + λ � ν(dz) ≤ (1 + λ)−1[(µ − ρ)u + (ρ + ∥ν∥)c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 16 Therefore we see that A0ψ(u) ≤ 0 for all u > ¯u ⇔ (α0 − ρ)u + (ρ + ∥ν∥)c ≤ 0 for all u > ¯u ⇔ (α0 − ρ)¯u + (ρ + ∥ν∥)c ≤ 0 ⇔ ¯u ≥ (ρ + ∥ν∥)c ρ − α0 ⇔ γ1c γ1 − 1 ≥ (ρ + ∥ν∥)c ρ − α0 ⇔ γ1 ≤ ρ + ∥ν∥ α0 + ∥ν∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Since F � ρ µ � ≥ −ρ + α0 ρ α0 + 1 2σ2 ρ α0 � ρ α0 − 1 � > 0, and F(γ1) = 0, γ1 > 1 we conclude that γ1 < ρ α0 and hence (vi) holds if ∥ν∥ is small enough, say ∥ν∥ ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Therefore, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='2 Suppose ∥ν∥ ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then the value function for Problem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1 is Φ(s, x, µ) = \uf8f1 \uf8f2 \uf8f3 e−ρsC1uγ1for 0 < u < ¯u e−ρs u − c 1 + λ for u ≥ ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' where u = ⟨µ, q⟩ = E[X(t)|F(1) t ] and ¯u = γ1c γ1 − 1 and C1 = ¯u − c 1 + λ ¯u−γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' and γ = γ1 > 1 is the positive solution of the equation −ρ + α0γ + 1 2σ2 1γ(γ − 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The optimal impulse control is to do nothing while u = E[X(t)|F(1) t ] < ¯u and take out immediately ˆζ(u) = u − c 1 + λ when u ≥ ¯u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' This brings E[X(t)|F(1) t ] down to 0, and the system stops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Hence the optimal impulse consists of at most one intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 17 5 Appendix: Double Fr´echet derivatives In this section we recall some basic facts we are using about the Fr´echet derivatives of a function f : V �→ W, where V, W are given Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='1 We say that f has a Fr´echet derivative ∇xf = Df(x) at x ∈ V if there exists a bounded linear map A : V �→ W such that lim h→0 ||f(x + h) − f(x) − A(h)||W ||h||V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then we call A the Fr´echet derivative of f at x and we put Df(x) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Note that Df(x) ∈ L(V, W) (the space of bounded linear functions from V to W), for each x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='2 We say that f has a double Fr´echet derivative D2f(x) at x if there exists a bounded bilinear map A(h, k) : V × V �→ W such that lim k→0 ||Df(x + k)(h) − Df(x)(h) − A(h, k)||W ||h||V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='3 Suppose f : M �→ R is given by f(µ) = ⟨µ, q⟩2, where q(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then f(µ + h) − f(µ) = ⟨µ + h, q⟩2 − ⟨µ, q⟩2 = 2⟨µ, q⟩⟨h, q⟩ + ⟨h, q⟩2, so we see that Df(µ)(h) = 2⟨µ, q⟩⟨h, q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' To find the double derivative we consider Df(µ + k)(h) − Df(µ)(h) = 2⟨µ + k, q⟩⟨h, q⟩ − 2⟨µ, q⟩⟨h, q⟩ = 2⟨k, q⟩⟨h, q⟩, and we conclude that D2f(µ)(h, k) = 2⟨k, q⟩⟨h, q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Next assume that g : M �→ R is given by g(µ) = ⟨µ, q⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Then, proceeding as above we find that Dg(µ)(h) = ⟨h, q⟩ (independent of µ) and D2g(µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 18 References [1] Agram, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', & Øksendal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Stochastic Fokker-Planck PIDE for conditional McKean-Vlasov jump diffusions and applications to optimal control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content='02193v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' [2] Basei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (2019).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=', & Huang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' The stochastic maximum principle for a jump-diffusion mean-field model involving impulse controls and applications in finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Journal of Systems Science and Complexity, 33(1), 26-42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Chicago [12] Øksendal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' & Sulem, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Applied Stochastic Control of Jump diffusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 3rd edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNAzT4oBgHgl3EQfiv3U/content/2301.01506v1.pdf'} diff --git a/FdE5T4oBgHgl3EQfVQ-f/content/tmp_files/2301.05550v1.pdf.txt b/FdE5T4oBgHgl3EQfVQ-f/content/tmp_files/2301.05550v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5918c928240c8c24b8e920d60abc897db91fe65a --- /dev/null +++ b/FdE5T4oBgHgl3EQfVQ-f/content/tmp_files/2301.05550v1.pdf.txt @@ -0,0 +1,406 @@ +Recognizing Unit Disk Graphs in Hyperbolic +Geometry is ∃R-Complete +Nicholas Bieker � +Karlsruhe Institute of Technology, Germany +Thomas Bläsius � +Karlsruhe Institute of Technology, Germany +Emil Dohse � +Karlsruhe Institute of Technology, Germany +Paul Jungeblut � +Karlsruhe Institute of Technology, Germany +Abstract +A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the +Euclidean plane R2. Recognizing them is known to be ∃R-complete, i.e., as hard as solving a system +of polynomial inequalities. In this note we describe a simple framework to translate ∃R-hardness +reductions from the Euclidean plane R2 to the hyperbolic plane H2. We apply our framework to +prove that the recognition of unit disk graphs in the hyperbolic plane is also ∃R-complete. +2012 ACM Subject Classification Theory of computation → Randomness, geometry and discrete +structures → Computational geometry; Theory of computation → Computational complexity and +cryptography → Complexity classes +Keywords and phrases Unit disk graphs, Hyperbolic geometry, Existential theory of the reals +1 +Introduction +A graph is a unit disk graph if its vertices can be represented by equally sized disk such +that two vertices are adjacent if and only if their corresponding disks intersect. The class of +unit disk graphs (UDG) is a well studied graph class due to its mathematical beauty and its +practical relevance, e.g., in the context of sensor networks. +Naturally, unit disk graphs are usually considered in the Euclidean plane R2. However, +in the past decade, research on intersection graphs of equally sized disks in the hyperbolic +plane H2 has gained traction. This is due to the fact that the hyperbolic geometry is well +suited to represent a wider range of graph structures, including complex scale-free networks +with heterogeneous degree distributions [6, 7, 11, 16, 22]; see Figure 1. Most research on such +graphs is driven by the network science community studying probabilistic network models, +i.e., hyperbolic random graphs. However, when omitting the probability distribution and +looking at hyperbolic unit disk graphs as a graph class, little is known so far. +The class of hyperbolic unit disk graphs (HUDG) has only been introduced recently [5]1. +When choosing disks of small radius, the difference between Euclidean and hyperbolic +geometry becomes negligible; also see our interactive visualization2 and Figure 1. +Arguably the most fundamental algorithmic question when it comes to studying graph +classes is the computational complexity of the recognition problem, i.e., Recog(HUDG) is +1 We note that there are earlier results on a related family of graph classes parameterized by the disk size +by Kisfaludi-Bak [14]. In a sense, the class HUDG is the union of all these classes. This subtle difference +is important when considering asymptotic behavior as it can be desirable to grow the disk size with the +graph size; see [5] for a detailed discussion. +2 https://thobl.github.io/hyperbolic-unit-disk-graph +arXiv:2301.05550v1 [cs.CG] 13 Jan 2023 + +2 +Recognizing Unit Disk Graphs in Hyperbolic Geometry is ∃R-Complete +Figure 1 Two hyperbolic unit disk graphs. Thee green circle indicates the threshold distance +below which vertices are connected. A small threshold (left) yields structures similar to Euclidean +unit disk graphs. A large threshold (right) facilitates heterogeneous vertex degrees. +the problem of testing whether a given graph is part of HUDG. In this paper we prove that +Recog(HUDG) is ∃R-complete. Containment in ∃R is less obvious than in the Euclidean +plane as distances are not (square roots of) a polynomial in hyperbolic geometry. Nonetheless, +containment is easy to show when using the hyperboloid model of the hyperbolic plane. For +∃R-hardness, our proof consists of five steps switching back and forth between Euclidean and +hyperbolic variants of problems in a particular way. Our proof has framework-character in +the sense that the first three steps are independent of the specific problem and the remaining +steps can probably be translated to other problems. Thus we believe that this can be +a template for proving ∃R-hardness for other hyperbolic problems that have an ∃R-hard +Euclidean counterpart. For our framework, we in particular use the Beltrami-Klein model +of the hyperbolic plane to observe that SimpleStretchability is equivalent in Euclidean +and hyperbolic geometry in the sense that a pseudoline arrangement is stretchable in the +Euclidean plane if and only if it is stretchable in the hyperbolic plane. +1.1 +Existential Theory of the Reals +The existential theory of the reals is the set of all true sentences of the form ∃X ∈ Rn : ϕ(X), +where ϕ(X) is a quantifier-free formula consisting of polynomial equations and inequalities, +e.g. ∃X, Y ∈ R : X · Y = 6 ∧ X + Y = 5. We denote the decision problem whether such +a sentence is true by ETR (which also stands for “existential theory of the reals”) and +define the complexity class ∃R to contain all decision problems that polynomial-time reduce +to ETR. It holds NP ⊆ ∃R ⊆ PSPACE [8]. The class ∃R has gained increasing attention +in the computational geometry community over the last years as it exactly captures the +complexity of many geometry problems like the art gallery problem [2], geometric packing [3] +or the recognition of many classes of geometric intersection graphs [15, 18, 24]. +1.2 +Hyperbolic Geometry +The are several ways to embed the hyperbolic plane into Euclidean space. In this paper we +use the Beltrami-Klein model and the hyperboloid model. + +N. Bieker, T. Bläsius, E. Dohse and P. Jungeblut +3 +−2 +−1 +1 +2 +−2 +2 +2 +D +ℓ1 +ℓ2 +ℓ3 +S+ +Figure 2 Left: The Beltrami-Klein disk with three hyperbolic lines. Right: The upper sheet S+ +used for the hyperboloid model. +Beltrami-Klein Model +In the Beltrami-Klein model the hyperbolic plane H2 is represented +by the interior of a unit disk D in R2 (the boundary of D is not part of the model). The set +of hyperbolic lines is exactly the set of chords of D. See Figure 2 (left). +Hyperboloid Model +Here the hyperbolic plane gets embedded into R3. The Minkowski +quadratic form Q(x, y, z) := z2 − x2 − y2 defines a two-sheeted hyperboloid S := {(x, y, z) ∈ +R3 | Q(x, y, z) = 1}, see Figure 2 (right). The hyperbolic plane is represented by all points +on the forward sheet S+ of S, obtained by additionally requiring that z > 0. The hyperbolic +distance between two points u, v ∈ S+ is +dh(u, v) = arcosh(B(u, v)) +with +B(u, v) := uzvz − uxvx − uyvy +where B(u, v) is known as the Minkowski bilinear form and arcosh(x) := ln +� +x + +√ +x2 − 1 +� +is +the inverse hyperbolic cosine. Note that the term inside the arcosh(·) is a polynomial. +2 +Simple Stretchability in the Euclidean and the Hyperbolic Plane +An pseudoline arrangement A is a collection of pseudolines (x-monotone curves in R2) such +that each pair of curves intersects at most once. We assume that each pseudoline ℓ ∈ A is +oriented and thus divides the plane R2 into two open half-planes ℓ− and ℓ+. Further, A +partitions the plane into cells, i.e., maximal connected components of R2 \ A not on any +pseudoline. We say that A is simple if any two lines intersect exactly once and no three lines +intersect in the same point. Given a pseudoline arrangement A = {ℓ1, . . . , ℓn} we assign to +each p ∈ R2 a sign vector σ(p) = (σi(p))n +i=1 ∈ {−, 0, +}n, where +σi(p) := +� +� +� +� +� +� +� +− +if p ∈ ℓ− +i +0 +if p ∈ ℓi ++ +if p ∈ ℓ+ +i +. +The combinatorial description D of A is then given by {σ(p) | p ∈ R2}. We say that A real- +izes D. A pseudoline arrangement is stretchable if there is a line arrangement with the same +combinatorial description. Not every pseudoline arrangement is stretchable and, given a com- +binatorial description D, deciding whether D is stretchable is known as the Stretchability +problem (or SimpleStretchability if D is simple). Stretchability and SimpleStretch- +ability are famously known to be ∃R-complete [21, 23, 27]. SimpleStretchability is the +starting problem for many ∃R-hardness reductions, e.g. [4, 12, 15, 24, 25]. + +4 +Recognizing Unit Disk Graphs in Hyperbolic Geometry is ∃R-Complete +↔ +↔ +Figure 3 Transforming line arrangements between Euclidean and hyperbolic geometry. +Apart from line arrangements in the Euclidean plane R2 one might also consider line +arrangements in the hyperbolic plane H2. The main result of this section is that Sim- +pleStretchability is equivalent in Euclidean and hyperbolic geometry. +▶ Proposition 1. Let D be a combinatorial description of a simple pseudoline arrangement. +Then there is a line arrangement realizing D in R2 if and only if there is one in H2. +Proof. The proof is an easy application of the Beltrami-Klein model of the hyperbolic plane. +Given a Euclidean line arrangement, we can obtain a hyperbolic line arrangement with the +same combinatorial description and vice versa, see Figure 3. +Let AR be a simple line arrangement in R2 and D be a disk strictly enclosing all +intersections of AR. For each line in AR, keep only its part inside D. We think of D as a +unit disk and obtain a representation of a hyperbolic line arrangement in the Beltrami-Klein +model. +For the other direction let AH be a simple hyperbolic line arrangement and take a repre- +sentation inside the Beltrami-Klein disk D, so all hyperbolic lines are chords D. Remove D +and extend all chords to lines. The resulting Euclidean line arrangement has the same +combinatorial description D because AH was simple: All possible intersections between two +lines were already inside the Beltrami-Klein disk D. +◀ +▶ Remark 2. Proposition 1 is only about simple (pseudo)line arrangements. There is no +corresponding result for the general (non-simple) Stretchability problem: For example, +given three lines ℓ1, ℓ2, ℓ3 ⊆ H2, lines ℓ2 and ℓ3 may cross each other while both being parallel +to ℓ1. However, Proposition 1 may be extended to line arrangements where each pair of lines +is still required to cross but multiple lines are allowed to cross at the same point. +3 +The Framework +Let ΠR be a geometric decision problem for which ∃R-hardness is shown in Euclidean geometry +by a polynomial-time reduction f from (Euclidean) SimpleStretchability. We denote +by ΠH the corresponding decision problem obtained by considering the hyperbolic plane H2 +instead of the Euclidean plane R2. Our framework below consists of several (hopefully) +simple steps that allow us to prove ∃R-hardness of ΠH by using the reduction for ΠR: +1. Let D be an instance of SimpleStretchability in H2, i.e., a combinatorial description +of a simple pseudoline arrangement. +2. Use Proposition 1 to consider D to be an instance of SimpleStretchability in R2. +3. Use the reduction f to obtain an instance I = f(D) of ΠR equivalent to D. +4. Prove that every yes-instance of ΠR is also a yes-instance of ΠH. + +N. Bieker, T. Bläsius, E. Dohse and P. Jungeblut +5 +5. Prove that a line arrangement realizing D can be extracted from a realization of I in H2. +Steps 1, 2 and 3 require no work when applying the framework. +Step 4 ensures that a stretchable instance D yields a yes-instance of ΠH. This step +requires to come up with a new argument but we expect it to be relatively simple because +locally R2 and H2 are very similar. A promising approach is to scale a Euclidean realization +of I to a tiny area and then interpret the Euclidean polar coordinates as hyperbolic ones. +Step 5 ensures correctness. By showing that a line arrangement realizing D can be +extracted from a realization of I in H2 we show that a no-instance D maps to a no-instance +of ΠH. Reduction f might help us again here (though not as a black box as in Step 3): If we +are lucky, the argument why a realization of I in R2 induces a Euclidean line arrangement +realizing D only uses the axioms of absolute geometry (the common “subset” of Euclidean +and hyperbolic geometry) and works without any adaptations for realizations in H2, too. +4 +Recognition of Hyperbolic Unit Disk Graphs +We apply our framework to prove that Recog(HUDG), the recognition problem of hyperbolic +unit disk graphs, is ∃R-hard. For Euclidean geometry this is shown in [13, 19, 20]. Let us +note that UDG and HUDG are not the same: For example, a star graph with six leaves is a +hyperbolic unit disk graph but not a Euclidean one. +For Step 1 of our framework let D be an instance of SimpleStretchability in H2. We +consider it to be an equivalent instance in R2 for Step 2. In Step 3 we use the reduction f +from the literature proving that Recog(UDG) in R2 is ∃R-hard [13, 19, 20]. We obtain a +graph GD that is a Euclidean unit disk graph if and only if D is stretchable. +Though not required for the framework, let us shortly summarize the reduction f to +construct GD from D as given in [19]. Let n be the number of pseudolines ℓ1, . . . , ℓn and +m = 1 + +�n+1 +2 +� +be the number of cells C1, . . . , Cm. The arrangement described by D has +exactly this number of cells, because it is simple. We define GD to be the graph with vertex +set V = A ∪ B ∪ C for A = {a1, . . . , an}, B = {b1, . . . , bn} and C = {c1, . . . , cm}. Here we +assume that vertex ci corresponds to cell Ci. For the edges, each of the sets A, B, C forms +a clique. Further, each ai ∈ A (for i ∈ {1, . . . , n}) is connected to cj (for j ∈ {1, . . . , m}) if +and only if Cj ∈ ℓ− +i . Similarly, each bi ∈ B is connected to cj if and only if Cj ∈ ℓ+ +i . +For Step 4 we have to show that every Euclidean unit disk graph is also a hyperbolic unit +disk graph. This has recently been proven by Bläsius, Friedrich, Katzmann and Stephan: +▶ Lemma 3 ([5]). Every Euclidean unit disk graph is also a hyperbolic one, so UDG ⊆ HUDG. +As foreshadowed above, the proof scales a Euclidean unit disk intersection representation +to a tiny area until the Euclidean and hyperbolic plane are “similar enough”. Then the polar +coordinates in R2 can be used as polar coordinates in H2 without changing any adjacencies. +For Step 5 it remains to prove how a line arrangement realizing D in H2 can be extracted +from a realization of GD in H2. +▶ Lemma 4 (adapted from [19, Lemma 1]). Given a realization of GD as the intersection +graph of equally sized disks in H2. Then the line arrangement L = {ℓ1, . . . , ℓn} defined by +ℓi := {p ∈ H2 | d(p, ai) = d(p, bi)} +has combinatorial description D. Here d(·, ·) denotes the hyperbolic distance. + +6 +Recognizing Unit Disk Graphs in Hyperbolic Geometry is ∃R-Complete +Proof. The proof is exactly the same as the proof of Lemma 1 in [19] where McDiarmid +and Müller prove that taking the perpendicular bisectors of the segments between any pair +of points ai and bi yields a Euclidean line arrangement realizing D. Their argument works +in H2 by just replacing Euclidean distances with hyperbolic distances. +◀ +At this point we proved ∃R-hardness of Recog(HUDG). To get ∃R-completeness we +prove ∃R-membership next. +▶ Lemma 5. Recognizing hyperbolic unit disk graphs is in ∃R. +Proof. By a result from Erickson, van der Hoog and Miltzow we can prove ∃R-membership +by describing a polynomial-time verification algorithm for a real RAM machine3 [10]. Given +a graph G = (V, E) and for each vertex v ∈ V a point (vx, vy, vz) in the hyperboloid +model of the hyperbolic plane representing the center of an equal-radius disk. Compute +dadj := maxuv∈E B(u, v) and dnon-adj := minuv̸∈E B(u, v) where B(·, ·) is the Minkowski +bilinear form. B(·, ·) is a polynomial, so it is computable on a real RAM. We can think of +dadj and dnon-adj as distances in hyperbolic space (actually they are the hyperbolic cosine of +a distance), but since arcosh is a monotone function, this view is justified. Now if and only +if dadj < dnon-adj, then there is a radius r such that G is a hyperbolic unit disk graph with +radius r (choose r such that dadj ≤ cosh(r) +2 +< dnon-adj). The algorithm takes O(|V |2) time. +This is polynomial in the input size, proving ∃R-membership. +◀ +We conclude with the following theorem: +▶ Theorem 6. Recognizing hyperbolic unit disk graphs is ∃R-complete. +5 +Conclusion and Outlook +We presented a simple framework that allows us to translate ∃R-hardness reductions for +geometric decision problems in R2 into reductions for their counterparts H2. As an application +we proved that Recog(HUDG) is ∃R-complete. Promising candidates for further applications +of our framework are the recognition of unit ball graphs (i.e., a generalization of our result to +higher dimensions) as already done in Rd in [13] or Recog(CONV), the recognition problem +for intersection graphs of convex sets (Euclidean reduction is in [24]). +Technically, the framework also works for the recognition problems Recog(HSEG) and +Recog(HDISK), where (H)SEG and (H)DISK denote the classes of intersection graphs of +(hyperbolic) segments and disks, respectively (Euclidean reductions are in [13, 15, 18, 20, 24]). +However, these are not really interesting as SEG = HSEG (easy to see in the Beltrami- +Klein model) and DISK = HDISK (easy to see in the Poincaré model, not considered here). +Therefore ∃R-completeness for Recog(HSEG) and Recog(HDISK) follows directly from the +Euclidean cases. Other interesting problems to consider in H2 are linkage realizability [1, 25], +simultaneous graph embeddings [9, 17] or RAC-drawings [26]. +Acknowledgements +We thank Torsten Ueckerdt for discussion on proving the membership +of Recog(HUDG) in ∃R. +3 The real RAM extends the classical word RAM by additional registers that contain real numbers (with +arbitrary precision). The basic arithmetic operations +, −, · and / are supported in constant time. +However, arbitrary analytic functions (like arcosh) are not supported. See [10] for a formal definition. + +N. Bieker, T. Bläsius, E. Dohse and P. Jungeblut +7 +References +1 +Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Jayson Lynch, and +Tao B. Schardl. Who Needs Crossings? Hardness of Plane Graph Rigidity. 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Gansner, editors, Graph Drawing (GD 2009), volume 5849 of Lecture Notes in +Computer Science, pages 334–344, 2010. doi:10.1007/978-3-642-11805-0_32. +25 +Marcus Schaefer. Realizability of Graphs and Linkages. In János Pach, editor, Thirty Essays on +Geometric Graph Theory, pages 461–482. Springer, 2013. doi:10.1007/978-1-4614-0110-0_ +24. +26 +Marcus Schaefer. RAC-Drawability is ∃R-Complete. In Helen C. Purchase and Ignaz Rutter, +editors, Graph Drawing and Network Visualization (GD 2021), volume 12868 of Lecture Notes +in Computer Science, pages 72–86, 2021. doi:10.1007/978-3-030-92931-2_5. +27 +Peter W. Shor. Stretchability of Pseudolines is NP-Hard. In Peter Gritzmann and Bernd +Sturmfels, editors, Applied Geometry And Discrete Mathematics, Proceedings of a DIMACS +Workshop, Providence, Rhode Island, USA, September 18, 1990, volume 4 of DIMACS Series +in Discrete Mathematics and Theoretical Computer Science, pages 531–554, 1991. +doi: +10.1090/dimacs/004/41. + diff --git a/FdE5T4oBgHgl3EQfVQ-f/content/tmp_files/load_file.txt b/FdE5T4oBgHgl3EQfVQ-f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28a4217cbbbdfd00feddce1eb1dd7abd29c87a39 --- /dev/null +++ b/FdE5T4oBgHgl3EQfVQ-f/content/tmp_files/load_file.txt @@ -0,0 +1,388 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf,len=387 +page_content='Recognizing Unit Disk Graphs in Hyperbolic Geometry is ∃R-Complete Nicholas Bieker � Karlsruhe Institute of Technology, Germany Thomas Bläsius � Karlsruhe Institute of Technology, Germany Emil Dohse � Karlsruhe Institute of Technology, Germany Paul Jungeblut � Karlsruhe Institute of Technology, Germany Abstract A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the Euclidean plane R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Recognizing them is known to be ∃R-complete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=', as hard as solving a system of polynomial inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' In this note we describe a simple framework to translate ∃R-hardness reductions from the Euclidean plane R2 to the hyperbolic plane H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We apply our framework to prove that the recognition of unit disk graphs in the hyperbolic plane is also ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 2012 ACM Subject Classification Theory of computation → Randomness, geometry and discrete structures → Computational geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Theory of computation → Computational complexity and cryptography → Complexity classes Keywords and phrases Unit disk graphs, Hyperbolic geometry, Existential theory of the reals 1 Introduction A graph is a unit disk graph if its vertices can be represented by equally sized disk such that two vertices are adjacent if and only if their corresponding disks intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The class of unit disk graphs (UDG) is a well studied graph class due to its mathematical beauty and its practical relevance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=', in the context of sensor networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Naturally, unit disk graphs are usually considered in the Euclidean plane R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' However, in the past decade, research on intersection graphs of equally sized disks in the hyperbolic plane H2 has gained traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' This is due to the fact that the hyperbolic geometry is well suited to represent a wider range of graph structures, including complex scale-free networks with heterogeneous degree distributions [6, 7, 11, 16, 22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Most research on such graphs is driven by the network science community studying probabilistic network models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=', hyperbolic random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' However, when omitting the probability distribution and looking at hyperbolic unit disk graphs as a graph class, little is known so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The class of hyperbolic unit disk graphs (HUDG) has only been introduced recently [5]1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' When choosing disks of small radius, the difference between Euclidean and hyperbolic geometry becomes negligible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' also see our interactive visualization2 and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Arguably the most fundamental algorithmic question when it comes to studying graph classes is the computational complexity of the recognition problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=', Recog(HUDG) is 1 We note that there are earlier results on a related family of graph classes parameterized by the disk size by Kisfaludi-Bak [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' In a sense, the class HUDG is the union of all these classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' This subtle difference is important when considering asymptotic behavior as it can be desirable to grow the disk size with the graph size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' see [5] for a detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 2 https://thobl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='io/hyperbolic-unit-disk-graph arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='05550v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='CG] 13 Jan 2023 2 Recognizing Unit Disk Graphs in Hyperbolic Geometry is ∃R-Complete Figure 1 Two hyperbolic unit disk graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Thee green circle indicates the threshold distance below which vertices are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' A small threshold (left) yields structures similar to Euclidean unit disk graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' A large threshold (right) facilitates heterogeneous vertex degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' the problem of testing whether a given graph is part of HUDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' In this paper we prove that Recog(HUDG) is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Containment in ∃R is less obvious than in the Euclidean plane as distances are not (square roots of) a polynomial in hyperbolic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Nonetheless, containment is easy to show when using the hyperboloid model of the hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For ∃R-hardness, our proof consists of five steps switching back and forth between Euclidean and hyperbolic variants of problems in a particular way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Our proof has framework-character in the sense that the first three steps are independent of the specific problem and the remaining steps can probably be translated to other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Thus we believe that this can be a template for proving ∃R-hardness for other hyperbolic problems that have an ∃R-hard Euclidean counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For our framework, we in particular use the Beltrami-Klein model of the hyperbolic plane to observe that SimpleStretchability is equivalent in Euclidean and hyperbolic geometry in the sense that a pseudoline arrangement is stretchable in the Euclidean plane if and only if it is stretchable in the hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='1 Existential Theory of the Reals The existential theory of the reals is the set of all true sentences of the form ∃X ∈ Rn : ϕ(X), where ϕ(X) is a quantifier-free formula consisting of polynomial equations and inequalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' ∃X, Y ∈ R : X · Y = 6 ∧ X + Y = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We denote the decision problem whether such a sentence is true by ETR (which also stands for “existential theory of the reals”) and define the complexity class ∃R to contain all decision problems that polynomial-time reduce to ETR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' It holds NP ⊆ ∃R ⊆ PSPACE [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The class ∃R has gained increasing attention in the computational geometry community over the last years as it exactly captures the complexity of many geometry problems like the art gallery problem [2], geometric packing [3] or the recognition of many classes of geometric intersection graphs [15, 18, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='2 Hyperbolic Geometry The are several ways to embed the hyperbolic plane into Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' In this paper we use the Beltrami-Klein model and the hyperboloid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Bieker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Bläsius, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Dohse and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Jungeblut 3 −2 −1 1 2 −2 2 2 D ℓ1 ℓ2 ℓ3 S+ Figure 2 Left: The Beltrami-Klein disk with three hyperbolic lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Right: The upper sheet S+ used for the hyperboloid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Beltrami-Klein Model In the Beltrami-Klein model the hyperbolic plane H2 is represented by the interior of a unit disk D in R2 (the boundary of D is not part of the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The set of hyperbolic lines is exactly the set of chords of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' See Figure 2 (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Hyperboloid Model Here the hyperbolic plane gets embedded into R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The Minkowski quadratic form Q(x, y, z) := z2 − x2 − y2 defines a two-sheeted hyperboloid S := {(x, y, z) ∈ R3 | Q(x, y, z) = 1}, see Figure 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The hyperbolic plane is represented by all points on the forward sheet S+ of S, obtained by additionally requiring that z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The hyperbolic distance between two points u, v ∈ S+ is dh(u, v) = arcosh(B(u, v)) with B(u, v) := uzvz − uxvx − uyvy where B(u, v) is known as the Minkowski bilinear form and arcosh(x) := ln � x + √ x2 − 1 � is the inverse hyperbolic cosine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Note that the term inside the arcosh(·) is a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 2 Simple Stretchability in the Euclidean and the Hyperbolic Plane An pseudoline arrangement A is a collection of pseudolines (x-monotone curves in R2) such that each pair of curves intersects at most once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We assume that each pseudoline ℓ ∈ A is oriented and thus divides the plane R2 into two open half-planes ℓ− and ℓ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Further, A partitions the plane into cells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=', maximal connected components of R2 \\ A not on any pseudoline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We say that A is simple if any two lines intersect exactly once and no three lines intersect in the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Given a pseudoline arrangement A = {ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , ℓn} we assign to each p ∈ R2 a sign vector σ(p) = (σi(p))n i=1 ∈ {−, 0, +}n, where σi(p) := � � � � � � � − if p ∈ ℓ− i 0 if p ∈ ℓi + if p ∈ ℓ+ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The combinatorial description D of A is then given by {σ(p) | p ∈ R2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We say that A real- izes D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' A pseudoline arrangement is stretchable if there is a line arrangement with the same combinatorial description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Not every pseudoline arrangement is stretchable and, given a com- binatorial description D, deciding whether D is stretchable is known as the Stretchability problem (or SimpleStretchability if D is simple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Stretchability and SimpleStretch- ability are famously known to be ∃R-complete [21, 23, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' SimpleStretchability is the starting problem for many ∃R-hardness reductions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' [4, 12, 15, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 4 Recognizing Unit Disk Graphs in Hyperbolic Geometry is ∃R-Complete ↔ ↔ Figure 3 Transforming line arrangements between Euclidean and hyperbolic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Apart from line arrangements in the Euclidean plane R2 one might also consider line arrangements in the hyperbolic plane H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The main result of this section is that Sim- pleStretchability is equivalent in Euclidean and hyperbolic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' ▶ Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Let D be a combinatorial description of a simple pseudoline arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Then there is a line arrangement realizing D in R2 if and only if there is one in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The proof is an easy application of the Beltrami-Klein model of the hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Given a Euclidean line arrangement, we can obtain a hyperbolic line arrangement with the same combinatorial description and vice versa, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Let AR be a simple line arrangement in R2 and D be a disk strictly enclosing all intersections of AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For each line in AR, keep only its part inside D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We think of D as a unit disk and obtain a representation of a hyperbolic line arrangement in the Beltrami-Klein model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For the other direction let AH be a simple hyperbolic line arrangement and take a repre- sentation inside the Beltrami-Klein disk D, so all hyperbolic lines are chords D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Remove D and extend all chords to lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The resulting Euclidean line arrangement has the same combinatorial description D because AH was simple: All possible intersections between two lines were already inside the Beltrami-Klein disk D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' ◀ ▶ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Proposition 1 is only about simple (pseudo)line arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' There is no corresponding result for the general (non-simple) Stretchability problem: For example, given three lines ℓ1, ℓ2, ℓ3 ⊆ H2, lines ℓ2 and ℓ3 may cross each other while both being parallel to ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' However, Proposition 1 may be extended to line arrangements where each pair of lines is still required to cross but multiple lines are allowed to cross at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 3 The Framework Let ΠR be a geometric decision problem for which ∃R-hardness is shown in Euclidean geometry by a polynomial-time reduction f from (Euclidean) SimpleStretchability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We denote by ΠH the corresponding decision problem obtained by considering the hyperbolic plane H2 instead of the Euclidean plane R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Our framework below consists of several (hopefully) simple steps that allow us to prove ∃R-hardness of ΠH by using the reduction for ΠR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Let D be an instance of SimpleStretchability in H2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=', a combinatorial description of a simple pseudoline arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Use Proposition 1 to consider D to be an instance of SimpleStretchability in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Use the reduction f to obtain an instance I = f(D) of ΠR equivalent to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Prove that every yes-instance of ΠR is also a yes-instance of ΠH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Bieker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Bläsius, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Dohse and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Jungeblut 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Prove that a line arrangement realizing D can be extracted from a realization of I in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Steps 1, 2 and 3 require no work when applying the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Step 4 ensures that a stretchable instance D yields a yes-instance of ΠH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' This step requires to come up with a new argument but we expect it to be relatively simple because locally R2 and H2 are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' A promising approach is to scale a Euclidean realization of I to a tiny area and then interpret the Euclidean polar coordinates as hyperbolic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Step 5 ensures correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' By showing that a line arrangement realizing D can be extracted from a realization of I in H2 we show that a no-instance D maps to a no-instance of ΠH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Reduction f might help us again here (though not as a black box as in Step 3): If we are lucky, the argument why a realization of I in R2 induces a Euclidean line arrangement realizing D only uses the axioms of absolute geometry (the common “subset” of Euclidean and hyperbolic geometry) and works without any adaptations for realizations in H2, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 4 Recognition of Hyperbolic Unit Disk Graphs We apply our framework to prove that Recog(HUDG), the recognition problem of hyperbolic unit disk graphs, is ∃R-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For Euclidean geometry this is shown in [13, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Let us note that UDG and HUDG are not the same: For example, a star graph with six leaves is a hyperbolic unit disk graph but not a Euclidean one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For Step 1 of our framework let D be an instance of SimpleStretchability in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We consider it to be an equivalent instance in R2 for Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' In Step 3 we use the reduction f from the literature proving that Recog(UDG) in R2 is ∃R-hard [13, 19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We obtain a graph GD that is a Euclidean unit disk graph if and only if D is stretchable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Though not required for the framework, let us shortly summarize the reduction f to construct GD from D as given in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Let n be the number of pseudolines ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , ℓn and m = 1 + �n+1 2 � be the number of cells C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The arrangement described by D has exactly this number of cells, because it is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We define GD to be the graph with vertex set V = A ∪ B ∪ C for A = {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , an}, B = {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , bn} and C = {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , cm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Here we assume that vertex ci corresponds to cell Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For the edges, each of the sets A, B, C forms a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Further, each ai ∈ A (for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , n}) is connected to cj (for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , m}) if and only if Cj ∈ ℓ− i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Similarly, each bi ∈ B is connected to cj if and only if Cj ∈ ℓ+ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For Step 4 we have to show that every Euclidean unit disk graph is also a hyperbolic unit disk graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' This has recently been proven by Bläsius, Friedrich, Katzmann and Stephan: ▶ Lemma 3 ([5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Every Euclidean unit disk graph is also a hyperbolic one, so UDG ⊆ HUDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' As foreshadowed above, the proof scales a Euclidean unit disk intersection representation to a tiny area until the Euclidean and hyperbolic plane are “similar enough”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Then the polar coordinates in R2 can be used as polar coordinates in H2 without changing any adjacencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' For Step 5 it remains to prove how a line arrangement realizing D in H2 can be extracted from a realization of GD in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' ▶ Lemma 4 (adapted from [19, Lemma 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Given a realization of GD as the intersection graph of equally sized disks in H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Then the line arrangement L = {ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' , ℓn} defined by ℓi := {p ∈ H2 | d(p, ai) = d(p, bi)} has combinatorial description D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Here d(·, ·) denotes the hyperbolic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 6 Recognizing Unit Disk Graphs in Hyperbolic Geometry is ∃R-Complete Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The proof is exactly the same as the proof of Lemma 1 in [19] where McDiarmid and Müller prove that taking the perpendicular bisectors of the segments between any pair of points ai and bi yields a Euclidean line arrangement realizing D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Their argument works in H2 by just replacing Euclidean distances with hyperbolic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' ◀ At this point we proved ∃R-hardness of Recog(HUDG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' To get ∃R-completeness we prove ∃R-membership next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' ▶ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Recognizing hyperbolic unit disk graphs is in ∃R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' By a result from Erickson, van der Hoog and Miltzow we can prove ∃R-membership by describing a polynomial-time verification algorithm for a real RAM machine3 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Given a graph G = (V, E) and for each vertex v ∈ V a point (vx, vy, vz) in the hyperboloid model of the hyperbolic plane representing the center of an equal-radius disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Compute dadj := maxuv∈E B(u, v) and dnon-adj := minuv̸∈E B(u, v) where B(·, ·) is the Minkowski bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' B(·, ·) is a polynomial, so it is computable on a real RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' We can think of dadj and dnon-adj as distances in hyperbolic space (actually they are the hyperbolic cosine of a distance), but since arcosh is a monotone function, this view is justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Now if and only if dadj < dnon-adj, then there is a radius r such that G is a hyperbolic unit disk graph with radius r (choose r such that dadj ≤ cosh(r) 2 < dnon-adj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The algorithm takes O(|V |2) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' This is polynomial in the input size, proving ∃R-membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' ◀ We conclude with the following theorem: ▶ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Recognizing hyperbolic unit disk graphs is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 5 Conclusion and Outlook We presented a simple framework that allows us to translate ∃R-hardness reductions for geometric decision problems in R2 into reductions for their counterparts H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' As an application we proved that Recog(HUDG) is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Promising candidates for further applications of our framework are the recognition of unit ball graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=', a generalization of our result to higher dimensions) as already done in Rd in [13] or Recog(CONV), the recognition problem for intersection graphs of convex sets (Euclidean reduction is in [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Technically, the framework also works for the recognition problems Recog(HSEG) and Recog(HDISK), where (H)SEG and (H)DISK denote the classes of intersection graphs of (hyperbolic) segments and disks, respectively (Euclidean reductions are in [13, 15, 18, 20, 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' However, these are not really interesting as SEG = HSEG (easy to see in the Beltrami- Klein model) and DISK = HDISK (easy to see in the Poincaré model, not considered here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Therefore ∃R-completeness for Recog(HSEG) and Recog(HDISK) follows directly from the Euclidean cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Other interesting problems to consider in H2 are linkage realizability [1, 25], simultaneous graph embeddings [9, 17] or RAC-drawings [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Acknowledgements We thank Torsten Ueckerdt for discussion on proving the membership of Recog(HUDG) in ∃R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 3 The real RAM extends the classical word RAM by additional registers that contain real numbers (with arbitrary precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The basic arithmetic operations +, −, · and / are supported in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' However, arbitrary analytic functions (like arcosh) are not supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' See [10] for a formal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Bieker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Bläsius, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Dohse and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Jungeblut 7 References 1 Zachary Abel, Erik D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Demaine, Martin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Demaine, Sarah Eisenstat, Jayson Lynch, and Tao B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Schardl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Who Needs Crossings?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Hardness of Plane Graph Rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' In Sándor Fekete and Anna Lubiw, editors, 32nd International Symposium on Computational Geometry (SoCG 2016), volume 51 of Leibniz International Proceedings in Informatics (LIPIcs), pages 3:1–3:15, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='SoCG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 2 Mikkel Abrahamsen, Anna Adamaszek, and Tillmann Miltzow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' The Art Gallery Problem is ∃R-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Journal of the ACM, 69(1):1–70, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='1145/3486220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 3 Mikkel Abrahamsen, 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Krohmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Cliques in hyperbolic random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Algorithmica, 80:2324–2344, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='1007/s00453-017-0323-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' 7 Michel Bode, Nikolaos Fountoulakis, and Tobias Müller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' On the largest component of a hyper- bolic model of complex networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' Electronic Journal of Combinatorics, 22(1–52):P3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='24, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content=' URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='org/ojs/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FdE5T4oBgHgl3EQfVQ-f/content/2301.05550v1.pdf'} +page_content='php/eljc/article/view/v22i3p24.' 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0000000000000000000000000000000000000000..16fd0054e281addf1241a5346d153591e8bd6016 --- /dev/null +++ b/IdAzT4oBgHgl3EQfjf31/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1a8cde3bc0814e20147b6d4d99a8e7e7231d21547f1f7404950eb3feabf3cdc1 +size 3276845 diff --git a/ItE2T4oBgHgl3EQfpAgB/content/tmp_files/2301.04023v1.pdf.txt b/ItE2T4oBgHgl3EQfpAgB/content/tmp_files/2301.04023v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f12dc8382ff2a2eed52a7be9fc4182e3118cab13 --- /dev/null +++ b/ItE2T4oBgHgl3EQfpAgB/content/tmp_files/2301.04023v1.pdf.txt @@ -0,0 +1,1440 @@ +arXiv:2301.04023v1 [gr-qc] 10 Jan 2023 +The energy–momentum complex in non-local gravity +Salvatore Capozziello ∗1,2,3, Maurizio Capriolo † 4,2, and Gaetano Lambiase ‡4,5 +1Dipartimento di Fisica "E. Pancini", Università di Napoli “Federico II”, Compl. +Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy, +2INFN Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via +Cinthia, I-80126, Napoli, Italy, +3Scuola Superiore Meridionale, Largo S. Marcellino 10, I-80138, Napoli, Italy, +4Dipartimento di Fisica Università di Salerno, via Giovanni Paolo II, 132, +Fisciano, SA I-84084, Italy. +5INFN Sezione di Napoli,Gruppo Collegato di Salerno, via Giovanni Paolo II, 132, +Fisciano, SA I-84084, Italy. +January 11, 2023 +Abstract +In General Relativity, the issue of defining the gravitational energy contained in a given +spatial region is still unresolved, except for particular cases of localized objects where the +asymptotic flatness holds for a given spacetime. In principle, a theory of gravity is not self- +consistent, if the whole energy content is not uniquely defined in a specific volume. Here we +generalize the Einstein gravitational energy-momentum pseudotensor to non-local theories of +gravity where analytic functions of the non-local integral operator □−1 are taken into account. +We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the +one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations. +The +invariance of non-local gravitational action under global translations leads to a locally conserved +Noether current, and thus, to the definition of a gravitational energy-momentum pseudotensor, +which is an affine object transforming like a tensor under affine transformations. Furthermore, +the energy-momentum complex remains locally conserved, thanks to the non-local contracted +Bianchi identities. The continuity equations for the gravitational pseudotensor and the energy- +momentum complex, taking into account both gravitational and matter components, can be +derived. Finally, the weak field limit of pseudotensor is performed to lowest order in metric +perturbation in view of astrophysical applications. +∗capozziello@unina.it +†mcapriolo@unisa.it +‡glambiase@unisa.it +1 + +1 +Introduction +Recently, non-local contributions to the gravitational action have been considered from various +points of view as possible solutions of the problem of renormalization and regularization of gravi- +tational field [1–3]. In this context, a non-local gravitational energy-momentum pseudo-tensor can +be proposed as a manifestation of non-locality of gravity and, therefore, as a possible manifestation +of quantum nature of gravity. +Theories of gravity can be endowed with non-local properties in three ways [4]. Firstly through +integral operators acting on functions whose value at a given point depends on the values of fields at +another point in spacetime [5–7]. Secondly, through gravitational Lagrangians involving an analytic +non-polynomial function F of the operator □, which can be expanded in convergent series with real +coefficients as +F(□) = +∞ +� +h=1 +ah□h , +(1.1) +known as Infinite Derivative Theories of Gravity (IDG) [8–14]. Thirdly, through a suitable consti- +tutive law where, like in electrodynamics, temporal dispersion, anisotropy and non-homogeneity +of medium, i.e. +the spatial dispersion, are due to temporal and spatial non-locality, respec- +tively [15–18]. +At infra-red scales, non-local models of gravity can naturally explain late-time acceleration +without introducing exotic material components such as dark matter and dark energy [5]. +In +addition, they can potentially fix some cosmological and astrophysical problems plaguing the ΛCDM +model [19–21], black hole stability [22], or stability and traversability of wormhole solutions [23]. +On the other hand, many authors such as Einstein, Tolman, Landau, Lifshitz, Papapetrou, +Møller and Weinberg have proposed definitions for gravitational pseudotensor [24–32], to describe +the energy and momentum of gravitational field in General Relativity. These prescriptions are based +either on the introduction of a super-potential or on expanding the Ricci tensor in metric pertur- +bation hµν or on manipulating the Einstein equations. Although these definitions are different, +it has been shown they coincide for Kerr-Schild metric [33]. Many prescriptions for gravitational +pseudotensor in higher-order curvature theories, in metric and Palatini approach, have been pro- +posed [34–41]. Also for teleparallel gravity, it is possible to formulate self-consistent definition of +gravitational pseudotensor [42,44]. +Here, we want to propose a generalization of Einstein gravitational pseudotensor to non-local +gravity models involving f(□−1) operators. It will be derived from a variational principle using the +Noether theorem applied to a gravitational Lagrangian invariant under global translations [43]. This +object remains an affine tensor, i.e. a pseudotensor, but it is a non-local quantity. Indeed, its non- +local corrections involve non-local □−1R terms, which assume, at a point x, a value depending on +the values assumed by the metric tensor gµν in all points of the integration domain. Then, we show +that the covariant conservation of the energy-momentum, associated to the gravitational and matter +fields, holds in non-local f(□−1R) gravity, thanks to the non-local contracted Bianchi identities. +Finally, we implement a lowest order expansion of the non-local pseudotensor, fundamental for +astrophysical calculations such as the power carried by gravitational waves. +The paper is organized as follows. In the Sec. 2, we firstly define the non-local integral operator +□−1, then we prove both that it is the inverse operator of d’Alembertian □ and a generalization +of the Green second identity to the □-operator on the manifold. +In addition, we perform the +total variation of non-local gravitational action with respect to both the metric tensor and the +coordinates. Then we derive the field equations from a variational principle. Sec. 3 is devoted to +2 + +the application of the Noether theorem to the non-local gravitational action for global translations. +The procedure allows us to derive the related Noether current, i.e., the locally conserved energy- +momentum pseudotensor of the gravitational field in non-local gravity. Hence, in Sec. 4, we prove +the non-local generalized contracted Bianchi identities and then analyze the energy-momentum +complex for gravitational and matter fields, in particular its non-local nature and its conservation. +In Sec. 5, we carry out the expansion to lower order in the metric perturbation hµν of non-local +gravitational energy-momentum pseudotensor. Finally, we discuss results and draw conclusions in +Sec. 6. +2 +Variational principle and field equations for non-local grav- +ity +Let the spacetime M be a differentiable 4-manifold endowed with a Lorentzian metric g and Ω +be a i four-dimensional region in M. We can define the integral operator □−1 as follows +Definition 2.1. Let G(x, x′) be the retarded Green function of the differential operator □, i.e., +the solution of the partial differential equation +� +−g(x) □xG(x, x′) = δ4(x − x′) , . +(2.1) +It is subject to retarded boundary condition, due to the causality principle. It is +G(x, x′) = 0 +∀t < t′ , +(2.2) +with the d’Alembert operator defined as +□ = ∇µ∇µ = +1 +√−g ∂σ +�√−ggσλ∂λ +� +. +(2.3) +If p ∈ C∞ +o (R4) is an element of the space of infinitely differentiable functions with compact support, +then the operator +□−1 : C∞ +o (R4) → C∞ +o (R4) , +(2.4) +is given by +(□−1p)(x) = +� +Ω +d4x′ � +−g(x′)G(x, x′)p(x′) , +(2.5) +where Ω ⊆ R4 and supp(p) = Ω. +From now on, we shell identify the region Ω of the manifold M with its image φ(Ω) through +the chart φ : Ω ⊆ M → φ(Ω) ⊆ R4. It always exists because the manifold is differentiable and +therefore covered by an atlas. Likewise, we can identify the boundary of the region ∂Ω with the +action of chart φ on it, i.e., φ(∂Ω). Therefore, let us consider functions, and more generally, vector +and tensor fields on the manifold, as defined on the open set of R4 by means of the graph φ. Thus +we have +Theorem 2.1. Let Ω ⊆ R4 be an open set and f, h ∈ C2(Ω) ∩ C1(Ω) be two twice continuously +differentiable functions in the open and once in its closure. If the boundary ∂Ω is a closed, regular +and orientable three-dimensional hypersurface, then +� +Ω +d4x√−g(f □h − h □f) = +� +∂Ω +dSµ +√−g(f∇µh − h∇µf) , +(2.6) +3 + +where dSµ is the infinitesimal hypersurface element. Under the further assumption that functions +f and h vanish on boundary, i.e., f = h = 0 on ∂Ω, we get +� +Ω +d4x√−g(f □h) = +� +Ω +d4x√−g(h □f) . +(2.7) +Proof. By means of the Leibniz rule applied to the functions f and h, we find the differential identity +f□h = h□f + ∇µ(f∇µh − h∇µf) . +(2.8) +So the integral (2.6) can be written as +� +Ω +d4x√−g(f □h) = +� +Ω +d4x√−g(h □f) + +� +Ω +d4x√−g∇µ(f∇µh − h∇µf) , +(2.9) +that, thanks to the Gauss theorem, transforms the second volume integral of Eq (2.9) into a surface +integral as +� +Ω +d4x√−g(f □h) = +� +Ω +d4x√−g(h □f) + +� +∂Ω +dSµ +√−g(f∇µh − h∇µf) . +If f and h are zero on ∂Ω, then the integral on the boundary ∂Ω vanishes and we get Eq (2.7). +Then, we show that □−1 operator (2.5) is the inverse operator of the d’Alembert operator □. +We can enunciate the following proposition +Theorem 2.2. For all p ∈ C∞ +o (R4), □−1 is the inverse of □, i.e., +(□□−1)p = (□−1□)p = 1p = p +(2.10) +Proof. From the definition of product between two operator, we have +(□□−1)p(x) ≡ □(□−1p)(x) = □x +� +Ω +d4x′ � +−g(x′)G(x, x′)p(x′) += +� +Ω +d4x′ � +−g(x′)□xG(x, x′)p(x′) += +1 +� +−g(x) +� +Ω +d4x′ � +−g(x′)δ4(x − x′)p(x′) = p(x) , +(2.11) +where we used definition (2.5) and the following identity involving the Dirac δ distribution function +f(x) = +� +Ω +d4x′ δ(x − x′)f(x′) , +(2.12) +non-null in x ∈ Ω and zero elsewhere. We have to prove now the second identity in Eq. (2.10), by +means of Theorem (2.1). Hence we have +(□−1□)p(x) ≡ □−1(□p)(x) = +� +Ω +d4x′ � +−g(x′)G(x, x′)□x′p(x′) += +� +Ω +d4x′ � +−g(x′)□x′G(x, x′)p(x′) += +� +Ω +d4x′ � +−g(x′)δ4(x′ − x) +� +−g(x′) +p(x′) = p(x) , +(2.13) +4 + +Let us now consider the following gravitational Lagrangian +Sg = 1 +2χ +� +Ω +d4x √−g +� +R + Rf(□−1R) +� +, +(2.14) +where f is an analytic function of □−1R and χ = 8πG/c4 is a dimensional constant that measures +the coupling between matter and geometry. The variation of gravitational action (2.14) with respect +to both metric tensor and coordinates, denoted by ˜δ, reads as +˜δSg = 1 +2χ +� +Ω +d4x +� +δ(√−gR) + δ(√−gR)f(□−1R) ++ √−gRδ +� +f(□−1R) +� ++ ∂µ(R + Rf(□−1R)δxµ) +� +, +(2.15) +where δ is the variation at fixed coordinates. Also, we have to introduce a further theorem useful for +the variation of gravitational action (2.15), which allows us, under suitable assumptions, to move +the □−1 operator from a factor to another of the product in the integral. +Theorem 2.3. Let f, h ∈ C∞(Ω) be two infinitely differentiable functions on Ω ⊆ R4, that is, +f, h : Ω → C. +If □−1 is the inverse integral operator of the d’Alembert operator □ as defined +in (2.5), then +� +Ω +d4x +� +−g(x)f(x) +� +□−1h +� +(x) = +� +Ω +d4x +� +−g(x)h(x) +� +□−1f +� +(x) . +(2.16) +Proof. Let us prove Theorem (2.3) considering the identity (2.12). It follows +� +Ω +d4x +� +−g(x)f(x) +� +□−1h +� +(x) += +� +Ω +d4x +� +−g(x) +� +Ω′′ d4x′′ f(x′′)δ(x − x′′) +� +Ω′ d4x′ � +−g(x′)G(x′, x)h(x′) += +� +Ω′ d4x′ � +−g(x′)h(x′) +� +Ω′′ d4x′′ +�� +Ω +d4x +� +−g(x)G(x′, x)δ(x − x′′) +� +f(x′′) += +� +Ω′ d4x′ � +−g(x′)h(x′) +� +Ω′′ d4x′′ � +−g(x′′)G(x′, x′′)f(x′′) += +� +Ω′ d4x′ � +−g(x′)h(x′) +� +□−1f +� +(x′) . +(2.17) +Here Ω, Ω′ and Ω′′ are the same region covered by different charts. +We establish, furthermore, a new relation that connects the variation of □ and that of □−1. +Theorem 2.4. Let □ be the d’Alembert operator with its inverse operator □−1 satisfying the identity +□ +� +□−1� += □−1(□) = 1 . +(2.18) +For all p ∈ C∞(R4), we get +� +δ □−1� +p = −□−1δ(□)□−1p , +(2.19) +where δ is the first variation of the operator part only. +5 + +Proof. Varying both sides of identity (2.18) and taking into account that variation of the Identity +operator 1 is zero, we have +δ(□□−1) = δ1 = 0 , +(2.20) +and then, from Eq. (2.18), we get +(δ□)□−1 + □δ(□−1) = 0 . +(2.21) +By means of the action of □−1 operator on the left side of Eq. (2.21), we obtain +□−1(δ□)□−1 + δ(□−1) = 0 , +(2.22) +from which follows the relation (2.19). +Thanks to the above theorems, we are ready to split Eq. (2.15) in three parts. The first part is +the same as in General Relativity +1 +2χ +� +Ω +d4x δ(√−gR) = 1 +2χ +� +Ω +d4x √−g Gµνδgµν + √−g ∇σ +� +gµν∇σδgµν − ∇λδgσλ� +, +(2.23) +while the second one is +1 +2χ +� +Ω +d4x +� +δ(√−gR)f(□−1R) +� += 1 +2χ +� +Ω +d4x +�√−g fGµνδgµν ++ √−gf∇σ +� +gµν∇σδgµν − ∇λδgσλ�� += 1 +2χ +� +Ω +d4x +�√−g +� +Gµν + gµν□ − ∇µ∇ν +� +fδgµν ++ √−g∇σ +�� +gµν∇σδgµν − ∇λδgσλ� +f − +� +gλσgµνδgµν − δgλσ� +∇λf +�� +, +(2.24) +where Gµν is the Einstein tensor +Gµν = Rµν − 1 +2gµνR . +(2.25) +Finally, we have for the third part of Eq. (2.15), from Eqs. (2.16) and (2.19), the following form +1 +2χ +� +Ω +d4x √−gRδ +� +f(□−1R) +� += 1 +2χ +� +Ω +d4x √−gRf ′δ +� +□−1R +� += 1 +2χ +� +Ω +d4x +�√−gRf ′ � +δ(□−1)R + □−1[δR] +�� += 1 +2χ +� +Ω +d4x +�√−gRf ′□−1[δR] − √−gRf ′□−1δ(□)□−1R +� +, +(2.26) +where f ′ = ∂f(□−1R) +∂(□−1R) . The first piece of Eq. (2.26) in the last line, from the identity (2.16), gives +1 +2χ +� +Ω +d4x √−gRf ′□−1[δR] = 1 +2χ +� +Ω +d4x √−g □−1[Rf ′]δR += 1 +2χ +� +Ω +d4x +�√−g □−1[Rf ′]Rµνδgµν + √−g(gµν□ − ∇µ∇ν)□−1[Rf ′]δgµν ++ √−g∇σ +�� +gµν∇σδgµν − ∇λδgσλ� +□−1[Rf ′] − +� +gλσgµνδgµν − δgλσ� +∇λ□−1[Rf ′] +�� +. +(2.27) +6 + +While the second piece of Eq. (2.26) in the last line, by means of the d’Alembert operator (2.3) and +from Eq. (2.16), yields +1 +2χ +� +Ω +d4x +� +−√−gRf ′□−1δ(□)□−1R +� += 1 +2χ +� +Ω +d4x +� +−√−g□−1[Rf ′]δ(□)□−1R +� += 1 +2χ +� +Ω +d4x +� +−√−g □−1[Rf ′]δ +� +1 +√−g +� +∂σ +�√−ggσλ∂λ +� +□−1R +− √−g □−1[Rf ′] +1 +√−g ∂σ +� +δ +�√−ggσλ� +∂λ +� +□−1R +� += 1 +2χ +� +Ω +d4x +� +√−g +� +−1 +2gµνR □−1[Rf ′]δgµν +� ++ ∂σ +� +□−1[Rf ′] +� +∂λ +� +□−1R +� +δ +�√−ggσλ� +− ∂σ +� +□−1[Rf ′]∂λ +� +□−1R +� +δ +�√−ggσλ��� +. +(2.28) +According to Eqs. (2.23), (2.24), (2.27), (2.28) and the following relation +δ +�√−ggσλ� += √−g +� +δ(σ +µ δλ) +ν − 1 +2gµνgσλ +� +, +(2.29) +the variation of the gravitational action (2.14) can be written as follows +˜δSg = 1 +2χ +� +Ω +d4x √−g +�� +Gµν + +� +Gµν + gµν□ − ∇µ∇ν +�� +f + □−1[Rf ′] +� ++ +� +δ(σ +µ δλ) +ν − 1 +2gµνgσλ +� +∂σ +� +□−1[Rf ′] +� +∂λ +� +□−1R +� � +δgµν ++ √−g∇σ +�� +gµν∇σδgµν − ∇λδgσλ� ++ +� +δgλσ − gλσgµνδgµν� +∇λ +� +f + □−1[Rf ′] +� ++ +� +gµν∇σδgµν − ∇λδgσλ� � +f + □−1[Rf ′] +� +− +� +δ(σ +µ δλ) +ν − 1 +2gµνgσλ +� +∇λ +� +□−1R +� +□−1[Rf ′]δgµν + +� +R + Rf +� +δxσ +�� +. +(2.30) +From the least action principle δSg = 0, if field variations and its derivatives vanish on boundary, +the field equations in vacuum are obtained, i.e., +Gµν + ∆Gµν = 0 , +(2.31) +with +∆Gµν = +� +Gµν + gµν□ − ∇µ∇ν +�� +f + □−1[Rf ′] +� ++ +� +δ(σ +µ δλ) +ν − 1 +2gµνgσλ +� +∂σ +� +□−1[Rf ′] +� +∂λ +� +□−1R +� +, +(2.32) +7 + +or if we define +G[P](x) = +� +□−1P +� +(x) , +(2.33) +Eq. (2.31) can be rewritten as +Gµν + +� +Gµν + gµν□ − ∇µ∇ν +�� +f + G[Rf ′] +� ++ +� +δ(σ +µ δλ) +ν − 1 +2gµνgσλ +� +∂σ (G[Rf ′]) ∂λ (G[R]) = 0 . (2.34) +We can find the field equations in presence of matter using the following action +Sm = 1 +2χ +� +Ω +d4x √−g Lm , +(2.35) +and imposing the stationarity of total action, i.e., +δ(Sg + Sm) = 0 , +(2.36) +with the matter energy-momentum tensor defined as +Tµν = − +2 +√−g +δ +�√−gLm +� +δgµν +. +(2.37) +Hence, the field equations in presence of matter are [7] +Gµν + ∆Gµν = χTµν , +(2.38) +or +Gµν+ +� +Gµν+gµν□−∇µ∇ν +�� +f+G[Rf ′] +� ++ +� +δ(σ +µ δλ) +ν − 1 +2gµνgσλ +� +∂σ (G[Rf ′]) ∂λ (G[R]) = χTµν . (2.39) +We shall use these considerations to derive the gravitational energy-momentum pseudotensor. +3 +Gravitational energy-momentum pseudotensor in non–local +gravity +Let us now use the Noether theorem to derive the non-local gravitational energy-momentum +pseudotensor. If the infinitesimal coordinate transformations +x′µ = xµ + δxµ , +(3.1) +leave the gravitational action (2.14) unchanged, ˜δSg = 0, and the domain of integration Ω can be +chosen arbitrarily, by means of the variation (2.30) and the assumption that the metric tensor gµν +is solution of the field equations in vacuum (2.34), we find a conserved current Jσ, i.e., the Noether +current [43], which reads as +2χJσ =Rδxσ − +� +gµνgλσ − gµλgσν� +∇λδgµν ++ +� +gµνgλσ − gµλgσν� +∇λ +� +f + □−1[Rf ′] +� +δgµν +− +� +gµνgλσ − gµλgσν� � +f + □−1[Rf ′] +� +∇λδgµν +− +�1 +2gµνgλσ − gµλgσν +� +∇λ +� +□−1R +� +□−1[Rf ′]δgµν + Rfδxσ , +(3.2) +8 + +that obeys the following local continuity equation +∂σ +�√−gJσ� += 0 . +(3.3) +Integrating the continuity equation (3.3) over a three-dimensional volume V at a given time x0, +from the Gauss theorem, we obtain +d +dx0 +� +V +d3x √−g J0 = − +� +∂V +dSi +√−g Ji . +(3.4) +If the fields with their derivatives vanish on the boundary ∂V , the surface integral on the right of +Eq. (3.4) vanishes, i.e., there is no current crossing the boundary, and we can derive the conserved +Noether charge in the volume V , associated to symmetries (3.1) +Q = +� +V +d3x √−g J0 . +(3.5) +So, if we consider the one-parameter group of diffeomorphisms for the global infinitesimal transla- +tions +x′µ = xµ + ǫµ , +(3.6) +the local variation δ of tensor metric gµν becomes +δgµν = g′ +µν(x) − gµν(x) = −gµν,αǫα . +(3.7) +Hence, the conserved Noether current, related to the translational symmetry (3.6), becomes the +energy-momentum density of the gravitational field, while, for isolated systems, where the spacetime +is asymptotically flat at spatial infinity, the conserved Noether charge becomes the energy and +momentum of the gravitational field. Therefore, the translation invariance of gravitational action, +from Eq. (3.2), gives +τ σ +α = τσ (GR) +α ++ ∆τ σ +α , +(3.8) +where τ σ (GR) +α +is the Einstein pseudotensor +2χτ σ (GR) +α += Rδσ +α + +� +gµνgλσ − gµλgσν� � +gµν,αλ − Γβ +λµgβν,α) , +(3.9) +while the correction ∆τ σ +α, is the gravitational energy-momentum pseudotensor of non-local part, +i.e., +2χ∆τ σ +α =Rfδσ +α + +� +gµνgλσ − gµλgσν� � +gµν,αλ − Γβ +λµgβν,α +� � +f + □−1[Rf ′] +� +− +�� +gµνgλσ − gµλgσν� +∇λ +� +f + □−1[Rf ′] +� +− +�1 +2gµνgλσ − gµλgσν +� +∇λ +� +□−1R +� +□−1[Rf ′] +� +gµν,α +. +(3.10) +The pseudotensor (3.10) has been obtained taking into account that the covariant derivative of +variation for the metric tensor is +∇λδgµν = ∂λδgµν − Γα +λµδgαν − Γα +λνδgαµ . +(3.11) +9 + +The symmetry of Levi Civita connection leads to +� +gµνgλσ − gµλgσν� +Γβ +λν = 0 , +(3.12) +and the local conservation of pseudotensor can be read as +∂α +�√−g τ σ +α +� += 0 , +(3.13) +being +Jα = τ σ +αǫα . +(3.14) +In terms of Eq. (2.33), in more compact form, one gets +2χ∆τ σ +α =Rfδσ +α + +� +gµνgλσ − gµλgσν� � +gµν,αλ − Γβ +λµgβν,α +� +(f + G[Rf ′]) +− +�� +gµνgλσ − gµλgσν� +∂λ (f + G[Rf ′]) +− +�1 +2gµνgλσ − gµλgσν +� +∂λ (G[R]) G[Rf ′] +� +gµν,α +. +(3.15) +It has to be emphasized that,from Eqs. (3.8), (3.9) and (3.10), it is clear that the geometric ob- +ject τ σ +α is a pseudotensor not a tensor. In other words, it transforms like a tensor under affine +transformations but not under generic transformations. So τ σ +α is at least an affine tensor. In an +asymptotically flat spacetime the tensoriality is recovered and the integral (3.5) returns to being a +four-vector for asymptotic linear coordinates, that is, +P α = +� +V +d3x √−g τ α +0 , +(3.16) +represents the energy and momentum in V of the gravitational field. Moreover the pseudotensor +τ σ +α is a non-local object because it involves non-local terms, such as □−1R or □−1[Rf ′], whose +value depends on the values assumed by the metric in the integration domain. +4 +The energy-momentum complex +The stationarity of gravitational action, ˜δSg = 0, with respect to the variation ˜δ, from Eqs. (2.30), +(2.31), (2.32), (3.8) and (3.10), gives +1 +2χ +√−g +� +Gµν + ∆Gµν +� +δgµν + ∂σ +�√−gτ σ +αǫα� += 0 . +(4.1) +Hence, inserting the field equations in presence of matter (2.38) into Eq. (4.1), we get +− 1 +2 +√−g T µνδgµν + ∂σ +�√−gτ σ +αǫα� += 0 . +(4.2) +From rigid translations and coordinates independence from ǫα, it yields +1 +2 +√−g T µνgµν,α + ∂σ +�√−gτ σ +α +� += −√−g∇σT σ +α + ∂σ +�√−gT σ +α +� ++ ∂σ +�√−gτ σ +α +� +, +(4.3) +10 + +where the identity +√−g∇σT σ +α = ∂σ +�√−gT σ +α +� +− 1 +2 +√−g gµν,αT µν , +(4.4) +has been taken into account. From Eq. (4.3), we obtain +∂σ +�√−g +� +T σ +α + τ σ +α +�� += √−g∇σT σ +α . +(4.5) +According to the previous considerations, it is possible to prove generalized contracted Bianchi +identities for non-local gravity [40,45,46]. They guarantee the conservation of energy–momentum +complex of gravitational and matter components. Let us first demonstrate a lemma useful for our +purpose. +Lemma 4.1. Let f ∈ C2(Ω) be a twice continuously differentiable function on an open set Ω of +R4, ∇ be the covariant derivative, □ be the d’Alembert operator and [, ] be the commutator, we +have +[∇ν, □]f = −Rµν∇µf . +(4.6) +Proof. From the commutator of two covariant derivatives ∇µ and ∇ν, which acts on the contravari- +ant vector field Aγ, we get +[∇µ, ∇ν]Aγ = Rγ +λµνAλ . +(4.7) +If we set Aγ = ∇γf and γ = ν in Eq. (4.7), we obtain +[∇µ, □]f = ∇µ∇ν∇νf − ∇ν∇ν∇µf += ∇ν[∇µ, ∇ν]f − [∇µ, ∇ν]∇νf = ∇ν[∇µ, ∇ν]f − Rµν∇νf . +(4.8) +Thus, the commutativity of covariant derivatives of a function, that is, +[∇µ, ∇ν]f = 0 , +(4.9) +inserted into Eq. (4.8), gives us the result (4.6). +Theorem 4.1 (Non-local generalized contracted Bianchi identities). Let Gµν be the Einstein tensor +and ∆Gµν be the corrections to the field equations due to non-local terms as in Eq. (2.38), then the +covariant 4-divergence of their sum vanishes, i.e., +∇µ� +Gµν + ∆Gµν +� += 0 . +(4.10) +Proof. We carry out the 4-divergence of Eq. (2.32) and we have +∇µ∆Gµν = +� +∇µGµν + ∇ν□ − □∇ν +�� +f + □−1[Rf ′] +� ++ Gµν∇µ(f + □−1[Rf ′]) ++ 1 +2 +� +δλ +ν ∇σ + δσ +ν ∇λ − gσλ∇ν +� +∇σ□−1[Rf ′]∇λ□−1R . +(4.11) +So, from the contracted Bianchi identities +∇µGµν = 0 , +(4.12) +11 + +and performing some calculations, Eq. (4.11) can be rewritten as follows +∇µ∆Gµν = [∇ν, □] +� +f + □−1[Rf ′] +� ++ Gµν∇µ(f + □−1[Rf ′]) ++ 1 +2 +� +□□−1[Rf ′]∇ν□−1R + ∇σ□−1[Rf ′]∇σ∇ν□−1R + ∇σ∇ν□−1[Rf ′]∇σ□−1R ++ ∇ν□−1[Rf ′]□□−1R − ∇ν∇σ[Rf ′]∇σ□−1R − ∇σ□−1[Rf ′]∇ν∇σ□−1R +� +. +(4.13) +Now, the relation (4.13) and the lemma (4.1) lead to Eq. (4.10), that is, we find +∇µ∆Gµν = −Rµν +� +f + □−1[Rf ′] +� ++ Gµν∇µ(f + □−1[Rf ′]) ++ 1 +2Rf ′∇ν□−1R + 1 +2R∇ν□−1[Rf ′] += −Rµνf ′∇µ□−1R − Rµν∇µ□−1[Rf ′] + Gµν∇µ(f + □−1[Rf ′]) ++ 1 +2gµνRf ′∇µ□−1R + 1 +2gµνR∇µ□−1[Rf ′] += −Gµν∇µ� +f + □−1[Rf ′] +� ++ Gµν∇µ(f + □−1[Rf ′]) = 0 . +(4.14) +According to the field equation in presence of matter (2.38), Eq. (4.10) leads to the standard +covariant conservation of matter energy-momentum tensor, that is, +∇µT µν = 0 . +(4.15) +It implicitly defines the trajectories of particles, that is, the time-like metric geodesics on the +spacetime manifold. Finally, Eq. (4.5) gives the local conservation of energy-momentum complex +T σ +α in non-local gravity, that is, the continuity equation for energy-momentum complex in non-local +gravity +∂σ +�√−g +� +T σ +α + τ σ +α +�� += 0 . +(4.16) +We can define +T σ +α = T σ +α + τ σ +α , +(4.17) +involving all gravitational and matter contributions. +5 +Weak field limit of non-local gravitaty energy-momentum +pseudotensor +Let us now develop the low energy limit perturbing the metric tensor gµν around the Minkowskian +metric ηµν. It is +gµν = ηµν + hµν , +(5.1) +and then, we can calculate the pseudotensor (3.10) or (3.15) to lowest order in the perturbation +hµν, that is, up to second ordear in hµν. Therefore we get, at the order h2, +(τ σ +α)(2) = +� +τ σ (GR) +α +�(2) ++ (∆τ σ +α)(2) , +(5.2) +12 + +where the Einstein pseudo-tensor is +2χ +� +τ σ (GR) +α +�(2) += R(2)δσ +α + +� +gµνgλσ − gµλgσν�(1) g(1) +µν,αλ , +(5.3) +and, from Eq. (3.15), the non-local perturbation of pseudotensor takes the form +2χ (∆τ σ +α)(2) = R(1)f (1)δσ +α + +� +gµνgλσ − gµλgσν�(0) � +f (1) + G(1)[Rf ′] +� +,λ g(1) +µν,α +− +� +gµνgλσ − gµλgσν�(0) � +f (1) + G(1)[Rf ′] +� +g(1) +µν,αλ . +(5.4) +Then, we expand f as +f (G[R]) (x) = f(0) + f ′(0)G[R](x) + . . . , +(5.5) +and imposing the case f(0) = 0, the relation (5.5) to the first order takes the form +f (1) (G[R]) (x) = f ′(0)G(1)[R](x) . +(5.6) +Taking into account the following first order perturbations in a generic coordinate system, the Ricci +scalar becomes +R(1) = +� +hβγ +,βγ − □(0)h +� +, +(5.7) +where +□(0) = ηαβ∂α∂β , +(5.8) +and the non-local operator □−1 at first order reads as +G(1)[R](x) = +� +□−1R +�(1) (x) = −h(x) + �G +� +hβγ +,βγ +� +(x) , +(5.9) +where +�G +� +hβγ +,βγ +� +(x) = +� +Ω +d4x′G(x, x′) hβγ +,βγ(x′) . +(5.10) +We have to prove the identity (5.9). Using Eqs. (2.1), (2.5), (5.7) and the theorem (2.1), it is +� +□−1R +�(1) (x) = +� +Ω′ d4x′� +−g(x′) +(0)G(x, x′)R(1)(x′) += +� +Ω +d4x′� +−g(x′) +(0)G(x, x′) +� +hβγ +,βγ(x′) − □x′h(x′) +� += − +� +Ω +d4x′□x′G(x, x′)h(x′) + +� +Ω +d4x′G(x, x′) hβγ +,βγ(x′) += − +� +Ω +d4x′δ(x − x′)h(x′) + �G +� +hβγ +,βγ +� +(x) = −h(x) + �G +� +hβγ +,βγ +� +(x) . +(5.11) +Furthermore, we perform the first-order perturbation of G[Rf ′], namely +G(1)[Rf ′](x) = +� +Ω +d4� +−g(x′) +(0)G(x, x′)R(1)(x′)f ′(0)[G](x′) += f ′(0) +� +Ω +d4� +−g(x′) +(0)G(x, x′)R(1)(x′) = f ′(0)G(1)[R](x) . +(5.12) +13 + +Finally substituting the Eqs. (5.7), (5.9) and (5.12) in the non-local perturbed gravitational energy– +momentum pseudotensor (5.4), we derive the non-local corrections of the gravitational pseudo-tensor +τ σ +α to the second order in hµν, that is, +2χ (∆τ σ +α)(2) = +�� +hβγ +,βγ − □h +�� +−h + �G +� +hβγ +,βγ +�� +δσ +α ++ 2 +� +ηµνηλσ − ηµληνσ� � +−h + �G +� +hβγ +,βγ +�� +,λhµν,α +− 2 +� +ηµνηλσ − ηµληνσ� � +−h + �G +� +hβγ +,βγ +�� +hµν,αλ +� +f ′(0) +. +(5.13) +The non-local contribution in Eq. (5.13) is evident and, as discussed in Refs. [47–49], it can con- +tribute to gravitational radiation representing a signature for non-local gravity. +6 +Discussion and Conclusions +In this paper, we investigated how non-locality gravity induces correction terms ∆τ σ +α into the +Einstein gravitational pseudotensor. Considering the Noether theorem and imposing the invariance +of gravitational action under rigid translations, we found the associated conserved Noether current +and charge. They can be interpreted as the gravitational density of the energy-momentum and +the energy and momentum of gravitational field present in a spatial volume enclosing localized +massive objects. The density and flux density of the gravitational energy and momentum expressed +in Eq. (3.10) are not described by a covariant tensor, which means that, under general coordinate +transformations, it does not transform like a tensor. The geometrical object (3.10) is an affine +tensor or pseudotensor because it transforms like a tensor only under affine transformations. The +non-tensorial character of Eq. (3.10) is closely linked to the non-localization of gravitational energy +which holds also in non-local gravity. The non-locality of the gravitational pseudotensor intervenes +through integral operators, like □−1, where its value, at a given point x, takes into account the value +assumed by the fields in other points x′ of the spacetime. Then, by generalizing the contracted +Bianchi identities to the non-local gravity, we have obtained an equation of continuity for the +energy-momentum complex that ensures its local conservation. Finally, we studied the behavior +at low energies of the non-local corrections of the gravitational pseudotensor (5.13), expanding it +up to the second order in hµν. The non-local gravitational energy-momentum pseudotensor is a +crucial physical quantity because, thanks to the gravitational waves obtained and analyzed in the +papers [47–49], it is possible to calculate the power emitted by a radiative system and transported +by the waves with all its polarizations and multipole terms. The presence, in the gravitational +radiation, of a scalar component with lower multipoles, in addition to the standard quadrupole +tensor component, can be investigated thanks to the gravitational pseudotensor. In this perspective, +it can give a relevant signature for the non-local gravity. In a forthcoming paper, we will investigate +possible observational constraints on these features. +Acknowledgements +This paper is based upon work from COST Action CA21136 Addressing observational tensions in +cosmology with systematics and fundamental physics (CosmoVerse) supported by COST (European +14 + +Cooperation in Science and Technology). Authors acknowledge the Istituto Nazionale di Fisica +Nucleare (INFN) Sez. di Napoli, Iniziative Specifiche QGSKY and MOONLIGHT, and the Istituto +Nazionale di Alta Matematica (INdAM), gruppo GNFM. +References +[1] L. Modesto, Super-renormalizable quantum gravity Phys. Rev. D 86, 044005 (2012). +[2] L. Modesto, L. Rachwał, Nonlocal quantum gravity: A review, Int. J. Mod. Phys. D 26, 1730020 +(2017). +[3] L. Modesto, L. Rachwał, I. L. Shapiro, Renormalization group in super-renormalizable quantum +gravity, Eur. Phys. J. C 78, 555 (2018). +[4] S. Capozziello and F. Bajardi, Nonlocal gravity cosmology: An overview, Int. J. Mod. 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B 810, 135821 (2020). +17 + diff --git a/ItE2T4oBgHgl3EQfpAgB/content/tmp_files/load_file.txt b/ItE2T4oBgHgl3EQfpAgB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..78240dc00689716f7891583dae251e76ea80c3c8 --- /dev/null +++ b/ItE2T4oBgHgl3EQfpAgB/content/tmp_files/load_file.txt @@ -0,0 +1,699 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf,len=698 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='04023v1 [gr-qc] 10 Jan 2023 The energy–momentum complex in non-local gravity Salvatore Capozziello ∗1,2,3, Maurizio Capriolo † 4,2, and Gaetano Lambiase ‡4,5 1Dipartimento di Fisica "E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Pancini", Università di Napoli “Federico II”, Compl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' di Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy, 2INFN Sezione di Napoli, Compl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' di Monte S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy, 3Scuola Superiore Meridionale, Largo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Marcellino 10, I-80138, Napoli, Italy, 4Dipartimento di Fisica Università di Salerno, via Giovanni Paolo II, 132, Fisciano, SA I-84084, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 5INFN Sezione di Napoli,Gruppo Collegato di Salerno, via Giovanni Paolo II, 132, Fisciano, SA I-84084, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' January 11, 2023 Abstract In General Relativity, the issue of defining the gravitational energy contained in a given spatial region is still unresolved, except for particular cases of localized objects where the asymptotic flatness holds for a given spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In principle, a theory of gravity is not self- consistent, if the whole energy content is not uniquely defined in a specific volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Here we generalize the Einstein gravitational energy-momentum pseudotensor to non-local theories of gravity where analytic functions of the non-local integral operator □−1 are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' We apply the Noether theorem to a gravitational Lagrangian, supposed invariant under the one-parameter group of diffeomorphisms, that is, the infinitesimal rigid translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The invariance of non-local gravitational action under global translations leads to a locally conserved Noether current, and thus, to the definition of a gravitational energy-momentum pseudotensor, which is an affine object transforming like a tensor under affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Furthermore, the energy-momentum complex remains locally conserved, thanks to the non-local contracted Bianchi identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The continuity equations for the gravitational pseudotensor and the energy- momentum complex, taking into account both gravitational and matter components, can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Finally, the weak field limit of pseudotensor is performed to lowest order in metric perturbation in view of astrophysical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' ∗capozziello@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='it †mcapriolo@unisa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='it ‡glambiase@unisa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='it 1 1 Introduction Recently, non-local contributions to the gravitational action have been considered from various points of view as possible solutions of the problem of renormalization and regularization of gravi- tational field [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In this context, a non-local gravitational energy-momentum pseudo-tensor can be proposed as a manifestation of non-locality of gravity and, therefore, as a possible manifestation of quantum nature of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Theories of gravity can be endowed with non-local properties in three ways [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Firstly through integral operators acting on functions whose value at a given point depends on the values of fields at another point in spacetime [5–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Secondly, through gravitational Lagrangians involving an analytic non-polynomial function F of the operator □, which can be expanded in convergent series with real coefficients as F(□) = ∞ � h=1 ah□h , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1) known as Infinite Derivative Theories of Gravity (IDG) [8–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Thirdly, through a suitable consti- tutive law where, like in electrodynamics, temporal dispersion, anisotropy and non-homogeneity of medium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' the spatial dispersion, are due to temporal and spatial non-locality, respec- tively [15–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' At infra-red scales, non-local models of gravity can naturally explain late-time acceleration without introducing exotic material components such as dark matter and dark energy [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In addition, they can potentially fix some cosmological and astrophysical problems plaguing the ΛCDM model [19–21], black hole stability [22], or stability and traversability of wormhole solutions [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' On the other hand, many authors such as Einstein, Tolman, Landau, Lifshitz, Papapetrou, Møller and Weinberg have proposed definitions for gravitational pseudotensor [24–32], to describe the energy and momentum of gravitational field in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' These prescriptions are based either on the introduction of a super-potential or on expanding the Ricci tensor in metric pertur- bation hµν or on manipulating the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Although these definitions are different, it has been shown they coincide for Kerr-Schild metric [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Many prescriptions for gravitational pseudotensor in higher-order curvature theories, in metric and Palatini approach, have been pro- posed [34–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Also for teleparallel gravity, it is possible to formulate self-consistent definition of gravitational pseudotensor [42,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Here, we want to propose a generalization of Einstein gravitational pseudotensor to non-local gravity models involving f(□−1) operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' It will be derived from a variational principle using the Noether theorem applied to a gravitational Lagrangian invariant under global translations [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' This object remains an affine tensor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' a pseudotensor, but it is a non-local quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Indeed, its non- local corrections involve non-local □−1R terms, which assume, at a point x, a value depending on the values assumed by the metric tensor gµν in all points of the integration domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Then, we show that the covariant conservation of the energy-momentum, associated to the gravitational and matter fields, holds in non-local f(□−1R) gravity, thanks to the non-local contracted Bianchi identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Finally, we implement a lowest order expansion of the non-local pseudotensor, fundamental for astrophysical calculations such as the power carried by gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In the Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 2, we firstly define the non-local integral operator □−1, then we prove both that it is the inverse operator of d’Alembertian □ and a generalization of the Green second identity to the □-operator on the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In addition, we perform the total variation of non-local gravitational action with respect to both the metric tensor and the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Then we derive the field equations from a variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 3 is devoted to 2 the application of the Noether theorem to the non-local gravitational action for global translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The procedure allows us to derive the related Noether current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', the locally conserved energy- momentum pseudotensor of the gravitational field in non-local gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Hence, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 4, we prove the non-local generalized contracted Bianchi identities and then analyze the energy-momentum complex for gravitational and matter fields, in particular its non-local nature and its conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 5, we carry out the expansion to lower order in the metric perturbation hµν of non-local gravitational energy-momentum pseudotensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Finally, we discuss results and draw conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 2 Variational principle and field equations for non-local grav- ity Let the spacetime M be a differentiable 4-manifold endowed with a Lorentzian metric g and Ω be a i four-dimensional region in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' We can define the integral operator □−1 as follows Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Let G(x, x′) be the retarded Green function of the differential operator □, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', the solution of the partial differential equation � −g(x) □xG(x, x′) = δ4(x − x′) , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1) It is subject to retarded boundary condition, due to the causality principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' It is G(x, x′) = 0 ∀t < t′ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='2) with the d’Alembert operator defined as □ = ∇µ∇µ = 1 √−g ∂σ �√−ggσλ∂λ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3) If p ∈ C∞ o (R4) is an element of the space of infinitely differentiable functions with compact support, then the operator □−1 : C∞ o (R4) → C∞ o (R4) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='4) is given by (□−1p)(x) = � Ω d4x′ � −g(x′)G(x, x′)p(x′) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) where Ω ⊆ R4 and supp(p) = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' From now on, we shell identify the region Ω of the manifold M with its image φ(Ω) through the chart φ : Ω ⊆ M → φ(Ω) ⊆ R4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' It always exists because the manifold is differentiable and therefore covered by an atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Likewise, we can identify the boundary of the region ∂Ω with the action of chart φ on it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', φ(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Therefore, let us consider functions, and more generally, vector and tensor fields on the manifold, as defined on the open set of R4 by means of the graph φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Thus we have Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Let Ω ⊆ R4 be an open set and f, h ∈ C2(Ω) ∩ C1(Ω) be two twice continuously differentiable functions in the open and once in its closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' If the boundary ∂Ω is a closed, regular and orientable three-dimensional hypersurface, then � Ω d4x√−g(f □h − h □f) = � ∂Ω dSµ √−g(f∇µh − h∇µf) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='6) 3 where dSµ is the infinitesimal hypersurface element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Under the further assumption that functions f and h vanish on boundary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', f = h = 0 on ∂Ω, we get � Ω d4x√−g(f □h) = � Ω d4x√−g(h □f) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' By means of the Leibniz rule applied to the functions f and h, we find the differential identity f□h = h□f + ∇µ(f∇µh − h∇µf) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='8) So the integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='6) can be written as � Ω d4x√−g(f □h) = � Ω d4x√−g(h □f) + � Ω d4x√−g∇µ(f∇µh − h∇µf) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='9) that, thanks to the Gauss theorem, transforms the second volume integral of Eq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='9) into a surface integral as � Ω d4x√−g(f □h) = � Ω d4x√−g(h □f) + � ∂Ω dSµ √−g(f∇µh − h∇µf) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' If f and h are zero on ∂Ω, then the integral on the boundary ∂Ω vanishes and we get Eq (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Then, we show that □−1 operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) is the inverse operator of the d’Alembert operator □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' We can enunciate the following proposition Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' For all p ∈ C∞ o (R4), □−1 is the inverse of □, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', (□□−1)p = (□−1□)p = 1p = p (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' From the definition of product between two operator, we have (□□−1)p(x) ≡ □(□−1p)(x) = □x � Ω d4x′ � −g(x′)G(x, x′)p(x′) = � Ω d4x′ � −g(x′)□xG(x, x′)p(x′) = 1 � −g(x) � Ω d4x′ � −g(x′)δ4(x − x′)p(x′) = p(x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='11) where we used definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) and the following identity involving the Dirac δ distribution function f(x) = � Ω d4x′ δ(x − x′)f(x′) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='12) non-null in x ∈ Ω and zero elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' We have to prove now the second identity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10), by means of Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Hence we have (□−1□)p(x) ≡ □−1(□p)(x) = � Ω d4x′ � −g(x′)G(x, x′)□x′p(x′) = � Ω d4x′ � −g(x′)□x′G(x, x′)p(x′) = � Ω d4x′ � −g(x′)δ4(x′ − x) � −g(x′) p(x′) = p(x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='13) 4 Let us now consider the following gravitational Lagrangian Sg = 1 2χ � Ω d4x √−g � R + Rf(□−1R) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='14) where f is an analytic function of □−1R and χ = 8πG/c4 is a dimensional constant that measures the coupling between matter and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The variation of gravitational action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='14) with respect to both metric tensor and coordinates, denoted by ˜δ, reads as ˜δSg = 1 2χ � Ω d4x � δ(√−gR) + δ(√−gR)f(□−1R) + √−gRδ � f(□−1R) � + ∂µ(R + Rf(□−1R)δxµ) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='15) where δ is the variation at fixed coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Also, we have to introduce a further theorem useful for the variation of gravitational action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='15), which allows us, under suitable assumptions, to move the □−1 operator from a factor to another of the product in the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Let f, h ∈ C∞(Ω) be two infinitely differentiable functions on Ω ⊆ R4, that is, f, h : Ω → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' If □−1 is the inverse integral operator of the d’Alembert operator □ as defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5), then � Ω d4x � −g(x)f(x) � □−1h � (x) = � Ω d4x � −g(x)h(x) � □−1f � (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Let us prove Theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3) considering the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' It follows � Ω d4x � −g(x)f(x) � □−1h � (x) = � Ω d4x � −g(x) � Ω′′ d4x′′ f(x′′)δ(x − x′′) � Ω′ d4x′ � −g(x′)G(x′, x)h(x′) = � Ω′ d4x′ � −g(x′)h(x′) � Ω′′ d4x′′ �� Ω d4x � −g(x)G(x′, x)δ(x − x′′) � f(x′′) = � Ω′ d4x′ � −g(x′)h(x′) � Ω′′ d4x′′ � −g(x′′)G(x′, x′′)f(x′′) = � Ω′ d4x′ � −g(x′)h(x′) � □−1f � (x′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='17) Here Ω, Ω′ and Ω′′ are the same region covered by different charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' We establish, furthermore, a new relation that connects the variation of □ and that of □−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Let □ be the d’Alembert operator with its inverse operator □−1 satisfying the identity □ � □−1� = □−1(□) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='18) For all p ∈ C∞(R4), we get � δ □−1� p = −□−1δ(□)□−1p , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='19) where δ is the first variation of the operator part only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Varying both sides of identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='18) and taking into account that variation of the Identity operator 1 is zero, we have δ(□□−1) = δ1 = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='20) and then, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='18), we get (δ□)□−1 + □δ(□−1) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='21) By means of the action of □−1 operator on the left side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='21), we obtain □−1(δ□)□−1 + δ(□−1) = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='22) from which follows the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Thanks to the above theorems, we are ready to split Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='15) in three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The first part is the same as in General Relativity 1 2χ � Ω d4x δ(√−gR) = 1 2χ � Ω d4x √−g Gµνδgµν + √−g ∇σ � gµν∇σδgµν − ∇λδgσλ� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='23) while the second one is 1 2χ � Ω d4x � δ(√−gR)f(□−1R) � = 1 2χ � Ω d4x �√−g fGµνδgµν + √−gf∇σ � gµν∇σδgµν − ∇λδgσλ�� = 1 2χ � Ω d4x �√−g � Gµν + gµν□ − ∇µ∇ν � fδgµν + √−g∇σ �� gµν∇σδgµν − ∇λδgσλ� f − � gλσgµνδgµν − δgλσ� ∇λf �� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='24) where Gµν is the Einstein tensor Gµν = Rµν − 1 2gµνR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='25) Finally, we have for the third part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='15), from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='19), the following form 1 2χ � Ω d4x √−gRδ � f(□−1R) � = 1 2χ � Ω d4x √−gRf ′δ � □−1R � = 1 2χ � Ω d4x �√−gRf ′ � δ(□−1)R + □−1[δR] �� = 1 2χ � Ω d4x �√−gRf ′□−1[δR] − √−gRf ′□−1δ(□)□−1R � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='26) where f ′ = ∂f(□−1R) ∂(□−1R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The first piece of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='26) in the last line, from the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='16), gives 1 2χ � Ω d4x √−gRf ′□−1[δR] = 1 2χ � Ω d4x √−g □−1[Rf ′]δR = 1 2χ � Ω d4x �√−g □−1[Rf ′]Rµνδgµν + √−g(gµν□ − ∇µ∇ν)□−1[Rf ′]δgµν + √−g∇σ �� gµν∇σδgµν − ∇λδgσλ� □−1[Rf ′] − � gλσgµνδgµν − δgλσ� ∇λ□−1[Rf ′] �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='27) 6 While the second piece of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='26) in the last line, by means of the d’Alembert operator (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3) and from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='16), yields 1 2χ � Ω d4x � −√−gRf ′□−1δ(□)□−1R � = 1 2χ � Ω d4x � −√−g□−1[Rf ′]δ(□)□−1R � = 1 2χ � Ω d4x � −√−g □−1[Rf ′]δ � 1 √−g � ∂σ �√−ggσλ∂λ � □−1R − √−g □−1[Rf ′] 1 √−g ∂σ � δ �√−ggσλ� ∂λ � □−1R � = 1 2χ � Ω d4x � √−g � −1 2gµνR □−1[Rf ′]δgµν � + ∂σ � □−1[Rf ′] � ∂λ � □−1R � δ �√−ggσλ� − ∂σ � □−1[Rf ′]∂λ � □−1R � δ �√−ggσλ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='28) According to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='23), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='24), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='27), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='28) and the following relation δ �√−ggσλ� = √−g � δ(σ µ δλ) ν − 1 2gµνgσλ � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='29) the variation of the gravitational action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='14) can be written as follows ˜δSg = 1 2χ � Ω d4x √−g �� Gµν + � Gµν + gµν□ − ∇µ∇ν �� f + □−1[Rf ′] � + � δ(σ µ δλ) ν − 1 2gµνgσλ � ∂σ � □−1[Rf ′] � ∂λ � □−1R � � δgµν + √−g∇σ �� gµν∇σδgµν − ∇λδgσλ� + � δgλσ − gλσgµνδgµν� ∇λ � f + □−1[Rf ′] � + � gµν∇σδgµν − ∇λδgσλ� � f + □−1[Rf ′] � − � δ(σ µ δλ) ν − 1 2gµνgσλ � ∇λ � □−1R � □−1[Rf ′]δgµν + � R + Rf � δxσ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='30) From the least action principle δSg = 0, if field variations and its derivatives vanish on boundary, the field equations in vacuum are obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', Gµν + ∆Gµν = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='31) with ∆Gµν = � Gµν + gµν□ − ∇µ∇ν �� f + □−1[Rf ′] � + � δ(σ µ δλ) ν − 1 2gµνgσλ � ∂σ � □−1[Rf ′] � ∂λ � □−1R � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='32) 7 or if we define G[P](x) = � □−1P � (x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='33) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='31) can be rewritten as Gµν + � Gµν + gµν□ − ∇µ∇ν �� f + G[Rf ′] � + � δ(σ µ δλ) ν − 1 2gµνgσλ � ∂σ (G[Rf ′]) ∂λ (G[R]) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='34) We can find the field equations in presence of matter using the following action Sm = 1 2χ � Ω d4x √−g Lm , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='35) and imposing the stationarity of total action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', δ(Sg + Sm) = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='36) with the matter energy-momentum tensor defined as Tµν = − 2 √−g δ �√−gLm � δgµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='37) Hence, the field equations in presence of matter are [7] Gµν + ∆Gµν = χTµν , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='38) or Gµν+ � Gµν+gµν□−∇µ∇ν �� f+G[Rf ′] � + � δ(σ µ δλ) ν − 1 2gµνgσλ � ∂σ (G[Rf ′]) ∂λ (G[R]) = χTµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='39) We shall use these considerations to derive the gravitational energy-momentum pseudotensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 3 Gravitational energy-momentum pseudotensor in non–local gravity Let us now use the Noether theorem to derive the non-local gravitational energy-momentum pseudotensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' If the infinitesimal coordinate transformations x′µ = xµ + δxµ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1) leave the gravitational action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='14) unchanged, ˜δSg = 0, and the domain of integration Ω can be chosen arbitrarily, by means of the variation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='30) and the assumption that the metric tensor gµν is solution of the field equations in vacuum (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='34), we find a conserved current Jσ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', the Noether current [43], which reads as 2χJσ =Rδxσ − � gµνgλσ − gµλgσν� ∇λδgµν + � gµνgλσ − gµλgσν� ∇λ � f + □−1[Rf ′] � δgµν − � gµνgλσ − gµλgσν� � f + □−1[Rf ′] � ∇λδgµν − �1 2gµνgλσ − gµλgσν � ∇λ � □−1R � □−1[Rf ′]δgµν + Rfδxσ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='2) 8 that obeys the following local continuity equation ∂σ �√−gJσ� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3) Integrating the continuity equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3) over a three-dimensional volume V at a given time x0, from the Gauss theorem, we obtain d dx0 � V d3x √−g J0 = − � ∂V dSi √−g Ji .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='4) If the fields with their derivatives vanish on the boundary ∂V , the surface integral on the right of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='4) vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', there is no current crossing the boundary, and we can derive the conserved Noether charge in the volume V , associated to symmetries (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1) Q = � V d3x √−g J0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) So, if we consider the one-parameter group of diffeomorphisms for the global infinitesimal transla- tions x′µ = xµ + ǫµ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='6) the local variation δ of tensor metric gµν becomes δgµν = g′ µν(x) − gµν(x) = −gµν,αǫα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='7) Hence, the conserved Noether current, related to the translational symmetry (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='6), becomes the energy-momentum density of the gravitational field, while, for isolated systems, where the spacetime is asymptotically flat at spatial infinity, the conserved Noether charge becomes the energy and momentum of the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Therefore, the translation invariance of gravitational action, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='2), gives τ σ α = τσ (GR) α + ∆τ σ α , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='8) where τ σ (GR) α is the Einstein pseudotensor 2χτ σ (GR) α = Rδσ α + � gµνgλσ − gµλgσν� � gµν,αλ − Γβ λµgβν,α) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='9) while the correction ∆τ σ α, is the gravitational energy-momentum pseudotensor of non-local part, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', 2χ∆τ σ α =Rfδσ α + � gµνgλσ − gµλgσν� � gµν,αλ − Γβ λµgβν,α � � f + □−1[Rf ′] � − �� gµνgλσ − gµλgσν� ∇λ � f + □−1[Rf ′] � − �1 2gµνgλσ − gµλgσν � ∇λ � □−1R � □−1[Rf ′] � gµν,α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) The pseudotensor (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) has been obtained taking into account that the covariant derivative of variation for the metric tensor is ∇λδgµν = ∂λδgµν − Γα λµδgαν − Γα λνδgαµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='11) 9 The symmetry of Levi Civita connection leads to � gµνgλσ − gµλgσν� Γβ λν = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='12) and the local conservation of pseudotensor can be read as ∂α �√−g τ σ α � = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='13) being Jα = τ σ αǫα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='14) In terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='33), in more compact form, one gets 2χ∆τ σ α =Rfδσ α + � gµνgλσ − gµλgσν� � gµν,αλ − Γβ λµgβν,α � (f + G[Rf ′]) − �� gµνgλσ − gµλgσν� ∂λ (f + G[Rf ′]) − �1 2gµνgλσ − gµλgσν � ∂λ (G[R]) G[Rf ′] � gµν,α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='15) It has to be emphasized that,from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10), it is clear that the geometric ob- ject τ σ α is a pseudotensor not a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In other words, it transforms like a tensor under affine transformations but not under generic transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' So τ σ α is at least an affine tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In an asymptotically flat spacetime the tensoriality is recovered and the integral (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) returns to being a four-vector for asymptotic linear coordinates, that is, P α = � V d3x √−g τ α 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='16) represents the energy and momentum in V of the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Moreover the pseudotensor τ σ α is a non-local object because it involves non-local terms, such as □−1R or □−1[Rf ′], whose value depends on the values assumed by the metric in the integration domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 4 The energy-momentum complex The stationarity of gravitational action, ˜δSg = 0, with respect to the variation ˜δ, from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='30), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='31), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='32), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10), gives 1 2χ √−g � Gµν + ∆Gµν � δgµν + ∂σ �√−gτ σ αǫα� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1) Hence, inserting the field equations in presence of matter (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='38) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1), we get − 1 2 √−g T µνδgµν + ∂σ �√−gτ σ αǫα� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='2) From rigid translations and coordinates independence from ǫα, it yields 1 2 √−g T µνgµν,α + ∂σ �√−gτ σ α � = −√−g∇σT σ α + ∂σ �√−gT σ α � + ∂σ �√−gτ σ α � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3) 10 where the identity √−g∇σT σ α = ∂σ �√−gT σ α � − 1 2 √−g gµν,αT µν , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='4) has been taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3), we obtain ∂σ �√−g � T σ α + τ σ α �� = √−g∇σT σ α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) According to the previous considerations, it is possible to prove generalized contracted Bianchi identities for non-local gravity [40,45,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' They guarantee the conservation of energy–momentum complex of gravitational and matter components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Let us first demonstrate a lemma useful for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Let f ∈ C2(Ω) be a twice continuously differentiable function on an open set Ω of R4, ∇ be the covariant derivative, □ be the d’Alembert operator and [, ] be the commutator, we have [∇ν, □]f = −Rµν∇µf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' From the commutator of two covariant derivatives ∇µ and ∇ν, which acts on the contravari- ant vector field Aγ, we get [∇µ, ∇ν]Aγ = Rγ λµνAλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='7) If we set Aγ = ∇γf and γ = ν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='7), we obtain [∇µ, □]f = ∇µ∇ν∇νf − ∇ν∇ν∇µf = ∇ν[∇µ, ∇ν]f − [∇µ, ∇ν]∇νf = ∇ν[∇µ, ∇ν]f − Rµν∇νf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='8) Thus, the commutativity of covariant derivatives of a function, that is, [∇µ, ∇ν]f = 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='9) inserted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='8), gives us the result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1 (Non-local generalized contracted Bianchi identities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Let Gµν be the Einstein tensor and ∆Gµν be the corrections to the field equations due to non-local terms as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='38), then the covariant 4-divergence of their sum vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=', ∇µ� Gµν + ∆Gµν � = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' We carry out the 4-divergence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='32) and we have ∇µ∆Gµν = � ∇µGµν + ∇ν□ − □∇ν �� f + □−1[Rf ′] � + Gµν∇µ(f + □−1[Rf ′]) + 1 2 � δλ ν ∇σ + δσ ν ∇λ − gσλ∇ν � ∇σ□−1[Rf ′]∇λ□−1R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='11) So, from the contracted Bianchi identities ∇µGµν = 0 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='12) 11 and performing some calculations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='11) can be rewritten as follows ∇µ∆Gµν = [∇ν, □] � f + □−1[Rf ′] � + Gµν∇µ(f + □−1[Rf ′]) + 1 2 � □□−1[Rf ′]∇ν□−1R + ∇σ□−1[Rf ′]∇σ∇ν□−1R + ∇σ∇ν□−1[Rf ′]∇σ□−1R + ∇ν□−1[Rf ′]□□−1R − ∇ν∇σ[Rf ′]∇σ□−1R − ∇σ□−1[Rf ′]∇ν∇σ□−1R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='13) Now, the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='13) and the lemma (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1) lead to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10), that is, we find ∇µ∆Gµν = −Rµν � f + □−1[Rf ′] � + Gµν∇µ(f + □−1[Rf ′]) + 1 2Rf ′∇ν□−1R + 1 2R∇ν□−1[Rf ′] = −Rµνf ′∇µ□−1R − Rµν∇µ□−1[Rf ′] + Gµν∇µ(f + □−1[Rf ′]) + 1 2gµνRf ′∇µ□−1R + 1 2gµνR∇µ□−1[Rf ′] = −Gµν∇µ� f + □−1[Rf ′] � + Gµν∇µ(f + □−1[Rf ′]) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='14) According to the field equation in presence of matter (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='38), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) leads to the standard covariant conservation of matter energy-momentum tensor, that is, ∇µT µν = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='15) It implicitly defines the trajectories of particles, that is, the time-like metric geodesics on the spacetime manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Finally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) gives the local conservation of energy-momentum complex T σ α in non-local gravity, that is, the continuity equation for energy-momentum complex in non-local gravity ∂σ �√−g � T σ α + τ σ α �� = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='16) We can define T σ α = T σ α + τ σ α , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='17) involving all gravitational and matter contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 5 Weak field limit of non-local gravitaty energy-momentum pseudotensor Let us now develop the low energy limit perturbing the metric tensor gµν around the Minkowskian metric ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' It is gµν = ηµν + hµν , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1) and then, we can calculate the pseudotensor (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) or (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='15) to lowest order in the perturbation hµν, that is, up to second ordear in hµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Therefore we get, at the order h2, (τ σ α)(2) = � τ σ (GR) α �(2) + (∆τ σ α)(2) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='2) 12 where the Einstein pseudo-tensor is 2χ � τ σ (GR) α �(2) = R(2)δσ α + � gµνgλσ − gµλgσν�(1) g(1) µν,αλ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='3) and, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='15), the non-local perturbation of pseudotensor takes the form 2χ (∆τ σ α)(2) = R(1)f (1)δσ α + � gµνgλσ − gµλgσν�(0) � f (1) + G(1)[Rf ′] � ,λ g(1) µν,α − � gµνgλσ − gµλgσν�(0) � f (1) + G(1)[Rf ′] � g(1) µν,αλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='4) Then, we expand f as f (G[R]) (x) = f(0) + f ′(0)G[R](x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) and imposing the case f(0) = 0, the relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5) to the first order takes the form f (1) (G[R]) (x) = f ′(0)G(1)[R](x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='6) Taking into account the following first order perturbations in a generic coordinate system, the Ricci scalar becomes R(1) = � hβγ ,βγ − □(0)h � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='7) where □(0) = ηαβ∂α∂β , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='8) and the non-local operator □−1 at first order reads as G(1)[R](x) = � □−1R �(1) (x) = −h(x) + �G � hβγ ,βγ � (x) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='9) where �G � hβγ ,βγ � (x) = � Ω d4x′G(x, x′) hβγ ,βγ(x′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) We have to prove the identity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='5), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='7) and the theorem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='1), it is � □−1R �(1) (x) = � Ω′ d4x′� −g(x′) (0)G(x, x′)R(1)(x′) = � Ω d4x′� −g(x′) (0)G(x, x′) � hβγ ,βγ(x′) − □x′h(x′) � = − � Ω d4x′□x′G(x, x′)h(x′) + � Ω d4x′G(x, x′) hβγ ,βγ(x′) = − � Ω d4x′δ(x − x′)h(x′) + �G � hβγ ,βγ � (x) = −h(x) + �G � hβγ ,βγ � (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='11) Furthermore, we perform the first-order perturbation of G[Rf ′], namely G(1)[Rf ′](x) = � Ω d4� −g(x′) (0)G(x, x′)R(1)(x′)f ′(0)[G](x′) = f ′(0) � Ω d4� −g(x′) (0)G(x, x′)R(1)(x′) = f ′(0)G(1)[R](x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='12) 13 Finally substituting the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='7), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='12) in the non-local perturbed gravitational energy– momentum pseudotensor (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='4), we derive the non-local corrections of the gravitational pseudo-tensor τ σ α to the second order in hµν, that is, 2χ (∆τ σ α)(2) = �� hβγ ,βγ − □h �� −h + �G � hβγ ,βγ �� δσ α + 2 � ηµνηλσ − ηµληνσ� � −h + �G � hβγ ,βγ �� ,λhµν,α − 2 � ηµνηλσ − ηµληνσ� � −h + �G � hβγ ,βγ �� hµν,αλ � f ′(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='13) The non-local contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='13) is evident and, as discussed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' [47–49], it can con- tribute to gravitational radiation representing a signature for non-local gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' 6 Discussion and Conclusions In this paper, we investigated how non-locality gravity induces correction terms ∆τ σ α into the Einstein gravitational pseudotensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Considering the Noether theorem and imposing the invariance of gravitational action under rigid translations, we found the associated conserved Noether current and charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' They can be interpreted as the gravitational density of the energy-momentum and the energy and momentum of gravitational field present in a spatial volume enclosing localized massive objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The density and flux density of the gravitational energy and momentum expressed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) are not described by a covariant tensor, which means that, under general coordinate transformations, it does not transform like a tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The geometrical object (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) is an affine tensor or pseudotensor because it transforms like a tensor only under affine transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The non-tensorial character of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='10) is closely linked to the non-localization of gravitational energy which holds also in non-local gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The non-locality of the gravitational pseudotensor intervenes through integral operators, like □−1, where its value, at a given point x, takes into account the value assumed by the fields in other points x′ of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Then, by generalizing the contracted Bianchi identities to the non-local gravity, we have obtained an equation of continuity for the energy-momentum complex that ensures its local conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Finally, we studied the behavior at low energies of the non-local corrections of the gravitational pseudotensor (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content='13), expanding it up to the second order in hµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The non-local gravitational energy-momentum pseudotensor is a crucial physical quantity because, thanks to the gravitational waves obtained and analyzed in the papers [47–49], it is possible to calculate the power emitted by a radiative system and transported by the waves with all its polarizations and multipole terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' The presence, in the gravitational radiation, of a scalar component with lower multipoles, in addition to the standard quadrupole tensor component, can be investigated thanks to the gravitational pseudotensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In this perspective, it can give a relevant signature for the non-local gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' In a forthcoming paper, we will investigate possible observational constraints on these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Acknowledgements This paper is based upon work from COST Action CA21136 Addressing observational tensions in cosmology with systematics and fundamental physics (CosmoVerse) supported by COST (European 14 Cooperation in Science and Technology).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Authors acknowledge the Istituto Nazionale di Fisica Nucleare (INFN) Sez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' di Napoli, Iniziative Specifiche QGSKY and MOONLIGHT, and the Istituto Nazionale di Alta Matematica (INdAM), gruppo GNFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' References [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE2T4oBgHgl3EQfpAgB/content/2301.04023v1.pdf'} +page_content=' Modesto, Super-renormalizable quantum gravity Phys.' metadata={'source': 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Williams∗, Olivia Zahn∗∗ and J. Nathan Kutz† +∗Department of Mechanical Engineering, University of Washington, Seattle, WA +∗∗ Department of Physics, University of Washington, Seattle, WA and +† Departments of Applied Mathematics and Electrical and Computer Engineering, University of Washington, Seattle, WA +Sensing is a universal task in science and engineering. Downstream tasks from sensing include +inferring full state estimates of a system (system identification), control decisions, and forecasting. +These tasks are exceptionally challenging to achieve with limited sensors, noisy measurements, and +corrupt or missing data. Existing techniques typically use current (static) sensor measurements to +perform such tasks and require principled sensor placement or an abundance of randomly placed +sensors. In contrast, we propose a SHallow REcurrent Decoder (SHRED) neural network structure +which incorporates (i) a recurrent neural network (LSTM) to learn a latent representation of the +temporal dynamics of the sensors, and (ii) a shallow decoder that learns a mapping between this +latent representation and the high-dimensional state space. By explicitly accounting for the time- +history, or trajectory, of the sensor measurements, SHRED enables accurate reconstructions with +far fewer sensors, outperforms existing techniques when more measurements are available, and is +agnostic towards sensor placement. In addition, a compressed representation of the high-dimensional +state is directly obtained from sensor measurements, which provides an on-the-fly compression for +modeling physical and engineering systems. Forecasting is also achieved from the sensor time-series +data alone, producing an efficient paradigm for predicting temporal evolution with an exceptionally +limited number of sensors. In the example cases explored, including turbulent flows, complex spatio- +temporal dynamics can be characterized with exceedingly limited sensors that can be randomly +placed with minimal loss of performance. +I. +INTRODUCTION +Emerging sensor technologies are transforming every +science and engineering domain, with the quantity and +quality of data collected also driving fundamental ad- +vances through data-science and machine learning meth- +ods. In many areas of interest, measurements of the full- +state are at best impractical and often impossible. Thus +sensors are commonly used to infer the current and fu- +ture behavior of high-dimensional systems with a lim- +ited number of sensor locations. With severely limited +and noisy sensor measurements, this task is exceptionally +difficult and frequently requires principled sensor place- +ment schemes to yield faithful reconstructions [1]. Ac- +curate and robust reconstruction techniques are vital in +enabling downstream tasks such as system identification, +forecasting, and control. While data-driven techniques +can learn mappings from sensor measurements to the +full-state space [2–5], this is typically done with the cur- +rent sensor values only. We advocate, instead, learning a +mapping from sensor trajectories, which contain the time +history of the sensors, to the full state-space by using a +recurrent neural network in partnership with a decoder +network. As will be shown, not only is there a significant +performance increase, but a minimal number of sensors, +randomly placed, can be used. +The reconstruction of spatio-temporal dynamics from +limited +sensors +relies +on +low-rank +features +of +the +data. +Computing low-rank embeddings of such high- +dimensional data is often achieved through the singular +value decomposition (SVD), also known as proper orthog- +onal decomposition (POD) [6, 7]. The coefficients of the +dominant correlated modes are determined by solving a +linear inverse problem and the modes themselves serve +as a linear map between measurements and the spatio- +temporal state space. Many of these linear methods are +built upon the mathematical framework of gappy POD +and have been successful across many disciplines [1, 8– +11]. More recently, shallow decoder networks (SDN) have +leveraged advances in machine learning and AI to learn +end-to-end, nonlinear maps between measurements and +high-dimensional state spaces. SDNs have been demon- +strated to outperform their linear counterparts, partic- +ularly when the number of available sensors is exceed- +ingly low [2, 12–14]. Both derivatives of gappy POD and +SDNs rely on measurements at a single snapshot in time +to reconstruct the corresponding high-dimensional state +at that time. +As a result of this static sensing, linear reconstruction +methods are heavily reliant on optimally placed sensors +for the inverse problem to be well-conditioned [1, 9]. In +general, determining optimal sensor locations is a combi- +natorially hard problem and is infeasible for large search +spaces. For smaller spaces, there exist a number of well- +known techniques for sensor placement [15–21], but they +become computationally intractable in high-dimensional +systems. Greedy algorithms, such as the QR decomposi- +tion with column pivoting, offer approximate solutions in +larger search spaces and are thus critical in enabling accu- +rate reconstructions for high-dimensional systems [1, 9]. +However, greedy algorithms suffer from the fact that +there is no guarantee that the sensor locations found are +physically implementable; for instance, measuring sea- +surface temperature in the middle of the Pacific Ocean is +significantly more challenging than taking measurements +along the coast. Variations of greedy search algorithms +arXiv:2301.12011v1 [math.DS] 27 Jan 2023 + +2 +A +B +C +D +FIG. 1: Summary diagram of SHallow REcurrent Decoder networks (SHRED) for flow reconstruction from sensor +measurements. A A graphical representation of the SHRED method. The mulitvariate time-series of sensor +measurements, {yi}t +t−k, is fed into a stacked long short-term memory layer. The final output of the recurrent layer, +ht, serves as an input to a fully-connected shallow decoder mapping from the hidden state to the high-dimensional +field. B The time series of sensor measurements fed into the SHRED model. C The evolution of the output hidden +state generated by the input sequence of sensor measurements. D Reconstruction errors of traditional methods +(QR/POD), shallow decoders (SDN), and SHRED on a turbulent flow when three sensors are available. +can incorporate additional constraints, such as cost and +sensor fidelity [22–25], but often at the expense of recon- +struction accuracy. The use of SDNs helps mitigate the +dependence on sensor location, but still is greatly aided +by principled sensor placement schemes [14]. +The data-driven method presented here incorporates +temporal trajectories of sensor measurements to improve +reconstruction accuracy, robustness to noise, and elim- +inate the need for optimally placed sensors. We use a +type of recurrent network layer, long short-term memory +networks (LSTM) [26], to process a time-series of sensor +measurements. The latent representation of the LSTM +is the input into a fully-connected, shallow decoder for +reconstruction. We demonstrate that these SHallow RE- +current Decoders (SHRED) outperform existing linear +and nonlinear techniques on three example datasets: a +forced isotropic turbulent flow [27], weekly sea-surface +temperature [28], and atmospheric ozone concentration +[29]. In all cases, SHRED with as few as three randomly +placed sensors achieves superior reconstruction accuracy +than existing techniques using a far greater number of +optimally placed sensors. Moreover, because the inputs +to a trained SHRED model consist of just a few sensor +measurements, SHRED offers an on-the-fly compression +for modeling physical and engineering systems. Fig. 1 +gives a graphical summary of SHRED. +II. +RECONSTRUCTION OF +HIGH-DIMENSIONAL SPATIO-TEMPORAL +FIELDS +A SHRED model is a neural network mapping from a +trajectory of sensor measurements to a high-dimensional, +spatio-temporal state. The architecture can be expressed +as +H +� +{yi}t +i=t−k +� += F +� +G({yi}t +i=t−k; WRN); WSD +� +(1) +where +yt +consists +of +measurements +of +the +high- +dimensional state xt, F is a fully-connected, feed-forward +neural network parameterized by weights WSD, and G is +a LSTM network parameterized by weights WRN. The + +LSTM +LSTM +LSTM +LSTM +LSTM +LSTM +Shallow +xt +Yt-k +Yt-1 +Yt +Decoder Network0.4 +0.2 +Pressure +0.0 +-0.2 +Sensor 1 +Sensor 2 +-0.4 +Sensor 3 +-100 +-75 +-50 +-25 +0 +At-t0.5 +Hidden States +0.0 +-0.5 +-100 +-75 +-50 +-25 +0 +At-t0.8 +0.6 +Error +0.4 +0.2 +0.0 +QR/POD +SDN +SHRED3 +A +B +C +(α) +FIG. 2: A Example reconstructions obtained via SHRED and QR/POD of a turbulent flow when three sensors are +available. Ground truth included for comparison. B Reconstruction errors of the current state-of-the-art methods +and SHRED with a varying number of available sensors. The solid lines denote the median error from 32 trained +estimators and the shaded region denotes the interquartile range. C Performance of reconstruction methods in the +presence of varying levels of added Gaussian white noise. The added noise has mean zero and standard deviation +equal to α times the mean absolute value of the field. The number in parentheses denotes the number of available +sensors. +desired network H minimizes the reconstruction loss, +H ∈ argmin +� +H∈H +N +� +i=1 +||xi − � +H +� +{yj}i +i−k +� +||2 +(2) +given a set of training states {xi}N +i=1 and corresponding +measurements {yi}N +i=1. We train the network to mini- +mize reconstruction loss using the ADAM optimizer [30]. +We demonstrate the performance of SHRED on three +example datasets. +In each case, we compare the re- +construction error obtained by SHRED with randomly +placed sensors to SHRED with QR placed sensors +(QR SHRED), shallow decoder networks with randomly +placed sensors (R-SDN), shallow decoder networks with +QR placed sensors (Q-SDN), and linear reconstructions +with QR placed sensors (QR/POD). The considered re- +construction error is defined to be the averaged mean +square error over each state in a test set, +Error = 1 +T +T +� +i=1 +||H +� +{yj}i +i−k +� +− xi||2 +||xi||2 +. +(3) +Because SHRED models rely on trajectories of sensor +measurements to perform state estimation, we truncate +each data set to reconstruct only the final N −k temporal +snapshots, where N is the initial number of samples and +k is the length of the utilized trajectories. This length +can be viewed as a hyper-parameter that can be tuned +according to the data. Detailed network parameters and +training protocols can be found in the SI. Finally, we +note that in this section, training, validation, and test +samples are temporally interspersed. In later sections, +we demonstrate results in the case that training and test +sets are temporally distinct. +A. +Forced isotropic turbulent flow +The first application we consider is that of a forced +isotropic turbulent flow from the Johns Hopkins Turbu- +lence Database [27]. The flow was generated by direct +numerical simulation using 10243 nodes and the pseudo- +spectral method. We select a 350 by 350 cutout of com- +puted pressure over 1667 evenly spaced temporal snap- +shots from the simulation. We seek to reconstruct these +high-dimensional states from trajectories of point mea- +surements of the states. In this case, the length of the +utilized trajectories is selected to be 100. Correspond- +ingly, the final 1567 temporal snapshots are randomly +split into training, validation, and test sets consisting + +Sample Snapshots +Ground Truth +SHRED (3 Sensors) +QR/POD (3 Sensors)0.4 +0.3 +QR/POD +R-SDN +Error +Q-SDN +0.2 +SHRED +QR SHRED +0.1 +0 +10 +20 +30 +40 +50 +Num. Sensors0.35 +0.30 +0.25 +QR/POD(100) +Error +Q-SDN (100) +0.20 +SHRED (3) +0.15 +SHRED (100) +0.10 +0.05 +0.05 +0.10 +0.15 +0.20 +Noise4 +A +B +C +(α) +FIG. 3: A Example reconstructions obtained via SHRED and QR/POD of sea-surface temperature when three +sensors are available. Ground truth included for comparison. B Reconstruction errors of the current state-of-the-art +methods and SHRED with a varying number of available sensors. The solid lines denote the median error from 32 +trained estimators and the shaded region denotes the interquartile range. C Performance of reconstruction methods +in the presence of varying levels of added Gaussian white noise. The added noise has mean zero and standard +deviation equal to α times the mean absolute value of the field. The number in parentheses denotes the number of +available sensors. +of 1100, 234, and 233 snapshots, respectively. For each +considered number of sensors, we generate 32 reconstruc- +tions with all considered methods. We plot the median +performance and denote the interquartile range by the +shaded region. +These results are shown in panel B of +Fig. 2. Even with only a single, randomly placed sen- +sor, the reconstructions obtained by SHRED yield signifi- +cantly lower error than competing methods with as many +as 50 sensors. Moreover, placement via the greedy QR +algorithm appears to have a negligible impact on recon- +structive performance. Panel A of Fig. 2 shows sample +reconstructions obtained by SHRED and QR/POD with +three sensors. While QR/POD is only able to identify +large scale features, SHRED accurately reconstructs fine +grain features as well. +In the vast majority of real world applications, and in +contrast to numerical simulations, data is often corrupted +by noise. As a result, methods for state estimation must +exhibit resilience to noise. To measure this resilience, we +corrupt the data with Gaussian noise of mean zero and +standard deviation of α×|¯x|, where |¯x| is the average ab- +solute value over all points in all snapshots of the training +set. We then generate 32 state estimates exactly as be- +fore using all considered methods. The resulting median +reconstruction error and interquartile range for varying +α is shown in panel C of Fig. 2. Again, SHRED outper- +forms competing techniques using a far greater number +of sensors. +B. +Sea-surface temperature +The second application we examine is that of sea- +surface temperature (SST). We consider weekly mean +sea-surface temperature from the years 1992 to 2019 as +reported by NOAA [28]. +Unlike the previous example +of a simulated turbulent flow, SST is a sensor generated +dataset for which governing equations are not known and +thus represents a more practical application of SHRED. +The data consists of 1400 snapshots of a 180 by 360 grid, +of which 44219 spatial locations correspond to the sea- +surface. We allow the input trajectories to SHRED to +have a length of 52, corresponding to one year of mea- +surements. The final 1348 samples are divided into train- +ing, test, and validation sets consisting of 1000, 174, +and 174 snapshots, respectively. Analogous to Fig. 2, +Fig. 3 shows example reconstructions and reconstruction +error distributions of SHRED and existing techniques +both in the presence and absence of added Gaussian +noise. Again, the reconstructions obtained by SHRED + +QR/POD (100) +0.20 +Q-SDN (3) +Q-SDN (100) +Error +0.15 +SHRED (3) +0.10 +0.05 +0.05 +0.10 +0.15 +0.20 +NoiseSample Snapshots +Ground Truth +SHRED (3 Sens.) +OR/POD (3 Sens.)0.150 +QR/POD +0.125 +R-SDN +0.100 +Q-SDN +Error +SHRED +0.075 +QR SHRED +0.050 +0.025 +0 +10 +20 +30 +40 +50 +Num. Sensors5 +A +B +C +(α) +FIG. 4: A Example reconstructions obtained via SHRED and QR/POD of atmospheric ozone concentration when +three sensors are available. Only one of thirty elevations is shown. Ground truth included for comparison. B +Reconstruction errors of the current state-of-the-art methods and SHRED with a varying number of available +sensors. The solid lines denote the median error from 32 trained estimators and the shaded region denotes the +interquartile range. C Performance of reconstruction methods in the presence of varying levels of added Gaussian +white noise. The added noise has mean zero and standard deviation equal to α times the mean absolute value of the +field. The number in parentheses denotes the number of available sensors. +are both visually and empirically superior, while requir- +ing far fewer sensors, which can be randomly placed with- +out a loss of accuracy. +C. +Atmospheric ozone concentration +The last system for which we consider the perfor- +mance of SHRED is a simulation of atmospheric chem- +istry. +Chemical transport models (CTM) simulate the +evolution of an ensemble of interacting chemical species +through a transport operator [29, 31] +∂ni +∂t = −∇ · (niU) +(4) +and a chemical operator +dni +dt = (Pi − Li)(n) + Ei − Di, +(5) +where each entry n = +�n1 n2 · · · nK +�T represents the +number density of a specific chemical species, U is the +wind vector, (Pi−Li)(n) is the local chemical production +and loss term, Ei the emission rate of a species, and +Di the deposition rate. +The output of a CTM, then, +consists of concentrations for K chemical species for a +grid of latitudes, longitudes, and elevations. The data +we consider is drawn from the work of Velegar et al. [31] +and contains simulated atmospheric ozone concentration +generated by the CTM software GEOS-Chem [29] over +the course of a year with dynamical time steps of 20 +minutes. The accessed data from [31] is a compressed +SVD representation using the first 50 POD modes. The +decompressed data matrix consists of 26,208 snapshots +of a 46 by 72 by 30 (latitude, longitude, elevation) grid, +from which we further downsample to obtain 2,600 evenly +temporally spaced global ozone concentration fields. To +account for the fact that the data matrix is inherently +rank 50, we add Gaussian noise with standard deviation +α = 0.05 to the data before performing any analyses. The +resulting snapshots are divided into training, validation, +and test sets consisting of 2000, 300, and 300 entries, +respectively. +Fig. 4 shows the performance of SHRED and exist- +ing techniques on this atmospheric ozone experiment. +SHRED still outperforms SDNs and QR/POD while re- +quiring fewer sensors and being agnostic towards sensor +placement. However, in this case when many sensors are +available (> 50) QR/POD performs comparably to non- +linear reconstructions. This performance is an artifact of + +Sample Snapshots +Ground Truth +SHRED (3 Sensors) +QR/POD (3 Sensors)0.30 +QR/POD +0.25 +R-SDN +Q-SDN +0.20 +Error +SHRED +0.15 +QR SHRED +0.10 +0.05 +0 +10 +20 +30 +40 +50 +Num. Sensors0.25 +QR/POD (50) +Q-SDN (3) +0.20 +Q-SDN (50) +Error +SHRED (3) +0.15 +0.10 +0.05 +0.05 +0.10 +0.15 +0.20 +Noise6 +the use of a compressed, rank 50 representation of the +data. +A +B +FIG. 5: Forecasting error for sea-surface temperature. +A LSTM is trained to forecast QR placed, panel A, or +randomly placed, panel B, sensor measurements which +are subsequently used to perform reconstructions using +SHRED and gappy POD. 16 forecasts are performed +and the median error is denoted by the solid lines, with +the 25th and 75th percentiles of reconstruction error +defining the shaded region. The dashed line represents +an ensembled forecast. +III. +FORECASTING SENSOR +MEASUREMENTS TO PREDICT THE +EVOLUTION OF SPATIO-TEMPORAL DATA +In this section, we explore how the expressiveness of +SHRED models with few sensors can be leveraged to per- +form forecasts of high-dimensional spatio-temporal states +using only a limited subsampling of the state. Forecast- +ing the evolution of high-dimensional states from sensor +measurements is an exceedingly challenging task, owing +to the fact that doing so combines the problem of sys- +tem identification with the difficulties of forecasting. We +propose a two step approach that leverages the success +of LSTMs for low-dimensional time-series forecasting [32] +and SHRED for state estimation. +Let {xi}Tt +1 +represent a training dataset of temporally +ordered states with corresponding sensor measurements +{yi}Tt +1 . +We train an LSTM network, G +� +{yi}t +t−k +� +, to +map between a trajectory of sensor measurements and +the subsequent measurement, yt+1, by finding +G ∈ argmin +� +G∈G +N +� +i=1 +||yt+1 − �G +� +{yj}i +i−k +� +||2 +(6) +using the ADAM optimizer on the training data. Do- +ing so, we then forecast beyond the time interval of the +training data to obtain forecasted sensor measurements +{ˆyi}Tt+p +Tt+1 for p > 1. The forecasted measurements are +used by a SHRED model, trained on {xi}Tt +1 , to obtain +forecasts of the high-dimensional state x. Detailed train- +ing protocols are included in SI. +The forecasted states {ˆxi}Tt+p +Tt+1 are evaluated against +the ground truth for each ∆t forecast rather than the +mean error across all forecasts to demonstrate perfor- +mance over forecasts of varying length. We also include +reconstructions from forecasted sensor measurements ob- +tained by gappy POD for comparison, akin to the work of +[33]. Finally, we only show results for SST and turbulent +flow data, as the compressed form of the atmospheric +ozone data is biased towards both QR placement and +POD reconstructions. +A. +Sea-surface temperature +For the SST example, we select the first 85% of the +dataset to act as training data, the subsequent 20 snap- +shots as validation data, and the remainder as the test +set. A LSTM for forecasting is trained on the training +data and sensor measurements are forecast beyond the +validation set. These forecasted sensor measurements are +then used to construct a forecast of the high-dimensional +spatio-temporal field using a trained SHRED model and +gappy POD. We consider the cases that the sensors are +placed via QR or randomly. We perform 16 runs for each +experiment, and consider an ensembled forecast found by +averaging the forecast of each run. The results for QR +placed and randomly placed sensors are shown in Fig. 5. +With QR placement, the forecasts obtained by SHRED +outperform that of POD in the short-term and are simi- +lar for longer forecast horizons. However, with randomly +placed sensors SHRED is still able to obtain comparably +accurate forecasts while POD fails to consistently yield +faithful reconstructions. +B. +Turbulent flow +Unlike sea-surface temperature, even medium-range +forecasts of a turbulent flow are impossible due to the +fact that the flow is not quasi-periodic or stationary in +nature. For this reason, we focus on achieving accurate +short-term forecasts in this application. +To do so, we +use the same scheme as before, with the exception that +the validation set is selected to occur earlier in the train- +ing data. Of the 1567 snapshots of the turbulent flow + +0.150 +QR/POD +SHRED +0.125 +QR SHRED +Error +0.100 +0.075 +0.050 +0.025 +0 +25 +50 +75 +100 +125 +At1.25 +POD +SHRED +1.00 +0.75 +0.50 +0.25 +0.00 +0 +25 +50 +75 +100 +125 +△t7 +with sufficiently long preceding measurement histories, +the first 1000 are selected as training, followed by 50 +samples selected for validation. The next 100 samples +are also used for training and the remainder constitute +the test set. This scheme allows for more accurate short- +term forecasts because the forecast occurs directly after +the training data, as opposed to directly after the vali- +dation set. Figure 6 shows the results for the cases that +sensors are placed via QR and randomly. In both cases, +the forecast obtained by SHRED deteriorates quickly but +greatly outperforms that obtained by POD reconstruc- +tions. +IV. +DISCUSSION +We have demonstrated SHRED as a method for sys- +tem identification/state estimation and the forecast- +ing of high-dimensional time-series data. +By includ- +ing sensor trajectories in the model for reconstruction, +SHRED outperformed competing state-of-the-art tech- +niques based on static reconstructions from gappy POD +or shallow decoder networks. We considered three high- +dimensional, spatio-temporal datasets, a synthetically +generated forced turbulent flow, a simulation of atmo- +spheric ozone concentration from a chemical transport +model, and weekly mean sea-surface temperature. +The superior performance of SHRED held across both +interpolatory (reconstruction) and extrapolatory (fore- +casting) regimes. In Section II, the task was purely an in- +terpolatory reconstruction. Training and test data, upon +which we evaluate reconstruction accuracy, were drawn +from the same distribution. As a result, overfitting to +the training data was not an issue, and SHRED was able +to capture fine grain details of complex flows with as +few as one or three sensors. +Applied in this manner, +SHRED offers an attractive method for compression of +large datasets as well as a framework for developing re- +duced order models of dynamical systems. The perfor- +mance of SHRED in this section is also indicative of the +promise of SHRED for high-dimensional data that is sta- +tionary or periodic. Notably, SHRED demonstrated re- +markable indifference towards sensor placement. In this +interpolatory regime, issues arising from the necessity of +optimal sensor placement can be mitigated by increas- +ing the information content of inputs to a neural network +through the use of sensor trajectories. These results show +that the sensor trajectories encode a significant amount +of information, just as is expected of time-delayed em- +beddings of dynamics [34–39]. +In Section III, the task at hand was forecasting the +evolution of high-dimensional spatio-temporal data. In +contrast to Section II, training and test sets are tem- +porally separated and thus the results are extrapolatory +in nature. High-dimensional forecasts were accomplished +by forecasting low-dimensional sensor measurements and +constructing states from the predicted measurements. +Forecasting is an inherently difficult task for complex +systems, especially in the high-dimensional fields often +encountered in the engineering and the physical sciences. +Still, SHRED’s use of sensor trajectories allows for im- +proved forecasting performance as compared to existing +methods. Furthermore, SHRED for forecasting exhibits +robustness to sensor placement not found in other state- +of-the-art methods. +We thus advocate for the use of +SHRED for both the tasks of state estimation and state +prediction from forecasted sensor measurements. +Fur- +ther work must be done to improve the accuracy of low- +dimensional sensor forecasts to be used in conjunction +with SHRED for state prediction. Improving the fore- +casts of high-dimensional data can enable on-the-fly com- +pression and novel sensing mechanisms. +A +B +FIG. 6: Forecasting error for turbulent flow. A LSTM is +trained to forecast QR placed, panel A, or randomly +placed, panel B, sensor measurements which are +subsequently used to perform reconstructions using +SHRED and gappy POD. In this experiment, the +validation set is selected from earlier in the data so that +the forecast occurs immediately after following training +data. 16 forecasts are performed and the median error +is denoted by the solid lines, with the 25th and 75th +percentiles of reconstruction error defining the shaded +region. The dashed line represents an ensembled +forecast. + +0.8 +QR/POD +SHRED +0.6 +QR SHRED +Error +0.4 +0.2 +0 +5 +10 +15 +20 +25 +△t3 +POD +SHRED +Error +2 +1 +0 +5 +10 +15 +20 +25 +△t8 +V. +MATERIALS AND METHODS +A. +Proper orthogonal decomposition and QR +Traditional techniques for high-dimensional state re- +construction from limited sensor measurements typically +rely on the SVD to compute a low-rank embedding of +the spatio-temporal dynamics. +Given N snapshots of +an m dimensional state, we construct a data matrix +X = [x1 x2 . . . xN] where xi ∈ Rm is the i-th temporal +snapshot of the state. The SVD factors the data matrix +into the product of an orthogonal matrix, U ∈ Rm×m, a +diagonal matrix with decreasing entries, Σ ∈ Rm×N, and +another orthogonal matrix V ∈ RN×N. An optimal rank +r approximation of X can be directly obtained through +the SVD by retaining only the first r columns of U, +X ≈ ˆX = UrΣrVT +r . +(7) +Thus, each temporal snapshot xi can be approximated +by a linear combination of the first r columns of U, oth- +erwise known as the dominant POD modes. +POD based methods estimate a high-dimensional +state, x, drawn from the same distribution as that of +the training data by solving for these r coefficients using +a subsampling of the state, +y = Cx ≈ CUrb, +(8) +where C is a “sensing” matrix consisting of rows of +the m × m identity matrix and b ∈ Rr. +A full-state +estimate from a set of sensor measurements, y, can then +be obtained by +x ≈ ˆx = Ur(CUr)−1y. +(9) +In practice, it is critical to design C such that the inver- +sion in (9) is well-conditioned. An indirect bound on the +condition number of CUr can be found by determining +C where | det(CUr)| is maximized. That is, we seek +C. = argmax +C +| det CUr| = argmax +C +� +i +|λi(CUr)| += argmax +C +� +i +σi(CUr). +(10) +Unfortunately, for large systems this NP-hard, combina- +torial search becomes computationally intractable. The +QR decomposition with column pivoting of UT +r offers an +approximate greedy solution. The decomposition utilizes +Householder transformations to find a column permuta- +tion matrix PT such that +UT +r PT = QR +(11) +where Q is orthogonal and R is upper triangular with +decreasing, positive entries on the diagonal. Because at +each iteration of the algorithm the diagonal entry of R +is selected to be as large as possible and Q is orthogo- +nal, we have a a greedy maximization of det(UT +r PT ) = +det(PUr). Thus, setting C = P can improve the recon- +structions obtained by POD based methods. We refer to +reconstructions obtained in this manner as QR/POD. +B. +Shallow decoder networks +Although QR/POD methods have demonstrated suc- +cess in a variety of fields, the method can, at most, es- +timate the coefficients of the first n POD modes if there +are n sensors available. The included POD modes are +insufficiently expressive to obtain accurate reconstruc- +tions when exceedingly few sensors are available. This +limitation motivates the development of more expressive, +nonlinear methods for reconstruction. Among these are +shallow decoder networks (SDN) [2]. +As before, let +y = Cx +(12) +be a point sampling of a high-dimensional state x. SDNs +are shallow, fully-connected neural networks that learn a +mapping from the space of sensor measurements, y, back +to x. SDNs can be denoted as +F(y; WSD) := R(WbR(Wb−1 · · · R(W1s))) +(13) +where y is the input sensor data, R is a scalar, nonlin- +ear activation function, WSD = {Wi}b +i=1 are trainable +weights, and k denotes the number of layers in the SDN. +Formally, we seek +F ∈ argmin +� +F∈F +N +� +i=1 +||xi − �F(yi)||2, +(14) +so that reconstruction error is minimized over a set of N +training states {xi}N +i=1. The parameters of the network +are randomly initialized and then trained by a gradient +descent method. Once the network is trained, reconstruc- +tions given subsequent sensor measurements are found by +ˆx = F(y). +(15) +SDNs can be trained for any set of sensor measure- +ments, and unlike QR/POD are not entirely reliant on +principled sensor placement schemes. Still, previous em- +pirical work suggests that even neural-network based re- +constructions can be improved through the use of QR +placement [14]. +C. +Recurrent neural networks and shallow +recurrent decoders +Both SDNs and QR/POD have been used successfully +in the reconstruction of high-dimensional dynamical sys- +tems, but also to broader classes of images [1]. +This + +9 +extension is possible because both methods rely only +on static, point measurements and each reconstruction +is performed individually. +Dynamical systems, on the +other hand, inherently depend on the temporal evolution +of the state, and incorporating this dependence into a +method for reconstruction offers a natural improvement +upon existing techniques. Our work includes these dy- +namics through the use of a trajectories of sensor mea- +surements +To do so, we rely on recurrent neural networks (RNN) +[40] and, more specifically, long short-term memory net- +works (LSTM). The most basic RNN accepts as an input +a sequence of vectors {vi}t +i=1 and outputs a single hidden +state, ht. This can be expressed through the recursion re- +lation +ht = R(Wcvt + Wrht−1 + br) +(16) +where R is a scalar, nonlinear activation function, Wc, +Wr and br are trainable weights, and h0 = +�0 · · · 0�T . +Generic RNNs are notorious for suffering from the van- +ishing gradient problem, causing them to have difficulty +in identifying long term dependencies. LSTMs address +this issue through the introduction of a so-called “gradi- +ent super-highway.” Instead of outputting a single hid- +den state, LSTMs incorporate an additional “cell state” +which only undergoes minor, pointwise operations, allow- +ing gradients to flow easily from many time steps in the +past. LSTMs are perhaps the most commonly used class +of RNN at the time of writing, due to their ability to +learn long and short term dependencies [32]. They are +frequently used in time-series forecasting, natural lan- +guage processing, and video analysis, among many other +domains [32]. For a more complete discussion of LSTMs, +we direct the reader to [26]. The recursion relation of an +LSTM can be written as +ht = σ +� +Wo +�ht−1, vt +� ++ bo +� +⊙ tanh(ct) +(17) +ct = σ +� +Wf +�ht−1, vt +� ++ bf +� +⊙ ct−1 +(18) ++σ +� +Wi +�ht−1, vt +� ++ bf +� +⊙ tanh +� +Wg +�ht−1, vt +� ++ bg +� +for +trainable +weights +and +biases +WRN += +{Wo, Wf, Wi, Wg, bo, bf, bi, bg}. Both σ, the sigmoid +function, and tanh operate pointwise. Note that ht can +be written as function in terms of only {vi}t +i=1 and +parameterized by the trainable weights and biases, +ht = G({vi}t +i=1; WRN). +(19) +We now introduce the SHallow REcurrent Decoder +(SHRED), which merges an LSTM with the SDN archi- +tecture. Suppose we have a multivariate time-series of a +high-dimensional state {xi}T +i=1 and corresponding mea- +surements of the system {yi}T +i=1. Rather than reconstruct +xi using only yi, as is done by QR/POD and SDN, we let +an LSTM learn a latent representation using the previous +k sets of sensor measurements. This latent representa- +tion is then used by a shallow decoder to reconstruct the +high-dimensional state. The SHRED architecture can be +written as +H +� +{yi}t +i=t−k +� += F +� +G({yi}t +i=t−k; WRN); WSD) +� +(20) +using eqs. 13 and 19. As in the case of SDNs, we use a +gradient descent method to train the network weights to +minimize reconstruction loss, +H ∈ argmin +� +H∈H +N +� +i=1 +||xi − � +H +� +{yi}t +i=t−k +� +||2. +(21) +H, so defined, represents a map from sensor trajectories +to the full-state. +ACKNOWLEDGEMENTS +The authors acknowledge support from the National +Science Foundation AI Institute in Dynamic Systems +(grant number 2112085). JNK further acknowledges sup- +port from the Air Force Office of Scientific Research +(FA9550-19-1-0011 and FA9550-19-1-0386). +REFERENCES +[1] K. Manohar, B. W. 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Conference Name: Parallel Distributed Processing: +Explorations in the Microstructure of Cognition: Foun- +dations. + diff --git a/KtFLT4oBgHgl3EQfLS8f/content/tmp_files/load_file.txt b/KtFLT4oBgHgl3EQfLS8f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92d92ed8a8241ecdea8634db901bf74be49d7f8b --- /dev/null +++ b/KtFLT4oBgHgl3EQfLS8f/content/tmp_files/load_file.txt @@ -0,0 +1,729 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf,len=728 +page_content='Sensing with Shallow Recurrent Decoder Networks Jan P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Williams∗, Olivia Zahn∗∗ and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Nathan Kutz† ∗Department of Mechanical Engineering, University of Washington, Seattle, WA ∗∗ Department of Physics, University of Washington, Seattle, WA and † Departments of Applied Mathematics and Electrical and Computer Engineering, University of Washington, Seattle, WA Sensing is a universal task in science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Downstream tasks from sensing include inferring full state estimates of a system (system identification), control decisions, and forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' These tasks are exceptionally challenging to achieve with limited sensors, noisy measurements, and corrupt or missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Existing techniques typically use current (static) sensor measurements to perform such tasks and require principled sensor placement or an abundance of randomly placed sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In contrast, we propose a SHallow REcurrent Decoder (SHRED) neural network structure which incorporates (i) a recurrent neural network (LSTM) to learn a latent representation of the temporal dynamics of the sensors, and (ii) a shallow decoder that learns a mapping between this latent representation and the high-dimensional state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' By explicitly accounting for the time- history, or trajectory, of the sensor measurements, SHRED enables accurate reconstructions with far fewer sensors, outperforms existing techniques when more measurements are available, and is agnostic towards sensor placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In addition, a compressed representation of the high-dimensional state is directly obtained from sensor measurements, which provides an on-the-fly compression for modeling physical and engineering systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Forecasting is also achieved from the sensor time-series data alone, producing an efficient paradigm for predicting temporal evolution with an exceptionally limited number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In the example cases explored, including turbulent flows, complex spatio- temporal dynamics can be characterized with exceedingly limited sensors that can be randomly placed with minimal loss of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' INTRODUCTION Emerging sensor technologies are transforming every science and engineering domain, with the quantity and quality of data collected also driving fundamental ad- vances through data-science and machine learning meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In many areas of interest, measurements of the full- state are at best impractical and often impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Thus sensors are commonly used to infer the current and fu- ture behavior of high-dimensional systems with a lim- ited number of sensor locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' With severely limited and noisy sensor measurements, this task is exceptionally difficult and frequently requires principled sensor place- ment schemes to yield faithful reconstructions [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Ac- curate and robust reconstruction techniques are vital in enabling downstream tasks such as system identification, forecasting, and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' While data-driven techniques can learn mappings from sensor measurements to the full-state space [2–5], this is typically done with the cur- rent sensor values only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We advocate, instead, learning a mapping from sensor trajectories, which contain the time history of the sensors, to the full state-space by using a recurrent neural network in partnership with a decoder network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' As will be shown, not only is there a significant performance increase, but a minimal number of sensors, randomly placed, can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The reconstruction of spatio-temporal dynamics from limited sensors relies on low-rank features of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Computing low-rank embeddings of such high- dimensional data is often achieved through the singular value decomposition (SVD), also known as proper orthog- onal decomposition (POD) [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The coefficients of the dominant correlated modes are determined by solving a linear inverse problem and the modes themselves serve as a linear map between measurements and the spatio- temporal state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Many of these linear methods are built upon the mathematical framework of gappy POD and have been successful across many disciplines [1, 8– 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' More recently, shallow decoder networks (SDN) have leveraged advances in machine learning and AI to learn end-to-end, nonlinear maps between measurements and high-dimensional state spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' SDNs have been demon- strated to outperform their linear counterparts, partic- ularly when the number of available sensors is exceed- ingly low [2, 12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Both derivatives of gappy POD and SDNs rely on measurements at a single snapshot in time to reconstruct the corresponding high-dimensional state at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' As a result of this static sensing, linear reconstruction methods are heavily reliant on optimally placed sensors for the inverse problem to be well-conditioned [1, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In general, determining optimal sensor locations is a combi- natorially hard problem and is infeasible for large search spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' For smaller spaces, there exist a number of well- known techniques for sensor placement [15–21], but they become computationally intractable in high-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Greedy algorithms, such as the QR decomposi- tion with column pivoting, offer approximate solutions in larger search spaces and are thus critical in enabling accu- rate reconstructions for high-dimensional systems [1, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' However, greedy algorithms suffer from the fact that there is no guarantee that the sensor locations found are physically implementable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' for instance, measuring sea- surface temperature in the middle of the Pacific Ocean is significantly more challenging than taking measurements along the coast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Variations of greedy search algorithms arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='12011v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='DS] 27 Jan 2023 2 A B C D FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 1: Summary diagram of SHallow REcurrent Decoder networks (SHRED) for flow reconstruction from sensor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A A graphical representation of the SHRED method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The mulitvariate time-series of sensor measurements, {yi}t t−k, is fed into a stacked long short-term memory layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The final output of the recurrent layer, ht, serves as an input to a fully-connected shallow decoder mapping from the hidden state to the high-dimensional field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' B The time series of sensor measurements fed into the SHRED model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' C The evolution of the output hidden state generated by the input sequence of sensor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' D Reconstruction errors of traditional methods (QR/POD), shallow decoders (SDN), and SHRED on a turbulent flow when three sensors are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' can incorporate additional constraints, such as cost and sensor fidelity [22–25], but often at the expense of recon- struction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The use of SDNs helps mitigate the dependence on sensor location, but still is greatly aided by principled sensor placement schemes [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The data-driven method presented here incorporates temporal trajectories of sensor measurements to improve reconstruction accuracy, robustness to noise, and elim- inate the need for optimally placed sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We use a type of recurrent network layer, long short-term memory networks (LSTM) [26], to process a time-series of sensor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The latent representation of the LSTM is the input into a fully-connected, shallow decoder for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We demonstrate that these SHallow RE- current Decoders (SHRED) outperform existing linear and nonlinear techniques on three example datasets: a forced isotropic turbulent flow [27], weekly sea-surface temperature [28], and atmospheric ozone concentration [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In all cases, SHRED with as few as three randomly placed sensors achieves superior reconstruction accuracy than existing techniques using a far greater number of optimally placed sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Moreover, because the inputs to a trained SHRED model consist of just a few sensor measurements, SHRED offers an on-the-fly compression for modeling physical and engineering systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 1 gives a graphical summary of SHRED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' RECONSTRUCTION OF HIGH-DIMENSIONAL SPATIO-TEMPORAL FIELDS A SHRED model is a neural network mapping from a trajectory of sensor measurements to a high-dimensional, spatio-temporal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The architecture can be expressed as H � {yi}t i=t−k � = F � G({yi}t i=t−k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' WRN);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' WSD � (1) where yt consists of measurements of the high- dimensional state xt, F is a fully-connected, feed-forward neural network parameterized by weights WSD, and G is a LSTM network parameterized by weights WRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The LSTM LSTM LSTM LSTM LSTM LSTM Shallow xt Yt-k Yt-1 Yt Decoder Network0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='2 Pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='2 Sensor 1 Sensor 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='4 Sensor 3 100 75 50 25 0 At-t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='5 Hidden States 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='5 100 75 50 25 0 At-t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='6 Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='0 QR/POD SDN SHRED3 A B C (α) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 2: A Example reconstructions obtained via SHRED and QR/POD of a turbulent flow when three sensors are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Ground truth included for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' B Reconstruction errors of the current state-of-the-art methods and SHRED with a varying number of available sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The solid lines denote the median error from 32 trained estimators and the shaded region denotes the interquartile range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' C Performance of reconstruction methods in the presence of varying levels of added Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The added noise has mean zero and standard deviation equal to α times the mean absolute value of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The number in parentheses denotes the number of available sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' desired network H minimizes the reconstruction loss, H ∈ argmin � H∈H N � i=1 ||xi − � H � {yj}i i−k � ||2 (2) given a set of training states {xi}N i=1 and corresponding measurements {yi}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We train the network to mini- mize reconstruction loss using the ADAM optimizer [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We demonstrate the performance of SHRED on three example datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In each case, we compare the re- construction error obtained by SHRED with randomly placed sensors to SHRED with QR placed sensors (QR SHRED), shallow decoder networks with randomly placed sensors (R-SDN), shallow decoder networks with QR placed sensors (Q-SDN), and linear reconstructions with QR placed sensors (QR/POD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The considered re- construction error is defined to be the averaged mean square error over each state in a test set, Error = 1 T T � i=1 ||H � {yj}i i−k � − xi||2 ||xi||2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' (3) Because SHRED models rely on trajectories of sensor measurements to perform state estimation, we truncate each data set to reconstruct only the final N −k temporal snapshots, where N is the initial number of samples and k is the length of the utilized trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' This length can be viewed as a hyper-parameter that can be tuned according to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Detailed network parameters and training protocols can be found in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Finally, we note that in this section, training, validation, and test samples are temporally interspersed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In later sections, we demonstrate results in the case that training and test sets are temporally distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Forced isotropic turbulent flow The first application we consider is that of a forced isotropic turbulent flow from the Johns Hopkins Turbu- lence Database [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The flow was generated by direct numerical simulation using 10243 nodes and the pseudo- spectral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We select a 350 by 350 cutout of com- puted pressure over 1667 evenly spaced temporal snap- shots from the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We seek to reconstruct these high-dimensional states from trajectories of point mea- surements of the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In this case, the length of the utilized trajectories is selected to be 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Correspond- ingly, the final 1567 temporal snapshots are randomly split into training, validation, and test sets consisting Sample Snapshots Ground Truth SHRED (3 Sensors) QR/POD (3 Sensors)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='3 QR/POD R-SDN Error Q-SDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='2 SHRED QR SHRED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='1 0 10 20 30 40 50 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Sensors0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='25 QR/POD(100) Error Q-SDN (100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='20 SHRED (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='15 SHRED (100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='20 Noise4 A B C (α) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 3: A Example reconstructions obtained via SHRED and QR/POD of sea-surface temperature when three sensors are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Ground truth included for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' B Reconstruction errors of the current state-of-the-art methods and SHRED with a varying number of available sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The solid lines denote the median error from 32 trained estimators and the shaded region denotes the interquartile range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' C Performance of reconstruction methods in the presence of varying levels of added Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The added noise has mean zero and standard deviation equal to α times the mean absolute value of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The number in parentheses denotes the number of available sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' of 1100, 234, and 233 snapshots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' For each considered number of sensors, we generate 32 reconstruc- tions with all considered methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We plot the median performance and denote the interquartile range by the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' These results are shown in panel B of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Even with only a single, randomly placed sen- sor, the reconstructions obtained by SHRED yield signifi- cantly lower error than competing methods with as many as 50 sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Moreover, placement via the greedy QR algorithm appears to have a negligible impact on recon- structive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Panel A of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 2 shows sample reconstructions obtained by SHRED and QR/POD with three sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' While QR/POD is only able to identify large scale features, SHRED accurately reconstructs fine grain features as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In the vast majority of real world applications, and in contrast to numerical simulations, data is often corrupted by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' As a result, methods for state estimation must exhibit resilience to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' To measure this resilience, we corrupt the data with Gaussian noise of mean zero and standard deviation of α×|¯x|, where |¯x| is the average ab- solute value over all points in all snapshots of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We then generate 32 state estimates exactly as be- fore using all considered methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The resulting median reconstruction error and interquartile range for varying α is shown in panel C of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Again, SHRED outper- forms competing techniques using a far greater number of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Sea-surface temperature The second application we examine is that of sea- surface temperature (SST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We consider weekly mean sea-surface temperature from the years 1992 to 2019 as reported by NOAA [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Unlike the previous example of a simulated turbulent flow, SST is a sensor generated dataset for which governing equations are not known and thus represents a more practical application of SHRED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The data consists of 1400 snapshots of a 180 by 360 grid, of which 44219 spatial locations correspond to the sea- surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We allow the input trajectories to SHRED to have a length of 52, corresponding to one year of mea- surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The final 1348 samples are divided into train- ing, test, and validation sets consisting of 1000, 174, and 174 snapshots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 2, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 3 shows example reconstructions and reconstruction error distributions of SHRED and existing techniques both in the presence and absence of added Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Again, the reconstructions obtained by SHRED QR/POD (100) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='20 Q-SDN (3) Q-SDN (100) Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='15 SHRED (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='20 NoiseSample Snapshots Ground Truth SHRED (3 Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=') OR/POD (3 Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' )0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='150 QR/POD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='125 R-SDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='100 Q-SDN Error SHRED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='075 QR SHRED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='025 0 10 20 30 40 50 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Sensors5 A B C (α) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 4: A Example reconstructions obtained via SHRED and QR/POD of atmospheric ozone concentration when three sensors are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Only one of thirty elevations is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Ground truth included for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' B Reconstruction errors of the current state-of-the-art methods and SHRED with a varying number of available sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The solid lines denote the median error from 32 trained estimators and the shaded region denotes the interquartile range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' C Performance of reconstruction methods in the presence of varying levels of added Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The added noise has mean zero and standard deviation equal to α times the mean absolute value of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The number in parentheses denotes the number of available sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' are both visually and empirically superior, while requir- ing far fewer sensors, which can be randomly placed with- out a loss of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Atmospheric ozone concentration The last system for which we consider the perfor- mance of SHRED is a simulation of atmospheric chem- istry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Chemical transport models (CTM) simulate the evolution of an ensemble of interacting chemical species through a transport operator [29, 31] ∂ni ∂t = −∇ · (niU) (4) and a chemical operator dni dt = (Pi − Li)(n) + Ei − Di, (5) where each entry n = �n1 n2 · · · nK �T represents the number density of a specific chemical species, U is the wind vector, (Pi−Li)(n) is the local chemical production and loss term, Ei the emission rate of a species, and Di the deposition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The output of a CTM, then, consists of concentrations for K chemical species for a grid of latitudes, longitudes, and elevations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The data we consider is drawn from the work of Velegar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' [31] and contains simulated atmospheric ozone concentration generated by the CTM software GEOS-Chem [29] over the course of a year with dynamical time steps of 20 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The accessed data from [31] is a compressed SVD representation using the first 50 POD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The decompressed data matrix consists of 26,208 snapshots of a 46 by 72 by 30 (latitude, longitude, elevation) grid, from which we further downsample to obtain 2,600 evenly temporally spaced global ozone concentration fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' To account for the fact that the data matrix is inherently rank 50, we add Gaussian noise with standard deviation α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='05 to the data before performing any analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The resulting snapshots are divided into training, validation, and test sets consisting of 2000, 300, and 300 entries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 4 shows the performance of SHRED and exist- ing techniques on this atmospheric ozone experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' SHRED still outperforms SDNs and QR/POD while re- quiring fewer sensors and being agnostic towards sensor placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' However, in this case when many sensors are available (> 50) QR/POD performs comparably to non- linear reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' This performance is an artifact of Sample Snapshots Ground Truth SHRED (3 Sensors) QR/POD (3 Sensors)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='30 QR/POD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='25 R-SDN Q-SDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='20 Error SHRED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='15 QR SHRED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='05 0 10 20 30 40 50 Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Sensors0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='25 QR/POD (50) Q-SDN (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='20 Q-SDN (50) Error SHRED (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='20 Noise6 the use of a compressed, rank 50 representation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 5: Forecasting error for sea-surface temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A LSTM is trained to forecast QR placed, panel A, or randomly placed, panel B, sensor measurements which are subsequently used to perform reconstructions using SHRED and gappy POD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 16 forecasts are performed and the median error is denoted by the solid lines, with the 25th and 75th percentiles of reconstruction error defining the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The dashed line represents an ensembled forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' FORECASTING SENSOR MEASUREMENTS TO PREDICT THE EVOLUTION OF SPATIO-TEMPORAL DATA In this section, we explore how the expressiveness of SHRED models with few sensors can be leveraged to per- form forecasts of high-dimensional spatio-temporal states using only a limited subsampling of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Forecast- ing the evolution of high-dimensional states from sensor measurements is an exceedingly challenging task, owing to the fact that doing so combines the problem of sys- tem identification with the difficulties of forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We propose a two step approach that leverages the success of LSTMs for low-dimensional time-series forecasting [32] and SHRED for state estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Let {xi}Tt 1 represent a training dataset of temporally ordered states with corresponding sensor measurements {yi}Tt 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We train an LSTM network, G � {yi}t t−k � , to map between a trajectory of sensor measurements and the subsequent measurement, yt+1, by finding G ∈ argmin � G∈G N � i=1 ||yt+1 − �G � {yj}i i−k � ||2 (6) using the ADAM optimizer on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Do- ing so, we then forecast beyond the time interval of the training data to obtain forecasted sensor measurements {ˆyi}Tt+p Tt+1 for p > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The forecasted measurements are used by a SHRED model, trained on {xi}Tt 1 , to obtain forecasts of the high-dimensional state x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Detailed train- ing protocols are included in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The forecasted states {ˆxi}Tt+p Tt+1 are evaluated against the ground truth for each ∆t forecast rather than the mean error across all forecasts to demonstrate perfor- mance over forecasts of varying length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We also include reconstructions from forecasted sensor measurements ob- tained by gappy POD for comparison, akin to the work of [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Finally, we only show results for SST and turbulent flow data, as the compressed form of the atmospheric ozone data is biased towards both QR placement and POD reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Sea-surface temperature For the SST example, we select the first 85% of the dataset to act as training data, the subsequent 20 snap- shots as validation data, and the remainder as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A LSTM for forecasting is trained on the training data and sensor measurements are forecast beyond the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' These forecasted sensor measurements are then used to construct a forecast of the high-dimensional spatio-temporal field using a trained SHRED model and gappy POD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We consider the cases that the sensors are placed via QR or randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We perform 16 runs for each experiment, and consider an ensembled forecast found by averaging the forecast of each run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The results for QR placed and randomly placed sensors are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' With QR placement, the forecasts obtained by SHRED outperform that of POD in the short-term and are simi- lar for longer forecast horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' However, with randomly placed sensors SHRED is still able to obtain comparably accurate forecasts while POD fails to consistently yield faithful reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Turbulent flow Unlike sea-surface temperature, even medium-range forecasts of a turbulent flow are impossible due to the fact that the flow is not quasi-periodic or stationary in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' For this reason, we focus on achieving accurate short-term forecasts in this application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' To do so, we use the same scheme as before, with the exception that the validation set is selected to occur earlier in the train- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Of the 1567 snapshots of the turbulent flow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='150 QR/POD SHRED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='125 QR SHRED Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='025 0 25 50 75 100 125 At1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='25 POD SHRED 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='00 0 25 50 75 100 125 △t7 with sufficiently long preceding measurement histories, the first 1000 are selected as training, followed by 50 samples selected for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The next 100 samples are also used for training and the remainder constitute the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' This scheme allows for more accurate short- term forecasts because the forecast occurs directly after the training data, as opposed to directly after the vali- dation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Figure 6 shows the results for the cases that sensors are placed via QR and randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In both cases, the forecast obtained by SHRED deteriorates quickly but greatly outperforms that obtained by POD reconstruc- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' DISCUSSION We have demonstrated SHRED as a method for sys- tem identification/state estimation and the forecast- ing of high-dimensional time-series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' By includ- ing sensor trajectories in the model for reconstruction, SHRED outperformed competing state-of-the-art tech- niques based on static reconstructions from gappy POD or shallow decoder networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We considered three high- dimensional, spatio-temporal datasets, a synthetically generated forced turbulent flow, a simulation of atmo- spheric ozone concentration from a chemical transport model, and weekly mean sea-surface temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The superior performance of SHRED held across both interpolatory (reconstruction) and extrapolatory (fore- casting) regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In Section II, the task was purely an in- terpolatory reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Training and test data, upon which we evaluate reconstruction accuracy, were drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' As a result, overfitting to the training data was not an issue, and SHRED was able to capture fine grain details of complex flows with as few as one or three sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Applied in this manner, SHRED offers an attractive method for compression of large datasets as well as a framework for developing re- duced order models of dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The perfor- mance of SHRED in this section is also indicative of the promise of SHRED for high-dimensional data that is sta- tionary or periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Notably, SHRED demonstrated re- markable indifference towards sensor placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In this interpolatory regime, issues arising from the necessity of optimal sensor placement can be mitigated by increas- ing the information content of inputs to a neural network through the use of sensor trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' These results show that the sensor trajectories encode a significant amount of information, just as is expected of time-delayed em- beddings of dynamics [34–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In Section III, the task at hand was forecasting the evolution of high-dimensional spatio-temporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In contrast to Section II, training and test sets are tem- porally separated and thus the results are extrapolatory in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' High-dimensional forecasts were accomplished by forecasting low-dimensional sensor measurements and constructing states from the predicted measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Forecasting is an inherently difficult task for complex systems, especially in the high-dimensional fields often encountered in the engineering and the physical sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Still, SHRED’s use of sensor trajectories allows for im- proved forecasting performance as compared to existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Furthermore, SHRED for forecasting exhibits robustness to sensor placement not found in other state- of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We thus advocate for the use of SHRED for both the tasks of state estimation and state prediction from forecasted sensor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Fur- ther work must be done to improve the accuracy of low- dimensional sensor forecasts to be used in conjunction with SHRED for state prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Improving the fore- casts of high-dimensional data can enable on-the-fly com- pression and novel sensing mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A B FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 6: Forecasting error for turbulent flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A LSTM is trained to forecast QR placed, panel A, or randomly placed, panel B, sensor measurements which are subsequently used to perform reconstructions using SHRED and gappy POD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' In this experiment, the validation set is selected from earlier in the data so that the forecast occurs immediately after following training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 16 forecasts are performed and the median error is denoted by the solid lines, with the 25th and 75th percentiles of reconstruction error defining the shaded region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The dashed line represents an ensembled forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='8 QR/POD SHRED 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='6 QR SHRED Error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content='2 0 5 10 15 20 25 △t3 POD SHRED Error 2 1 0 5 10 15 20 25 △t8 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' MATERIALS AND METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Proper orthogonal decomposition and QR Traditional techniques for high-dimensional state re- construction from limited sensor measurements typically rely on the SVD to compute a low-rank embedding of the spatio-temporal dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Given N snapshots of an m dimensional state, we construct a data matrix X = [x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' xN] where xi ∈ Rm is the i-th temporal snapshot of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The SVD factors the data matrix into the product of an orthogonal matrix, U ∈ Rm×m, a diagonal matrix with decreasing entries, Σ ∈ Rm×N, and another orthogonal matrix V ∈ RN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' An optimal rank r approximation of X can be directly obtained through the SVD by retaining only the first r columns of U, X ≈ ˆX = UrΣrVT r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' (7) Thus, each temporal snapshot xi can be approximated by a linear combination of the first r columns of U, oth- erwise known as the dominant POD modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' POD based methods estimate a high-dimensional state, x, drawn from the same distribution as that of the training data by solving for these r coefficients using a subsampling of the state, y = Cx ≈ CUrb, (8) where C is a “sensing” matrix consisting of rows of the m × m identity matrix and b ∈ Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' A full-state estimate from a set of sensor measurements, y, can then be obtained by x ≈ ˆx = Ur(CUr)−1y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' (9) In practice, it is critical to design C such that the inver- sion in (9) is well-conditioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' An indirect bound on the condition number of CUr can be found by determining C where | det(CUr)| is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' That is, we seek C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' = argmax C | det CUr| = argmax C � i |λi(CUr)| = argmax C � i σi(CUr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' (10) Unfortunately, for large systems this NP-hard, combina- torial search becomes computationally intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The QR decomposition with column pivoting of UT r offers an approximate greedy solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The decomposition utilizes Householder transformations to find a column permuta- tion matrix PT such that UT r PT = QR (11) where Q is orthogonal and R is upper triangular with decreasing, positive entries on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Because at each iteration of the algorithm the diagonal entry of R is selected to be as large as possible and Q is orthogo- nal, we have a a greedy maximization of det(UT r PT ) = det(PUr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Thus, setting C = P can improve the recon- structions obtained by POD based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' We refer to reconstructions obtained in this manner as QR/POD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Shallow decoder networks Although QR/POD methods have demonstrated suc- cess in a variety of fields, the method can, at most, es- timate the coefficients of the first n POD modes if there are n sensors available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The included POD modes are insufficiently expressive to obtain accurate reconstruc- tions when exceedingly few sensors are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' This limitation motivates the development of more expressive, nonlinear methods for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Among these are shallow decoder networks (SDN) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' As before, let y = Cx (12) be a point sampling of a high-dimensional state x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' SDNs are shallow, fully-connected neural networks that learn a mapping from the space of sensor measurements, y, back to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' SDNs can be denoted as F(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' WSD) := R(WbR(Wb−1 · · · R(W1s))) (13) where y is the input sensor data, R is a scalar, nonlin- ear activation function, WSD = {Wi}b i=1 are trainable weights, and k denotes the number of layers in the SDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Formally, we seek F ∈ argmin � F∈F N � i=1 ||xi − �F(yi)||2, (14) so that reconstruction error is minimized over a set of N training states {xi}N i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The parameters of the network are randomly initialized and then trained by a gradient descent method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Once the network is trained, reconstruc- tions given subsequent sensor measurements are found by ˆx = F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' (15) SDNs can be trained for any set of sensor measure- ments, and unlike QR/POD are not entirely reliant on principled sensor placement schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Still, previous em- pirical work suggests that even neural-network based re- constructions can be improved through the use of QR placement [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Recurrent neural networks and shallow recurrent decoders Both SDNs and QR/POD have been used successfully in the reconstruction of high-dimensional dynamical sys- tems, but also to broader classes of images [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' This 9 extension is possible because both methods rely only on static, point measurements and each reconstruction is performed individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Dynamical systems, on the other hand, inherently depend on the temporal evolution of the state, and incorporating this dependence into a method for reconstruction offers a natural improvement upon existing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Our work includes these dy- namics through the use of a trajectories of sensor mea- surements To do so, we rely on recurrent neural networks (RNN) [40] and, more specifically, long short-term memory net- works (LSTM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The most basic RNN accepts as an input a sequence of vectors {vi}t i=1 and outputs a single hidden state, ht.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' This can be expressed through the recursion re- lation ht = R(Wcvt + Wrht−1 + br) (16) where R is a scalar, nonlinear activation function, Wc, Wr and br are trainable weights, and h0 = �0 · · · 0�T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Generic RNNs are notorious for suffering from the van- ishing gradient problem, causing them to have difficulty in identifying long term dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' LSTMs address this issue through the introduction of a so-called “gradi- ent super-highway.” Instead of outputting a single hid- den state, LSTMs incorporate an additional “cell state” which only undergoes minor, pointwise operations, allow- ing gradients to flow easily from many time steps in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' LSTMs are perhaps the most commonly used class of RNN at the time of writing, due to their ability to learn long and short term dependencies [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' They are frequently used in time-series forecasting, natural lan- guage processing, and video analysis, among many other domains [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' For a more complete discussion of LSTMs, we direct the reader to [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The recursion relation of an LSTM can be written as ht = σ � Wo �ht−1, vt � + bo � ⊙ tanh(ct) (17) ct = σ � Wf �ht−1, vt � + bf � ⊙ ct−1 (18) +σ � Wi �ht−1, vt � + bf � ⊙ tanh � Wg �ht−1, vt � + bg � for trainable weights and biases WRN = {Wo, Wf, Wi, Wg, bo, bf, bi, bg}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Both σ, the sigmoid function, and tanh operate pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Note that ht can be written as function in terms of only {vi}t i=1 and parameterized by the trainable weights and biases, ht = G({vi}t i=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' WRN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' (19) We now introduce the SHallow REcurrent Decoder (SHRED), which merges an LSTM with the SDN archi- tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Suppose we have a multivariate time-series of a high-dimensional state {xi}T i=1 and corresponding mea- surements of the system {yi}T i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Rather than reconstruct xi using only yi, as is done by QR/POD and SDN, we let an LSTM learn a latent representation using the previous k sets of sensor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' This latent representa- tion is then used by a shallow decoder to reconstruct the high-dimensional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' The SHRED architecture can be written as H � {yi}t i=t−k � = F � G({yi}t i=t−k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' WRN);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' WSD) � (20) using eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' 13 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' As in the case of SDNs, we use a gradient descent method to train the network weights to minimize reconstruction loss, H ∈ argmin � H∈H N � i=1 ||xi − � H � {yi}t i=t−k � ||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' (21) H, so defined, represents a map from sensor trajectories to the full-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors acknowledge support from the National Science Foundation AI Institute in Dynamic Systems (grant number 2112085).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' JNK further acknowledges sup- port from the Air Force Office of Scientific Research (FA9550-19-1-0011 and FA9550-19-1-0386).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Manohar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtFLT4oBgHgl3EQfLS8f/content/2301.12011v1.pdf'} +page_content=' Brunton, J.' metadata={'source': 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b/OdAyT4oBgHgl3EQftPnB/content/tmp_files/2301.00593v1.pdf.txt @@ -0,0 +1,2445 @@ +Arxiv +vol.1, no.1, pp.1– 21, 2022 +https://doi.org/xxxxxx/xxxx-xxx-xxx- +xxxx +Original ariticle +Reconfigurable Metasurface: A Systematic +Categorization and Recent Advances +Changhao Liu1,2, Fan Yang1,2, Shenheng Xu1,2, and Maokun Li1,2 +1. Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China. +2. Beijing National Research Center for Information Science and Technology (BNRist), Beijing, 100084, China. +Corresponding author: Fan Yang; Email: fan yang@tsinghua.edu.cn. +Received March 22, 2022; Accepted March 22, 2022; Published March 22, 2022. +Copyright © 2022 C. Liu et al. This is a gold open access article under a Creative Commons Attribution License (CC BY 4.0). +Abstract — Considering the rapid progress of theory, design, fabrication and applications, metasurface (MTS) has become a new research frontier in +microwave, terahertz and optical bands. Reconfigurable metasurface (R-MTS) can dynamically modulate electromagnetic (EM) wave with unparalleled +flexibility, which leads to great research tide in recent years. Numerous R-MTSs with powerful capabilities and various functions are presented +explosively. In light of the five dimensions of EM wave, this review proposes a unified model to describe the interactions among R-MTS, EM wave +and EM information, and suggests information bit allocation strategy to categorize different types of R-MTSs systematically. As recent advances of +R-MTS, 1-bit and 2-bit elements manipulating different wave dimensions are reviewed respectively in detail. Finally, this review discusses the future +research trends of R-MTS. Hopefully, R-MTSs with diverse dimensions and functions can propel the next generation of communication, detection, +sensing, imaging and computing applications. +Keywords — Antenna, electromagnetic surface, electromagnetic wave, information bits, information metasurface, metasurface, multi-dimension, multi- +function, reconfigurable, reflectarray, switch, transmitarray. +I. Introduction +Electromagnetic (EM) wave is the cornerstone of modern so- +ciety. To manipulate EM wave, metasurface (MTS), an arti- +ficial array composed of ultra-thin subwavelength elements, +is invented and investigated over the decades. MTS can ma- +nipulate multi-dimensional EM wave with unparalleled de- +gree of freedom, which has great advantages compared with +conventional bulk device manipulating EM wave, like low +cost, low profile, light weight, high efficiency and easy fab- +rication. Therefore, MTS is attracting tremendous interest of +researchers nowadays, and applications around MTS are ex- +periencing explosive growth, such as high performance an- +tennas [1], radar cross section (RCS) reducing radomes [2], +invisibility cloaks [3], holograms [4, 5], ultra-thin lenses [6], +to name a few. +Evolving from conventional MTS, reconfigurable meta- +surface (R-MTS), integrated with tunable materials on each +element, shows dynamic capability to control EM wave. Scat- +tered patterns of EM wave after encountering R-MTS can be +tuned by applying input stimuli to R-MTS. A variety of R- +MTS are proposed with the reconfigurable functions of tuning +different dimensions of EM wave. Moreover, some state-of- +the-art prototypes are emerging with multiplexed functions +on one R-MTS, which shows that the research of R-MTS are +evolving toward multi-dimensions and multi-functions. +Digital R-MTS, a reconfigurable form of coding meta- +surface [7], brings about huge research tide. Loading lumped +switches with discrete controlling signals on MTS shows +great advantages over conventional analog R-MTS, due to its +low cost, easy manipulation, scalability, high reliability, and +compatibility with information world. Therefore, this review +mainly focuses on digital R-MTS, which is characterized by +information bit. +Digital R-MTS not only shapes the physical EM wave +and guides the EM energy flow, but also modulates the inci- +dent wave and generates information. Information is loaded +on EM wave, and has various forms, depending on the +various dimensions of EM wave. Consequently, R-MTS is +given more functions and capabilities to manipulate multi- +dimensions of EM wave, which also brings the great boom of +R-MTS investigation. +With tremendous advantages and revolutionary applica- +tions, numerous R-MTSs with diverse functions are just un- +folding. However, various terminologies based on their func- +tions appear correspondingly, which make the architecture of +R-MTS research seem complex and confusing. Therefore, a +arXiv:2301.00593v1 [physics.app-ph] 2 Jan 2023 + +Reconfigurable Metasurface: A Systematic Categorization and Recent Advances +2 +systematic categorization of R-MTSs is in need. Here, a uni- +fied mathematical model of R-MTSs is propose based on five +dimensions of EM wave to describe the interactions among +R-MTS, EM wave and EM information, and then we suggest +a method called information allocation strategy, attempting +to reorganize these R-MTSs under one terminology system. +Based on this systematic catergorization, recent advances of +R-MTSs are reviewed within the same terminology system. +This strategy could contribute to the architecture of surface +electromagnetics theory [8]. Hopefully, this review could pro- +vide a comprehensive view of R-MTS, EM wave, EM infor- +mation and their interactions, and inspire researchers to dis- +cover more fascinating functions of R-MTS. +This review is organized as follows. Firstly, we focus on +the interactions among EM wave, EM information and R- +MTS, and propose an information allocation strategy in Sec- +tion 2. Next, under the information allocating frame, different +forms of 1-bit allocation elements are reviewed in Section 3. +Section 4 continues the point of view, and it mainly reviews +the 2-bit allocation elements. In Section 5, the future trends of +R-MTS are discussed, and some challenges are also outlined. +Conclusion is drawn in Section 6 as the final part. +II. Information Allocation Strategy +1. Dimensions of EM Wave +To begin with, it is necessary to review the dimensions of +plane EM wave. A plane EM wave can be mathematically +described in the following formula +⃗E(t,⃗r) = E0ej(ωt−⃗k·⃗r+ϕ)|p⟩. +(1) +This formula contains five dimensions of plane EM wave: +phase (ϕ), amplitude (E0), polarization (|p⟩), direction (⃗k) +and frequency (ω). Note that field has polarization, so ⃗E is a +vector, also as a function of space (⃗r) and time (t). +2. Interactions among R-MTS, Wave and Information +To understand the functions of R-MTS, the physical point of +view is interpreted at first. When EM wave encounters R- +MTS, each element on R-MTS scatters local EM wave. Then +the scattered EM waves propagate in free space. Summing up +every local scattered EM wave, the receiver obtains the final +EM response. R-MTS can manipulate EM wave dynamically +in five dimensions. Accordingly, R-MTS enables the dynamic +control of wave-metasurface interactions. +If information viewpoint is taken into consideration, an- +other understanding of the functions of R-MTS can be ob- +tained. We mainly discuss digital information in this review. +Digital information is fundamental in modern world, which is +characterized by bit. Compared with analog information, dig- +ital information is robust, fast, low-cost and easy to process, +which is more and more important nowadays. To generate +and transmit digital EM information in free space, the physi- +cal information medium, EM wave, is indispensable. In other +words, EM wave carries EM information. Since EM wave has +five dimensions, EM information has five dimensions corre- +spondingly. Besides, each dimension of EM wave can carry +EM information respectively. R-MTS is invented to manip- +ulate EM wave, so using R-MTS to manipulate EM wave is +to modulate EM information. Since EM information and EM +wave have five dimensions, R-MTS also has to support the +function of modulating five dimensions of EM information. +Therefore, such type of R-MTS is multi-dimensional. Figure. +1 illustrates interactions among R-MTS, EM wave and EM +information, linked by EM dimensions. +EM elements compose R-MTS, and tunable material en- +ables the reconfiguration of EM element. Consequently, the +key to modulating digital EM information lies in designing +EM elements with tunable materials. Since the information +to be generated is discrete and digital, the tuning material is +specified as digital switch, which generates discrete states of +elements, and is also characterized by bit in physical world +correspondingly. Compared with general tunable materials, +digital switches are cost-effective, easy to integrate, easy to +control and compatible with digital information. Therefore, +R-MTSs using digital switches attract enormous research in- +terest in recent years. +Based on the information point of view, the R-MTS pro- +vides dynamic control of information-metasurface interac- +tions. R-MTS can not only manipulate physical EM wave, +but can also generate and modulate EM information. R-MTS +links physical world and information world, which is named +as information metasurface [9]. +3. Mathematical Model of R-MTS +Here, we propose a mathematical model of R-MTS to de- +scribe the interactions among R-MTS, EM wave and EM in- +formation. Suppose an R-MTS is composed of M elements. +When an EM wave encounters R-MTS, the scattered wave is +expressed as the following formula +⃗Esca(t,⃗r) = +M +� +i=1 +Pi(⃗r) · Mi(t,⃗r, Ri) · ⃗Einc +i +(t,⃗ri), +(2) +where inc represents the incident wave, and sca means the +scattered wave. ⃗r is the location of the receiver, and ⃗ri is the +position of the ith element. Here, ⃗Einc +i +is the incident wave +at ith element. Note that the incident wave is not necessarily +plane wave, but it is assumed at small local region of each +element, the incident wave is plane wave. According to (1), +⃗Einc +i +can be expressed as +⃗Einc +i +(t,⃗ri) = Einc +0i ej(ωinc +i +t−⃗kinc +i +·⃗ri+ϕinc +i +)|pinc +i +⟩, +(3) +with variable values of incident dimensions at each element +of different positions. +Mi(t,⃗r, Ri) is the ith element modulation matrix of the +incident wave, and the notation +Ri = ( ⃗Einc +i +, Switchi) +(4) + +3 +Arxiv, vol.1, no.1, pp.1-21, https://doi.org/xxxxxx/xxxx-xxx-xxx-xxxx +𝐸𝑖𝑛𝑐 +𝕄 +Frequency: 𝝎 +𝒑 +𝝎 +𝝋 +𝒌 +𝑬𝟎 +𝐸𝑠𝑐𝑎 +ℙ +𝒑 +𝝎 +𝒌 +𝝋 +𝑬𝟎 +𝑨 +𝕁 +𝚫𝝎 +𝑭 +𝚫𝝋 +𝐸 = 𝐸0𝑒𝑗(𝜔𝑡−𝑘⋅ Ԧ𝑟+𝜑) 𝑝 +𝐸𝑠𝑐𝑎 = ෍ +𝑖=1 +𝑀 +ℙ𝑖𝕄𝑖𝐸𝑖 +𝑖𝑛𝑐 +EM wave +Metasurface +EM information +Bit +𝝎 +𝒌 +𝒑 +𝝋 +𝑬𝟎 +Dimensions +Figure 1 An illustration of information allocation strategy and the interactions among EM information, EM wave and R-MTS, which are linked by EM +dimensions. Since EM wave has five dimensions, EM information bits can be allocated to these dimensions. When an incident EM wave containing +five dimensions encounters an R-MTS, it is modulated by a modulation matrix M of each element, which also manipulates five EM dimensions +and loads EM information on EM wave. After propagating in free space, the total scattered wave in far field is a summation of all scattered waves +from each element. In summary, information to be transmitted is allocated to dimensions of scattered EM wave, and modulated dynamically by +R-MTS. +is used to represent the response of elements. Ri contains the +incident dimensions of EM wave and the element configura- +tions determined by switch. We call the former excitation and +the latter control. It is also remarked that Ri is also a funtion +of time (t), because Switchi can be modulated temporally, +which is actually the origin of reconfiguration. +Mi(t,⃗r, Ri) is expressed as +Mi(t,⃗r, Ri) = Ai(Ri)Fi(⃗r, Ri)ej(∆ωi(Ri)t+∆ϕi(Ri))Ji(Ri), +(5) +where the amplitude response (A) describes the energy am- +plifying, attenuating or maintaining states; element radiation +pattern (F) is a function of the receiving location (⃗r), and can +shape the beam as well as determine the beam direction; addi- +tional frequency (∆ω) can shift the incident frequency, which +is a kind of nonlinear effect; additional phase (∆ϕ) describes +the phase shifting effect when the incident wave encounters +EM element; Jones polarization matrix (J) is a 2 × 2 matrix, +usually anisotropic, which can modulate the incident polar- +ization (|pinc⟩), and M is also a 2 × 2 matrix accordingly. +These five element modulation dimensions can exactly cover +and modulate the five dimensions of EM wave. +Pi(⃗r) describes the path from R-MTS to the information +receiver. Suppose the path is linear, time-invariant, isotropic, +homogeneous and single-path, so the mathematical model is +Pi(⃗r) = Li(⃗r)e−j⃗k·(⃗r−⃗ri)I, +(6) +where Li is the path loss from the ith element to the receiver, +and I is the polarization identity matrix. Obviously, the path +model is determined by the location of information receiver +(⃗r) and the position of each element (⃗ri). +Each element may have different responses to different +dimensions of incident wave, and since the elements can be +reconfigurable in several dimensions, these dimensions of +EM wave can be dynamically modulated. Summing up each +response of element, the EM wave scattered by R-MTS is ob- +tained according to (2). + += += += += +=Reconfigurable Metasurface: A Systematic Categorization and Recent Advances +4 +4. Information Allocation Strategy +As shown in (2), the characteristics of R-MTS are determined +by EM elements. Suppose N-bit information (2N states) +needs to be generated by an EM element. Since EM informa- +tion rests in EM wave, the N-bit information can be allocated +to different dimensions of EM wave. Correspondingly, those +dimensions of EM wave need to be independently modulated +by the element and each element has to respond to the de- +sired dimensions of EM wave. Meanwhile, loading N inde- +pendent switches enables N-bit element. The key to transmit- +ting multi-dimensional EM information is designing multi- +dimensional reconfigurable elements correspondingly. This +is why this strategy is named as information bit allocation, +which means allocating multi-dimensional EM information +bits to multi-dimensional reconfigurable EM elements. This +strategy can help researchers to understand the functions of +each type of reconfigurable element, and find the category of +various reconfigurable elements. +Strictly speaking, the total EM response is the summation +of each element of different dimensions. Since there are M +independent elements in an R-MTS, and each element manip- +ulates N independent bits, the total modulated bits of the en- +tire R-MTS are M × N = K, which can greatly multiply the +amount of information. Considering the effect of array, the +arrangements and states of each element at array level also +worth careful design, which is called array-level bit alloca- +tion. But we mainly focus on the element level in this review +due to its importance and fundamentality. The array-level bit +allocation strategy is more complex and flexible, which is be- +yond the scope of this review. +In the next sections, the information allocation strategy is +applied to review the reconfigurable elements of different di- +mensions. We first review 1-bit elements which have only one +reconfigurable dimension, and it is usually enabled by one +independent switch. Then the 2-bit elements are reviewed, +which may have two reconfigurable independent dimensions +enabled by two independent switches. Following this path, +more-bit reconfigurable elements of different dimensions can +be designed to realize more functions. +III. 1-Bit Element +In this section, 1-bit reconfigurable elements which have only +one independent switch are reviewed. Limited by one bit, +each element has two states. So 1-bit element only manip- +ulates one dimension of EM wave, or several associated di- +mensions but with only 1-bit information or two states totally. +Since there are five dimensions of EM wave, there are +five types of information bit allocation, specified as phase- +only, amplitude-only, polarization-only, direction-only and +frequency-only elements (Figure. 2). Researchers have de- +voted significant efforts to design each type of reconfigurable +elements, and have achieved major breakthrough in the past +decades. Some typical achievements of their works are re- +Phase +Direction +Amplitude +Frequency +Polarization +(a) +(b) +Figure 2 An illustration of 1-bit allocation. (a) 1 bit information can be al- +located to one of five dimensions at a time. So there are five types +of 1-bit elements. (b) The function of the 1-bit reconfigurable ele- +ments, where the element can manipulate one incident wave with +two different scattering states. +viewed in the following parts respectively. +1. Phase-Only Element +Phase-only elements are most intensively studied type among +the five types of 1-bit R-MTS. Considering phase-only ele- +ments, the formula (2) is simplified into the following form +⃗Esca(⃗r, t) = +M +� +i=1 +⃗Ci(⃗r)ej(−⃗k·(⃗r−⃗ri)+∆ϕi(Ri)+ϕinc +i +) · ejωt, +(7) +with time-invariant coefficient term +⃗Ci(⃗r) = Li(⃗r)AiEinc +0i Fi(⃗r)Ji|pinc +i +⟩. +(8) +Phase-only elements only change the ith additional phase +∆ϕi, and other dimensions are usually invariant with switch +configurations and time. Formula (7) indicates that the radi- +ation pattern of R-MTS is determined by ∆ϕ of each ele- +ment, which means the near field phase influencies far field +beam patterns. Therefore, various applications are realized +by phase-only R-MTSs, such as dynamic beamforming and +beam scanning [10], reconfigurable hologram [11], recon- +figurable orbit angular momentum (OAM) beam generation +[12], and so on. +1-bit phase is the quantization of continuous phase, which +has only two phase states. Usually, researchers choose the +phase difference of two configurations of near 180◦, because +the phase quantization loss can be minimized to about 3 dB - +3.9 dB [13, 14]. +Various 1-bit elements are designed, with different con- +figurations, switches, frequency bands and functions. Ac- +cording to reflectarray theory, additional phase of elements +can be tuned by time-delay lines, variable sizes or variable +rotating angles [1, 15, 16]. In light of these phase tuning ap- +proaches, ON/OFF states of RF switch on element can be +used to mimic the size changing or angle rotating. Therefore, +the additional phase of elements can be changed. + +State 1 +State 2 +Ip) +k +p) +k +Eo +0 += +=5 +Arxiv, vol.1, no.1, pp.1-21, https://doi.org/xxxxxx/xxxx-xxx-xxx-xxxx +a +b +Phase-only elements +e +f +d +c +g +h +i +Amplitude-only elements +j +k +l +m +Polarization-only +elements +n +o +Direction-only elements +p +Frequency-only elements +Figure 3 1-bit reconfigurable elements manipulating one of five dimensions. (a)-(f) Phase-only elements. (a) 1-bit phase reconfiguration with single switch +on each element [17]. (b) 1-bit phase reconfigurable reflective element with two switches. It can realize polarization conversion [23]. (c) 1-bit +element with two switches controlled by one bias signal can respond to dual LPs and CPs [28]. (d) Reconfigurable transmitarray element based on +receive-transmit structure. Two PIN diodes are turned ON/OFF alternately, leading to 180◦ current reversal on transmitting structure [30]. (e) A +dual-layer reconfigurable transmitarray element with one switch on one layer to tune the phase simultaneously [39]. (f) Single layer bidirectional +reconfigurable element [40]. (g)-(j) Amplitude-only elements. (g) Amplifying grid array based on differential amplifier [45]. (h) Amplifying +reflectarray based on FET transistors [46]. (i) An amplifying element using the principle of parametric amplification [47]. (j) A switchable +absorber with four PIN diodes on different directions has the ability to respond to full polarizations [52]. (k)-(m) Polarization-only elements. +(k) LP-LP switching element [10]. (l) LP-CP switching element [59]. (m) CP-CP switching element [60]. (n)-(o) Direction-only elements. (n) +Cylindrical active dipole elements on an FSS [61]. (o) Complementary #-shaped 1-bit reconfigurable planar FSS in full polarization [64]. (p) +Frequency-conversion element based on time-modulation [70]. + +capacitance +PIN diode 2 +PIN diodel +py +PIN diode4 +PIN diode3 +capacitanceDiode 1 +:D1.ID +D +Ti +X +T2 +R1 +Diode 2PIN1 +MetalLayer +PIN2 +Via +BiasLineLayerActive microstrip patch +RO4403 film +d, +dpias +-Bias line +di +Ground plane +Passive microstrip patchPIN +DiodesGrounded-vias +DC-viaGate leads +Drain leads +WHH1-INFOMW +www.ainfoinc.com +aight Waveguide +P/N:340WAL-100MEMS +0 +45 +X +p厦Z +h +xp +PIN diode +py +px +XVaractor +Varactor +Patch radiator +Varactor +Grounding vias +Varactor +Grounding vias +RFthrough +Timemodulation +signalinput +Patch radiatorReconfigurable Metasurface: A Systematic Categorization and Recent Advances +6 +1) Basic Phase-Only Element +One switch with 1-bit controlling signal can enable phase- +only reconfiguration for single polarization EM wave. In mi- +crowave band, single PIN diode is often used as the switch +to tune the phase [7, 10, 17–22]. In [10, 17, 18], indepen- +dently addressable reconfigurable reflectarray antenna (RRA) +is realized with high efficiency by carefully designing the el- +ements, switches and bias lines (Figure. 3(a)). Besides, vari- +ous RRAs are brought out with different bands. In millime- +ter wave band, a 60 GHz RRA prototype using commercial +PIN diodes is fabricated [19]. Wideband RRAs are always a +research focus, and up to 38.4% bandwidth is reported in a +recent work [22]. In THz and optical bands, the modulation +principles are similar. However, the immaturity of tunable de- +vices limits the development in this research field. In recent +years, numerous tunable materials are also introduced to tune +the phase at such high frequency bands. +Manipulating several switches synchronously using one +controlling signal with two states is also considered as a 1-bit +element. More functions can be integrated onto one element +with more associated switches. +2) Phase-Only Element Related With Polarization Dimen- +sion +Polarization conversion with stable 180◦ phase modulation in +full band can be realized by loading more than one associated +switch. Jones matrix with polarization conversion function is +derived if elements and switches are designed properly with +J = +�0 +1 +1 +0 +� +, +(9) +which means this Jones matrix can swap the incident polar- +ization, and in other words, the reflection polarization is ro- +tated by 90◦. Applying controlling signal to alternately tune +the ON/OFF states of switches can realize accurate 180◦ +phase difference. Lots of works have been done using the +principle of polarization conversion in RRA designs [23– +27]. For instance, literature [23] demonstrates that alternately +switching two PIN diodes controlled by one signal line can +realize stable 180◦ reflection phase difference based on the +principle of polarization conversion (Figure. 3(b)), which can +manipulate circular polarization (CP) wave with 1-bit phase +difference. +Dual-switch elements can respond to dual LP and dual CP +waves. Polarization conversion elements with two or more +switches can respond to multi-polarizations. Besides, litera- +ture [28] reports that a patch loading two associated switches +on the orthogonal sides can tune phase with response to dual- +linear and dual-circular polarizations (Figure. 3(c)). +3) Phase-Only Element Related With Direction Dimension +Apart from reflective metasurfaces, transmissive metasur- +faces with 1-bit phase tuning ability are also investigated +comprehensively, which are called reconfigurable transmi- +tarray antenna (RTA). It has been proved that at least two +switches are necessary to realize a high-performance 1-bit +RTA element [29]. Based on two switches, using current re- +versal method to realize 1-bit phase control is a useful ap- +proach when designing RTAs [30–36]. For example, liter- +ature [30] proposes the receive-transmit structure with two +PIN diodes changing the element configuration, which can +reverse current by 180◦ mutually, and then the transmitted +phases are reversed by 180◦ (Figure. 3(d)). Unlike RTAs with +all switches on a single layer, multi-layer RTAs with switches +on different layers are proposed in recent years [37–39]. For +instance, a dual-layer Huygens’ element is designed in litera- +ture [39], with two associated switches loading on each layer +respectively to realize 1-bit phase control (Figure. 3(e)). +Single layer R-MTS can also implement bidirectional +beam scanning functions with polarization conversion [40], +where half of the energy is reflected and the other half is trans- +mitted (Figure. 3(f)). +2. Amplitude-Only Element +According to formula (5), amplitude modulation of element +can be represented by A. When an EM wave encounters +MTS, energy can be amplified (A > 1), attenuated (A < 1) +or maintained (A ≈ 1), so the amplitude-only element should +be active, lossy or near transparent. Amplitude modulation +element can apply for channel modulation [41], relay ampli- +fying [42] and so on. Some microwave band 1-bit amplitude +R-MTS prototypes are demonstrated in [42–52]. +Amplifying transmitarrays based on receive-transmit +structure with active transistors as energy amplifier are pro- +posed in 1990s [43–45]. As Figure. 3(g) illustrates, a pair of +vertical gate leads are employed to receive the energy from +free space, and two transistors compose a differential am- +plifier with their sources connected [45]. The gate energy is +amplified and radiated horizontally by a pair of drain leads. +Meanwhile, amplifying/maintaining states can be switched +by tuning the ON/OFF states of power source of transistors. +Amplifying reflectarray based on FET transistors is pro- +posed in [46], as shown in Figure. 3(h). The patch receives +LP wave and couples the energy into microstrip line through +H-shaped slot. Then the energy is amplified in the circuit and +re-radiates into free space through the patch in orthogonal po- +larization. +Recently, amplifying reflectarray based on parametric +amplifier is investigated in [42, 47]. 2.36 GHz incident sig- +nal is amplified and reflected based on the nonlinear effect of +varactor on circuit with energy provided by 4.72 GHz pump +(Figure. 3(i)). The amplifying gain can be tuned by changing +the pump energy, which enables the amplitude reconfigura- +tion. +Switchable absorber-reflectors or absorber-transmitters +based on active frequency selective surface (AFSS) have been +developed in recent years [48–53]. Imperfect PIN diodes with +large insertion loss at ON state can absorb energy at reso- +nance frequency, while at OFF state, the wave is scattered + +7 +Arxiv, vol.1, no.1, pp.1-21, https://doi.org/xxxxxx/xxxx-xxx-xxx-xxxx +y +x +x’ +y’ +Figure 4 Illustration of coordinate system definitions considering polariza- +tion. +by the R-MTS without attenuation. Switching the ON/OFF +states can modulate the amplitude. Elements can respond to +multiple polarizations with several PIN diodes on them, from +LP [49, 50], dual-LP [51], to full polarizations (Figure. 3(j)) +[52, 53]. +3. Polarization-Only Element +Polarization |p⟩ is one of the intrinsic dimensions of spatial +EM wave. 1-bit polarization reconfigurable element switches +between two polarization states, and mathematically, Jones +matrix J changes between two forms. Polarization bits are +also exploited to transmit information [54]. +Here, the principle of 1-bit polarization reconfiguration +is reviewed in advance. Suppose the incident polarization is +along x axis. One switch is loaded along x′ axis, where x′- +y′ coordinate system is rotated by 45◦ from x-y coordinate +system, as Figure. 4 shows. Suppose the scattering responses +along x′ and y′ axes are isolated. If the additional phase along +y′ axis is ϕy′ and the phase along x′ axis is ϕx′, under x′-y′ +coordinate system, the element modulation matrix is simply +written as +M′ = +�ejϕx′ +0 +0 +ejϕy′ +� +, +(10) +and the incident wave is +|p′⟩ = +1 +√ +2 +�1 +1 +� +. +(11) +Hence, the scattered wave is +⃗E′ = M′|p′⟩ = +1 +√ +2 +�ejϕx′ +ejϕy′ +� +. +(12) +Under x-y coordinate system, the scattered wave is +⃗E = 1 +2 +�ejϕx′ + ejϕy′ +ejϕx′ − ejϕy′ +� +. +(13) +Since the phase along y′ is fixed, suppose ϕy′ is fixed as +180◦; the phase along x′ is reconfigurable. Let us consider +three special cases. (1) ϕON +x′ += 180◦, ϕOFF +x′ += 0◦. Accord- +ing to (13), the scattered wave switches between LP(x) and +Table 1 A summary of 1-bit polarization manipulation types. +ϕy′ = 180◦ +Incident polairzation +Scattered polarization +ϕON +x′ += 180◦ +LP(x) +LP(x) - LP(y) +ϕOFF +x′ += 0◦ +CP +RHCP - LHCP +ϕON +x′ += 90◦ +LP(x) +LP-CP +ϕOFF +x′ += 0◦ +CP +ϕON +x′ += 90◦ +LP(x) +RHCP - LHCP +ϕOFF +x′ += −90◦ +CP +LP(x) - LP(y) +LP(y). (2) ϕON +x′ += 90◦, ϕOFF +x′ += 0◦. ⃗EON is circular po- +larization, and ⃗EOFF is linear polarization. Hence, polariza- +tion modulation between LP and CP waves is obtained. (3) +ϕON +x′ += 90◦, ϕOFF +x′ += −90◦. The scattered waves are two +spins of CP. Therefore, dual-CP transition can also be gener- +ated and switched. The discussions are summarized in Table. +1. +Some prototypes verify the transitions of LP(x)-LP(y), +LP-CP and LHCP-RHCP in literatures [10, 55–60]. LP-LP +switching based on transmitarray is investigated in [10, 57, +58]. By tuning the ON/OFF states of PIN diodes, the re- +flection linear polarization is converted or remained (Figure. +3(k)). Literature [59] studies the LP-CP switching type with +LP incidence (Figure. 3(l)). When the switch is ON, the re- +flection phase difference along x′ axis and y′ axis is 180◦, +so the reflection polarization is converted. When the switch +is OFF, the phase difference is 90◦, so CP is generated and +reflected. Polarization-only R-MTS can also convert LP inci- +dent wave to RHCP/LHCP dynamically [60], if phase differ- +ence between x′ axis and y′ axis is 90◦ or -90◦ modulated by +states of switches (Figure. 3(m)). We note that other states of +polarizations can also be switched as summarized in Table. 1. +4. Direction-Only Element +In the formula (5), F denotes the element radiation pattern, +which determines the direction of the main beam scattered by +a single element. It is worth mentioning that element radiation +pattern is not exactly the array radiation pattern. The latter +not only considers the element radiation pattern, but is also +influenced by other dimensions of elements like phase and +amplitude, as derived in (7) and (8). +Reflection-transmission pattern reconfigurable AFSSs are +comprehensively studied in recent years [58, 61–64]. A +dipole FSS with tunable PIN diodes at the center is reported +in [61], which can switch the reflection/transmission states of +EM wave (Figure. 3(n)). If the PIN diode is at ON state, the +dipole is resonant, so the EM wave is blocked by the FSS and +then reflected. Otherwise, if the PIN diode is at OFF state, the +dipole is not excited, so EM wave can pass through the FSS. +Later, elements with complementary reflection and transmis- +sion responses in full polarization are reported in [64], as +shown in Figure. 3(o). The #-shaped AFSS element can re- +spond to full polarizations. +Note that ⃗k is a vector in full space, which means beyond + +Reconfigurable Metasurface: A Systematic Categorization and Recent Advances +8 +propagating forward or backward. Propagating along differ- +ent angles based on pattern reconfiguration is also direction- +only types. Though reflection- or transmission-type direction +reconfigurable elements are well studied, to the best of our +knowledge, direction-only elements dedicated to pattern re- +configuration at one side (angle reconfiguration) are not re- +ported in published literatures, and more related works could +be expected in the future. +5. Frequency-Only Element +The inherent operating mechanism of new frequency gen- +eration is nonlinear effect. Hence, EM elements should be +nonlinear if the function of R-MTS is converting frequency. +Frequency-only R-MTS has been under investigation since +1980s [65]. However, because of the complexity of elements +and the lack of applications, researchers have paid little at- +tention to this topic in last decades. Recently, the rapid devel- +opment of digital controlling circuits like field programmable +gate array (FPGA) makes it possible to use time dimension +to generate new frequency based on R-MTS [66–70]. Fig- +ure. 3(p) shows a frequency-conversion element, which can +mix frequencies of spatial EM wave and guided wave, and +re-radiate the mixed wave into free space [70]. +As shown in (4), notation R enables the reconfiguration +of elment, as a function of time t. Suppose periodic control- +ling signal is applied to a phase-only element with period T. +Then the additional phase of elements can be modulated peri- +odically. According to formula (5), and considering the only +variable is phase (∆ϕ), the main term of element modulation +function is +M(t) = ej∆ϕ(t). +(14) +M(t) is also a periodic funtion of time t, so applying the +Fourier expansion, M(t) is expressed as +M(t) = +∞ +� +h=−∞ +ahej 2πh +T t, +(15) +where {ah} is the Fourier series, and +ah = 1 +T +� +T +2 +− T +2 +ej∆ϕ(t)e−j 2πh +T tdt. +(16) +Harmonic frequency components are generated, provided that +the ∆ϕ(t) is not a constant number. According to formula (2), +when the incident frequency is fc, the core term of scattered +wave is +E(t) = +∞ +� +h=−∞ +ahej(2πfc+ 2πh +T )t. +(17) +As (17) illustrates, new frequencies are added to the incident +frequency, and the harmonic frequency components of spatial +wave are reconfigurable if the controlling signal is reconfig- +urable or the switching period is changeable. +Essentially, EM elements act as a frequency mixer, mix- +ing the frequency in free space and the switching frequency +of FPGA on board. The reason why rapidly switching ele- +ment can work as mixer results from the fact that the lumped +switch is a nonlinear component. Temporal modulation of the +ON/OFF states of switch generates square wave of phase, +which is the inherent source of new frequency generation. We +remark that the utmost frequency shift is determined by the +switching speed of switches and FPGA, which is low so far +compared to the frequency of spatial EM wave (1/T ≪ fc). +So the significance of frequency shift realized by rapidly +switching states on elements needs to be improved. +IV. 2-Bit Element +This section reviews the advances of 2-bit elements with +two independent switches. With more information bits ma- +nipulated by a single element, R-MTS goes to be multi- +dimensional, and a variety of functions can be realized by +R-MTS. +Note that the response notation R in (4) is determined by +two parts: control and excitation. Following the information +allocation strategy, there are two types of allocating 2 bits: +both two bits go to control, or one bit goes to control and +another bit goes to excitation. +In the first type, each element can actively manipu- +late 2-bit information at one time, which is called 2-bit- +manipulating type. Since there are five dimensions of EM +wave, 2 bits go to these five dimensions respectively, as +shown in Figure. 5(a). There are C2 +5 ways to allocate 2 bits +to two different dimensions, and 5 ways to allocate all the 2 +bits to one dimension. Totally, there are C2 +5 +5 = 15 ways of +bit allocation of the 2-bit-manipulating type elements. +In the second type, element can be designed to actively +control 1 bit information on one dimension, while respond- +ing to excitations of two incident waves on one dimension +independently using the other 1 bit. Two incident waves can +be independently manipulated by multiplexing one element, +so we call these elements multiplexed-manipulating type el- +ements. Theoretically, there are 5 × 5 = 25 combinations of +this type, as shown in Figure. 5(b). +Here, some existing combinations are reviewed respec- +tively, and the others still need to be further studied. +1. 2-Bit-Manipulating Type Element in One Dimension +1) 2-Bit-Phase Reconfigurable Element +2-bit-phase element with 2 bits both allocated to phase di- +mension is a well studied type [71–76]. Four states of phase, +0◦, 90◦, 180◦ and 270◦, are available in 2-bit-phase elements, +which can improve the phase resolution and reduce the quan- +tization loss of RRA by 2.4 dB - 3 dB compared to 1-bit phase +[13, 14]. +Two switches are minimum for 2-bit phase modulation. +Both reflective and transmissive 2-bit-phase R-MTSs are real- +ized. A reflective asymmetric pentagon-shaped element based + +9 +Arxiv, vol.1, no.1, pp.1-21, https://doi.org/xxxxxx/xxxx-xxx-xxx-xxxx +(a) +(b) +Figure 5 Illustrations of two types of 2-bit allocation. (a) 2-bit-manipulating +type, which can actively manipulate one or two dimensions under +one incident wave. Two bits are allocated to five dimensions, so +there are C2 +5 + 5 = 15 combinations of this type of bit allocation. +(b) Multiplexed-manipulating type, which can manipulate one di- +mension (pink arrow) while responding to two incident waves dif- +fering in one dimension (green arrow). Theoretically, there are +5 × 5 = 25 ways of bit allocation of this type. +on resonance method is proposed in [71], as shown in Figure. +6(a), which can generate 0◦, 90◦, 180◦ and 270◦ reflection +phase states. For 2-bit-phase transmitarrays, more switches +are needed in receive-transmit structure based transmitarrays +[73]. 0◦/180◦ phase shift is realized in receiving patch, and +0◦/90◦ phase shift is achieved in transmitting structure, and +then four states of phase are generated (Figure. 6(b)). +2) 2-Bit-Amplitude Reconfigurable Element +A 2-bit-amplitude R-MTS with amplitude amplifying, main- +taining and attenuating functions is implemented by control- +ling supply voltages of two amplifiers on each side of receive- +transmit structure based transmitarray [77], as shown in Fig- +ure. 6(c). In fact, continuous amplitude reconfiguration can +also be achieved by continuously tuning supply voltages. +3) 2-Bit-Polarization Reconfigurable Element +2-bit-polarization reconfigurable elements are also reported +in recent literatures. Three states of polarizations are actively +generated and modulated using two independent switches +[78]. PIN diodes are soldered along x, y axis respectively +(Figure. 6(d)), with 90◦ ON/OFF phase difference along both +axes. So 90◦, -90◦ and 0◦ phase differences between x, y axes +are available by independently tuning the two switches. If the +incident LP is along x′ axis, RHCP, LHCP and LP reflection +waves are obtained respectively. It is worth mentioning that +the rest state of four states is the 90◦ phase of LP which is not +used in this design. +Furthermore,to realize four-state polarization reconfigu- +ration of RHCP, LHCP, LP(x′) and LP(y′), we propose two +approaches. It can be obtained with incident polarization +along x′ if the additional phase is 0◦/180◦ along x axis, and +0◦/90◦ along y axis. Besides, another approach is that, if the +reflection phase along y axis is fixed such as 180◦, 2-bit-phase +reconfigurable element with tunable phase along x axis can +also act as a 2-bit-polarization reconfigurable element, when +incident polarization is along x′ axis. +4) 2-Bit-Frequency Reconfigurable Element +Frequency reconfigurable elements are usually enabled by +time modulation, which will be discussed in Section 4.3. +2. 2-Bit-Manipulating Type Element in Two Different Di- +mensions +1) Phase-Amplitude Reconfigurable Element +If one bit is allocated to phase, and the other bit goes to ampli- +tude, it is called phase-amplitude reconfigurable element. Lit- +erature [79] reports a dual-layer structure, with the top layer +formed by graphene which manipulates the reflection ampli- +tude, and the bottom layer composed of PIN diodes which +tunes the reflection phase. In [80], single-layer elements in- +tegrated with two types of PIN diodes are proposed (Figure. +6(e)). Here, one PIN diode has little insertion loss at ON state +and high isolation at OFF state, which is employed to tune +reflection phase while maintaining the reflection amplitude. +The other PIN diode also has little insertion loss at ON state, +the, but at OFF state, it can absorb a large part of energy while +impact little influence on the phase. Hence, a 2-bit phase- +amplitude reconfigurable element is realized via these careful +designs. +Amplifying reflectarray while modulating reflection +phase is proposed in [81], as shown in Figure. 6(f). A patch +and I-shaped slot convert y-polarized spatial wave into guided +wave. Varactors are loaded on transmission line, which can +modulate the transmission phase. Lumped amplifier is em- +ployed to amplify the energy on transmission line. Then the +x-polaried energy is re-radiated through slot and patch. +2) Phase-Polarization Reconfigurable Element +Phase-polarization reconfigurable elements have been de- +signed and fabricated in early years. In 2010, literature [82] +introduced the idea of realizing phase modulation while +controlling polarizations in reflectarrays. It can be achieved +by dual-polarized reconfigurable element (we classify it as +polarization-multiplexed phase-manipulating type in Section +4.4.2) with incident polarization along x′ axis. The lossless +modulation matrix of dual-polarized reconfigurable element +in x-y coordinate system is +M = +� +ejϕx +0 +0 +ejϕy +� +. +(18) +Following the similar derivation in (11)-(13), the reflection +wave in x′-y′ coordinate system is +⃗E′ = 1 +2 +�ejϕx + ejϕy +ejϕx − ejϕy +� +, +(19) + +Phase +Amplitude +Direction +Polarization +Frequency +State 1 +State 3 +State 2 +(m +p) +State 4 +k +Eo +p)Phase +Amplitude +Direction +Polarization +Frequency +Incidence 2 +1: State 1 +Incidence 1 +1: State +2 +2: State 1 +m +[p) +2: State 2 +k +Eo +(p) +(0 +EOReconfigurable Metasurface: A Systematic Categorization and Recent Advances +10 +Phase +Polarization +Amplitude +Direction +Frequency +a +b +c +d +e +f +g +h +i +j +k +2 Bits +2 Bits +ab +c +ef +d +g +h +i +j +k +k +NA +l +m +n +NA +l +m +n +x +y' +y +x' +Figure 6 2-bit reconfigurable elements actively manipulating one or two of five dimensions. (a) 2-bit-phase reconfigurable reflectarray based on four +switchable resonant states [71]. (b) 2-bit-phase reconfigurable transmitarray based on receive-transmit structure with two sets of switches on +each side [73]. (c) A 2-bit-amplitude reconfigurable element realizing amplitude amplifying, maintaining and attenuating functions [77]. (d) 2- +bit-polarization R-MTS with the ability to switch among RHCP, LHCP and LP states [78]. (e) A phase-amplitude R-MTS with one PIN diode +modulating phase and the other controlling amplitude attenuating or not [80]. (f) A phase-amplitude reconfigurable reflective element with varac- +tor tuning the phase and transistor controlling the amplitude [81]. (g) A phase-polarization reconfigurable element with two independent switches +operating synchronously [83]. (h) Phase-direction reconfigurable element with two PIN diodes modulating phase and two PIN diodes manipu- +lating direction [87]. (i) A direction-amplitude reconfigurable prototype [89]. (j) Direction-polarization reconfigurable element using two SPDT +switches [91]. Polarization and direction are flexibly controlled. (k) An illustration of frequency-phase reconfiguration [92]. The beam directions +of harmonic frequency waves are individually controlled and the harmonic frequencies are tuned. (l) The frequency-amplitude R-MTS, where the +amplification level is tunable at harmonic frequencies based on space-time modulation [97]. (m) The frequency-polarization reconfiguration based +on time modulation strategy, which can generate arbitrary polarization at harmonic frequencies [99]. (n) A frequency-direction reconfigurable +Huygens’ element based on dynamic space-time modulation, where the ratio between forward and backward energy is tunable [100]. + +Port2 +RF +RF +Port1 +Front view +Backview +Metal +Front +ground +Back +layer +layer +Metal +F4B +Metal +F4B +Metal +SpatialEM +SpatialEM +energy +PA +PA +energy +Via hole +EMenergyPIN diode-1 +PIN +diode-2Bias lines +Amplifier +Extratrans- +missionline +-Patch +*Slot2 +-Slot 1 +y +x +TunableloadAPTSTCM +Forward +Backward +fe+fo +Space-timeCoding Matrix +te +f-2fo +c-3fo +0 +O +Time +0 +0 +EqurvalentCircunt +O +T +axis +0 +0 +0 +FPGA +Xaxisfc-fo +fo +ICP +fc+fo +fc+fo +(LP) +fc +EP) +[CP) +fc +fc-fo +(EP) +fc+fo +(LP) +fc-fo +fo +FPGAXM +XE +YM +ds/2 +V +SW +SW +D +p11 +W1 +W2 +W4 +12Bus +BiaslineGroundPlane +MetalProbe +SPDTSwitch +Via Hole +MetalProbef-3fo +Programmable metasurface +fc-2fo +fe-fo +fc+fo +Space-time-coding +fa+2fo +L +pq +fa+3fo +bd +FPGA +2 +Equivalent +pq +space-coding +pq +Space:X axis +Space:epin2 +pinl +3n +DCbiaP-i-n diode +M1 +M2 +RT/Duroid6002 +Bias-lines Tx +M3 +RO4450F +M4 +RT/Duroid6002 +DC con.to GND +M5 +Ground plane +RO4450F +Bias-lines Rx +RT/Duroid6002 +M6 +P-i-n diode +Dea +D211 +Arxiv, vol.1, no.1, pp.1-21, https://doi.org/xxxxxx/xxxx-xxx-xxx-xxxx +with 1-bit reconfigurable phase along x and y axes inde- +pendently. Suppose the phases switch between 0◦/180◦ at +OFF/ON states, four states of reflection wave are obtained +as follows +⃗E′ +ON−ON = +� +ej180◦ +0 +� +, ⃗E′ +OFF−OFF = +� +ej0◦ +0 +� +, +⃗E′ +OFF−ON = +� 0 +ej0◦ +� +, ⃗E′ +ON−OFF = +� +0 +ej180◦ +� +. +(20) +As it demonstrates, polarization switches dynamically be- +tween x′ and y′ axes, with independent modulation of the +reflection phase. +A +simplified +reflective +element +with +2-bit +phase- +polarization reconfiguration is further designed and simulated +in [83], as shown in Figure. 6(g). In recent years, RTAs based +on receive-transmit structure also demonstrate the capability +of 2-bit phase-polarization modulation for LP(x)-LP(y) tran- +sition [84]. Furthermore, if the phase difference along x′ and +y′ axes is designed as 90◦/-90◦, RHCP/LHCP can be gener- +ated and modulated, as demonstrated in [85]. +3) Phase-Direction Reconfigurable Element +Phase-direction reconfigurable type emerges recently [86– +88]. For example, as shown in Figure. 6(h), the element is +composed of two layers, with the top layer controlling phase +and the bottom layer controlling direction. Each element is +controlled independently with 1 bit for phase reconfiguration +and 1 bit for direction reconfiguration. +4) Direction-Amplitude Reconfigurable Element +Direction-amplitude manipulation type is another combina- +tion of 2-bit elements for reflection, transmission or absorp- +tion switching applications [89, 90]. In literature [89], two +layers of the substrate are both soldered with PIN diodes (Fig- +ure. 6(i)). PIN diodes on the top layer can determine whether +the EM wave passes through the first layer or is absorbed. +PIN diodes on the bottom layer can control the reflection or +transmission states. Note that there are only three states of +the 2-bit element, because if the EM energy is absorbed on +the top layer, it is meaningless to consider the propagation +directions of EM wave. +5) Direction-Polarization Reconfigurable Element +Direction-polarization reconfigurable element is also re- +ported in a recent literature. A reflecting-transmitting R-MTS +controlled by incident polarization is proposed in [91]. Two +identical structures are designed on the top and bottom layers +with a single-pole double-throw (SPDT) switch on each layer +(Figure. 6(j)). The top SPDT switch can control which po- +larization state could be transmitted to the bottom layer, and +the bottom SPDT switch can determine which polarization +state of transmitting wave could be excited. Hence, if the in- +cident polarization is along x axis, three states are obtained: +x-polarized reflection wave, x-polarized transmission wave +and y-polarized transmission wave, with polarization and di- +rection manipulated independently. +3. 2-Bit-Manipulating Type Element Related with Fre- +quency Reconfiguration +Frequency reconfiguration has been discussed in Section 3.5. +Here, exploiting properties of controlling signals can modu- +late frequency as well as other dimensions of EM wave. Ac- +cording to (16), ah is a complex number representing the am- +plitude and phase responses of the hth harmonic frequency, +determined by controlling signal waveform. Each ah can +be designed with great degrees of freedom. Therefore, var- +ious wave dimensions at harmonic frequencies can be inde- +pendently modulated by applying different switchable signal +waveforms. +1) 2-Bit-Frequency and Frequency-Phase Reconfigurable +Element +Literature [92–96] demonstrate that phases at different har- +monic frequencies can be independently modulated using 1- +bit or 2-bit phase reconfigurable elements by carefully de- +signing the phase waveform of elements. Hence, scanning +beams at different frequencies can be generated simultane- +ously (Figure. 6(k)). Besides, the signal period T can modu- +late the harmonic frequency, so the frequency is also recon- +figurable. +2) Frequency-Ampitude Reconfigurable Element +Apart from frequency-phase reconfiguration, frequency- +amplitude reconfiguration type is also reported recently [97]. +As shown in Figure. 6(l), an amplifier and a FET are loaded +on the passive structure, and by dynamically controlling the +FET, the R-MTS can generate tunable amplified EM wave at +different harmonic frequencies. +3) Frequency-Polarization Reconfigurable Element +Recent years have also witnessed the development of arbi- +trary polarization reconfigration based on time-modulation +strategy [98, 99]. For example, orthogonal dipoles with two +sets of PIN diodes are modulated dynamically to change the +transmitted amplitude and phase of orthogonal LP waves at +harmonic frequenies (Figure. 6(m)), and thus the arbitrary +transmitted polarization is generated by composing the two +modulated LP waves. +4) Frequency-Direction Reconfigurable Element +Frequency-direction reconfiguration type appears in [100], as +shown in Figure. 6(n). A time-varying Huygens’ metasurface +is designed to change the energy ratio between reflection and +transmission at harmonic frequencies. By applying space- +time modulation strategy, the beam scanning angles can also +be manipulated. +5) Discussion +According to information allocation strategy, a single recon- +figurable element with at least N-bit physical devices can +modulate N-bit information. Here, by rapidly switching el- +ement states, some extra tunable bits are provided by time- +domain modulation. Hence, this type of R-MTS is also called +time-modulated metasurface. Besides, when extending ele- +ments to form a periodic array, with different states at dif- + +Reconfigurable Metasurface: A Systematic Categorization and Recent Advances +12 +ferent element locations, the amount of information is also +multiplexed, which is provided by space-domain modulation. +This type is the space-modulated R-MTS, which is the ar- +ray level reconfiguration as discussed previously. Moreover, +by combining space modulation and time modulation, space- +time-modulated R-MTSs can exploit additional dimensions +in both space and time domain, so they can implement novel +functions, which is a promising topic for future studies. +4. Multiplexed-Manipulating Type Element +Apart from active manipulation of two dimensions, actively +manipulating one dimension while independently responding +to two different incident waves is also a large category of 2- +bit elements. There are 25 combinations of the multiplexed- +manipulating type theoretically, but some are not practical. +Here, we review the existing combinations, and propose some +ideas of the unrealized combinations. +1) Frequency-Multiplexed Phase-Manipulating Element +Frequency-multiplexed phase control is a natural idea for full +duplex communication using a single R-MTS. To indepen- +dently modulate phase in response to two frequency bands +with the same polarization, one can design a supercell con- +sisted of two independent elements working at two bands +[101, 102], as shown in Figure. 7(a) for an example. +2) Polarization-Multiplexed Phase-Manipulating Element +Polarization-multiplexed phase-manipulation is a well stud- +ied type [103–107]. Elements based on isolation between +two orthogonal directions can independently manipulate two +polarizations. So a single element can work for two inci- +dent waves distinguished by orthogonal polarizations simul- +taneously. As shown in Figure. 7(b), a cross-shaped element +with two sets of independent MEMSs can modulate reflection +phase of dual-LP waves [103]. +Additionally, +dual-LP-multiplexed +transmitarray +is +demonstrated in [107]. Two 1-bit phase reconfigurable +dipoles are placed orthogonally to form an element, thus the +transmitting polarization is multiplexed (Figure. 7(c)). +The above mentioned literatures are all based on LP in- +cidence. We notice that fixed MTSs for independent dual- +CP multiplexing are sufficiently studied in [108–111]. How- +ever, as far as we know, independent phase reconfiguration +for dual-CP incident wave is not reported in literatures, which +might be a future research topic. +3) Polarization-Multiplexed +Direction-Manipulating +Ele- +ment +Polarization-multiplexed +direction-manipulation +evolves +from 1-bit dierction reconfiguration as demonstrated in +Section 3.4. Literature [112, 113] load orthogonal switches +on two layers to manipulate propagating directions of two +incident polarizations independently, shown in Figure. 7(d) +as an example. Furthermore, polarization conversion layer is +added on the top of these layers to implement polarization +conversion functions [114]. Besides, fixed-phase layers are +added on the top and bottom, and different holographic +images are obtained from different incident polarizations +with controllable directions [115]. +4) Direction-Multiplexed Direction-Manipulating Element +A 2-bit R-MTS that modulates direction of wave while re- +sponding to two directions of incident wave appears in [116]. +It exploits the inherent nonreciprocity of amplifier to real- +ize the independent direction modulation of both sides. A +supercell combined by two opposite amplifying elements +can respond to forward and backward wave independently +(Figure. 7(e)). If the forward element operates at ON state, +the forward EM energy is amplified and transmitted; oth- +erwise, the forward energy is blocked and reflected back- +ward. The backward element can modulate the direction of +the backward-incident EM wave similarly. Consequently, the +incident waves from two directions are modulated indepen- +dently. +5) Unrealized Multiplexed-Manipulating Types to be Ex- +plored +As +demonstrated +above, +multi-functional +R-MTSs +of +multiplexed-manipulating types are emerging nowadays. +Here, some ideas of other multiplexed-manipulating type re- +configurable elements will be discussed in the next few para- +graphs. +Direction-multiplexed phase-modulation is also a promis- +ing topic. Here, we discuss two types of direction multiplex- +ing. The first type is called Janus metasurface with reference +to [117, 118]. With Janus metasurface responding to bidirec- +tional incident waves respectively, two unrelated transmitting +holographic images are obtained from both sides of the MTS +[118]. +Besides, the term direction is not only for the opposite +directions, but also for the incident angles. Considering this, +another type is angle-multiplexed phase modulation, where +independent designed beam patterns are generated with dif- +ferent incident angles on the MTS. Angle-sensitive elements +are designed in [119–123], which can respond to different in- +cident angles independently. +Direction-multiplexed MTSs are a new research trend +nowadays, and all the above mentioned works focus on fixed +MTSs. Reconfigurable direction-multiplexed phase-tuning +could have great potential in the future. +Polarization-multiplexed reconfiguration is also an attrac- +tive topic. Besides phase reconfiguration and direction recon- +figuration as reviewed above, polarization-multiplexed ampli- +tude or frequency reconfiguration are unexplored topics. Re- +lated works could be carried out in the future. +V. Emerging Topics and Future Trends +1. N-Bit Element +This paper mainly reviews the 1-bit and 2-bit reconfigurable +elements. Besides, elements with more than 2-bit reconfig- +uration are also one of the research highlights [124, 125]. +Evolving from 1-bit and 2-bit allocation, an illustration of N- + +13 +Arxiv, vol.1, no.1, pp.1-21, https://doi.org/xxxxxx/xxxx-xxx-xxx-xxxx +Multiplexed +Dimension +Manipulating +Dimension +Samples +Frequency +Phase +Polarization +Phase +Polarization +Direction +Direction +Direction +b +c +d +e +a +Figure 7 2-bit reconfigurable elements with one multiplexed dimension and one manipulating dimension. (a) Frequency-multiplexed phase-manipulating +element with two reconfigurable elements in one supercell operating at the two frequency bands [102]. (b) Polarization-multiplexed phase- +manipulating reflective element [103]. (c) Polarization-multiplexed phase-manipulating transmitarray [107]. (d) A polarization-multiplexed +direction-manipulating prototype [113]. (e) A direction-multiplexed direction-reconfigurable design [116]. The supercell can determine whether +EM wave from two sides can pass through the MTS or not. +Phase +Direction +Amplitude +Frequency +Polarization +Figure 8 An illustration of N-bit allocation. N bits can be allocated to five +dimensions independently with various combinations of multi- +plexed bits and manipulating bits. More dimensions are linked by +a single element with higher resolution, so the functions of R-MTS +can be greatly enriched compared with 1-bit or 2-bit R-MTS. +bit allocation is shown in Figure. 8. +Allocating N bits to one dimension can improve the res- +olution on the dimension, thus improving the performance. +Furthermore, full-dimensional element with the ability to +modulate five dimensions of EM wave can increase the func- +tions of R-MTS, which is also a promising topic. We have +known that the quantization loss of 2-bit phase reconfig- +uration is 0.6 dB - 0.9 dB [13, 14], which is acceptable +in most applications. Conditions are similar for other di- +mensions. Therefore, allocating N bits to diverse dimen- +sions is generally more rewarding than just increasing the +resolution of only one dimension. Besides, multiplexing a +single element in multi-dimensions, while manipulating the +multi-dimensional EM wave calles for N-bit multiplexed- +manipulating type R-MTSs. Nonetheless, elements with more +than 2-bit reconfigurable dimensions are insufficiently stud- +ied. +To realize N-bit element, integrating more lumped +switches on a single element is a natural approach. How- +ever, suffered from the bulky size of switches and the com- +plexity of bias lines, the number of switches on a single +element is limited. Applying continuously tunable switches +like varactors can enable multi-bit reconfiguration too. How- +ever, this approach requires precise voltage control, which is +not robust and increases the complexity of controlling cir- +cuit. As discussed in Section 4.3, time-modulated MTS has +potential for N-bit modulation and multi-dimensional ele- + +Viato +GND +Via to +GND1 0: +0 1: +1 1: +Forward +Backward +Nonreciprocal +Reciprocal +Nonreciprocal +Transmission +Transmission +Transmission +Digital Power +Supply +1 +Digitalpower.control +ON&OFF +0 +States: +1 +114mm +1.08mm +=3.2mm +=0.88mm +t2 +14mm +12 +=3.8mm +vias +resistors +V +varactordiodes +silver-plated +X +coppertracesGround +surface +MEMSReconfigurable Metasurface: A Systematic Categorization and Recent Advances +14 +ments. Nevertheless, the limited radiation efficiency, complex +controlling voltages, the narrow bandwidth and low modula- +tion speed are the bottlenecks of this approach. Accordingly, +multi-dimensional N-bit elements are still a challenging topic +for future investigations. +2. Terahertz and Optical R-MTS +Since high-performance lumped switches such as PIN diodes +and varactors are available in microwave band, and the main- +stream fabrication process like printed circuit board (PCB) +technology is mature and low-cost, microwave band R-MTSs +have experienced great breakthrough in recent decades. The +demands for THz and optical R-MTSs are also emerging +rapidly. However, in THz and optical band, commercial +lumped switches are too large to be soldered and ON/OFF +ratio is reduced. Scientists are seeking for new switches +and new architectures with higher frequency and higher per- +formance. With the great revolution of micro- and nano- +fabrication technology in recent years, R-MTSs in THz and +optical bands are experiencing rapid development nowadays. +In THz band, numerous switches are introduced, like +Schottky diode [126, 127] (Figure. 9(a)), high electron +mobility transistor (HEMT) [128–132] (Figure. 9(b)-(c)), +graphene [133, 134], complementary metal oxide semicon- +ductor (CMOS) [125, 135, 136] (Figure. 9(d)), vanadium +dioxide (VO2) [137–139], to name a few. For instance, +HEMT switch has high ON/OFF ratio and low controlling +complexity, which is promised to provide 1-bit phase control +in THz band [128, 129]. Besides, spatial THz amplitude mod- +ulators based on HEMT switches are reported in [130, 131], +with 93% amplitude modulation depth and 1 GHz modulation +rate. +In optical frequency band, no lumped switch is avail- +able neither, so element acts as both nanoantenna and +switch [140]. These switches are made of indium tin ox- +ide (ITO) [141] (Figure. 9(e)), graphene [142], chalco- +genide compound germanium-antimony-tellurium (GST) +[143–148] +(Figure. +9(f)), +liquid +crystall +[149], +poly- +mer poly(3,4-ethylenedioxythiophene):polystyrene sulfonate +(PEDOT:PSS) [150] (Figure. 9(g)) and so on. For example, +phase-change materials like GST can provide obvious switch- +ing performance in optical band. Femtosecond pulses induce +the amorphous-crystalline transition of GST [143], so the ele- +ment resonance states are changed, and then direction, phase +or amplitude can be modulated. +However, THz and optical R-MTSs still have many chal- +lenges to be handled, like improving efficiency, increasing +switching speed, eliminating grating lobes and addressing +independently. Additionally, multi-dimensional and multi- +functional R-MTS in THz and optical bands are also the fu- +ture trends. +(a) +(b) +(c) +(d) +(e) +(f) +(g) +Figure 9 (a)-(d) Terahertz and (e)-(g) optical R-MTSs. (a) Schottky diode +based THz amplitude reconfigurable array [126]. (b) HEMT based +THz amplitude reconfigurable transmitarray [131]. (c) HEMT +based THz phase reconfigurable prototype [129]. (d) CMOS based +THz phase reconfigurable prototype [125]. (e) ITO based optical +phase reconfigurable prototype [141]. (f) GST based optical am- +plitude reconfigurable design [143]. (g) PEDOT:PSS based optical +phase reconfigurable design [150]. +3. Surface-Wave R-MTS +Above mentioned most MTSs manipulate spatial wave, +where EM wave propagates in free space. Moreover, MTS +also shows the ability of manipulating surface wave, where +EM wave propagates along the tangential direction of MTS. +In optical frequency band, 2-D photonic crystals have +been proposed to manipulate surface light since 1990s [151– +154], which is still a hot topic nowadays. To guide surface +wave with more flexibility, photonic topological insulators +[155–157] are made artificially, which can control the light + +Bias +H+ +Schottky +Ohmic +// +Incident +会/会/ +Ohmic +Schottky +Split gap +Depletion +Transmitted +n-GaAs +SI-GaAsCollective state +Individualstate10umIncidence +Dynamic +beam-steering +Phase(V)Read channel +ZnS-SiO2 +Ge2Sb,Tes +ZnS-SiO2 +Glass substrate +Phase change canvas +0.59 μm +Write channel-1V ++1V +Polymer +metasurface +IR camera +Transmitted +Transmitted +Diffracted +beam +beam +beam15 +Arxiv, vol.1, no.1, pp.1-21, https://doi.org/xxxxxx/xxxx-xxx-xxx-xxxx +Figure 10 Reprogrammable topological insulator in microwave band [165]. +The propagating directions on surface are programmed by inde- +pendently addressing the configurations of elements. +propagation directions on surface. +Periodic structures can also manipulate surface wave in +microwave band. High impedance surfaces (HIS) form elec- +tromagnetic band gap (EBG) [158–161] to forbid wave prop- +agation along MTS. +MTSs inducing the transition between surface wave and +spatial wave are also studied. Phase-gradient metasurface can +convert spatial EM wave to surface wave [162]. Besides, ar- +tificial impedance surface (AIS) can transform surface wave +into directional spatial beams [163]. +Despite great breakthroughs in surface wave manipula- +tion, R-MTSs for surface wave modulation are less investi- +gated than the fixed surface-wave MTSs. A few R-MTSs are +proposed to manipulate surface waves dynamically. Litera- +ture [164] proposed tunable EBG structures based on varac- +tors in early years. Recently, a reprogrammable topological +insulator operating in microwave band is proposed [165]. As +Figure. 10 illustrates, configurations of elements are tuned by +two states of PIN diodes, and the propagating direction of sur- +face wave are modulated dynamically by switching states of +elements. +The high flexibility of R-MTS for manipulating surface +wave shows it is a promising topic for future investigations. +4. Nonlinear R-MTS +Linear MTSs have experienced intensive research in past +decades. To realize more functions, nonlinear effects need to +be taken into consideration when designing MTSs. Strictly +speaking, reconfiguration is a kind of nonlinearity, which oc- +curs at the state-change moment. Nevertheless, at each stable +working state, the response to incident EM wave is still linear. +Therefore, the R-MTS is quasi-linear essentially. +For real nonlinear MTS, the response is always nonlin- +ear. Generally, energy amplification, new frequency genera- +tion and some magnet-free nonreciprocity effects are certain +forms of nonlinear effects. Based on these nonlinear effects, +some nonlinear MTSs are designed over the years. +Energy amplification is based on the nonlinearity of ma- +terials. MTS amplifier based on transistors are implemented +Figure 11 Optical nonlinear MTS for second harmonic generation [170]. +in early years [43–46, 81], and parametric amplification MTS +element based on varactor is realized recently [47]. +Frequency generation MTSs have been investigated since +1980s [47, 65]. Quasi-optical grid arrays are introduced with +lumped nonlinear components to realize functions of spa- +tial wave processing in microwave band, such as oscillator +[166, 167], frequency doubler [168] and frequency mixer +[169]. In optical frequency band, nonlinear materials are +employed to form nonlinear MTS [170–173], with second +harmonic generation (SHG) or third harmonic generation +(THG), as shown in Figure. 11. In recent years, with the de- +velopment of time-modulated MTS, frequency reconfigura- +tion using time dimension has become a new research trend +[67–70, 92–100]. +Magnet-free nonreciprocal MTSs are emerging nowa- +days. To realize nonreciprocity without magnetic effects, +nonlinear effects are taken into consideration. Based on the +nonreciprocity of amplifier, nonreciprocal MTS can amplify +the forward wave and block the backward wave [116, 174, +175]. Besides, time-modulated R-MTS can also modulate fre- +quency and beam direction without reciprocity [176–178]. +Apart from these applications, nonlinear MTS could also +carry out other functions, such as all-optical logic gates [179], +spatial wave limiters [180], waveform-dependent absorbers +[181] and so on, which are interesting topics for future re- +search. Furthermore, reconfigurable nonlinear MTS can em- +ploy nonlinear effects with dynamic functions, which may +lead a larger research tide in the future. +VI. Conclusion +As the advanced form of MTS, R-MTS is proposed to ma- +nipulate the scattered wave dynamically. In recent years, nu- +merous R-MTSs emerge with different manipulating dimen- +sions and functions. In this review, we start with the interac- +tions among R-MTS, EM wave and EM information in five +dimensions, and propose the mathematical model of R-MTS +in response to five dimensions of incident EM wave. Then +we suggest a frame called information allocation strategy to +categorize different types of R-MTSs systematically. Based +on this strategy, 1-bit reconfigurable elements manipulating +one of five dimensions are firstly reviewed and categorized. +The advances of multi-dimensional and multi-functional 2- + +2222222 +2222222 +1111111 +1111111 +control network +FPGA +Programmable.PT +23 +Near-field scanning20 +MReconfigurable Metasurface: A Systematic Categorization and Recent Advances +16 +bit elements are reviewed and categorized in the following +section. Various 2-bit elements are divided into two large cat- +egories, 2-bit-manipulating types (15 kinds) and multiplexed- +manipulating types (25 kinds), and the existing ones are re- +viewed respectively in detail. +Due to the rapid evolving speed of R-MTS, diverse ter- +minologies appear, making the whole research architecture +confusing. Hopefully, R-MTSs reorganized and reunited un- +der information allocation strategy might provide a helpful +viewpoint for researchers to find out the development thread +and future research trends. In this paper, we find some types +of the multi-bit reconfigurable elements are unrealized, like +some 2-bit-manipulating elements and many multiplexed- +manipulating elements, and the functions of R-MTS are not +fully utilized. Besides, we have discussed some possible evo- +lution clues of R-MTS such as multi-bit R-MTS, THz/optical +R-MTS, surface-wave R-MTS, nonlinear R-MTS, and so on. +As the recent researching highlight in both science and +engineering fields, the exploration of R-MTS is just unfold- +ing. R-MTS shows grand opportunities for critical appli- +cations in communication, detection, sensing, imaging and +computing areas. It is believed that R-MTS will bring about +a technological revolution in the near future. +Acknowledgements +This work was supported in part by the National Natural Sci- +ence Foundation of China under Grant U2141233, in part by +ZTE Industry-Academia-Research Cooperation Funds, and in +part by THE XPLORER PRIZE. +References +[1] P. Nayeri, F. Yang, and A. Z. Elsherbeni, Reflectarray antennas: the- +ory, designs, and applications, John Wiley & Sons, 2018. +[2] N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. 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Sievenpiper, +“Waveform-dependent absorbing metasurfaces”, Physical review let- +ters, vol.111, no.24, artilce no.245501, 2013. + diff --git a/PdFRT4oBgHgl3EQfIzcw/vector_store/index.pkl b/PdFRT4oBgHgl3EQfIzcw/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..6196e1f1494b3cec157c38423cacf99bd900b1f5 --- /dev/null +++ b/PdFRT4oBgHgl3EQfIzcw/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6e1d6b808c11f55b8043b8c54713bdcd582e72b8cae22b9b81960a036953367d +size 101546 diff --git a/RNAzT4oBgHgl3EQfI_uT/content/tmp_files/2301.01072v1.pdf.txt b/RNAzT4oBgHgl3EQfI_uT/content/tmp_files/2301.01072v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..034477cf12ef0f0dfce5e70000ac36f9c8cce54c --- /dev/null +++ b/RNAzT4oBgHgl3EQfI_uT/content/tmp_files/2301.01072v1.pdf.txt @@ -0,0 +1,1550 @@ +arXiv:2301.01072v1 [math.AP] 3 Jan 2023 +Splitting-type variational problems with +asymmetrical growth conditions +Michael Bildhauer & Martin Fuchs +Abstract +1 Splitting-type variational problems +� +Ω +n +� +i=1 +fi(∂iw) dx → min +with superlinear growth conditions are studied by assuming +hi(t) ≤ f ′′ +i (t) ≤ Hi(t) +with suitable functions hi, Hi: R → R+, i = 1, . . . , n, measuring the +growth and ellipticity of the energy density. Here, as the main feature, +a symmetric behaviour like hi(t) ≈ hi(−t) and Hi(t) ≈ Hi(−t) for +large |t| is not supposed. +Assuming quite weak hypotheses as above, we establish higher +integrability of |∇u| for local minimizers u ∈ L∞(Ω) by using a +Caccioppoli-type inequality with some power weights of negative ex- +ponent. +1 +Introduction +Suppose that Ω ⊂ Rn is a bounded Lipschitz domain and consider the vari- +ational integral +J[w] := +� +Ω +f +� +∇w +� +dx +(1.1) +of splitting-type, i.e. +f : Rn → R , +f(Z) = +n +� +i=1 +fi(Zi) +(1.2) +1AMS subject classification: 49N60, 49N99, 35J45 +Keywords: +splitting-type variational problems, asymmetrical growth conditions, non- +uniform ellipticity +1 + +with strictly convex functions fi: R → R of class C2(R), i = 1, . . . , n, sat- +isfying in addition some suitable superlinear growth and ellipticity conditions. +Problem (1.1), (1.2) serves as a prototype for non-uniformly elliptic varia- +tional problems and it is well known that the ratio of the highest and the +lowest eigenvalue of D2f is the crucial quantity for proving the regularity of +solutions (see, e.g., [1], [2], [3]). The reader will find an extensive overview +including different settings of non-uniformly elliptic variational problems in +the recent paper [4]. Without going into further details we refer to the series +of references given in this paper. +In Section 1.3 of [4], the authors consider general growth conditions which, +roughly speaking, means that the energy density f is controlled in the sense +of +g(|Z|)|ξ|2 ≤ D2f(Z)(ξ, ξ) , +|D2f(Z)| ≤ G(|Z|) , +(1.3) +with suitable functions g(t), G(t): R+ +0 → R+. Then, under appropriate as- +sumptions on g, G, a general approach to regularity theory is given in [4]. +Our note is motivated by the observation, that in (1.2) there is no obvious +reason to assume some kind of symmetry for the functions fi, i.e. in general +we have fi(t) ̸= fi(−t) and, as one model case, we just consider (q± +i +≥ 1, +i = 1, . . . , n) +fi(t) ≈ |t|q− +i +if +t ≪ −1 , +fi(t) ≈ |t|q+ +i +if +t ≫ 1 . +(1.4) +Then, both for t ≪ 1 and for t ≫ 1, the functions fi just behave like a +uniform power of |t|. Nevertheless, the power q− +i enters the left-hand side of +(1.3) and q+ +i is needed on the right-hand side of (1.3). +This motivates to study the model case (1.2) and to establish regularity +results for solutions under the weaker assumption +hi(t) ≤ f ′′ +i (t) ≤ Hi(t) +t ∈ R , +(1.5) +with suitable functions hi, Hi: R → R+, i = 1, . . . n. +There is another quite subtle difficulty in studying regularity of solutions to +splitting-type variational problems: in [5] the authors consider variational +integrals of the form (1 ≤ k < n) +I[w, Ω] = +� +Ω +� +f(∂1w, . . . , ∂kw) + g(∂k+1w, . . . , ∂nw) +� +dx , +(1.6) +where f and g are of p and q-growth, respectively (p, q > 1). Then the regu- +larity of bounded solutions follows in the sense of [5], Theorem 1.1, without +2 + +any further condition relating p and q. The proof argues step by step and +works since the energy density splits into two parts. If, as supposed in (1.2), +the energy density splits in more than two components, then one has to be +more careful dealing with the exponents and some more restrictive (but still +quite weak) assumptions have to be made. In this sense Remark 1.3 of [5] +might be a little bit misleading. We note that a splitting structure into two +components as supposed in (1.6) is also assumed, e.g., in [6] and related pa- +pers. +In the following we consider the variational integral (1.1), (1.2) defined on +the energy class +Ef(Ω) := +� +w ∈ W 1,1(Ω) : +� +Ω +f(∇w) dx < ∞ +� +. +We are interested in local minimizers u: Ω → R of class Ef(Ω), i.e. it holds +that +� +Ω +f(∇u) dx ≤ +� +Ω +f(∇w) dx +(1.7) +for all w ∈ Ef(Ω) such that spt(u − w) ⋐ Ω. +Notation. We will always denote by q+ +i +> 1, q− +i +> 1, 1 ≤ i ≤ n, real +exponents and we let for fixed 1 ≤ i ≤ n +qi := min{q± +i } , +qi := max{q± +i } . +(1.8) +Moreover, we let +Γ : [0, ∞) → R , +Γ(t) = 1 + t2 . +Recalling the idea sketched in (1.4), (1.5) we denote by hi and Hi, i = 1, . . . , +n, functions R → R+ such that with positive constants ai, ai +aiΓ +q− +i −2 +2 (|t|) +if t < −1 +aiΓ +q+ +i −2 +2 (|t|) +if t > 1 + + + + + +≤ hi(t) +(1.9) +and +Hi(t) ≤ + + + + + +aiΓ +q− +i −2 +2 (|t|) +if t < −1 +aiΓ +q+ +i −2 +2 (|t|) +if t > 1 +. +(1.10) +As a general assumption we consider functions fi: R → [0, ∞) of class C2(R), +i = 1, . . . , n, such that for all t ∈ R +hi(t) ≤ f ′′ +i (t) ≤ Hi(t) +(1.11) +3 + +and note that (1.11) immediately implies for all i ∈ {1, . . . , n} with constants +bi > 0 +|f ′ +i(t)| ≤ bi + + + + + +Γ +q− +i −1 +2 (|t|) if t < −1 +Γ +q+ +i −1 +2 (|t|) if t > 1 + + + + + +. +(1.12) +Moreover we obtain for all i = 1, . . . , n with constants ci, ci > 0 +ci + + + + + +Γ +q− +i +2 (|t|) if t < −1 +Γ +q+ +i +2 (|t|) if t > 1 + + + + + +≤ fi(t) ≤ ci + + + + + +Γ +q− +i +2 (|t|); if t < −1 +Γ +q+ +i +2 (|t|) if t > 1 + + + + + +. +(1.13) +With this notation our main result reads as follows. +Theorem 1.1. Suppose that for i = 1, . . . , n the functions fi: R → [0, ∞) +are of class C2(R) and satisfy (1.11) with hi, Hi given in (1.9), (1.10). +With the notation (1.8) we assume in addition that we have for every fixed +1 ≤ i ≤ n +qj +< +2qi + 2 +for all +i < j ≤ n , +(1.14) +qj +< +3qi + 2 +for all +1 ≤ j < i . +(1.15) +If u ∈ L∞(Ω) ∩ Ef(Ω) denotes a local minimizer of (1.1), (1.2), i.e. of +J[w] = +� +Ω +� +n +� +i=1 +fi(∂iw) +� +dx , +then there exists a real number δ > −1/2 such that for every 1 ≤ i ≤ n +� +B +fi(∂iu)Γ1+δ(|∂iu|)η2k dx ≤ c . +(1.16) +Remark 1.1. As outlined in Remark 5.1 and Remark 5.2 below, we recover +the results of [5] in the sense that (1.14) and (1.15) are superfluous in the +case n = 2 (or related situations) and +q+ +1 = q− +1 , +q+ +2 = q− +2 . +Theorem 1.1 describes the typical situation we have in mind. +The proof +however is not limited to this particular case which leads to the generalized +version stated in Theorem 2.1 below. +In Section 3 we shortly sketch a regularization procedure via Hilbert-Haar +solutions while Section 4 presents the main inequalities for the iteration pro- +cedure of Section 5. This completes the proof ot Theorem 2.1 and hence +Theorem 1.1. +4 + +2 +Precise assumptions on f +The suitable larger class of admissible energy densities is given by the fol- +lowing assumption. +Assumption 2.1. The energy density f, +f : Rn → R , +f(Z) = +n +� +i=1 +fi(Zi) , +introduced in (1.2) is supposed to satisfy the following hypotheses. +i) The function fi: R → [0, ∞), i = 1, . . . , n, is of class C2(R) and for +all t ∈ R we have f ′′ +i (t) > 0. +For 1 ≤ i ≤ n we suppose superlinear growth in the sense of +lim +t→±∞ |f ′ +i(t)| = ∞ +and at most of polynomial growth in the sense that for some s > 0 we +have for |t| sufficiently large +f(t) ≤ c|t|s +with a finte constant c . +ii) For i ∈ {1, . . . , n} with exponents δi ≥ 0, θi ≥ 0 satisfying +θi < 1 − δi +(2.1) +we suppose that for all |t| sufficiently large +c1Γ1−δi(|t|)f ′′ +i (t) +≤ +fi(t) ≤ c2f ′′ +i (t)Γ1+θi(|t|) , +(2.2) +|f ′ +i(t)|2 +≤ +c3f ′′ +i (t)fi(t)Γθi(|t|) , +(2.3) +where c1, c2 and c3 denote positive constants. +iii) We let +Γ +q± +i +2 (t) = + + + + + +Γ +q− +i +2 (|t|) if t < 0 +Γ +q+ +i +2 (|t|) if t ≥ 0 + + + + + +. +and suppose that fi, i = 1, . . . ,n, satisfies with positive constants c4, +c5 and for |t| sufficiently large +c4Γ +q± +i +2 (|t|) ≤ fi(t) ≤ c5Γ +q± +i +2 (|t|) . +(2.4) +5 + +Remark 2.1. +i) If fi is a power growth function like, e.g., fi(t) = (1 + +t2)pi/2, pi > 1 fixed, then we have +cΓ(|t|)f ′′ +i (t) ≤ fi(t) ≤ cΓ(|t|)f ′′ +i (t) , +i.e. (2.2). The same is true for our asymmetric model case given by +(1.9) – (1.13). +ii) By convexity it is well known (see, e.g., [7], exercise 1.5.9, p. 53) that +the right-hand side of (2.3) and the right-hand side of (1.13) imply +(2.4). +iii) The condition (2.1) formally corresponds with the condition q < p + 2 +in the standard (p, q)-case (see, e.g., [8], Chapter 5, and the references +quoted therein). +iv) Assumption 2.1, iii) is assumed w.l.o.g. In fact, on account of f ′′ +i > 0 +we know that for any i ∈ {1, . . . , n} the function f ′ +i is an increasing +function, and by Assumption 2.1, i), we let +s+ := inf +s lim +t→∞ +f ′(t) +ts +< ∞ , +s− := inf +s +lim +t→−∞ +|f ′(t)| +|t|s +< ∞ . +Then for arbitrary small ε > 0 we have the right-hand side of (2.4) with +exponent s± + 1 + ε and the left-hand side with exponent s± + 1 − ε. +Going through the proof of Theorem 2.1 we may suppose (2.4) w.l.o.g. +Theorem 2.1. Suppose that we have Assumption 2.1. With the above nota- +tion we assume in addition that we have for every fixed 1 ≤ i ≤ n +qj +< +2qi(1 − δi) +1 + 2θi ++ 2(1 − δi) +for all +i < j ≤ n , +(2.5) +qj +< +2 +1 + 2θi +� +qi +2 (1 − δi) +� +2 + +1 +1 − δi +� +− θi(1 + qj) +� ++ 2(1 − δi) +for all +1 ≤ j < i . +(2.6) +If u ∈ L∞(Ω) ∩ Ef(Ω) denotes a local minimizer of (1.1), (1.2), i.e. of +J[w] = +� +Ω +� +n +� +i=1 +fi(∂iw) +� +dx , +then there exists a real number δ > −1/2 such that for every 1 ≤ i ≤ n +� +B +fi(∂iu)Γ1+δ(|∂iu|)η2k dx ≤ c . +(2.7) +Remark 2.2. In particular we note that (2.5), (2.6) reduce to (1.14), (1.15) +for δi, θi sufficiently small. +6 + +3 +Some remarks on regularization +We have to start with a regularization procedure such that the expressions +given below are well defined. We follow Section 2 of [5] and fix a ball D ⋐ Ω. +If u denotes the local minimizer under in the sense of (1.7) and if ε > 0 is +sufficiently small, we consider the mollification (u)ε of u w.r.t. the radius ε. +We consider the Dirichlet-problem +� +D +n +� +i=1 +fi(∂iw) dx → min +among all Lipschitz mappings D → R with boundary data (u)ε. According +to, e.g., [9], there exits a unique (Hilbert-Haar) solution uε to this problem. +Exactly as outlined in [5] Lemma 2.1 and Lemma 2.2 we obtain: +Lemma 3.1. Let q := min1≤i≤n qi +i) We have as ε → 0 +uε ⇁ u +in W 1,q(D) , +� +D +n +� +i=1 +fi(∂iuε) dx → +� +D +n +� +i=1 +fi(∂iu) dx . +ii) There is a finite constant c > 0 not depending on ε such that +∥uε∥L∞(D) ≤ c . +iii) For any α < 1 we have uε ∈ C1,α(D) ∩ W 2,2 +loc (D). +We then argue as follows: consider a local minimizer u of (1.1), (1.2) and the +approximating sequence {uε} minimizing the functional +J[w, D] := +� +D +n +� +i=1 +fi(∂iwi) dx +(3.1) +w.r.t. the data (u)ε. +In particular we have a sequence of local J[w, D]- +minimizers. +We apply the a priori results of the next section to uε and +Theorem 1.1 follows from Lemma 3.1 passing to the limit ε → 0. +4 +General inequalities +The main result of this section is Proposition 4.2 which is not depending on +the particular structure (1.9). +7 + +We will rely on the following variant of Caccioppoli’s inequality which was +first introduced in [10]. We also refer to Section 6 of [11] on Caccioppoli-type +inequalities involving powers with negative exponents, in particular we refer +to Proposition 6.1. +Lemma 4.1. Fix l ∈ N and suppose that η ∈ C∞ +0 (D), 0 ≤ η ≤ 1. +If +we consider a local minimizer u ∈ W 1,∞ +loc (D) ∩ W 2,2 +loc (D) of the variational +functional +I[w] = +� +D +g(∇w) dx +with energy density g: Rn → R of class C2 satisfying D2g(Z)(Y, Y ) > 0 for +all Y , Z ∈ Rn, then for any fixed i ∈ {1, . . . , n} we have +� +D +D2g(∇u) +� +∇∂iu, ∇∂iu +� +η2lΓβ(|∂iu|) dx +≤ c +� +D +D2g(∇u)(∇η, ∇η)η2l−2Γ1+β(|∂iu|) dx +for any β > −1/2. +To the end of our note we always consider a fixed ball B = B2r(x0) ⋐ D. +With this notation we have the following auxiliary proposition. +Proposition 4.1. Suppose that we have i) of Assumption 2.1 and let η ∈ +C∞ +0 (B), 0 ≤ η ≤ 1, η ≡ 1 on Br(x0), |∇η| ≤ c/r. Moreover, we assume that +u ∈ L∞(D) ∩ W 1,∞ +loc (D) ∩ W 2,2 +loc (D). +Then we have for fixed γ ∈ R, for all k > 0 sufficiently large and for i = 1, +. . . , n the starting inequalities (no summation w.r.t. i) +� +B +fi(∂iu)Γ1+γ(|∂iu|)η2k dx +≤ +c +� +1 + +� +B +|∂i∂iu|Γγ(|∂iu|)fi(∂iu)η2k dx ++ +� +B +|∂i∂iu| |f ′ +i|(∂iu) Γ +1 +2 +γ(|∂iu|)η2k dx +� +(4.1) +Remark 4.1. +i) The idea of the proof of Proposition 4.1 is based on an +integration by parts using the boundedness of u. +An Ansatz of this +kind was already made by Choe [12], where all relevant quantities are +depending on |∇u|. +Here the main new feature is to work with the +energy density f which is not depending on the modulus of ∇u. +8 + +ii) We note that for the proof of Proposition 4.1 no minimizing property +of u is needed. +Proof of Proposition 4.1. +With i ∈ {1, . . . , n} fixed we obtain using an +integration by parts +� +B +fi(∂iu)Γ1+γ(|∂iu|)η2k dx += +� +B +|∂iu|2fi(∂iu)Γγ(|∂iu|)η2k dx + +� +B +fi(∂iu)Γγ(|∂iu|)η2k dx += +− +� +B +u∂i +� +∂iufi(∂iu)Γγ(|∂iu|)η2k� +dx + +� +B +fi(∂iu)Γγ(|∂iu|)η2k dx +≤ +c +� +B +|∂i∂iu|Γγ(|∂iu|)fi(∂iu)η2k dx ++c +� +B +|∂i∂iu| |∂iu| |f ′ +i|(∂iu) Γγ(|∂iu|)η2k dx ++c +� +B +|∂iu|fi(∂iu)Γγ(|∂iu|)η2k−1|∂iη| dx + +� +B +fi(∂iu)Γγ(|∂iu|)η2k dx += +I1,i + I2,i + I3,i + I4,i . +(4.2) +In (4.2) we discuss I3,i: for ε > 0 sufficiently small we estimate +I3,i +≤ +� +B +|∂iu|f +1 +2 +i (∂iu)Γ +γ +2 (|∂iu|)ηkf +1 +2 +i (∂iu)Γ +γ +2 (|∂iu|)ηk−1|∇η| dx +≤ +ε +� +B +|∂iu|2fi(∂iu)Γγ(|∂iu|)η2kdx ++c(ε, r) +� +B +fi(∂iu)Γγ(|∂iu|)η2k−2 dx . +(4.3) +The first integral on the right-hand side of (4.3) is absorbed in the left-hand +9 + +side of (4.2), i.e. +� +B +fi(∂iu)Γ1+γ(|∂iu|)η2k dx +≤ +I1,i + I2,i + c(ε, r) +� +B +fi(∂iu)Γγ(|∂iu|)η2k−2 dx ++ +� +B +fi(∂iu)Γγ(|∂iu|)η2k dx +≤ +I1,i + I2,i + c(ε, r) +� +B +fi(∂iu)Γγ(|∂iu|)η2k−2 dx . +(4.4) +Discussing the remaining integral we recall that the function fi(t)Γ1+γ(|t|) is +at most of polynomial growth, hence we may apply the auxiliary Lemma 4.2 +below to the functions ϕ(t) = fi(t)Γγ(|t|) and ψ(t) := fi(t)Γ1+γ(|t|) with the +result that for some ρ > 0 and for all t ∈ R +fi(t)Γγ(|t|) ≤ c +� +fi(t)Γ1+γ(|t|) +� 1 +ρ + c . +(4.5) +With (4.5) we estimate for ˜ε > 0 sufficiently small and for k > ρ∗ = ρ/(ρ−1) +c(ε, r) +� +B +fi(∂iu)Γγ(|∂iu|)η2k−2 dx +≤ +c(ε, r) +� +B +� +fi(∂iu)Γ1+γ(|∂iu|) +� 1 +ρη +2k +ρ η +2k +ρ∗ −2 dx + c +≤ +˜ε +� +B +fi(∂iu)Γ1+γ(|∂iu|)η2k dx + c(˜ε, ε, r) +� +B +η2(k−ρ∗) dx + c . (4.6) +The inequalities (4.4) and (4.6) complete the proof of the proposition by ab- +sorbing the first integral on the right-hand side of (4.6) in the left-hand side +of (4.4). +It remains to give an elementary proof of the following auxiliary Lemma. +Lemma 4.2. For m ∈ N we consider functions ϕ, ψ: Rm → [0, ∞) such +that ψ(X) ≤ cΓτ(|X|) for some τ > 0 and for all X ∈ Rm. Suppose that we +have for some ε > 0 and for all X ∈ Rn +ϕ(X) ≤ cΓ−ε(|X|)ψ(X) . +Then there exists a real number ρ > 1 and a constant C > 0 such that +ϕ(X) ≤ +� +ψ(X) +� 1 +ρ + C . +10 + +Proof. Let δ := ε/τ, i.e. for all X ∈ Rm +1 + ψδ ≤ 1 + Γε ≤ 2Γε , +hence we have by assumption +ϕ(X) +≤ +c +� +1 + ψδ(X) +�−1ψ(X) +≤ +� +c +if +ψδ(X) ≤ 1 +cψ1−δ(X) +if +ψδ(X) > 1 +� +. +The lemma follows with the choice ρ = 1/(1 − δ). +With the help of Proposition 4.1 we now establish the main inequality of this +section. +Proposition 4.2. Suppose that we have Assumption 2.1 and let η ∈ C∞ +0 (B), +0 ≤ η ≤ 1, η ≡ 1 on Br(x0), |∇η| ≤ c/r. +Moreover, we assume that +u ∈ L∞(D) ∩ W 1,∞ +loc (D) ∩ W 2,2 +loc (D) is a local minimizer of (3.1). +For i ∈ {1, . . . , n} we choose εi satisfying θi < εi < 1 − δi (recall (2.1)) and +let γi + εi =: βi, where we always suppose in the following that βi > −1/2. +Then we have for any sufficiently large real number k > 0 +� +B +fi(∂iu)Γ1+γi(|∂iu|)η2k dx ≤ c +� +j̸=i +� +B +f ′′ +j (∂ju)Γ1+βi(|∂iu|)η2k−2 dx . +(4.7) +Proof. We recall the starting inequality (4.1), +� +B +fi(∇u)Γ1+γi(|∂iu|)η2k dx ≤ c +� +1 + I1,i + I2,i +� +, +(4.8) +where we fix i ∈ {1, . . . , n}. We estimate for fixed βi as above +I1,i += +� +B +|∂i∂iu|f ′′ +1 +2 +i (∂iu)Γ +βi +2 (|∂iu|)(f ′′ +i )− 1 +2(∂iu)Γ− βi +2 (|∂iu|) +·Γγi(|∂iu|)fi(∂iu)η2k dx +≤ +c +� +B +f ′′ +i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx ++c +� +B +(f ′′ +i )−1(∂iu)Γγi−εi(|∂iu|)f 2 +i (∂iu)η2k dx . +(4.9) +11 + +The second integral on the right-hand side of (4.9) is handled with the help +of the right-hand side of (2.2) using in addition Lemma 4.2 (recalling εi > θi) +� +B +(f ′′ +i )−1(∂iu)Γγi−εi(|∂iu|)f 2 +i (∂iu)η2k dx +≤ +� +B +� +fi(∂iu)Γ1+γi−(εi−θi)(|∂iu|) +� +η2k dx +≤ +� +B +� +fi(∂iu)Γ1+γi(|∂iu|) +� 1 +ρη +2k +ρ η +2k +ρ∗ dx + c +≤ +ε +� +B +fi(∂iu)Γ1+γi(|∂iu|)η2k dx + c(ε, r) . +(4.10) +Absorbing terms it is shown up to now (using (4.8) - (4.10)) +� +B +fi(∂iu)Γ1+γi(|∂iu|)η2k dx +≤ +c +� +1 + +� +B +f ′′ +i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx + I2,i +� +. +(4.11) +Let us consider I2,i, i ∈ {1, . . . , n}. With βi > −1/2 as above we have +I2,i += +� +B +|∂i∂iu|f ′′ +i +1 +2(∂iu)Γ +βi +2 (|∂iu)(f ′′ +i )− 1 +2(∂iu)Γ− βi +2 (|∂iu|) +·Γ +1 +2+γi(|∂iu|)|f ′ +i|(∂iu)η2k dx +≤ +c +� +B +f ′′ +i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx ++c +� +B +(f ′′ +i )−1(∂iu)Γ1+γi−εi(|∂iu|)|f ′ +i|2(∂iu)η2k dx . +(4.12) +The first integral on the right-hand side of (4.12) already occurs in (4.11) +12 + +and the second one is handled with (2.3) and Lemma 4.2 (recalling εi > θi) +� +B +(f ′′ +i )−1(∂iu)Γ1+γi−εi(|∂iu|)|f ′ +i|2(∂iu)η2k dx +≤ +� +B +fi(∂iu)Γ1+γi−(εi−θi)(|∂iu|)η2k dx +≤ +� +B +� +fi(∂iu)Γ1+γi(|∂iu|) +� 1 +ρη +2k +ρ η +2k +ρ∗ dx + c +≤ +ε +� +B +fi(∂iu)Γ1+γi(|∂iu|)η2k dx + c(ε, r) +(4.13) +and once more the integral on the right-hand side is absorbed. +To sum up, (4.11) implies with the help of (4.12) and (4.13) for i = 1, . . . ,n +� +B +fi(∂iu)Γ1+γi(|∂iu|)η2k dx +≤ +c +� +1 + +� +B +f ′′ +i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx +� +. +(4.14) +Discussing the right-hand side of (4.14) we apply Lemma 4.1, where we let +f(Z) = �n +j=1 fj(Zj) and fix i ∈ {1, . . . , n}: +� +B +f ′′ +i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx +≤ +c +� +B +D2f(∇u) +� +∂i∇u, ∂i∇u +� +Γβi(|∂iu|)η2k dx +≤ +c +� +B +D2f(∇u) +� +∇η, ∇η)Γ1+βi(|∂iu|)η2k−2 dx +≤ +c(r) +n +� +j=1 +� +B +f ′′ +j (∂ju)Γ1+βi(|∂iu|)η2k−2 dx . +(4.15) +For j = i on the right-hand side of (4.15) we now apply the left-hand side of +13 + +(2.2) and again Lemma 4.2 with the result (recall δi + εi < 1) +� +B +f ′′ +i (∂iu)Γ1+βi(|∂iu|)η2k−2 dx +≤ +� +B +fi(∂iu)Γγi+δi+εi(|∂iu|)η2k−2 dx +≤ +� +B +� +fi(∂iu)Γ1+γi(|∂iu|)) +� 1 +ρη +2k +ρ η +2k +ρ∗ −2 dx + c +≤ +ε +� +B +fi(∂iu)Γ1+γi(|∂iu|)η2k dx + c(ε, r) . +(4.16) +Note that the integral on the right-hand side of (4.16) can be absorbed in +the left-hand side of (4.14). This proves Proposition 4.2. +5 +Iteration +We start with an elementary proposition recalling and relating the relevant +parameters of the problem. +Proposition 5.1. With q± +i , qi, qi, δi, θi, βi, γi, εi, i = 1, . . . , n as above +we further let ϑi =: 1 − δi and +ω± +i := q± +i +2 + γi , +i ∈ {1, . . . , n} . +We fix τ ≥ 0, i, j ∈ {1, . . . , n} and choose γi such that (M > 0 denoting an +arbitrary fixed number) +1 + γi < + + + + + +qiϑi +2 +2 + τ +qj − 2ϑi +− εiϑi +τ + qj/ϑi +qj − 2ϑi +if +qj > 2ϑi +M +if +qj ≤ 2ϑi + + + + + +. +(5.1) +This yields (for any combination of q± +j and q± +i ) +q± +j +1 + βi +ω± +i − βi +< 2ϑi +1 + q± +i +2 + γi +ω± +i − βi ++ τϑi , +(5.2) +Proof. In the case qj > 2ϑi we note that +1 + γi < +qiϑi +2 +2 + τ +qj − 2ϑi +− εiϑi +τ + qj/ϑi +qj − 2ϑi +, +14 + +which is equivalent to +(1 + γi) +� +qj − 2ϑi +� +< qiϑi + τ +� +qiϑi +2 +− εiϑi +� +− εiqj . +Writing this in the form +qj(1 + βi) < 2ϑi +� +1 + γi + +qi +2 +� ++ τϑi +� +qi +2 − εi +� +and recalling that we have by definition ω± +i − βi = (q± +i /2) − εi we obtain as +an equivalent inequality +qj +1 + βi +ω± +i − βi +< 2ϑi +1 + +qi +2 + γi +ω± +i − βi ++ τϑi +qi − 2εi +q± +i − 2εi +. +Up to now no relation between q+ +i and q− +i was needed due to our particular +Ansatz depending on t instead of |t|. +To complete the proof of Theorem 2.1 it remains to handle the mixed terms +on the right-hand side of (4.7). Here, of course, it is no longer possible to ar- +gue with the structure conditions for fixed i, i.e. to argue with q± +i separated +from each other in disjoint regions. +Throughout the rest of this section we suppose that the assumptions of The- +orem 2.1 are satisfied. +Consider a set U ⊂ Ω and a C1-function v: Ω → R. We let for any i ∈ +{1, . . . , n} +U ∩ [∂iv ≥ 0] =: U+ +i [v] =: U+ +i , +U ∩ [∂iv < 0] =: U− +i [v] =: U− +i , +in particular u can be written as the disjoint union +U = U+ +i ∪ U− +i . +and for every 1 ≤ i ≤ n. +Using this notation, recalling Proposition 4.2 and the left-hand side of (2.2) +15 + +we have for every 1 ≤ i ≤ n +� +B +fi(∂iu)Γ1+γi(|∂iu|)η2k dx +≤ +c +� +j̸=i +� +B +f ′′ +j (∂ju)Γ1+βi(|∂iu|)η2k−2 dx +≤ +c +� +j̸=i +� +B +fj(∂ju)Γδi−1(|∂ju|)Γ1+βi(|∂iu|)η2k−2 dx +(5.3) +Fix i ∈ {1, . . . , n}. For any j ∈ {1, . . . , n} we let +κ± +i = 1 + ω± +i +1 + βi +, +ˆκ± +i = 1 + ω± +i +ω± +i − βi +. +This gives for fixed 1 ≤ i ≤ n and for ε > 0 sufficiently small (note that the +ball B is divided into two parts w.r.t. the function ∂iu) +� +j̸=i +� +B +fj(∂ju)Γδi−1(|∂ju|)Γ1+βi(|∂iu|)η2k−2 dx +≤ c +� +j̸=i +� +± +� +Bi,± +� +1 + fj(∂ju) +� +Γδi−1(|∂ju|)Γ1+βi(|∂iu|)η2k−2 dx +≤ +� +j̸=i +� +± +� +ε +� +Bi,± +Γ(|∂iu|)1+ω± +i η2k dx ++c(ε) +� +Bi,± +� +1 + fj(∂ju) +� +1+ω± +i +ω± +i −βi Γ +(δi−1) +1+ω± +i +ω± +i −βi (|∂ju|) dx +� +. +(5.4) +By (2.4) we have on Bi,± for |∂iu| sufficiently large Γ(|∂iu|)q± +i /2 ≤ cfi(∂iu), +hence by the definition of ω± +i +ε +� +j̸=i +� +± +� +Bi,± +Γ(|∂iu|)1+ω± +i η2k dx +≤ (n − 1)ε +� +B +fi(∂iu)Γ1+γi(|∂iu|)η2k dx + c +and, as usual, the integral on the right-hand side can be absorbed in (5.3). +16 + +We will finally show with the help of an iteration procedure that for every +1 ≤ i ≤ n +� +j̸=i +� +± +� +Bi,± +� +1 + fj(∂ju) +� +1+ω± +i +ω± +i −βi Γ +(δi−1) +1+ω± +i +ω± +i −βi (|∂ju|) dx ≤ c , +(5.5) +which completes the proof of Theorem 1.1. +If fact, let us suppose that (5.1) is true with the choice ϑi = 1 − δi. Then we +may apply Proposition 5.1 and (5.2) implies in the case q > 2ϑi +Γ +(δi−1) +1+ω± +i +ω± +i −βi ++(δi−1) τ +2 (|∂ju|) ≤ c +� +1 + fj(∂ju) +�− +1+βi +ω± +i −βi . +Thus we obtain +� +1 + fj(∂ju) +� +1+ω± +i +ω± +i −βi Γ +(δi−1) +1+ω± +i +ω± +i −βi (|∂ju|) ≤ c +� +1 + fj(∂ju) +� +Γ(1−δi) τ +2 (|∂ju|) . (5.6) +In the case qj ≤ 2ϑi we have +−(1 + fj(∂ju) ≤ cΓ1−δi(|∂ju|) , +hence +� +1 + fj(∂ju) +� +1+ω± +i +ω± +i −βi Γ +(δi−1) +1+ω± +i +ω± +i −βi (|∂ju|) ≤ c +and (5.6) holds as well. +We note that (5.6) is formulated uniformly w.r.t. the index j and the symbol +± is just related to ∂iu. +Inequality (5.6) is the main tool for the following iteration leading to the +claim (5.5). +i = 1. +Choosing γi > −1/2 + θi sufficiently close to −1/2 + θi, (5.1) is valid with +the choice τ = 0 if we have +qj < +2qi(1 − δi) +1 + 2θi ++ 2(1 − δi) +for all +2 ≤ j ≤ n , +(5.7) +and (5.7) is just assumption (2.5) for i = 1. +17 + +From (5.1) we deduce (5.6) for i = 1 and (5.5) follows from (5.6) for i = 1 +and for all 2 ≤ j ≤ n with the choice τ = 0 +� +j̸=i +� +± +� +Bi,± +� +1 + fj(∂ju) +� +1+ω± +1 +ω± +1 −β1 Γ +(δi−1) +1+ω± +1 +ω± +1 −β1 (|∂ju|) +≤ c +� +B +� +1 + fj(∂ju) +� +dx ≤ c . +(5.8) +Returning to (5.3) and (5.4) we insert (5.8) and on account of 1 + γi > 1/2 +we have +� +B +f1(∂1u)Γ +1 +2(|∂1u|)η2k dx ≤ c . +(5.9) +Remark 5.1. In [5] we have δi = θi = 0, i = 1, 2, and w.l.o.g. the case +p = q2 ≤ q1 = q is considered. Moreover, qj = qj = qj, j = 1, 2. In this case +we trivially have (5.7). +1 < i ≤ n. +Suppose that we have in addition to (5.7) (again compare (2.5)) +qj < qj < +2qi(1 − δi) +1 + 2θi ++ 2(1 − δi) +for +i + 1 ≤ j ≤ n . +(5.10) +With the same argument leading to (5.8) we have for all i + 1 ≤ j ≤ n +� +j>i +� +± +� +Bi,± +� +1 + fj(∂ju) +� +1+ω± +i +ω± +i −βi Γ +(δi−1) +1+ω± +i +ω± +i −βi (|∂ju|) ≤ c . +(5.11) +Moreover, we suppose that by iteration we have (5.9) for 1 ≤ j < i, i.e. +� +B +fj(∂ju)Γ +1 +2(|∂ju|)η2k dx ≤ c , +1 ≤ j < i . +(5.12) +Then we return to (5.1) with the choice τ = (1 − δi)−1. For γi > −1/2 + θi +and γi sufficiently close to −1/2 + θi we are lead to the condition +qj < +2 +1 + 2θi +� +qi +2 (1 − δi) +� +2 + +1 +1 − δi +� +− θi(1 + qj) +� ++ 2(1 − δi) , +(5.13) +1 ≤ j < i, and (5.13) is just the assumption (2.6). +18 + +With (5.1) we again have (5.6), now with τ = (1 − δi)−1, hence +� +j 3 +2 +we may choose γ1 = 1/2 without imposing any condition relating p and q, +thus the results presented in [5] immediately follow as a corollary. +References +[1] Giaquinta, M. +Growth conditions and regularity, a counterexample. +Manuscripta Math., 59(2):245–248, 1987. +[2] Marcellini, P. Regularity of minimizers of integrals of the calculus of vari- +ations with nonstandard growth conditions. Arch. Rational Mech. Anal., +3:267–284, 1989. +[3] Marcellini, P. Everywhere regularity for a class of elliptic systems with- +out growth conditions. +Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), +23(1):1–25, 1996. +19 + +[4] Beck, L.; Mingione, G. Lipschitz bounds and nonuniform ellipticity. +Comm. Pure Appl. Math., LXXIII:944–1034, 2020. +[5] Bildhauer, M.; Fuchs, M.; Zhong, X. A regularity theory for scalar local +minimizers of splitting-type variational integrals. Ann. Sc. Norm. Super. +Pisa Cl. Sci. (5), 6(3):385–404, 2007. +[6] Breit, D. +A note on splitting-type variational problems with sub- +quadratic growth. Arch. Math. (Basel), 94(5):467–476, 2010. +[7] Dacorogna, B. Introduction to the calculus of variations. Imperial Col- +lege Press, London, 3rd edition, 2015. +[8] Bildhauer, M. Convex variational problems. Linear, nearly linear and +anisotropic growth conditions, volume 1818 of Lecture Notes in Mathe- +matics. Springer, Berlin, 2003. +[9] Massari, U., Miranda, M. Minimal surfaces of codimension one, vol- +ume 91 of North-Holland Mathematics Studies. North-Holland Publish- +ing Co., Amsterdam, 1984. +[10] Bildhauer, M.; Fuchs, M. Splitting type variational problems with linear +growth conditions. +J. Math. Sci. (N.Y.), Problems in mathematical +analysis. No. 105, 250(2):45–58, 2020. +[11] Bildhauer, M.; Fuchs, M. On the global regularity for minimizers of +variational integrals: splitting-type problems in 2D and extensions to +the general anisotropic setting. J. Elliptic Parabol. Equ., 8(2):853–884, +2022. +[12] Choe, H.J. Interior behaviour of minimizers for certain functionals with +nonstandard growth. Nonlinear Anal., 19(10):933–945, 1992. +20 + diff --git a/RNAzT4oBgHgl3EQfI_uT/content/tmp_files/load_file.txt b/RNAzT4oBgHgl3EQfI_uT/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..933b223baafd22a5033250ad944a201b60d00af2 --- /dev/null +++ b/RNAzT4oBgHgl3EQfI_uT/content/tmp_files/load_file.txt @@ -0,0 +1,610 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf,len=609 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='01072v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='AP] 3 Jan 2023 Splitting-type variational problems with asymmetrical growth conditions Michael Bildhauer & Martin Fuchs Abstract 1 Splitting-type variational problems � Ω n � i=1 fi(∂iw) dx → min with superlinear growth conditions are studied by assuming hi(t) ≤ f ′′ i (t) ≤ Hi(t) with suitable functions hi, Hi: R → R+, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n, measuring the growth and ellipticity of the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Here, as the main feature, a symmetric behaviour like hi(t) ≈ hi(−t) and Hi(t) ≈ Hi(−t) for large |t| is not supposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Assuming quite weak hypotheses as above, we establish higher integrability of |∇u| for local minimizers u ∈ L∞(Ω) by using a Caccioppoli-type inequality with some power weights of negative ex- ponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 1 Introduction Suppose that Ω ⊂ Rn is a bounded Lipschitz domain and consider the vari- ational integral J[w] := � Ω f � ∇w � dx (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) of splitting-type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' f : Rn → R , f(Z) = n � i=1 fi(Zi) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) 1AMS subject classification: 49N60, 49N99, 35J45 Keywords: splitting-type variational problems, asymmetrical growth conditions, non- uniform ellipticity 1 with strictly convex functions fi: R → R of class C2(R), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n, sat- isfying in addition some suitable superlinear growth and ellipticity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) serves as a prototype for non-uniformly elliptic varia- tional problems and it is well known that the ratio of the highest and the lowest eigenvalue of D2f is the crucial quantity for proving the regularity of solutions (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=', [1], [2], [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' The reader will find an extensive overview including different settings of non-uniformly elliptic variational problems in the recent paper [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Without going into further details we refer to the series of references given in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3 of [4], the authors consider general growth conditions which, roughly speaking, means that the energy density f is controlled in the sense of g(|Z|)|ξ|2 ≤ D2f(Z)(ξ, ξ) , |D2f(Z)| ≤ G(|Z|) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) with suitable functions g(t), G(t): R+ 0 → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Then, under appropriate as- sumptions on g, G, a general approach to regularity theory is given in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Our note is motivated by the observation, that in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) there is no obvious reason to assume some kind of symmetry for the functions fi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' in general we have fi(t) ̸= fi(−t) and, as one model case, we just consider (q± i ≥ 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n) fi(t) ≈ |t|q− i if t ≪ −1 , fi(t) ≈ |t|q+ i if t ≫ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) Then, both for t ≪ 1 and for t ≫ 1, the functions fi just behave like a uniform power of |t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Nevertheless, the power q− i enters the left-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) and q+ i is needed on the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' This motivates to study the model case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) and to establish regularity results for solutions under the weaker assumption hi(t) ≤ f ′′ i (t) ≤ Hi(t) t ∈ R , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5) with suitable functions hi, Hi: R → R+, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' There is another quite subtle difficulty in studying regularity of solutions to splitting-type variational problems: in [5] the authors consider variational integrals of the form (1 ≤ k < n) I[w, Ω] = � Ω � f(∂1w, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , ∂kw) + g(∂k+1w, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , ∂nw) � dx , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) where f and g are of p and q-growth, respectively (p, q > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Then the regu- larity of bounded solutions follows in the sense of [5], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1, without 2 any further condition relating p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' The proof argues step by step and works since the energy density splits into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' If, as supposed in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2), the energy density splits in more than two components, then one has to be more careful dealing with the exponents and some more restrictive (but still quite weak) assumptions have to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In this sense Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3 of [5] might be a little bit misleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We note that a splitting structure into two components as supposed in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) is also assumed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=', in [6] and related pa- pers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In the following we consider the variational integral (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) defined on the energy class Ef(Ω) := � w ∈ W 1,1(Ω) : � Ω f(∇w) dx < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We are interested in local minimizers u: Ω → R of class Ef(Ω), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' it holds that � Ω f(∇u) dx ≤ � Ω f(∇w) dx (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7) for all w ∈ Ef(Ω) such that spt(u − w) ⋐ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We will always denote by q+ i > 1, q− i > 1, 1 ≤ i ≤ n, real exponents and we let for fixed 1 ≤ i ≤ n qi := min{q± i } , qi := max{q± i } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='8) Moreover, we let Γ : [0, ∞) → R , Γ(t) = 1 + t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Recalling the idea sketched in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5) we denote by hi and Hi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n, functions R → R+ such that with positive constants ai, ai aiΓ q− i −2 2 (|t|) if t < −1 aiΓ q+ i −2 2 (|t|) if t > 1 \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe ≤ hi(t) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9) and Hi(t) ≤ \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 aiΓ q− i −2 2 (|t|) if t < −1 aiΓ q+ i −2 2 (|t|) if t > 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='10) As a general assumption we consider functions fi: R → [0, ∞) of class C2(R), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n, such that for all t ∈ R hi(t) ≤ f ′′ i (t) ≤ Hi(t) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='11) 3 and note that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='11) immediately implies for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} with constants bi > 0 |f ′ i(t)| ≤ bi \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Γ q− i −1 2 (|t|) if t < −1 Γ q+ i −1 2 (|t|) if t > 1 \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='12) Moreover we obtain for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n with constants ci, ci > 0 ci \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Γ q− i 2 (|t|) if t < −1 Γ q+ i 2 (|t|) if t > 1 \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe ≤ fi(t) ≤ ci \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Γ q− i 2 (|t|);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' if t < −1 Γ q+ i 2 (|t|) if t > 1 \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='13) With this notation our main result reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Suppose that for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n the functions fi: R → [0, ∞) are of class C2(R) and satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='11) with hi, Hi given in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' With the notation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='8) we assume in addition that we have for every fixed 1 ≤ i ≤ n qj < 2qi + 2 for all i < j ≤ n , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='14) qj < 3qi + 2 for all 1 ≤ j < i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='15) If u ∈ L∞(Ω) ∩ Ef(Ω) denotes a local minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' of J[w] = � Ω � n � i=1 fi(∂iw) � dx , then there exists a real number δ > −1/2 such that for every 1 ≤ i ≤ n � B fi(∂iu)Γ1+δ(|∂iu|)η2k dx ≤ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='16) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' As outlined in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 and Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2 below, we recover the results of [5] in the sense that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='14) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='15) are superfluous in the case n = 2 (or related situations) and q+ 1 = q− 1 , q+ 2 = q− 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 describes the typical situation we have in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' The proof however is not limited to this particular case which leads to the generalized version stated in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In Section 3 we shortly sketch a regularization procedure via Hilbert-Haar solutions while Section 4 presents the main inequalities for the iteration pro- cedure of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' This completes the proof ot Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 and hence Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 4 2 Precise assumptions on f The suitable larger class of admissible energy densities is given by the fol- lowing assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' The energy density f, f : Rn → R , f(Z) = n � i=1 fi(Zi) , introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) is supposed to satisfy the following hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' i) The function fi: R → [0, ∞), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n, is of class C2(R) and for all t ∈ R we have f ′′ i (t) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' For 1 ≤ i ≤ n we suppose superlinear growth in the sense of lim t→±∞ |f ′ i(t)| = ∞ and at most of polynomial growth in the sense that for some s > 0 we have for |t| sufficiently large f(t) ≤ c|t|s with a finte constant c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' ii) For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} with exponents δi ≥ 0, θi ≥ 0 satisfying θi < 1 − δi (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) we suppose that for all |t| sufficiently large c1Γ1−δi(|t|)f ′′ i (t) ≤ fi(t) ≤ c2f ′′ i (t)Γ1+θi(|t|) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) |f ′ i(t)|2 ≤ c3f ′′ i (t)fi(t)Γθi(|t|) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) where c1, c2 and c3 denote positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' iii) We let Γ q± i 2 (t) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Γ q− i 2 (|t|) if t < 0 Γ q+ i 2 (|t|) if t ≥ 0 \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' and suppose that fi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' ,n, satisfies with positive constants c4, c5 and for |t| sufficiently large c4Γ q± i 2 (|t|) ≤ fi(t) ≤ c5Γ q± i 2 (|t|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) 5 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' i) If fi is a power growth function like, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=', fi(t) = (1 + t2)pi/2, pi > 1 fixed, then we have cΓ(|t|)f ′′ i (t) ≤ fi(t) ≤ cΓ(|t|)f ′′ i (t) , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' The same is true for our asymmetric model case given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9) – (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' ii) By convexity it is well known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=', [7], exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 53) that the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) and the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='13) imply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' iii) The condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) formally corresponds with the condition q < p + 2 in the standard (p, q)-case (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=', [8], Chapter 5, and the references quoted therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' iv) Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1, iii) is assumed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In fact, on account of f ′′ i > 0 we know that for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} the function f ′ i is an increasing function, and by Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1, i), we let s+ := inf s lim t→∞ f ′(t) ts < ∞ , s− := inf s lim t→−∞ |f ′(t)| |t|s < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Then for arbitrary small ε > 0 we have the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) with exponent s± + 1 + ε and the left-hand side with exponent s± + 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Going through the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 we may suppose (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Suppose that we have Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' With the above nota- tion we assume in addition that we have for every fixed 1 ≤ i ≤ n qj < 2qi(1 − δi) 1 + 2θi + 2(1 − δi) for all i < j ≤ n , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5) qj < 2 1 + 2θi � qi 2 (1 − δi) � 2 + 1 1 − δi � − θi(1 + qj) � + 2(1 − δi) for all 1 ≤ j < i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) If u ∈ L∞(Ω) ∩ Ef(Ω) denotes a local minimizer of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' of J[w] = � Ω � n � i=1 fi(∂iw) � dx , then there exists a real number δ > −1/2 such that for every 1 ≤ i ≤ n � B fi(∂iu)Γ1+δ(|∂iu|)η2k dx ≤ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In particular we note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) reduce to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='14), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='15) for δi, θi sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 6 3 Some remarks on regularization We have to start with a regularization procedure such that the expressions given below are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We follow Section 2 of [5] and fix a ball D ⋐ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' If u denotes the local minimizer under in the sense of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7) and if ε > 0 is sufficiently small, we consider the mollification (u)ε of u w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' the radius ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We consider the Dirichlet-problem � D n � i=1 fi(∂iw) dx → min among all Lipschitz mappings D → R with boundary data (u)ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' According to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=', [9], there exits a unique (Hilbert-Haar) solution uε to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Exactly as outlined in [5] Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2 we obtain: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Let q := min1≤i≤n qi i) We have as ε → 0 uε ⇁ u in W 1,q(D) , � D n � i=1 fi(∂iuε) dx → � D n � i=1 fi(∂iu) dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' ii) There is a finite constant c > 0 not depending on ε such that ∥uε∥L∞(D) ≤ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' iii) For any α < 1 we have uε ∈ C1,α(D) ∩ W 2,2 loc (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We then argue as follows: consider a local minimizer u of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) and the approximating sequence {uε} minimizing the functional J[w, D] := � D n � i=1 fi(∂iwi) dx (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' the data (u)ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In particular we have a sequence of local J[w, D]- minimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We apply the a priori results of the next section to uε and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 passing to the limit ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 4 General inequalities The main result of this section is Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2 which is not depending on the particular structure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 7 We will rely on the following variant of Caccioppoli’s inequality which was first introduced in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We also refer to Section 6 of [11] on Caccioppoli-type inequalities involving powers with negative exponents, in particular we refer to Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Fix l ∈ N and suppose that η ∈ C∞ 0 (D), 0 ≤ η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' If we consider a local minimizer u ∈ W 1,∞ loc (D) ∩ W 2,2 loc (D) of the variational functional I[w] = � D g(∇w) dx with energy density g: Rn → R of class C2 satisfying D2g(Z)(Y, Y ) > 0 for all Y , Z ∈ Rn, then for any fixed i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} we have � D D2g(∇u) � ∇∂iu, ∇∂iu � η2lΓβ(|∂iu|) dx ≤ c � D D2g(∇u)(∇η, ∇η)η2l−2Γ1+β(|∂iu|) dx for any β > −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' To the end of our note we always consider a fixed ball B = B2r(x0) ⋐ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' With this notation we have the following auxiliary proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Suppose that we have i) of Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 and let η ∈ C∞ 0 (B), 0 ≤ η ≤ 1, η ≡ 1 on Br(x0), |∇η| ≤ c/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Moreover, we assume that u ∈ L∞(D) ∩ W 1,∞ loc (D) ∩ W 2,2 loc (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Then we have for fixed γ ∈ R, for all k > 0 sufficiently large and for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n the starting inequalities (no summation w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' i) � B fi(∂iu)Γ1+γ(|∂iu|)η2k dx ≤ c � 1 + � B |∂i∂iu|Γγ(|∂iu|)fi(∂iu)η2k dx + � B |∂i∂iu| |f ′ i|(∂iu) Γ 1 2 +γ(|∂iu|)η2k dx � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' i) The idea of the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 is based on an integration by parts using the boundedness of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' An Ansatz of this kind was already made by Choe [12], where all relevant quantities are depending on |∇u|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Here the main new feature is to work with the energy density f which is not depending on the modulus of ∇u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 8 ii) We note that for the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 no minimizing property of u is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' With i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} fixed we obtain using an integration by parts � B fi(∂iu)Γ1+γ(|∂iu|)η2k dx = � B |∂iu|2fi(∂iu)Γγ(|∂iu|)η2k dx + � B fi(∂iu)Γγ(|∂iu|)η2k dx = − � B u∂i � ∂iufi(∂iu)Γγ(|∂iu|)η2k� dx + � B fi(∂iu)Γγ(|∂iu|)η2k dx ≤ c � B |∂i∂iu|Γγ(|∂iu|)fi(∂iu)η2k dx +c � B |∂i∂iu| |∂iu| |f ′ i|(∂iu) Γγ(|∂iu|)η2k dx +c � B |∂iu|fi(∂iu)Γγ(|∂iu|)η2k−1|∂iη| dx + � B fi(∂iu)Γγ(|∂iu|)η2k dx = I1,i + I2,i + I3,i + I4,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) In (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) we discuss I3,i: for ε > 0 sufficiently small we estimate I3,i ≤ � B |∂iu|f 1 2 i (∂iu)Γ γ 2 (|∂iu|)ηkf 1 2 i (∂iu)Γ γ 2 (|∂iu|)ηk−1|∇η| dx ≤ ε � B |∂iu|2fi(∂iu)Γγ(|∂iu|)η2kdx +c(ε, r) � B fi(∂iu)Γγ(|∂iu|)η2k−2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) The first integral on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) is absorbed in the left-hand 9 side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' � B fi(∂iu)Γ1+γ(|∂iu|)η2k dx ≤ I1,i + I2,i + c(ε, r) � B fi(∂iu)Γγ(|∂iu|)η2k−2 dx + � B fi(∂iu)Γγ(|∂iu|)η2k dx ≤ I1,i + I2,i + c(ε, r) � B fi(∂iu)Γγ(|∂iu|)η2k−2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) Discussing the remaining integral we recall that the function fi(t)Γ1+γ(|t|) is at most of polynomial growth, hence we may apply the auxiliary Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2 below to the functions ϕ(t) = fi(t)Γγ(|t|) and ψ(t) := fi(t)Γ1+γ(|t|) with the result that for some ρ > 0 and for all t ∈ R fi(t)Γγ(|t|) ≤ c � fi(t)Γ1+γ(|t|) � 1 ρ + c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5) With (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5) we estimate for ˜ε > 0 sufficiently small and for k > ρ∗ = ρ/(ρ−1) c(ε, r) � B fi(∂iu)Γγ(|∂iu|)η2k−2 dx ≤ c(ε, r) � B � fi(∂iu)Γ1+γ(|∂iu|) � 1 ρη 2k ρ η 2k ρ∗ −2 dx + c ≤ ˜ε � B fi(∂iu)Γ1+γ(|∂iu|)η2k dx + c(˜ε, ε, r) � B η2(k−ρ∗) dx + c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) The inequalities (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) complete the proof of the proposition by ab- sorbing the first integral on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) in the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' It remains to give an elementary proof of the following auxiliary Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' For m ∈ N we consider functions ϕ, ψ: Rm → [0, ∞) such that ψ(X) ≤ cΓτ(|X|) for some τ > 0 and for all X ∈ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Suppose that we have for some ε > 0 and for all X ∈ Rn ϕ(X) ≤ cΓ−ε(|X|)ψ(X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Then there exists a real number ρ > 1 and a constant C > 0 such that ϕ(X) ≤ � ψ(X) � 1 ρ + C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Let δ := ε/τ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' for all X ∈ Rm 1 + ψδ ≤ 1 + Γε ≤ 2Γε , hence we have by assumption ϕ(X) ≤ c � 1 + ψδ(X) �−1ψ(X) ≤ � c if ψδ(X) ≤ 1 cψ1−δ(X) if ψδ(X) > 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' The lemma follows with the choice ρ = 1/(1 − δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' With the help of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 we now establish the main inequality of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Suppose that we have Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 and let η ∈ C∞ 0 (B), 0 ≤ η ≤ 1, η ≡ 1 on Br(x0), |∇η| ≤ c/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Moreover, we assume that u ∈ L∞(D) ∩ W 1,∞ loc (D) ∩ W 2,2 loc (D) is a local minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} we choose εi satisfying θi < εi < 1 − δi (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1)) and let γi + εi =: βi, where we always suppose in the following that βi > −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Then we have for any sufficiently large real number k > 0 � B fi(∂iu)Γ1+γi(|∂iu|)η2k dx ≤ c � j̸=i � B f ′′ j (∂ju)Γ1+βi(|∂iu|)η2k−2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We recall the starting inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1), � B fi(∇u)Γ1+γi(|∂iu|)η2k dx ≤ c � 1 + I1,i + I2,i � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='8) where we fix i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We estimate for fixed βi as above I1,i = � B |∂i∂iu|f ′′ 1 2 i (∂iu)Γ βi 2 (|∂iu|)(f ′′ i )− 1 2(∂iu)Γ− βi 2 (|∂iu|) Γγi(|∂iu|)fi(∂iu)η2k dx ≤ c � B f ′′ i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx +c � B (f ′′ i )−1(∂iu)Γγi−εi(|∂iu|)f 2 i (∂iu)η2k dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9) 11 The second integral on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9) is handled with the help of the right-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) using in addition Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2 (recalling εi > θi) � B (f ′′ i )−1(∂iu)Γγi−εi(|∂iu|)f 2 i (∂iu)η2k dx ≤ � B � fi(∂iu)Γ1+γi−(εi−θi)(|∂iu|) � η2k dx ≤ � B � fi(∂iu)Γ1+γi(|∂iu|) � 1 ρη 2k ρ η 2k ρ∗ dx + c ≤ ε � B fi(∂iu)Γ1+γi(|∂iu|)η2k dx + c(ε, r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='10) Absorbing terms it is shown up to now (using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='8) - (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='10)) � B fi(∂iu)Γ1+γi(|∂iu|)η2k dx ≤ c � 1 + � B f ′′ i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx + I2,i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='11) Let us consider I2,i, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' With βi > −1/2 as above we have I2,i = � B |∂i∂iu|f ′′ i 1 2(∂iu)Γ βi 2 (|∂iu)(f ′′ i )− 1 2(∂iu)Γ− βi 2 (|∂iu|) Γ 1 2+γi(|∂iu|)|f ′ i|(∂iu)η2k dx ≤ c � B f ′′ i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx +c � B (f ′′ i )−1(∂iu)Γ1+γi−εi(|∂iu|)|f ′ i|2(∂iu)η2k dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='12) The first integral on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='12) already occurs in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='11) 12 and the second one is handled with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2 (recalling εi > θi) � B (f ′′ i )−1(∂iu)Γ1+γi−εi(|∂iu|)|f ′ i|2(∂iu)η2k dx ≤ � B fi(∂iu)Γ1+γi−(εi−θi)(|∂iu|)η2k dx ≤ � B � fi(∂iu)Γ1+γi(|∂iu|) � 1 ρη 2k ρ η 2k ρ∗ dx + c ≤ ε � B fi(∂iu)Γ1+γi(|∂iu|)η2k dx + c(ε, r) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='13) and once more the integral on the right-hand side is absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' To sum up, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='11) implies with the help of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='13) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' ,n � B fi(∂iu)Γ1+γi(|∂iu|)η2k dx ≤ c � 1 + � B f ′′ i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='14) Discussing the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='14) we apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1, where we let f(Z) = �n j=1 fj(Zj) and fix i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n}: � B f ′′ i (∂iu)|∂i∂iu|2Γβi(|∂iu|)η2k dx ≤ c � B D2f(∇u) � ∂i∇u, ∂i∇u � Γβi(|∂iu|)η2k dx ≤ c � B D2f(∇u) � ∇η, ∇η)Γ1+βi(|∂iu|)η2k−2 dx ≤ c(r) n � j=1 � B f ′′ j (∂ju)Γ1+βi(|∂iu|)η2k−2 dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='15) For j = i on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='15) we now apply the left-hand side of 13 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) and again Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2 with the result (recall δi + εi < 1) � B f ′′ i (∂iu)Γ1+βi(|∂iu|)η2k−2 dx ≤ � B fi(∂iu)Γγi+δi+εi(|∂iu|)η2k−2 dx ≤ � B � fi(∂iu)Γ1+γi(|∂iu|)) � 1 ρη 2k ρ η 2k ρ∗ −2 dx + c ≤ ε � B fi(∂iu)Γ1+γi(|∂iu|)η2k dx + c(ε, r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='16) Note that the integral on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='16) can be absorbed in the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' This proves Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 5 Iteration We start with an elementary proposition recalling and relating the relevant parameters of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' With q± i , qi, qi, δi, θi, βi, γi, εi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n as above we further let ϑi =: 1 − δi and ω± i := q± i 2 + γi , i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We fix τ ≥ 0, i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} and choose γi such that (M > 0 denoting an arbitrary fixed number) 1 + γi < \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 qiϑi 2 2 + τ qj − 2ϑi − εiϑi τ + qj/ϑi qj − 2ϑi if qj > 2ϑi M if qj ≤ 2ϑi \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) This yields (for any combination of q± j and q± i ) q± j 1 + βi ω± i − βi < 2ϑi 1 + q± i 2 + γi ω± i − βi + τϑi , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In the case qj > 2ϑi we note that 1 + γi < qiϑi 2 2 + τ qj − 2ϑi − εiϑi τ + qj/ϑi qj − 2ϑi , 14 which is equivalent to (1 + γi) � qj − 2ϑi � < qiϑi + τ � qiϑi 2 − εiϑi � − εiqj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Writing this in the form qj(1 + βi) < 2ϑi � 1 + γi + qi 2 � + τϑi � qi 2 − εi � and recalling that we have by definition ω± i − βi = (q± i /2) − εi we obtain as an equivalent inequality qj 1 + βi ω± i − βi < 2ϑi 1 + qi 2 + γi ω± i − βi + τϑi qi − 2εi q± i − 2εi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Up to now no relation between q+ i and q− i was needed due to our particular Ansatz depending on t instead of |t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' To complete the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 it remains to handle the mixed terms on the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Here, of course, it is no longer possible to ar- gue with the structure conditions for fixed i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' to argue with q± i separated from each other in disjoint regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Throughout the rest of this section we suppose that the assumptions of The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Consider a set U ⊂ Ω and a C1-function v: Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We let for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} U ∩ [∂iv ≥ 0] =: U+ i [v] =: U+ i , U ∩ [∂iv < 0] =: U− i [v] =: U− i , in particular u can be written as the disjoint union U = U+ i ∪ U− i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' and for every 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Using this notation, recalling Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2 and the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) 15 we have for every 1 ≤ i ≤ n � B fi(∂iu)Γ1+γi(|∂iu|)η2k dx ≤ c � j̸=i � B f ′′ j (∂ju)Γ1+βi(|∂iu|)η2k−2 dx ≤ c � j̸=i � B fj(∂ju)Γδi−1(|∂ju|)Γ1+βi(|∂iu|)η2k−2 dx (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) Fix i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' For any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' , n} we let κ± i = 1 + ω± i 1 + βi , ˆκ± i = 1 + ω± i ω± i − βi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' This gives for fixed 1 ≤ i ≤ n and for ε > 0 sufficiently small (note that the ball B is divided into two parts w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' the function ∂iu) � j̸=i � B fj(∂ju)Γδi−1(|∂ju|)Γ1+βi(|∂iu|)η2k−2 dx ≤ c � j̸=i � ± � Bi,± � 1 + fj(∂ju) � Γδi−1(|∂ju|)Γ1+βi(|∂iu|)η2k−2 dx ≤ � j̸=i � ± � ε � Bi,± Γ(|∂iu|)1+ω± i η2k dx +c(ε) � Bi,± � 1 + fj(∂ju) � 1+ω± i ω± i −βi Γ (δi−1) 1+ω± i ω± i −βi (|∂ju|) dx � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) we have on Bi,± for |∂iu| sufficiently large Γ(|∂iu|)q± i /2 ≤ cfi(∂iu), hence by the definition of ω± i ε � j̸=i � ± � Bi,± Γ(|∂iu|)1+ω± i η2k dx ≤ (n − 1)ε � B fi(∂iu)Γ1+γi(|∂iu|)η2k dx + c and, as usual, the integral on the right-hand side can be absorbed in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 16 We will finally show with the help of an iteration procedure that for every 1 ≤ i ≤ n � j̸=i � ± � Bi,± � 1 + fj(∂ju) � 1+ω± i ω± i −βi Γ (δi−1) 1+ω± i ω± i −βi (|∂ju|) dx ≤ c , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5) which completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' If fact, let us suppose that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) is true with the choice ϑi = 1 − δi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Then we may apply Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='2) implies in the case q > 2ϑi Γ (δi−1) 1+ω± i ω± i −βi +(δi−1) τ 2 (|∂ju|) ≤ c � 1 + fj(∂ju) �− 1+βi ω± i −βi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Thus we obtain � 1 + fj(∂ju) � 1+ω± i ω± i −βi Γ (δi−1) 1+ω± i ω± i −βi (|∂ju|) ≤ c � 1 + fj(∂ju) � Γ(1−δi) τ 2 (|∂ju|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) In the case qj ≤ 2ϑi we have −(1 + fj(∂ju) ≤ cΓ1−δi(|∂ju|) , hence � 1 + fj(∂ju) � 1+ω± i ω± i −βi Γ (δi−1) 1+ω± i ω± i −βi (|∂ju|) ≤ c and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' We note that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) is formulated uniformly w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' the index j and the symbol ± is just related to ∂iu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) is the main tool for the following iteration leading to the claim (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Choosing γi > −1/2 + θi sufficiently close to −1/2 + θi, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) is valid with the choice τ = 0 if we have qj < 2qi(1 − δi) 1 + 2θi + 2(1 − δi) for all 2 ≤ j ≤ n , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7) is just assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5) for i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 17 From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) we deduce (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) for i = 1 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6) for i = 1 and for all 2 ≤ j ≤ n with the choice τ = 0 � j̸=i � ± � Bi,± � 1 + fj(∂ju) � 1+ω± 1 ω± 1 −β1 Γ (δi−1) 1+ω± 1 ω± 1 −β1 (|∂ju|) ≤ c � B � 1 + fj(∂ju) � dx ≤ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='8) Returning to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='4) we insert (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='8) and on account of 1 + γi > 1/2 we have � B f1(∂1u)Γ 1 2(|∂1u|)η2k dx ≤ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In [5] we have δi = θi = 0, i = 1, 2, and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' the case p = q2 ≤ q1 = q is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Moreover, qj = qj = qj, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' In this case we trivially have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 1 < i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' Suppose that we have in addition to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='7) (again compare (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='5)) qj < qj < 2qi(1 − δi) 1 + 2θi + 2(1 − δi) for i + 1 ≤ j ≤ n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='10) With the same argument leading to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='8) we have for all i + 1 ≤ j ≤ n � j>i � ± � Bi,± � 1 + fj(∂ju) � 1+ω± i ω± i −βi Γ (δi−1) 1+ω± i ω± i −βi (|∂ju|) ≤ c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='11) Moreover, we suppose that by iteration we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='9) for 1 ≤ j < i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' � B fj(∂ju)Γ 1 2(|∂ju|)η2k dx ≤ c , 1 ≤ j < i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='12) Then we return to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) with the choice τ = (1 − δi)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' For γi > −1/2 + θi and γi sufficiently close to −1/2 + θi we are lead to the condition qj < 2 1 + 2θi � qi 2 (1 − δi) � 2 + 1 1 − δi � − θi(1 + qj) � + 2(1 − δi) , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='13) 1 ≤ j < i, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='13) is just the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content=' 18 With (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='1) we again have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNAzT4oBgHgl3EQfI_uT/content/2301.01072v1.pdf'} +page_content='6), now with τ = (1 − δi)−1, hence � j 1 is the number of photons +emitted or absorbed [3]. +These processes also impart +large phonon momentum transfers which the form factor +in the electron-phonon matrix element suppresses. +The total rate equation is then: +˙F (ξ) +K+ = ˙F (ξ) +K+|ph,in − ˙F (ξ) +K+|ph,out + ˙F (ξ) +K+|r,in − ˙F (ξ) +K+|r,out. +(7) +At steady-state ˙F (ξ) +K+ = 0, and we find +F (ξ) +K+ = +RinF (ξ) +in + Γin +RinF (ξ) +in + Rout + Γin + Γout +. +(8) +Since Rin ∝ Ain, it decreases as a function of E and even- +tually shrinks to zero for E ≥ E∗ (see Fig. 3(b) for numer- +ical verification). We expect a similar E-dependence of +F (ξ) +K− and thus σxy, yet smeared by additional scattering +rates appearing in Eq. 8, as is demonstrated numerically +in Fig. 1(b). We define the visibility of the sharp feature +in the Hall conductivity in Eq. 9, below. +Scattering rates about K′ and K points, related by a +π/3 rotation, are equal, and the two contribute equally +to σxy. The resonance ring vicinity, SR (see Fig. 2(c)), +yields a similar E-dependence, with a much lower critical +field E∗ +R due to different effective electronic velocities in +near the resonance ring. Indeed, in Figs. 1(b) and 3(b), +E∗ +R is not visible for the drive strengths plotted. Finally, +we find the contribution of the lower Floquet band to +σxy using particle-hole symmetry, F (ξ) +k− = 1 − F (ξ) +−k+ and +B(ξ) +k− = −B(ξ) +−k+. The arguments above allow us to repro- +duce qualitatively the numerically obtained result (Fig. +3) of the suppression of σxy with drive strength. +Numerical analysis.—The results in Fig. 3(b-d) were +derived from a simplified toy model describing TBG as a +tight-binding hexagonal lattice, similar to graphene [49], +but with parameters tuned to match the Fermi velocity +and the Brillouin zone size of TBG. This model misses +some subtle details, but captures well the interplay be- +tween the electron and phonon velocities, and the large + +4 +0.96 +0.99 +1.03 +(a) +(b) +(c) +(d) +FIG. 3. +(a) Schematics of the Floquet spectrum and one +of the phonon light-cones that originates from the area SK +in the α = + band. +The intersection between the α = + +(α = −) band and all cones centered in SK form Sin (Sout). +As E → E∗, the area of Sin shrinks to zero. (b-d) Numerical +verification of the phenomenological model. (b) The area of +Sin, Ain, as a function of E for three values of cph/v0 +eff. (c) +The average occupation of states in SK. (d) Anomalous Hall +conductivity σxy for the same parameters as in (b, c). At the +critical drive E∗ (dashed lines), Ain, F (ξ) +K+, and σxy plateau. +Berry curvature at the Dirac points and resonance ring. +The model represents only the central ν = ±1 bands of +the undriven bandstructure, but since the low drive an- +gular frequency Ω is only resonant to these flat bands, +we can ignore the |ν| > 1 dispersive bands. +This ap- +proximation is valid only when θ is near the magic an- +gle where the |ν| > 1 bands are far from the ν = ±1. +In the Supp. +Mat. +[39], we also present the numer- +ical analysis of a continuum model without electron- +inc interactions [20, 32], which yields qualitatively sim- +ilar results. +In the toy model, v0 +eff = 18.9 km/s, and +we select cph ∈ [17.9 km/s, 19.4 km/s] (see Fig. +1). +In the range cph < v0 +eff, the drive induces the regime +cph > veff(E) for E > E∗. +Note that calculating the +form factor ⟨ξν′k + q + G|ξνk⟩, relies on the contin- +uum model [37, 39], and thus needs to be introduced +by hand in the toy model. +We therefore approximate +⟨ξν′k + q|ξνk⟩ ≈ δν,ν′e−l2 +wq2/4, with lw ≈ LM/(5 +√ +3) +representing the spatial extent of the Wannier orbitals +localized to TBG layer alignment sites [37]. +First, we show how solving the FBE (Eq. 3) for the +steady-state distribution verifies the phenomenological +model. +For this purpose, we need only take the non- +interacting limit by solving Eq. 3 for F (ξ) +kα with ϵ → ∞, +such that Iel-el +kα += 0. The left two quadrants of Fig. 4(a) +show the non-interacting steady-state distributions for a +low phonon bath temperature of 1 K and phonon speed +cph = 0.98v0 +eff in the E > E∗ and E < E∗ cases. +In +the E > E∗ case (left top quadrant), the K, K′ points, +painted blue, have a reduced occupation relative to the +E < E∗ case (left bottom quadrant), due to the suppres- +sion of the incoming scattering rates into SK,K′ (as in the +phenomenological model). Fig. 3(c) shows the occupa- +tion near the K point, F (ξ) +K+, as a function of E for three +different values of cph/v0 +eff and verifies that Ain, F (ξ) +K+, +and σxy plateau at the same critical amplitude E = E∗. +Next, we numerically quantify the strength of Coulomb +screening necessary to stabilize the steady state distribu- +tion. Now, we include Iel-el +kα +̸= 0, setting the dielectric ϵ +to a finite value. On the right top and bottom quadrants +of Fig. 4(a), we show the resulting steady-state occupa- +tions, which have noticeably higher entropy than the non- +interacting case. The entropy of the steady-state occu- +pation depends on the balance between electron-phonon +cooling processes and electron-electron heating processes +and is therefore sensitive to ϵ and the screening gate dis- +tance d. Fig. 1(b) shows the steady-state conductivity as +a function of E for various ϵ (see Eq. 2), with deformation +potential D = 25 eV and d = 1 nm. The slope ∂Eσxy in +the E < E∗ regime where σxy is less steep as ϵ decreases. +To quantify this, we define a visibility parameter +V = −∂Eσxy/[(e2/h)/E0] +(9) +where ∂Eσxy denotes an average of ∂Eσxy across a window +E∗ − δE < E < E∗ where δE/E0 = 1.45 denotes a range +of amplitudes in which σxy decreases rapidly (see shaded +region in Fig. 1(b)). In Fig. 4(b), we show V as a func- +tion of d and ϵ, again keeping D = 25 eV and keeping δE +constant. The dark red region, where ϵ and d are small, +represents a regime where electron-electron interactions +dominate, and the system reaches a hot steady-state with +low visibility; in the blue and green regions, phonon cool- +ing dominates and a low-entropy steady-state appears. +We draw lines comparing the electron-phonon energy +scale Vph ≈ D +� +ℏcph|K|/(√2AMρcph)e−|K|2l2 +w/4 to the +electron-electron energy scale Vel ≈ e2/(2πϵAM|K|)(1 − +e−2|K|d)e−|K|2l2 +w/2 (see Eqs. 1 and 2) as evaluated for +a characteristic momentum transfer magnitude |K| = +4π/(3LM) which we choose to be that of the K point. +Conclusion—TBG is a remarkable system where the +Fermi velocity is comparable to the speed of sound. Upon +THz-laser driving, the electronic population dynamics +exhibits bottlenecks for electron-phonon scattering into +high-Berry curvature Floquet states, which strongly af- +fects the anomalous Hall transport. These bottlenecks, +we show, can be sensitively controlled by the drive am- +plitude. If the undriven effective electron speed is faster +than sound v0 +eff > cph, a drive with E > E∗ induces the +opposite regime veff(E) < cph, decouples the electrons +from the phonons, and suppresses the Hall conductivity +(Fig. 1(b)). We also find that screening gates affect the +electronic steady-state and the anomalous transport. A +strong drive field-dependence of σxy arises for efficient +Coulomb screening, e.g., by a close-by gate or a strong +dielectric. Recent experimental advances in Floquet en- +gineering [23], and THz laser sources [50], show that our + +5 +AB/3icbVDLSsNAFL3xWesrKrhxM1g +EVzUpRV0W3bisYB+QxDKZTtqhkwczE6HELvwVNy4UcetvuPNvnLRZaOuBgcM593LPHD/hTCrL+jaWldW19ZLG+XNre2dXNvy3jVBDaIjGPRdfHknIW0ZitNuIigOfU47/ug69zsPVEgWR3dqnFAvxIOIBYxgpaWe +eiGWA0J5l7glzk1Ol97Wzo9cyKVbWmQIvELkgFCjR75pfbj0ka0kgRjqV0bCtRXoaFYoTSdlNJU0wGeEBdTSNcEil03zT9CJVvoiIV+kUJT9fdGhkMpx6GvJ/O0ct7Lxf8J1XBpZexKEkVjcjsUJBypGKUl4H6T +FCi+FgTATWREZYoGJ0pWVdQn2/JcXSbtWtc+r9dt6pXFV1FGCIziGU7DhAhpwA01oAYFHeIZXeDOejBfj3fiYjS4Zxc4B/IHx+QONLJUo +V [4e2/h] +(b) +(a) +FIG. 4. (a) Left column: electronic steady-state occupation of +the upper Floquet band in the non-interacting case (ϵ = ∞). +Right column: steady-state occupation for the same parame- +ters but with finite interactions (ϵ = 24). Top row: the E > E∗ +case where the population at K is depleted. Bottom row: the +E < E∗ case. (b) The visibility V defined in Eq. 9 as a function +of ϵ and the screening gate distance d. The blue/green region +represents a regime where electron-phonon interactions dom- +inate. Dashed lines represent points with fixed ratio Vph/Vel +of electron-phonon Vph to electron-electron Vel characteristic +interaction strengths (see text for definition). +predicitions should be accessible experimentally. +Higher frequency and stronger drives, such as those +in the UV-visible or X-ray regimes, are a subject of fu- +ture work. Such a theoretical analysis must account for +dispersive bands, which mix significantly with the flat +bands under high-frequency drives [17]. +These high- +frequency drives could further reduce heating by fa- +cilitating fewer electron-electron Floquet-Umklapp pro- +cesses, thus lowering the necessary dielectric constant +for the formation of a non-trivial steady-state [3]. The +drives would also make use of the enhanced electron- +phonon Umklapp cooling processes which arise due to +the tightly-localized electronic Wannier orbitals in TBG +[37]. In the low-frequency, resonant drive limit considered +in this work, the momentum electron-phonon Umklapp +processes are also Floquet-Umklapp processes that are +suppressed. Another interesting direction involves sym- +metry broken phases that can arise in the steady-state +of a strongly coupled TBG [2]. We leave these exciting +directions to future studies. +We thank Netanel Lindner, Mark Rudner, Or Katz, +Gaurav Gupta, +Seamus O’Hara, +Jason Alicea and +Alex Thomson for valuable discussions. +C.Y. grate- +fully acknowledges support from the DOE NNSA Stew- +ardship Science Graduate Fellowship program, which +is provided under cooperative agreement number DE- +NA0003960. C.L. acknowledges support by the Gordon +and Betty Moore Foundation’s EPiQS Initiative, Grant +GBMF8682, start-up funds from Florida State University +and the National High Magnetic Field Laboratory. The +National High Magnetic Field Laboratory is supported +by the National Science Foundation through NSF/DMR- +1644779 and the state of Florida. G.R. and I.E. are grate- +ful for support from the Simons Foundation and the In- +stitute of Quantum Information and Matter, as well as +support from the NSF DMR grant number 1839271. This +work is supported by ARO MURI Grant No. W911NF- +16-1-0361, and was performed in part at Aspen Center for +Physics, which is supported by National Science Founda- +tion grant PHY-1607611. +[1] I. Esin, M. S. Rudner, and N. H. Lindner, Floquet +metal-to-insulator phase transitions in semiconductor +nanowires, Science Advances 6, eaay4922 (2020). +[2] I. Esin, G. Gupta, E. Berg, M. Rudner, and N. 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We can expand the time-dependent eigenstates of +the Hamiltonian in a Floquet-Bloch basis [1]: +|ψkα(t)⟩ = e−iε(ξ) +α t/ℏ|Φm +kα(t)⟩, +(S2) +where r is the position vector, |Φm +kα(t)⟩ is periodic in +time (|Φm +kα(t)⟩ = |Φm +kα(t+2π/Ω)⟩), ε(ξ) +α are the quasiener- +gies plotted in Fig. 2(d), and α enumerates the Floquet +quasienergy bands. To determine the Floquet-Bloch ba- +sis, it is easiest to expand the time-dependent |Φm +kα(t)⟩ +in terms of time-independent Fourier harmonics |φm +kα⟩, +|Φm +kα(t)⟩ = +� +m +e−imΩt|φm +kα⟩, +(S3) +take a Fourier transform the Hamiltonian, +H(k, t) = +� +m +e−imΩtH(m)(k), +(S4) +and solve the Schr¨odinger equation in the basis of Floquet +harmonics: +(ε(ξ) +α + mℏΩ)|φm +kα⟩ = +� +m′ +H(m−m′)(k)|φm′ +kα⟩. +(S5) +In the following subsections, we detail the exact form of +the Floquet Hamiltonians. +A. +Tight binding Floquet toy Hamiltonian +We use a rescaled, two-band tight binding model for +graphene to replicate the flat conduction and valence +bands of TBG. In the rescaled Hamiltonian +Htoy(k) = +� 0 +hk +h∗ +k +0 +� +, +(S6) +hk = W +3 +� +j +eik·δj, +(S7) +we choose long hopping vectors +δj = LM/ +√ +3[sin(2πm/3)ˆx + cos(2πm/3)ˆy], +(S8) +with LM = 0.246 nm/(2 sin θ/2), and a narrow band- +width W. The corresponding rescaled eigenenergies and +Bloch states are +Eν(k) = ν|hk|, +(S9) +and +|νk⟩ = +1 +√ +2 +� +νeiarg(hk) +1 +� +, +(S10) +respectively, with ν = ±1 enumerating the flat Bloch +bands. +Following [2], we perform minimal coupling, which +turns the functions hk into time-dependent quantities +with Fourier transforms +h(n) +k += +1 +2π/Ω +� 2π/Ω +0 +hk+eA(t)/ℏe−inΩtdt += +� +j +teik·δjeinφjJn(− ˜E), +(S11) +where ˜E is the dimensionless drive strength +˜E = eLM +√ +3ℏ A = eLM +√ +3ℏ +E +Ω; +(S12) +the phase angles are φ0 = π/2, φ1 = −5π/6, and φ2 = +−π/6; and +Jn(z) = +1 +2πin +� 2π +0 +eiz cos θeinθdθ. +(S13) +The Fourier-transformed Hamiltonian is +H(n) +toy(k) = +� +0 +h(n) +k +h∗(n) +k +0 +� +. +(S14) +Note that +h∗(n) +k += +� +j +te−ik·δjeinφjJn( ˜E) +(S15) +is the Fourier transform of the conjugate of hk. In sim- +ulations, we generally truncate the Fourier Hamiltonian +(Eq. +S5) to −12 ≤ m ≤ 12, so that we account for +a sufficient number of high-order Floquet-Umklapp pro- +cesses in the Floquet-Boltzmann equation. For both the +undriven and Floquet Hamiltonians, we also perform a +gauge transformation, replacing h(n) +k +→ ie−ik·δ0h(n) +k +and +hk → −ieik·δ0hk to make the Hamiltonians periodic by +shifts of k → k+G, where G is a reciprocal lattice vector. +arXiv:2301.02248v1 [cond-mat.mes-hall] 5 Jan 2023 + +2 +FIG. S1. (a) The quasienergy band structure of the toy model +with for the parameters used in the main text. +(b) The +quasienergy band structure of the continuum model at val- +ley ξ = +1. In both panels, the first Floquet Brillouin zone +is shaded. +See Sec. +II for details and justification for the +parameters we have used. +B. +Continuum Model Floquet Hamiltonian +The undriven continuum model for TBG [3] describes +the bandstructure of TBG near the valley ξ = ±1 of the +monolayer graphene Brillouin zone. Its Hamiltonian +Hξ = +� +H1 U † +U +H2 +� +(S16) +is diagonalized in the basis ψnk = (ψA1 +nk, ψB1 +nk, ψA2 +nk, ψB2 +nk)T +with +ψX +nk(r) = eik·r � +G +CX +nk(G)eiG·r +(S17) +where X = Al, Bl represents sublattice A or B degree +of freedom in layer index l = ±1. In Eq. S16, Hl are +the monolayer graphene Hamiltonians, which, in close +vicinity of the ξ = ±1 valleys, resemble Dirac cones: +Hl = −ℏvml +F +� +R(lθ/2)(k − K(l) +ξ ) +� +· (ξσx, σy) +(S18) +where R(ϕ) is the 2×2 rotation matrix, vml +F is the mono- +layer Graphene Fermi velocity, and Kl +ξ is the Dirac point +of layer l at valley ξ. The interlayer coupling is +U = +�u u′ +u′ u +� ++ +� u +u′ν−ξ +u′νξ +u +� +eiξG1·r ++ +� +u +u′νξ +u′ν−ξ +u +� +eiξ(G2+G3)·r +(S19) +Using minimal coupling, +we obtain time-dependent +monolayer graphene Hamiltonians, with Fourier trans- +form +H(n) +l += −ℏv +� +R(lθ/2) +� +k + e +ℏ +1 +2E[(δn,1 + δn,−1)ˆy +− i(δn,−1 − δn,1)ˆx] − K(l) +ξ +�� +· (ξσx, σy). +(S20) +Then, +H(n) +ξ += +� +H(n) +1 +U †δn,0 +Uδn,0 +H(n) +2 +� +(S21) +is the Fourier transform of the continuum model Hamilto- +nian. For the continuum model, we truncate the Floquet +Hamiltonian (Eq. S16) to −6 ≤ m ≤ 6. +Upon diagonalizing the Floquet Hamiltonian, we ob- +tain a large number of Floquet states per energy interval +[−ℏΩ/2, ℏΩ/2]. We select two states per k-point whose +spectral weights A0 +α(k) = |⟨φ0 +kα|φ0 +kα⟩|2 are large (which +makes their contribution to the Floquet-Boltzmann equa- +tion most important, see Sec. X). +C. +Quasienergy Bands +In Sec. +II, we provide and motivate the choices of +physical parameters that we use in the main text. +In +Fig. S1, we preview the quasienergy bands for our choice +of toy and continuum model parameters. +II. +CHOICE OF PHYSICAL PARAMETERS +First, we present the physical parameters we use for +the electronic Hamiltonian in the TBG continuum model +(see Sec. I B for the Hamiltonian). We consider the non- +interacting continuum model [3, 4] at a near-magic twist +angle of θ = 1.13◦. The bandwidth of the central bands +at this angle is W ≈ 5 meV, and a perturbative expan- +sion of the Hamiltonian around the Brillouin zone Dirac +points [4] estimates the Fermi velocity as +vF (θ) = vml +F (1 − 3β2)/(1 + 3β2(1 + η2)), +(S22) +where β = u′/(ℏkθvml +F ) and η = u/u′ with vml +F += +8 × 105 m/s, kθ = 4π/(3LM), u = 0.0797 eV, and +u′ = 0.0975 eV [3, 4]. Eq. S22 predicts that the Fermi +velocity at the chosen twist angle is vF = 27 km/s. How- +ever, the derivation of Eq. S22 approximates that H(n) +l +is +roughly θ-independent and tends to overestimate vF (see +Fig. 4 inset in [4]). We can obtain a better estimate by +numerically calculating the Fermi velocity along the path +K-M in k-space of the ν = +1 band in the ξ = +1 valley. +(This is the direction of maximum Fermi velocity.) The +estimate yields vF = 17.5 km/s, and we hereafter use +this value. In our Floquet Hamiltonian, we use a laser +angular-frequency of Ω ≈ W/ℏ ≈ 5 meV/ℏ. +Second, we present the parameters we use for the +electronic Hamiltonian of the TBG two-band toy tight +binding model (see Sec. I A for the Hamiltonian). We +choose our toy model Fermi velocity, frequency, and +twist angle to roughly match those of the continuum +model. Specifically, we use a twist angle of θ = 1.13◦ + +3 +and choose W = 3.1 meV so that the Fermi velocity +vF = WLM/(2 +√ +3ℏ) = 17 km/s roughly matches that +of the continuum model at the same angle. In the toy +model Floquet Hamiltonian, we choose Ω ≈ 5 meV/ℏ. +Third, we discuss the parameters we use for the +TBG phonons. +For both the continuum and toy +models, we consider phonons speeds in the range of +cph +∈ +[17.9 km/s, 19.4 km/s]. +In the toy model, +v0 +eff = 18.9 km/s, and, in the continuum model, v0 +eff = +19.5 km/s, so the range of cph we choose covers the +regime cph < v0 +eff, in which the drive induces the op- +posite regime cph > veff(E) when E > E∗. We also use +the same phonon bath temperature of Tph = 1 K and +Wannier orbital width lw = LM/(5 +√ +3) for the toy and +continuum model calculations. +Please see Sec. X for details of the numerical k-point +grid. +III. +ANOMALOUS HALL CONDUCTIVITY +CALCULATIONS FOR THE CONTINUUM +MODEL +In this section, we repeat the calculations in the main +text on the TBG continuum model [3, 4]. We consider the +non-interacting limit, setting ϵ → ∞ so that Iel-el +kα += 0. +First, we discuss differences in the bandstructure and +topology at valleys ξ = +1 and ξ = −1. The circularly +polarized laser opens a gap at the Dirac points, ∆K, ef- +fectively adding a mass term ξ∆Kσz to the Hamiltonian +(see Sec. I B and [5] for a derivation) in the vicinity of +the Dirac points. Because the sign of the mass term de- +pends on ξ, the ξ = ±1 superlattice valley contributions +to σxy do not trivially cancel to zero. In fact, in recip- +rocal space, the Berry curvature and occupations near +(a) +(b) +[km/s] +18.5 +19.5 +FIG. S2. (a) Left: the steady-state occupation of the upper +Floquet band in valley ξ = +1 of the continuum model [3, 4]. +Right: the Berry curvature of the same band, which peaks +near the Dirac points and the resonance ring. (b) The anoma- +lous Hall conductivity σxy as a function of drive strength E. +ξ = +1 are simple π/3 rotations of those in ξ = −1, so +σxy = 4e2 +h +� +α=± +� +MBZ +d2k +(2π)2 B(+1) +kα F (+1) +kα +. +(S23) +In Fig. +S2, we show the steady-state and σxy for the +continuum model calculation. Note that to simplify the +calculations, we use the same effective form factor ⟨ξν′k+ +q|ξνk⟩ ≈ δν,ν′e−l2 +wq2/4 as the toy model. +IV. +DIRECT VARIATION OF THE PHONON +SPEED cph +Throughout the main text, we use the drive strength +E to control electron speeds. We could achieve similar +results by keeping E fixed and varying cph instead. Fig. +S3 shows the variation of σxy as a function of cph. The +curves resemble the dependence of σxy on E in the main +text (see, for e.g., Fig. 1(b)). +V. +FORMAL DEFINITION, NUMERICAL +EVALUATION, AND PHENOMENOLOGICAL +MODEL OF Ain +As described in the main text, an patch Sin shaped +as an elliptical annulus (see Fig. 3(a)) with area Ain in +momentum space vanishes as E → E∗. Here, we provide +a formal definition of Ain and explain how we estimate +its dependence on E numerically and analytically. +A. +Formal Definition +Let us first define Ain formally. Consider a family of +phonon cones centered throughout SK, the circular patch +enclosing a K-point in the quasienergy spectrum (see Fig. +3(a)). Suppose that a subset of the phonon cones are +centered throughout a small quasienergy window dεk+. +FIG. S3. Anomalous Hall conductivity of the toy model as +a function of the ratio cph/v0 +eff for three different drive field +strengths E/E0. The same electron-phonon decoupling pro- +cess is visible as σa +xy plateaus. + +4 +The k-space area of states dAin containing intersections +of the cones with the upper Floquet band is +dAin = dεk+ +� +s=± +� +d2k′ δ(εk+ − εk′+ + sℏcph|k′ − k|). +(S24) +Next, we integrate over εk+ contained in SK to obtain +Ain = +� +dA = +� +k∈SK +d2k +1 +D(εk+)× +× +�� +s=± +� +d2k′ δ(εk+ − εk′+ + sℏcph|k′ − k|) +� +, +(S25) +where +D(ε) = +� +α +� +d2k +(2π)2 δ(ε − εkα) +(S26) +is the density of states in the quasienergy band structure. +Exploiting the circular shape of SK, +� +k∈SK +d2k ≈ +� +d2k Θ(|k − K| − kp) +(S27) +where kp is the radius of the circular area AK of SK. +Lastly, we calculate an approximate expression for kp, +the radius of AK. In the vicinity of the Dirac cone, the +Hamiltonian is +HK(k, t) = d · σ, +(S28) +where d = ℏvF ξkxˆx+ℏvF ky ˆy +ξ∆KE2ˆz. (See Sec. VIII +for a detailed derivation.) The z-component of the Berry +curvature is +Bz +kα = α dz +2|d|3 = αξ +∆K +[(ℏvF k)2 + ∆2 +K]3/2 +(S29) +where dz = ξ∆K. At the half-maximum, Bz +kpα = 0.5Bz +0α, +so +kp = (22/3 − 1)1/2 ∆K +ℏvF +. +(S30) +B. +Numerical Estimate +To generate the values of Ain we present in Fig. 3(b), +we evaluate the integrals in Eq. +S25 on a finite-sized +grid of k-points, smearing the step function by replac- +ing Θ(|k − K| − kp) → [e(|k−K|−kp)/σk + 1]−1, where +σk = 2π/(LMN) is the grid spacing between k-points on +an N × N Monkhorst-Pack grid (see Sec. X). Thus, we +approximate +Ain ≈ +� +k +[e(|k−K|−kp)/σk + 1]−1 +1 +D(εk+)× +× +�� +s=± +� +k′ +δ(εk+ − εk′+ + sℏcph|k′ − k|) +� +. +(S31) +(b) +(a) +FIG. S4. (a) The intersection SK (Fig. 3(a)) as viewed on +the Brillouin zone. The outer radius along the path KR is +hb(E). (b) Quasienergy (pink) along the path KR, with the +phonon light cone (grey) that determines the outer radius of +Ain. The intersections k+ and k− between the cone and the +upper Floquet band determines hb(E) = k+ − k−. +For more information on how we approximate the Dirac +Delta function on the grid, please see Sec. X A. +C. +Phenomenological Model +In this section, we prove that the intersection area +Ain ∝ max(E∗ − E, 0) as E → E∗. +The shape of Ain +is an elliptical annulus as shown in Fig. 3(a). Let us use +hb(E) and wb(E) respectively to denote the outer major +and minor axis radii of the elliptical annulus (see Figs. +S4(a) and S5(a)). In the following sections, we begin by +generating analytical estimates of hb(E) and wb(E). +1. +Estimate of hb +First, let us consider a slice of the upper Floquet band +in k-space from the K to the resonance ring (R) along +the direction of hb(E), as we show in Fig. S4(b). Let us +define a one-dimensional momentum component q along +the path K-R. +We sketch a phonon light cone (grey) +originating from a point (yellow) in SK that determines +the outer radius of Ain. The phonon cone intersects with +the quasienergy at points k+ and k−, and the outer radius +of Ain is therefore hb(E) = k+ − k−. First, consider the +undriven limit E = 0, where the gaps ∆R = 0 and ∆K = +0. We choose some point qm such that k− < qm < k+ +and series expand the energy E(q) of the undriven system +around qm: +E(q) ≈ E(qm) + E′(qm)(q − qm) + 1 +2E′′(qm)(q − qm)2 += a2q2 + a1q + a0, +(S32) + +hs +25 +(a) +(b) +FIG. S5. (a) Width of the intersection SK, wb(E). (b) Circu- +lar coordinate system with arc length w (increasing counter- +clockwise) that we use to determine wb(E) = w+ − w−. +where a2 = E′′(qm)/2, a1 = E′(qm) − E′′(qm)qm, and +a0 = E(qm)−E′(qm)qm+E′′(qm)q2 +m/2. As we increase E, +the gaps ∆K and ∆R widen. Let us write the quasienergy +in the vicinity of qm as +ε(q) ≈ f(E)E(q) + ∆K +2 +(S33) +where f(E) ≤ 1 is a scaling factor that decreases as E +increases and accounts for band flattening due to ∆K +and ∆R. Let +f −1 = 1 − b1 ˜E − b2 ˜E2, +(S34) +where b1 ≥ 0 and b2 ≥ 0 are constants dependent on +the exact bandstructure (i.e., how the widening of ∆K +and ∆R with E affects the bandstructure near qm). The +roots of the equation E(q) = ∆K/2 + ℏcphq are k±, and +we may write the equation as +a2q2 + a1q + a0 = fℏcsq, +(S35) +from which we find that +hb = k+ − k− = +� +(a1 − fℏcs)2 − 4a2a0. +(S36) +Solving for E∗ through the equation hb = 0, and then +series expanding the expression (a1 − fℏcs)2 − 4a2a0 in +powers of small E−E∗, we find that (a1−fℏcs)2−4a2a0 ∼ +E∗ − E, so hb ∼ +√ +E∗ − E. +2. +Estimate of wb +To estimate wb (see Fig. S5(a)), we define a circular +coordinate system shown in Fig. S5(b) whose origin is +the K point and arc length w is zero along the KR slice, +increasing counterclockwise. The quasienergy ε(w) along +the circle perimeter varies with w; let us approximate +ε(w) ≈ ℏΩ/2 − (d0 + d2w2), +(S37) +using some fitting parameters d0 and d2. (We assume +that w = 0 is at local maximum of ϵ(w), so there is no +linear term in Eq. S37.) Roughly, wb = w+ − w−, where +we find w+ and w− by finding the roots of the equation +ℏΩ/2 − ∆(w) = fℏcsqm. +(S38) +Here, once again, we use the factor f in Eq. +S34 to +account for band flattening as E increases from zero. So, +wb = w+ − w− = 2 +� +(fℏcsqm + ℏΩ/2 − d0)/d2. (S39) +Solving for E∗ by setting wb = 0 and series expanding +fℏcsqm + ℏΩ/2 − d0 in powers of E, we find that wb ∼ +√ +E∗ − E. +3. +Estimate of Ain +In the limit E → E∗, the elliptical annulus with fi- +nite thickness collapses into a filled ellipse. Thus, in the +limit E → E∗, we estimate that Ain = πhb(E)wb(E) ∝ +max(E∗ − E, 0). +VI. +PREDICTING E∗ FOR THE TOY MODEL +Here, we use the quasienergy dispersion of the toy +model to predict E∗. By writing an approximate, ana- +lytic expression for veff(E) (see Eq. 6), we can find E∗ +using the relation veff(E∗) = cph. +From Eq. +6, veff(E) = (εk∗+ − εK+)/|k∗ − K| for +some appropriately-chosen k∗ (dropping the superlattice +valley index for notational simplicity). One can find nu- +merically that k∗ does not shift significantly with Ω or +E. We write an ansatz +εk∗+ ≈ ℏv0 +eff|k∗ − K| − ℏvF +LM +� +f ′ +1 ˜E + f ′ +2 ˜E2� |k∗ − K| +Ω/(2v0 +eff), +(S40) +where f ′ +1 and f ′ +2 are fitting constants dependent on the +quasienergy bandstructure. Here, ℏvF /LM is the order +of magnitude energy scale of the resonance ring gap ∆R. +The dependence of εk∗+ on E arises predominantly from +∆R. +The dependence is stronger when k∗ is close to +the resonance ring, and we encode this behavior in the +ratio |k∗ − K|/Ω/(2v0 +eff), where Ω/(2v0 +eff) is the k-space +distance between the K point and the resonance ring. +Separately, we know that εK+ = ∆K/2. We use Eq. 6 +to infer +v0 +eff(E) = v0 +eff − +∆K +2ℏ|k∗ − K| − 2ℏvF v0 +eff +LMΩ +� +f ′ +1 ˜E + f ′ +2 ˜E2� +. +(S41) +We know that v0 +eff ∝ vF . We also assume that |k∗ − K| +does not change significantly with E, so it is independent +of the drive and only dependent on the superlattice scale: + +6 +(b) +(a) +FIG. S6. Comparing numerical evaluation of E∗ (points) to +an analytic fit to Eq. S44. We use the same fitting parameters +f2 = 0.778, f1 = 0, and δ(N) = 0.006 for both panels. +|k∗ − K| ∝ L−1 +M . Thus, we can absorb some unknown +coefficients into new coefficients f ′′ +1 and f ′′ +2 to obtain +v0 +eff(E) = v0 +eff − ℏv2 +F +LMΩ +� +f ′′ +1 ˜E + f ′′ +2 ˜E2� +. +(S42) +Upon solving for ˜E∗ from cs = veff(E∗), we find that +˜E∗ ≈ +� +LMΩ +3f2vF +�� +1 − cph/v0 +eff + f 2 +1 − f1 +� +, +(S43) +where f1 and f2 are new, rescaled fitting constants. Using +the relation ˜E = eLME/( +√ +3ℏΩ), we find +E∗ ≈ +ℏΩ3/2 +f2eL1/2 +M v1/2 +F +�� +1 − cph/v0 +eff + f 2 +1 − f1 +� +, +(S44) +As cph → v0 +eff, E∗ ∝ (1 − cph/v0 +eff)γ where γ = 1 (1/2) +if f1 ̸= 0 (= 0). See Fig. S6 for a fit for two different +frequencies Ω. +Finite grid size effects on an N × N Monkhorst-Pack +grid (see Sec. X) generate a small numerical error δ(N) +that enters S44 as +E∗ ≈ ℏL1/2 +M Ω3/2 +f2eLMv1/2 +F +�� +1 − cph/v0 +eff + δ(N) + f 2 +1 − f1 +� +. +(S45) +To see this, let us consider the details of the finite-sized +grid. We impose energy conservation through a broad- +ened Dirac Delta function (see Sec. X A), which we model +as a Gaussian function in energy with a tiny width +√ +2σ ≈ 0.1 · +√ +2 · +W +2N/3. +(S46) +(We motivate the choice of the prefactor of 0.1 in Sec. +X A.) Since we avoid the high symmetry K point in our +grids, the k-point with largest Berry curvature is, in fact, +a point knear point shifted away from K by a small dis- +tance in momentum space of +|δk| = |knear−K| ≈ 1 +2 +Ω/(2ℏv0 +eff) +2N/3 += +Ω +4v0 +eff(2N/3). (S47) +(b) +(a) +0.99 +0.96 +FIG. S7. Comparing the dependence of σxy on E for (a) the +frequency considered in the main text and (b) a lower fre- +quency where Floquet-Umklapp processes are stronger. Note +that the frequency in panel (b) is inaccessible without gener- +ating two-photon resonances in the continuum model due to +the peaked shape of the ν = ±1 bands near the Γ point. +FIG. S8. Comparison of the fitted ∆R and predicted ∆K in +Equations S51 and S55 (solid lines) to those obtained from +numerics (points), using ℏΩ = 5 meV in the toy model. Here, +we fit ∆R with factors of f R +1 = 0.04 and f R +2 = 0.0184 (see Eq. +S51). +This point is shifted in quasienergy by ℏvF |δk| relative +to the actual K point. We can account for both of these +effects by shifting εK+ → εK+ + δε, with δε = +√ +2σ + +ℏvF |δk| and solve veff(E∗) = cph to find Eq. S45 with +δ(N) = δε/(ℏv0 +eff|k∗ − K|). +VII. +DIFFERENT FREQUENCIES +Reducing Ω below the value considered above will in- +crease the ratio (vF eE/Ω2)2 and in turn strengthen Flo- +quet Umklapp processes, modifying the shape of the σxy +curve. We demonstrate this in Fig. S7(b) for an angu- +lar frequency Ω = 4.135 meV/ℏ. However, such a low- +frequency regime is inaccessible in the continuum model +(without generating two-photon resonances) due to the +peaked shape of the continuum model ν = ±1 band near +the Γ point, so we do not consider this lower (doubly- +resonant) frequency regime in the main text. + +7 +VIII. +GAP SIZES +In this section, we estimate the size of the Floquet- +induced gaps ∆K and ∆R. By the rotating wave approx- +imation, the Floquet-induced gap at the resonance ring, +∆R, is roughly proportional to the drive energy [6]. For +a resonant drive that couples electronic states near the +Dirac points, the drive energy is roughly +vF eA/ℏ, +(S48) +as predicted by minimal coupling q → q+eA(t)/ℏ in the +Dirac cone Hamiltonian +HK(q) = ℏvF q · (ξσx, σy) +(S49) +with vF = WLM/(2 +√ +3ℏ). (We always use perturbative +drives that generally fall in the range of ˜E < 1.) +We +expect that +∆R ≈ ℏvF +LM +˜E. +(S50) +Such an approximation works well for low-frequency res- +onant drives that couple states near the Dirac points. +However, resonant drives with higher frequencies, like +those used in the main text, couple states closer to the Γ- +points of the TBG energy dispersion where the bands are +nonlinear in q. In such a case, higher order (e.g., O( ˜E2)) +contributions (from O(q2) contributions of the band- +structure) to ∆R become dominant. In the present exam- +ple, the energy of the tight binding model for graphene +is quadratic in momentum near the Γ point, so we write +an ansatz +∆R ≈ ℏvF +LM +(f R +1 ˜E + f R +2 ˜E2), +(S51) +and fit f R +1 and f R +2 to match ∆R obtained by numerically +diagonalizing the Floquet Hamiltonian, as shown in Fig. +S8. +We can estimate the Floquet-induced K-point gap, +∆K, by considering the time-dependent Dirac Hamilto- +nian +HK(q, t) = ℏvF (ξqxσx + qyσy) ++ vF eA[ξ cos(Ωt)σx − sin(Ωt)σy]. +(S52) +and performing a Van Vleck expansion [6–8] to obtain an +effective Floquet Hamiltonian +HK,eff(q) = H(0) +K + [H(−1) +K +, H(1) +K ] +ℏΩ += HK + ξ e2v2 +F A2 +ℏΩ +σz +(S53) +with +H(n) +K (q) = +1 +2π/Ω +� 2π/Ω +0 +HK(q, t)e−inΩtdt. +(S54) +From Eq. S53, we can extract +∆K = 2e2v2 +F +ℏΩ A2 = 6ℏv2 +F +L2 +MΩ +˜E2. +(S55) +IX. +FLOQUET BOLTZMANN EQUATION +Here, we present the full expression for the Floquet- +Boltzmann equation [9], which we copy below for conve- +nience: +∂tFkα(t) = Iel-ph +kα +[{Fkα(t)}] + Iel-el +kα [{Fkα(t)}]. +(S56) +The electron-phonon collision integral is +Iel-ph +kα +[{Fkα}] = 2π +ℏ +� +k′∈BZ +� +α′ +� +j +� +n +|Gk′α′ +kα (n, j)|2 +× +� � +Fk′α′(1 − Fkα)N(ℏωj(k′ − k)) − Fkα(1 − Fk′α′)[1 + N(ℏωj(k′ − k))] +� +× δ(εk′α′ − εkα + ℏωj(q) + nℏΩ) ++ +� +Fk′α′(1 − Fkα)[1 + N(ℏωj(k′ − k))] − Fkα(1 − Fk′α′)N(ℏωj(k′ − k)) +� +× δ(εk′α′ − εkα − ℏωj(q) + nℏΩ) +� +(S57) +Gk′α′ +kα (n, j) = +� +ν +1 +√AMoir´e +D +√2ρcph +� +ℏωj(k′ − k)e−|k′−k+Gj|2l2 +w/4 � +m +⟨φn+m +k′α′ |νk′⟩⟨νk|φm +kα⟩ +(S58) +where ρ = 1.52 × 10−6 kg/m2 is the 2D density of the +graphene layers, D is the deformation potential, and the +acoustic phonon mode j has frequency ωj(q) = ℏcph|q + +Gj| with {Gj} being the set of all possible reciprocal + +8 +lattice vectors. The function N(ε) = (e−ε/kBTph − 1)−1 +is the Bose-Einstein occupation of the phonon bath at +temperature Tph. The electron-electron collision integral +is +Iel-el +kα [{Fkα}] = 4π +ℏ +1 +N 2 +� +k2∈BZ +� +k3∈BZ +� +α2,α3,α4 +� +n +� +G +|V(k3,α3),(k1+k2−k3,α4) +(k,α),(k2,α2) +(n, G)|2× +× δ(εkα + εk2α2 − εk3α3 − εk+k2−k3,α4 + nℏΩ)× +× [(1 − Fkα)(1 − Fk2α2)Fk3α3Fk1+k2−k3,α4 − FkαFk2α2(1 − Fk3α3)(1 − Fk1+k2−k3,α4)] +(S59) +V(k3,α3),(k1+k2−k3,α4) +(k,α),(k2,α2) +(n) = +� +ν1,ν2 +� +ν3,ν4 +� +n2,n3,n4 +V (k2 − k3 + G)e−|q+G|2l2 +w/2δν,ν3δν2,ν4⟨φn−n2+n3+n4 +kα +|ν1k⟩⟨φn2 +k2α2|ν2k2⟩× +× ⟨ν3k3|φn3 +k3α3⟩⟨ν4k4|φn4 +k+k2−k3,α4⟩. +(S60) +We solve for ∂tFkα = 0 using the Newton-Raphson al- +gorithm. +To ensure charge neutrality, we add a La- +grange multiplier term λ(� +kα Fkα − N) to the Floquet- +Boltzmann equation, choosing some large constant λ. +X. +MONKHORST-PACK GRID, NUMERICAL +INTEGRATION, AND CONVERGENCE +In this section, we describe the methods we use to dis- +cretize the momentum Brillouin zone. We perform the +Boltzmann equation integrals, introduced in Equations +S57 and S59, over an N × N Monkhorst-Pack (MP) set +of grid points [10], with k-points +km,n = mG1 + nG2 +N +, +(S61) +odd N, and m, n = 0, . . . , N − 1. Specifically, we avoid +values of N(mod 3) = 0 that generate a k-point ex- +actly at the high-symmetry point of K, because such +grids converge poorly when the drive strength is weak +and Floquet-induced gap ∆K is small. +A. +Energy and Momentum Conservation +Here, we discuss in detail how we impose momentum +and energy conservation on this MP grid. The space of +MP k vectors are closed under addition and subtraction +(modulo a reciprocal lattice vector), so conservation of +momentum (e.g., k+k2−k3 in Eq. S59), is simple to im- +plement. We impose energy conservation via a smeared +Dirac Delta function +δ(ε) = +� +1.04766e−ε2/2σ2/(2.5066283σ), +if |ε| < 2σ, +0, +otherwise, +(S62) +(a) +(b) +FIG. S9. Convergence of anomalous conductivity with grid +size for (a) N(mod 3) = 1 and (b) N(mod 3) = 2. Due to +the positioning of grid points near the K point, the results +at low grid resolutions show significant disagreement. Here, +E0 = 4.41 × 104 V/m. +where we have chosen numerical factors so that +� ∞ +−∞ +δ(ε)dε = 1. +(S63) +The smearing parameter σ is one-tenth of the maxi- +mum quasienergy spacing between nearest-neighbor MP +k-points +σ = 0.1 max +⟨k,k′⟩,α |ε(ξ) +kα − ε(ξ) +k′α|, +(S64) +where ⟨k, k′⟩ restricts k′ to be a nearest-neighbor of k, +and we have tuned the prefactor of 0.1 so that upon +calculating the steady-state without Floquet-Umklapp +processes, we obtain a Fermi-Dirac distribution, F (ξ) +kα = +(eεkα/kBTph + 1)−1 with temperature Tph of the phonon +bath [11]. + +9 +1.05 +1.10 +1.15 +θ [◦] +0 +5 +10 +E∗ [105 V/m] +12 +15 +18 +21 +[km/s] +FIG. S10. The requirement that the laser drive strength E is +perturbative, i.e. a fraction of electron bandwidth eELM < +W, narrows the range of E values that can be used. +As a +result, the range of cph whose E∗ is visible is limited as well +- we postulate that they are pushed to higher drive strengths +E. +B. +Convergence of Conductivities +In Fig. S9, we show the convergence of the Hall con- +ductivity σxy with grid size, using ℏΩ = 5 meV. In the +main text, we use a 163×163 MP grid for non-interacting +calculations, and a 73 × 73 grid for interacting calcula- +tions. +XI. +BERRY CURVATURE CALCULATIONS +We follow the Berry curvature calculation presented in +[12], defining U(1) link variables +Uµ(k, t) = ⟨α(k, t)|α(k + ˆµ, t)⟩ +|⟨α(k, t)|α(k + ˆµ, t)⟩| +(S65) +where µ = x, y, ˆµ = Gµ/N, and |α(k, t)⟩ are the Bloch +vectors (i.e., |ψkα(t)⟩ = e−ik·r|α(k, t)⟩). The Berry cur- +vature is +Bkα(t) = (2π)2 +N 2AM +arg +�Ux(k, t)Uy(k + ˆx, t) +Ux(k + ˆy, t)Uy(k, t) +� +(S66) +and we use the time-averaged Berry curvature +Bkα ≡ +1 +2π/Ω +� 2π/Ω +0 +Bkα(t)dt +(S67) +in transport calculations. +XII. +THE PERTURBATIVE REGIME AT +DIFFERENT TWIST ANGLES +We have treated the laser drive as a perturbation to +the undriven TBG Hamiltonian, which restricts the range +of field strengths E we can use to a weak perturbative +regime. This also narrows the range of phonon speeds +cph that will generate a critical field strength E∗ in the +perturbative regime, hence the narrow range of cph we +have considered in, e.g., Fig. +1(c). +For various twist +angles, we estimate the range of drive strengths E that +are perturbative in the unshaded region of Fig. S10 and +overlap in solid lines the predicted value of E∗ for different +speeds of sound. The shaded, non-perturbative regime +corresponds to drive energy scales vF eE/Ω greater than +a fraction, e.g., 0.3, of the bandwidth W. Here, we follow +the analysis in [4] to estimate the undriven Fermi velocity +vF (θ) = +�� +(1 − 3α2)/(1 + 3α2(1 + η2)) × vml +F +�2 + v2 +min, +(S68) +where vmin = 104 m/s is a manually set minimum Fermi +velocity of the undriven flat bands, and we use the same +parameters as in Sec. +II. We also adjust Ω such that +Ω/vF (θ) is constant and equal to those considered in +Figs. 1-4. +[1] M. S. Rudner and N. H. Lindner, The floquet engineer’s +handbook (2020), arXiv:2003.08252 [cond-mat]. +[2] Q. Chen, L. Du, and G. A. Fiete, Phys. Rev. B 97, +035422 (2018). +[3] M. Koshino, N. F. Q. Yuan, T. Koretsune, M. Ochi, +K. Kuroki, and L. Fu, Phys. Rev. X 8, 031087 (2018). +[4] R. Bistritzer and A. Macdonald, Proceedings of the Na- +tional Academy of Sciences of the United States of Amer- +ica 108 (2010). +[5] O. Katz, G. Refael, and N. H. Lindner, Phys. Rev. B +102, 155123 (2020). +[6] M. Rudner and N. Lindner, Nature Reviews Physics 2, +1 (2020). +[7] M. Rodriguez-Vega, M. Vogl, and G. Fiete, Annals of +Physics 435, 168434 (2021). +[8] T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, +Phys. Rev. B 84, 235108 (2011). +[9] K. I. Seetharam, C.-E. Bardyn, N. H. Lindner, M. S. +Rudner, and G. Refael, Phys. Rev. B 99, 014307 (2019). +[10] H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188 +(1976). +[11] V. M. Galitskii and V. F. Elesin, Journal of Experimental +and Theoretical Physics 30 (1970). +[12] T. +Fukui, +Y. +Hatsugai, +and +H. +Suzuki, +Journal +of the Physical Society of Japan 74, 1674 (2005), +https://doi.org/10.1143/JPSJ.74.1674 + diff --git a/RdE0T4oBgHgl3EQfUQCb/content/tmp_files/load_file.txt b/RdE0T4oBgHgl3EQfUQCb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f240975e56ad85ef05268025443f61fef3c0742e --- /dev/null +++ b/RdE0T4oBgHgl3EQfUQCb/content/tmp_files/load_file.txt @@ -0,0 +1,1125 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf,len=1124 +page_content='Optical Control of Slow Topological Electrons in Moir´e Systems Christopher Yang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1 Iliya Esin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1 Cyprian Lewandowski,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3 and Gil Refael1 1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' IQIM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' CA 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' USA 2National High Magnetic Field Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Tallahassee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Florida,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 32310,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Florida State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Tallahassee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Florida 32306,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' USA Floquet moir´e materials possess optically-induced flat-electron bands with steady-states sensitive to drive parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Within this regime, we show that strong interaction screening and phonon bath coupling can overcome enhanced drive-induced heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In twisted bilayer graphene (TBG) irra- diated by a terahertz-frequency continuous circularly polarized laser, the extremely slow electronic states enable the drive to control the steady state occupation of high-Berry curvature electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In particular, above a critical field amplitude, high-Berry-curvature states exhibit a slow regime where they decouple from acoustic phonons, allowing the drive to control the anomalous Hall response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Our work shows that the laser-induced control of topological and transport physics in Floquet TBG are measurable using experimentally available probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Introduction— Time-periodic fields can drive materials into exotic non-equilibrium phases [1–9], with unconven- tional transport and optical characteristics [10–16] con- trollable by external parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In laser-driven twisted bilayer graphene (TBG) [17–19], a flat-band regime with pronounced electron-electron interaction effects is acces- sible even away from the magic angles [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Generat- ing low-temperature Floquet states in such a regime re- quires cooling processes that compensate for strong drive- induced electron-electron heating processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A common cooling solution involves the coupling of Floquet system to a low-temperature phonon bath [3, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' This work demonstrates that the intrinsic electron- phonon coupling in TBG and strong Coulomb screening can stabilize a low-temperature steady-state in Floquet TBG under a terahertz (THz) frequency, circularly po- larized laser drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In this steady-state, the drive am- plitude controls the filling of electronic states with large Berry curvature, resulting in a highly tunable anomalous conductivity σxy [10, 23–26] (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(a-b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The ability to tune the Floquet steady-state results from the unique slow electron regime in TBG in which phonons travel faster than—and decouple from—many flat band elec- tronic states [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The drive strength controls the electron velocities, tuning between an equilibrated elec- tronic gas with a finite σxy at low drive amplitudes, and a decoupled electronic gas with reduced electron velocities and suppressed σxy at high drive amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='—We begin our analysis by construct- ing the time-periodic, interacting Hamiltonian for laser- driven TBG near the charge neutrality point and at a twist angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The single-particle effective Hamiltonian of an undriven TBG is given by ˆH0 = � kνξ E(ξ) kν ˆc(ξ)† kν ˆc(ξ) kν , where ˆc(ξ)† kν creates a Bloch state |ξνk⟩ of crystal momen- tum k, band ν, and energy E(ξ) kν , in the vicinity of the valley index ξ = ±1 of the single-layer graphene Bril- louin zone [20, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The index ν = ± labels the cen- tral particle and hole bands (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2(a, b)) with a total bandwidth of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' These central bands are sepa- (c) (a) laser d (b) 16 24 40 56 72 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (a) Schematic experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Circularly po- larized laser induces non-trivial Berry-curvature in the driven flat-bands, resulting in an anomalous Hall conductivity σxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A dielectric and metallic gate screen Coulomb interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b) Anomalous Hall conductivity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' drive amplitude E for various dielectric constants ϵ and a gate distance of d = 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The conductivity features a rapid drop (shaded) with E be- low the critical amplitude E∗ (dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Lower panel focuses on the strongly interacting ϵ = 16, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, E0 = ℏvF L−1 M /(eLM) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='177 × 104 V/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (c) The critical ampli- tude vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' cph/v0 eff, where v0 eff = veff(0) is an effective electron velocity defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The enlarged red circle denotes E∗ corresponding to the dashed line in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' rated by a large energy gap from all other bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We consider a circularly polarized laser beam of vector po- tential A(t) = (E/Ω)[cos(Ωt)ˆx − sin(Ωt)ˆy] with electric field amplitude E and angular-frequency Ω that couples to the electrons by minimal coupling k → k + eA(t)/ℏ, giving rise to the time-periodic Hamiltonian ˆH0(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The periodic Hamiltonian ˆH0(t) gives rise to Flo- quet eigenstates |Φ(ξ) kα(t)⟩ with quasienergies ε(ξ) kα with |ε(ξ) kα| < 1 2ℏΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We consider circularly polarized THz laser with W ≤ ℏΩ < 2W corresponding to a single photon resonance within the central TBG bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In particular, we consider TBG at a near-magic twist an- gle of θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='13◦, whose Fermi velocity vF ≈ 17 km/s (corresponding to a bandwidth of W = 5 meV in the Bistritzer-MacDonald model [20, 32]) is comparable to phonon speeds in TBG [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The drive with an angular- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='02248v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='mes-hall] 5 Jan 2023 2 (a) (b) (c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='25 32 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (a) Schematic of the flat bands in a moir´e system as used in the numerical calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A drive of frequency Ω resonantly couples states along resonance rings (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b) Undriven spectrum of TBG along a line in the BZ (orange) demonstrated in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The dashed frame indicates the optically-active central bands corresponding to ν = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (c) Berry curvature B(ξ) k+ in the upper Floquet band, with the blue color intensity proportional to tanh(2B(ξ) k+/L2 M) (see color bar) so that the B(ξ) k+ peaks are more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Dashed lines indicate areas enclosing B(ξ) k+ peaks at the Dirac points and resonance ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (d) The periodic quasienergy Floquet spectrum of the driven system, having two central bands shown in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The Floquet spectrum exhibits the upper (α = +) and lower (α = −) Floquet bands, separated by off-resonant gaps ∆K at the Dirac K and K′ points, and a Rabi-like gap ∆R along the resonance ring [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' frequency Ω = 5 meV/ℏ mixes the two central bands ν = ±1, resulting in the quasienergies ε(ξ) kα, with an up- per and lower Floquet bands denoted by α = ± (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2(d)) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The drive opens off-resonant gaps of size ∆K ≈ 2e2v2 F E2/ℏΩ3 at the Dirac points K and K′ of the moir´e Brillouin zone and a Rabi-like gap of ∆R ∼ V along the resonance ring, which is the ring on the k-plane satisfying E(ξ) k+ − E(ξ) k− = ℏΩ (see the green rings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2(a, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, vF is the Fermi velocity of the undriven band structure, V is the energy scale of the drive, and the expression for ∆K comes from the Van-Vleck pertur- bative expansion [24, 29, 31, 33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The key components for stabilizing Floquet many- body states are phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Electrons interact with a low- temperature bath of longitudinal TBG acoustic phonons: ˆHel-ph = � k,q,G ν,ν′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='ξ M νν′ξ k (q, G)ˆc(ξ)† k+q+G,ν′ˆc(ξ) kν (ˆb† q + ˆb−q) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (1) Here, G is a moir´e Brillouin zone reciprocal lattice vector, and M νν′ξ k (q, G) = D � ℏcphq/(√2AMρcph)⟨ξν′k + q + G|ξνk⟩ is the phonon-electron matrix element with defor- mation potential D = 25 eV, moir´e unit cell area AM = √ 3L2 M/2, lattice vector length LM = a/[2 sin(θ/2)], monolayer graphene density ρ, and monolayer lattice vec- tor length a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='246 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The operator ˆb† q creates an acoustic phonon mode of momentum q with amplitude q, speed cph, and energy ℏcphq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Note that the speed of sound cph in TBG is roughly the same as that in monolayer graphene [36], but the small Brillouin zone size in TBG folds the acoustic phonon dispersion into many branches [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The form-factor ⟨ξν′k + q + G|ξνk⟩ in the matrix element captures the decreasing coupling of electrons to the folded phonon branches with large G [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We also include screened electronic interactions: ˆHel-el = � k1,k2 q,G ν1,ν2,ξ V ν1,ν2,ξ k1,k2 (q, G)ˆc(ξ)† k1+q,ν1ˆc(ξ)† k2−q,ν2ˆc(ξ) k1,ν1ˆc(ξ) k2,ν2, (2) where V ν1,ν2,ξ k1,k2 (q, G) = Vb(q+G)(1−e−|q+G|qd)⟨ξν1k1 + q+G|ξν1k1⟩⟨ξν2k2 −q−G|ξν2k2⟩ contains the screened Coulomb potential that accounts for a dielectric of thick- ness d = 1 nm and dielectric constant ϵ which separates a metallic gate from the TBG layers (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, Vg(q) = e2/(2πϵqAM) is the bare Coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Our analysis focuses on the electron dynamics in its flo- quet basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We treat the interactions ˆHel-ph and ˆHel-el as weak perturbations that scatter electrons between single- particle Floquet states [1–3, 25, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The occupation probability F (ξ) kα (t) = ⟨ ˆf (ξ)† kα (t) ˆf (ξ) kα (t)⟩ is then described by the Floquet-Boltzmann Equation (FBE) [3, 38], ˙F (ξ) kα (t) = Iel-ph kα [{F (ξ) kα (t)}] + Iel-el kα [{F (ξ) kα (t)}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (3) Here, ˆf (ξ)† kα (t) creates a single-particle electron state |Φ(ξ) kα(t)⟩, and Iel-ph kα and Iel-el kα are respectively the electron-phonon and electron-electron collision integrals, evaluated by the Fermi golden rule (see Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' for FBE details [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The steady-state distribution yields ˙F (ξ) kα = 0, and the steady-state coherences between the Floquet states ⟨ ˆf (ξ)† kα (t) ˆf (ξ) kα′(t)⟩ for α ̸= α′ are suppressed as long as τ el scat, τ ph scat ≫ ℏ/∆R, ℏ/∆K, where τ el scat and τ ph scat are the electronic and electron-phonon scattering times, respectively [38, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='—To probe the electronic dynam- ics induced by the laser, we study the anomalous conduc- tivity in the steady-state of the system [10, 22–26, 41–43] σxy = 2e2 h � α,ξ=± � d2k (2π)2 B(ξ) kαF (ξ) kα , (4) which averages the product of Berry curvature [10, 31, 44] B(ξ) kα(t) = Ω π � 2π/Ω 0 dt Im⟨∂kxΦ(ξ) kα(t)|∂kyΦ(ξ) kα(t)⟩, (5) and the steady-state fillings, F (ξ) kα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Without the drive, TBG has only fragile topology with no anomalous trans- port and σxy = 0 at charge neutrality [45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A finite σxy at the charge neutrality point, therefore, indicates a laser-induced change in the band topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3 Our main finding is that σxy can be controlled by the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' It features a rapid drop as a function of the amplitude of the drive, E, near the critical amplitude E∗, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' This strong dependence on the ex- ternal field indicates profound changes in the electronic steady-state distribution as the drive amplitude changes across E = E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Furthermore, as we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(c), this strong amplitude-dependence arises only when the undriven effective electronic velocity v0 eff is close to the speed of sound cph in TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Such a condition is unique to TBG near the “slow-electron” regime [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We also expect electrons in TBG to be decoupled from phonons in the screening medium, since typical screening media such as hBN have speeds of sound much larger both cph and v0 eff in TBG [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Phenomenological analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='—The origin of the strong dependence of σxy on the drive amplitude near E = E∗ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(b)), can be understood by focusing on key pro- cesses affecting σxy in the steady-state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' These processes involve momentum states with large Berry curvature B(ξ) kα near the Dirac points (K, K′) and the resonance ring (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We assume that the steady-state occupation of the upper Floquet band (α = +) and valley index ξ near K, are uniform F (ξ) k+ = F (ξ) K+, for k ∈ SK, which is a small circle enclosing the full-width half maximum of the Berry curvature peak at K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The steady-state occupation emerges as a bal- ance between the total incoming and outgoing rate ˙F (ξ) K+|in, ˙F (ξ) K+|out, into SK from Sin and into Sout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Sin- gle phonon emission connects the upper band Sin with SK which is the dominant contribution to ˙F (ξ) K+|in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In- deed, the two regions are connected by the phonon light- cone (see Fig 3(a)) This rate is roughly ˙F (ξ) K+|ph,in = Rin(1 − F (ξ) K+)F (ξ) in , where F (ξ) in is the average upper Flo- quet band occupation in Sin, and Rin is the average in- trinsic scattering rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Importantly, Rin is proportional to the momentum-space area of Sin, denoted as Ain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We estimate Ain by counting the upper Floquet band states that may scatter to SK by electron-phonon inter- actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Hence, it is the intersection between the upper Floquet band and phonon light-cones originating any- where within SK (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' As the Floquet gaps ∆R and ∆K widen with E, the Floquet bands become nar- rower [13, 17, 18], and the intersection area Ain shrinks, eventually vanishing at E = E∗ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The criti- cal strength E∗ is defined by the condition veff(E∗) = cph, where veff(E) = max k′ (ε(ξ) k′+ − ε(ξ) K+)/|k′ − K| (6) is the electronic velocity near the K point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' By estimating veff(E), one can show that E∗ ∝ [1 − cph/v0 eff]γ for small 1−cph/v0 eff, where γ depends on the quasienergy structure and v0 eff ≡ veff(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' One can also show that Ain ∝ max(E− E∗, 0) as E → E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (See the Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=') Similarly, the phonon-mediated outgoing rate is roughly ˙F (ξ) K+|ph,out = RoutF (ξ) K+(1 − F (ξ) out), where F (ξ) out is the upper Floquet band average occupation in Sout, and Rout is the average intrinsic rate, which is proportional to Aout = � Sout d2k, with Sout the momentum region enclosing intersections between the α = − quasienergy band with phonon light cones originating from states in SK (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' However, unlike Ain, Aout does not vanish as E → E∗ and instead expands as E increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Electronic interactions and photon-mediated Floquet- Umklapp (FU) processes introduce additional terms in the rate equation that depend smoothly on E and are roughly uniformly spread in momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We thus add an incoming rate ˙F (ξ) K+|r,in = Γin(1 − F (ξ) K+) and outgoing rate ˙F (ξ) K+|r,out = ΓoutF (ξ) K+ with constant Γin and Γout to the rate equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Note the strength of FU proce- ses is weaker than Rin and Rout by factors of roughly (vF eE/Ω2)2n, where |n| > 1 is the number of photons emitted or absorbed [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' These processes also impart large phonon momentum transfers which the form factor in the electron-phonon matrix element suppresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The total rate equation is then: ˙F (ξ) K+ = ˙F (ξ) K+|ph,in − ˙F (ξ) K+|ph,out + ˙F (ξ) K+|r,in − ˙F (ξ) K+|r,out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (7) At steady-state ˙F (ξ) K+ = 0, and we find F (ξ) K+ = RinF (ξ) in + Γin RinF (ξ) in + Rout + Γin + Γout .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (8) Since Rin ∝ Ain, it decreases as a function of E and even- tually shrinks to zero for E ≥ E∗ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(b) for numer- ical verification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We expect a similar E-dependence of F (ξ) K− and thus σxy, yet smeared by additional scattering rates appearing in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 8, as is demonstrated numerically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We define the visibility of the sharp feature in the Hall conductivity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 9, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Scattering rates about K′ and K points, related by a π/3 rotation, are equal, and the two contribute equally to σxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The resonance ring vicinity, SR (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2(c)), yields a similar E-dependence, with a much lower critical field E∗ R due to different effective electronic velocities in near the resonance ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Indeed, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(b) and 3(b), E∗ R is not visible for the drive strengths plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Finally, we find the contribution of the lower Floquet band to σxy using particle-hole symmetry, F (ξ) k− = 1 − F (ξ) −k+ and B(ξ) k− = −B(ξ) −k+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The arguments above allow us to repro- duce qualitatively the numerically obtained result (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3) of the suppression of σxy with drive strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='—The results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(b-d) were derived from a simplified toy model describing TBG as a tight-binding hexagonal lattice, similar to graphene [49], but with parameters tuned to match the Fermi velocity and the Brillouin zone size of TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' This model misses some subtle details, but captures well the interplay be- tween the electron and phonon velocities, and the large 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='99 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='03 (a) (b) (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (a) Schematics of the Floquet spectrum and one of the phonon light-cones that originates from the area SK in the α = + band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The intersection between the α = + (α = −) band and all cones centered in SK form Sin (Sout).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' As E → E∗, the area of Sin shrinks to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b-d) Numerical verification of the phenomenological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b) The area of Sin, Ain, as a function of E for three values of cph/v0 eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (c) The average occupation of states in SK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (d) Anomalous Hall conductivity σxy for the same parameters as in (b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' At the critical drive E∗ (dashed lines), Ain, F (ξ) K+, and σxy plateau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Berry curvature at the Dirac points and resonance ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The model represents only the central ν = ±1 bands of the undriven bandstructure, but since the low drive an- gular frequency Ω is only resonant to these flat bands, we can ignore the |ν| > 1 dispersive bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' This ap- proximation is valid only when θ is near the magic an- gle where the |ν| > 1 bands are far from the ν = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the Supp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' [39], we also present the numer- ical analysis of a continuum model without electron- inc interactions [20, 32], which yields qualitatively sim- ilar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the toy model, v0 eff = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='9 km/s, and we select cph ∈ [17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='9 km/s, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='4 km/s] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the range cph < v0 eff, the drive induces the regime cph > veff(E) for E > E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Note that calculating the form factor ⟨ξν′k + q + G|ξνk⟩, relies on the contin- uum model [37, 39], and thus needs to be introduced by hand in the toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We therefore approximate ⟨ξν′k + q|ξνk⟩ ≈ δν,ν′e−l2 wq2/4, with lw ≈ LM/(5 √ 3) representing the spatial extent of the Wannier orbitals localized to TBG layer alignment sites [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' First, we show how solving the FBE (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3) for the steady-state distribution verifies the phenomenological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' For this purpose, we need only take the non- interacting limit by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3 for F (ξ) kα with ϵ → ∞, such that Iel-el kα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The left two quadrants of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 4(a) show the non-interacting steady-state distributions for a low phonon bath temperature of 1 K and phonon speed cph = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='98v0 eff in the E > E∗ and E < E∗ cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the E > E∗ case (left top quadrant), the K, K′ points, painted blue, have a reduced occupation relative to the E < E∗ case (left bottom quadrant), due to the suppres- sion of the incoming scattering rates into SK,K′ (as in the phenomenological model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(c) shows the occupa- tion near the K point, F (ξ) K+, as a function of E for three different values of cph/v0 eff and verifies that Ain, F (ξ) K+, and σxy plateau at the same critical amplitude E = E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Next, we numerically quantify the strength of Coulomb screening necessary to stabilize the steady state distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Now, we include Iel-el kα ̸= 0, setting the dielectric ϵ to a finite value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' On the right top and bottom quadrants of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 4(a), we show the resulting steady-state occupa- tions, which have noticeably higher entropy than the non- interacting case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The entropy of the steady-state occu- pation depends on the balance between electron-phonon cooling processes and electron-electron heating processes and is therefore sensitive to ϵ and the screening gate dis- tance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(b) shows the steady-state conductivity as a function of E for various ϵ (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2), with deformation potential D = 25 eV and d = 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The slope ∂Eσxy in the E < E∗ regime where σxy is less steep as ϵ decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' To quantify this, we define a visibility parameter V = −∂Eσxy/[(e2/h)/E0] (9) where ∂Eσxy denotes an average of ∂Eσxy across a window E∗ − δE < E < E∗ where δE/E0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='45 denotes a range of amplitudes in which σxy decreases rapidly (see shaded region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 4(b), we show V as a func- tion of d and ϵ, again keeping D = 25 eV and keeping δE constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The dark red region, where ϵ and d are small, represents a regime where electron-electron interactions dominate, and the system reaches a hot steady-state with low visibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' in the blue and green regions, phonon cool- ing dominates and a low-entropy steady-state appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We draw lines comparing the electron-phonon energy scale Vph ≈ D � ℏcph|K|/(√2AMρcph)e−|K|2l2 w/4 to the electron-electron energy scale Vel ≈ e2/(2πϵAM|K|)(1 − e−2|K|d)e−|K|2l2 w/2 (see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1 and 2) as evaluated for a characteristic momentum transfer magnitude |K| = 4π/(3LM) which we choose to be that of the K point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Conclusion—TBG is a remarkable system where the Fermi velocity is comparable to the speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Upon THz-laser driving, the electronic population dynamics exhibits bottlenecks for electron-phonon scattering into high-Berry curvature Floquet states, which strongly af- fects the anomalous Hall transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' These bottlenecks, we show, can be sensitively controlled by the drive am- plitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' If the undriven effective electron speed is faster than sound v0 eff > cph, a drive with E > E∗ induces the opposite regime veff(E) < cph, decouples the electrons from the phonons, and suppresses the Hall conductivity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We also find that screening gates affect the electronic steady-state and the anomalous transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A strong drive field-dependence of σxy arises for efficient Coulomb screening, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=', by a close-by gate or a strong dielectric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Recent experimental advances in Floquet en- gineering [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' and THz laser sources [50],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' show that our ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='AB/3icbVDLSsNAFL3xWesrKrhxM1g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='EVzUpRV0W3bisYB+QxDKZTtqhkwczE6HELvwVNy4UcetvuPNvnLRZaOuBgcM593LPHD/hTCrL+jaWldW19ZLG+XNre2dXNvy3jVBDaIjGPRdfHknIW0ZitNuIigOfU47/ug69zsPVEgWR3dqnFAvxIOIBYxgpaWe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='eiGWA0J5l7glzk1Ol97Wzo9cyKVbWmQIvELkgFCjR75pfbj0ka0kgRjqV0bCtRXoaFYoTSdlNJU0wGeEBdTSNcEil03zT9CJVvoiIV+kUJT9fdGhkMpx6GvJ/O0ct7Lxf8J1XBpZexKEkVjcjsUJBypGKUl4H6T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='FCi+FgTATWREZYoGJ0pWVdQn2/JcXSbtWtc+r9dt6pXFV1FGCIziGU7DhAhpwA01oAYFHeIZXeDOejBfj3fiYjS4Zxc4B/IHx+QONLJUo ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='V [4e2/h] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (a) Left column: electronic steady-state occupation of the upper Floquet band in the non-interacting case (ϵ = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Right column: steady-state occupation for the same parame- ters but with finite interactions (ϵ = 24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Top row: the E > E∗ case where the population at K is depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Bottom row: the E < E∗ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b) The visibility V defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 9 as a function of ϵ and the screening gate distance d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The blue/green region represents a regime where electron-phonon interactions dom- inate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Dashed lines represent points with fixed ratio Vph/Vel of electron-phonon Vph to electron-electron Vel characteristic interaction strengths (see text for definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' predicitions should be accessible experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Higher frequency and stronger drives, such as those in the UV-visible or X-ray regimes, are a subject of fu- ture work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Such a theoretical analysis must account for dispersive bands, which mix significantly with the flat bands under high-frequency drives [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' These high- frequency drives could further reduce heating by fa- cilitating fewer electron-electron Floquet-Umklapp pro- cesses, thus lowering the necessary dielectric constant for the formation of a non-trivial steady-state [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The drives would also make use of the enhanced electron- phonon Umklapp cooling processes which arise due to the tightly-localized electronic Wannier orbitals in TBG [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the low-frequency, resonant drive limit considered in this work, the momentum electron-phonon Umklapp processes are also Floquet-Umklapp processes that are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Another interesting direction involves sym- metry broken phases that can arise in the steady-state of a strongly coupled TBG [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We leave these exciting directions to future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We thank Netanel Lindner, Mark Rudner, Or Katz, Gaurav Gupta, Seamus O’Hara, Jason Alicea and Alex Thomson for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' grate- fully acknowledges support from the DOE NNSA Stew- ardship Science Graduate Fellowship program, which is provided under cooperative agreement number DE- NA0003960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' acknowledges support by the Gordon and Betty Moore Foundation’s EPiQS Initiative, Grant GBMF8682, start-up funds from Florida State University and the National High Magnetic Field Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The National High Magnetic Field Laboratory is supported by the National Science Foundation through NSF/DMR- 1644779 and the state of Florida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' are grate- ful for support from the Simons Foundation and the In- stitute of Quantum Information and Matter, as well as support from the NSF DMR grant number 1839271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' This work is supported by ARO MURI Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' W911NF- 16-1-0361, and was performed in part at Aspen Center for Physics, which is supported by National Science Founda- tion grant PHY-1607611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Esin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Rudner, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Lindner, Floquet metal-to-insulator phase transitions in semiconductor nanowires, Science Advances 6, eaay4922 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' [2] I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1 Supplemental Material: Optical Control of Slow Topological Electrons in Moir´e Systems Christopher Yang, Iliya Esin, Cyprian Lewandowski, and Gil Refael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' DETAILS OF THE MODELS In both the toy and continuum models, we take the undriven Hamiltonians H(k) and obtain the time- dependent Hamiltonian H(k, t) via minimal coupling k → k + eA(t)/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, A(t) = A[cos(Ωt)ˆx − sin(Ωt)ˆy] (S1) is the magnetic vector potential of circularly polarized laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We can expand the time-dependent eigenstates of the Hamiltonian in a Floquet-Bloch basis [1]: |ψkα(t)⟩ = e−iε(ξ) α t/ℏ|Φm kα(t)⟩, (S2) where r is the position vector, |Φm kα(t)⟩ is periodic in time (|Φm kα(t)⟩ = |Φm kα(t+2π/Ω)⟩), ε(ξ) α are the quasiener- gies plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2(d), and α enumerates the Floquet quasienergy bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' To determine the Floquet-Bloch ba- sis, it is easiest to expand the time-dependent |Φm kα(t)⟩ in terms of time-independent Fourier harmonics |φm kα⟩, |Φm kα(t)⟩ = � m e−imΩt|φm kα⟩, (S3) take a Fourier transform the Hamiltonian, H(k, t) = � m e−imΩtH(m)(k), (S4) and solve the Schr¨odinger equation in the basis of Floquet harmonics: (ε(ξ) α + mℏΩ)|φm kα⟩ = � m′ H(m−m′)(k)|φm′ kα⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S5) In the following subsections, we detail the exact form of the Floquet Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Tight binding Floquet toy Hamiltonian We use a rescaled, two-band tight binding model for graphene to replicate the flat conduction and valence bands of TBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the rescaled Hamiltonian Htoy(k) = � 0 hk h∗ k 0 � , (S6) hk = W 3 � j eik·δj, (S7) we choose long hopping vectors δj = LM/ √ 3[sin(2πm/3)ˆx + cos(2πm/3)ˆy], (S8) with LM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='246 nm/(2 sin θ/2), and a narrow band- width W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The corresponding rescaled eigenenergies and Bloch states are Eν(k) = ν|hk|, (S9) and |νk⟩ = 1 √ 2 � νeiarg(hk) 1 � , (S10) respectively, with ν = ±1 enumerating the flat Bloch bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Following [2], we perform minimal coupling, which turns the functions hk into time-dependent quantities with Fourier transforms h(n) k = 1 2π/Ω � 2π/Ω 0 hk+eA(t)/ℏe−inΩtdt = � j teik·δjeinφjJn(− ˜E), (S11) where ˜E is the dimensionless drive strength ˜E = eLM √ 3ℏ A = eLM √ 3ℏ E Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S12) the phase angles are φ0 = π/2, φ1 = −5π/6, and φ2 = −π/6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' and Jn(z) = 1 2πin � 2π 0 eiz cos θeinθdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S13) The Fourier-transformed Hamiltonian is H(n) toy(k) = � 0 h(n) k h∗(n) k 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S14) Note that h∗(n) k = � j te−ik·δjeinφjJn( ˜E) (S15) is the Fourier transform of the conjugate of hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In sim- ulations, we generally truncate the Fourier Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S5) to −12 ≤ m ≤ 12, so that we account for a sufficient number of high-order Floquet-Umklapp pro- cesses in the Floquet-Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' For both the undriven and Floquet Hamiltonians, we also perform a gauge transformation, replacing h(n) k → ie−ik·δ0h(n) k and hk → −ieik·δ0hk to make the Hamiltonians periodic by shifts of k → k+G, where G is a reciprocal lattice vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='02248v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='mes-hall] 5 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (a) The quasienergy band structure of the toy model with for the parameters used in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b) The quasienergy band structure of the continuum model at val- ley ξ = +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In both panels, the first Floquet Brillouin zone is shaded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' II for details and justification for the parameters we have used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Continuum Model Floquet Hamiltonian The undriven continuum model for TBG [3] describes the bandstructure of TBG near the valley ξ = ±1 of the monolayer graphene Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Its Hamiltonian Hξ = � H1 U † U H2 � (S16) is diagonalized in the basis ψnk = (ψA1 nk, ψB1 nk, ψA2 nk, ψB2 nk)T with ψX nk(r) = eik·r � G CX nk(G)eiG·r (S17) where X = Al, Bl represents sublattice A or B degree of freedom in layer index l = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S16, Hl are the monolayer graphene Hamiltonians, which, in close vicinity of the ξ = ±1 valleys, resemble Dirac cones: Hl = −ℏvml F � R(lθ/2)(k − K(l) ξ ) � (ξσx, σy) (S18) where R(ϕ) is the 2×2 rotation matrix, vml F is the mono- layer Graphene Fermi velocity, and Kl ξ is the Dirac point of layer l at valley ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The interlayer coupling is U = �u u′ u′ u � + � u u′ν−ξ u′νξ u � eiξG1·r + � u u′νξ u′ν−ξ u � eiξ(G2+G3)·r (S19) Using minimal coupling, we obtain time-dependent monolayer graphene Hamiltonians, with Fourier trans- form H(n) l = −ℏv � R(lθ/2) � k + e ℏ 1 2E[(δn,1 + δn,−1)ˆy − i(δn,−1 − δn,1)ˆx] − K(l) ξ �� (ξσx, σy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S20) Then, H(n) ξ = � H(n) 1 U †δn,0 Uδn,0 H(n) 2 � (S21) is the Fourier transform of the continuum model Hamilto- nian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' For the continuum model, we truncate the Floquet Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S16) to −6 ≤ m ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Upon diagonalizing the Floquet Hamiltonian, we ob- tain a large number of Floquet states per energy interval [−ℏΩ/2, ℏΩ/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We select two states per k-point whose spectral weights A0 α(k) = |⟨φ0 kα|φ0 kα⟩|2 are large (which makes their contribution to the Floquet-Boltzmann equa- tion most important, see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Quasienergy Bands In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' II, we provide and motivate the choices of physical parameters that we use in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S1, we preview the quasienergy bands for our choice of toy and continuum model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' CHOICE OF PHYSICAL PARAMETERS First, we present the physical parameters we use for the electronic Hamiltonian in the TBG continuum model (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' I B for the Hamiltonian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We consider the non- interacting continuum model [3, 4] at a near-magic twist angle of θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='13◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The bandwidth of the central bands at this angle is W ≈ 5 meV, and a perturbative expan- sion of the Hamiltonian around the Brillouin zone Dirac points [4] estimates the Fermi velocity as vF (θ) = vml F (1 − 3β2)/(1 + 3β2(1 + η2)), (S22) where β = u′/(ℏkθvml F ) and η = u/u′ with vml F = 8 × 105 m/s, kθ = 4π/(3LM), u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='0797 eV, and u′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='0975 eV [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S22 predicts that the Fermi velocity at the chosen twist angle is vF = 27 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' How- ever, the derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S22 approximates that H(n) l is roughly θ-independent and tends to overestimate vF (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 4 inset in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We can obtain a better estimate by numerically calculating the Fermi velocity along the path K-M in k-space of the ν = +1 band in the ξ = +1 valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (This is the direction of maximum Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=') The estimate yields vF = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='5 km/s, and we hereafter use this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In our Floquet Hamiltonian, we use a laser angular-frequency of Ω ≈ W/ℏ ≈ 5 meV/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Second, we present the parameters we use for the electronic Hamiltonian of the TBG two-band toy tight binding model (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' I A for the Hamiltonian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We choose our toy model Fermi velocity, frequency, and twist angle to roughly match those of the continuum model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Specifically, we use a twist angle of θ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='13◦ 3 and choose W = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1 meV so that the Fermi velocity vF = WLM/(2 √ 3ℏ) = 17 km/s roughly matches that of the continuum model at the same angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the toy model Floquet Hamiltonian, we choose Ω ≈ 5 meV/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Third, we discuss the parameters we use for the TBG phonons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' For both the continuum and toy models, we consider phonons speeds in the range of cph ∈ [17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='9 km/s, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='4 km/s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the toy model, v0 eff = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='9 km/s, and, in the continuum model, v0 eff = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='5 km/s, so the range of cph we choose covers the regime cph < v0 eff, in which the drive induces the op- posite regime cph > veff(E) when E > E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We also use the same phonon bath temperature of Tph = 1 K and Wannier orbital width lw = LM/(5 √ 3) for the toy and continuum model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Please see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' X for details of the numerical k-point grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' ANOMALOUS HALL CONDUCTIVITY CALCULATIONS FOR THE CONTINUUM MODEL In this section, we repeat the calculations in the main text on the TBG continuum model [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We consider the non-interacting limit, setting ϵ → ∞ so that Iel-el kα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' First, we discuss differences in the bandstructure and topology at valleys ξ = +1 and ξ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The circularly polarized laser opens a gap at the Dirac points, ∆K, ef- fectively adding a mass term ξ∆Kσz to the Hamiltonian (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' I B and [5] for a derivation) in the vicinity of the Dirac points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Because the sign of the mass term de- pends on ξ, the ξ = ±1 superlattice valley contributions to σxy do not trivially cancel to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In fact, in recip- rocal space, the Berry curvature and occupations near (a) (b) [km/s] 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (a) Left: the steady-state occupation of the upper Floquet band in valley ξ = +1 of the continuum model [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Right: the Berry curvature of the same band, which peaks near the Dirac points and the resonance ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b) The anoma- lous Hall conductivity σxy as a function of drive strength E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' ξ = +1 are simple π/3 rotations of those in ξ = −1, so σxy = 4e2 h � α=± � MBZ d2k (2π)2 B(+1) kα F (+1) kα .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S23) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S2, we show the steady-state and σxy for the continuum model calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Note that to simplify the calculations, we use the same effective form factor ⟨ξν′k+ q|ξνk⟩ ≈ δν,ν′e−l2 wq2/4 as the toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' DIRECT VARIATION OF THE PHONON SPEED cph Throughout the main text, we use the drive strength E to control electron speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We could achieve similar results by keeping E fixed and varying cph instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S3 shows the variation of σxy as a function of cph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The curves resemble the dependence of σxy on E in the main text (see, for e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' FORMAL DEFINITION, NUMERICAL EVALUATION, AND PHENOMENOLOGICAL MODEL OF Ain As described in the main text, an patch Sin shaped as an elliptical annulus (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(a)) with area Ain in momentum space vanishes as E → E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, we provide a formal definition of Ain and explain how we estimate its dependence on E numerically and analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Formal Definition Let us first define Ain formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Consider a family of phonon cones centered throughout SK, the circular patch enclosing a K-point in the quasienergy spectrum (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Suppose that a subset of the phonon cones are centered throughout a small quasienergy window dεk+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Anomalous Hall conductivity of the toy model as a function of the ratio cph/v0 eff for three different drive field strengths E/E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The same electron-phonon decoupling pro- cess is visible as σa xy plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 4 The k-space area of states dAin containing intersections of the cones with the upper Floquet band is dAin = dεk+ � s=± � d2k′ δ(εk+ − εk′+ + sℏcph|k′ − k|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S24) Next, we integrate over εk+ contained in SK to obtain Ain = � dA = � k∈SK d2k 1 D(εk+)× × �� s=± � d2k′ δ(εk+ − εk′+ + sℏcph|k′ − k|) � , (S25) where D(ε) = � α � d2k (2π)2 δ(ε − εkα) (S26) is the density of states in the quasienergy band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Exploiting the circular shape of SK, � k∈SK d2k ≈ � d2k Θ(|k − K| − kp) (S27) where kp is the radius of the circular area AK of SK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Lastly, we calculate an approximate expression for kp, the radius of AK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the vicinity of the Dirac cone, the Hamiltonian is HK(k, t) = d · σ, (S28) where d = ℏvF ξkxˆx+ℏvF ky ˆy +ξ∆KE2ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' VIII for a detailed derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=') The z-component of the Berry curvature is Bz kα = α dz 2|d|3 = αξ ∆K [(ℏvF k)2 + ∆2 K]3/2 (S29) where dz = ξ∆K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' At the half-maximum, Bz kpα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='5Bz 0α, so kp = (22/3 − 1)1/2 ∆K ℏvF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S30) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Numerical Estimate To generate the values of Ain we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(b), we evaluate the integrals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S25 on a finite-sized grid of k-points, smearing the step function by replac- ing Θ(|k − K| − kp) → [e(|k−K|−kp)/σk + 1]−1, where σk = 2π/(LMN) is the grid spacing between k-points on an N × N Monkhorst-Pack grid (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Thus, we approximate Ain ≈ � k [e(|k−K|−kp)/σk + 1]−1 1 D(εk+)× × �� s=± � k′ δ(εk+ − εk′+ + sℏcph|k′ − k|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S31) (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (a) The intersection SK (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(a)) as viewed on the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The outer radius along the path KR is hb(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b) Quasienergy (pink) along the path KR, with the phonon light cone (grey) that determines the outer radius of Ain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The intersections k+ and k− between the cone and the upper Floquet band determines hb(E) = k+ − k−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' For more information on how we approximate the Dirac Delta function on the grid, please see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' X A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Phenomenological Model In this section, we prove that the intersection area Ain ∝ max(E∗ − E, 0) as E → E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The shape of Ain is an elliptical annulus as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Let us use hb(E) and wb(E) respectively to denote the outer major and minor axis radii of the elliptical annulus (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S4(a) and S5(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the following sections, we begin by generating analytical estimates of hb(E) and wb(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Estimate of hb First, let us consider a slice of the upper Floquet band in k-space from the K to the resonance ring (R) along the direction of hb(E), as we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Let us define a one-dimensional momentum component q along the path K-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We sketch a phonon light cone (grey) originating from a point (yellow) in SK that determines the outer radius of Ain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The phonon cone intersects with the quasienergy at points k+ and k−, and the outer radius of Ain is therefore hb(E) = k+ − k−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' First, consider the undriven limit E = 0, where the gaps ∆R = 0 and ∆K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We choose some point qm such that k− < qm < k+ and series expand the energy E(q) of the undriven system around qm: E(q) ≈ E(qm) + E′(qm)(q − qm) + 1 2E′′(qm)(q − qm)2 = a2q2 + a1q + a0, (S32) hs 25 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (a) Width of the intersection SK, wb(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (b) Circu- lar coordinate system with arc length w (increasing counter- clockwise) that we use to determine wb(E) = w+ − w−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' where a2 = E′′(qm)/2, a1 = E′(qm) − E′′(qm)qm, and a0 = E(qm)−E′(qm)qm+E′′(qm)q2 m/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' As we increase E, the gaps ∆K and ∆R widen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Let us write the quasienergy in the vicinity of qm as ε(q) ≈ f(E)E(q) + ∆K 2 (S33) where f(E) ≤ 1 is a scaling factor that decreases as E increases and accounts for band flattening due to ∆K and ∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Let f −1 = 1 − b1 ˜E − b2 ˜E2, (S34) where b1 ≥ 0 and b2 ≥ 0 are constants dependent on the exact bandstructure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=', how the widening of ∆K and ∆R with E affects the bandstructure near qm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The roots of the equation E(q) = ∆K/2 + ℏcphq are k±, and we may write the equation as a2q2 + a1q + a0 = fℏcsq, (S35) from which we find that hb = k+ − k− = � (a1 − fℏcs)2 − 4a2a0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S36) Solving for E∗ through the equation hb = 0, and then series expanding the expression (a1 − fℏcs)2 − 4a2a0 in powers of small E−E∗, we find that (a1−fℏcs)2−4a2a0 ∼ E∗ − E, so hb ∼ √ E∗ − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Estimate of wb To estimate wb (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S5(a)), we define a circular coordinate system shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S5(b) whose origin is the K point and arc length w is zero along the KR slice, increasing counterclockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The quasienergy ε(w) along the circle perimeter varies with w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' let us approximate ε(w) ≈ ℏΩ/2 − (d0 + d2w2), (S37) using some fitting parameters d0 and d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (We assume that w = 0 is at local maximum of ϵ(w), so there is no linear term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=') Roughly, wb = w+ − w−, where we find w+ and w− by finding the roots of the equation ℏΩ/2 − ∆(w) = fℏcsqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S38) Here, once again, we use the factor f in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S34 to account for band flattening as E increases from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' So, wb = w+ − w− = 2 � (fℏcsqm + ℏΩ/2 − d0)/d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S39) Solving for E∗ by setting wb = 0 and series expanding fℏcsqm + ℏΩ/2 − d0 in powers of E, we find that wb ∼ √ E∗ − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Estimate of Ain In the limit E → E∗, the elliptical annulus with fi- nite thickness collapses into a filled ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Thus, in the limit E → E∗, we estimate that Ain = πhb(E)wb(E) ∝ max(E∗ − E, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' PREDICTING E∗ FOR THE TOY MODEL Here, we use the quasienergy dispersion of the toy model to predict E∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' By writing an approximate, ana- lytic expression for veff(E) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 6), we can find E∗ using the relation veff(E∗) = cph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 6, veff(E) = (εk∗+ − εK+)/|k∗ − K| for some appropriately-chosen k∗ (dropping the superlattice valley index for notational simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' One can find nu- merically that k∗ does not shift significantly with Ω or E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We write an ansatz εk∗+ ≈ ℏv0 eff|k∗ − K| − ℏvF LM � f ′ 1 ˜E + f ′ 2 ˜E2� |k∗ − K| Ω/(2v0 eff), (S40) where f ′ 1 and f ′ 2 are fitting constants dependent on the quasienergy bandstructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, ℏvF /LM is the order of magnitude energy scale of the resonance ring gap ∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The dependence of εk∗+ on E arises predominantly from ∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The dependence is stronger when k∗ is close to the resonance ring, and we encode this behavior in the ratio |k∗ − K|/Ω/(2v0 eff), where Ω/(2v0 eff) is the k-space distance between the K point and the resonance ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Separately, we know that εK+ = ∆K/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 6 to infer v0 eff(E) = v0 eff − ∆K 2ℏ|k∗ − K| − 2ℏvF v0 eff LMΩ � f ′ 1 ˜E + f ′ 2 ˜E2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S41) We know that v0 eff ∝ vF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We also assume that |k∗ − K| does not change significantly with E, so it is independent of the drive and only dependent on the superlattice scale: 6 (b) (a) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Comparing numerical evaluation of E∗ (points) to an analytic fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We use the same fitting parameters f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='778, f1 = 0, and δ(N) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='006 for both panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' |k∗ − K| ∝ L−1 M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Thus, we can absorb some unknown coefficients into new coefficients f ′′ 1 and f ′′ 2 to obtain v0 eff(E) = v0 eff − ℏv2 F LMΩ � f ′′ 1 ˜E + f ′′ 2 ˜E2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S42) Upon solving for ˜E∗ from cs = veff(E∗), we find that ˜E∗ ≈ � LMΩ 3f2vF �� 1 − cph/v0 eff + f 2 1 − f1 � , (S43) where f1 and f2 are new, rescaled fitting constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Using the relation ˜E = eLME/( √ 3ℏΩ), we find E∗ ≈ ℏΩ3/2 f2eL1/2 M v1/2 F �� 1 − cph/v0 eff + f 2 1 − f1 � , (S44) As cph → v0 eff, E∗ ∝ (1 − cph/v0 eff)γ where γ = 1 (1/2) if f1 ̸= 0 (= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S6 for a fit for two different frequencies Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Finite grid size effects on an N × N Monkhorst-Pack grid (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' X) generate a small numerical error δ(N) that enters S44 as E∗ ≈ ℏL1/2 M Ω3/2 f2eLMv1/2 F �� 1 − cph/v0 eff + δ(N) + f 2 1 − f1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S45) To see this, let us consider the details of the finite-sized grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We impose energy conservation through a broad- ened Dirac Delta function (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' X A), which we model as a Gaussian function in energy with a tiny width √ 2σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1 · √ 2 · W 2N/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S46) (We motivate the choice of the prefactor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1 in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' X A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=') Since we avoid the high symmetry K point in our grids, the k-point with largest Berry curvature is, in fact, a point knear point shifted away from K by a small dis- tance in momentum space of |δk| = |knear−K| ≈ 1 2 Ω/(2ℏv0 eff) 2N/3 = Ω 4v0 eff(2N/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S47) (b) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='96 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Comparing the dependence of σxy on E for (a) the frequency considered in the main text and (b) a lower fre- quency where Floquet-Umklapp processes are stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Note that the frequency in panel (b) is inaccessible without gener- ating two-photon resonances in the continuum model due to the peaked shape of the ν = ±1 bands near the Γ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Comparison of the fitted ∆R and predicted ∆K in Equations S51 and S55 (solid lines) to those obtained from numerics (points), using ℏΩ = 5 meV in the toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, we fit ∆R with factors of f R 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='04 and f R 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='0184 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' This point is shifted in quasienergy by ℏvF |δk| relative to the actual K point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We can account for both of these effects by shifting εK+ → εK+ + δε, with δε = √ 2σ + ℏvF |δk| and solve veff(E∗) = cph to find Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S45 with δ(N) = δε/(ℏv0 eff|k∗ − K|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' DIFFERENT FREQUENCIES Reducing Ω below the value considered above will in- crease the ratio (vF eE/Ω2)2 and in turn strengthen Flo- quet Umklapp processes, modifying the shape of the σxy curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We demonstrate this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S7(b) for an angu- lar frequency Ω = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='135 meV/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' However, such a low- frequency regime is inaccessible in the continuum model (without generating two-photon resonances) due to the peaked shape of the continuum model ν = ±1 band near the Γ point, so we do not consider this lower (doubly- resonant) frequency regime in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 7 VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' GAP SIZES In this section, we estimate the size of the Floquet- induced gaps ∆K and ∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' By the rotating wave approx- imation, the Floquet-induced gap at the resonance ring, ∆R, is roughly proportional to the drive energy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' For a resonant drive that couples electronic states near the Dirac points, the drive energy is roughly vF eA/ℏ, (S48) as predicted by minimal coupling q → q+eA(t)/ℏ in the Dirac cone Hamiltonian HK(q) = ℏvF q · (ξσx, σy) (S49) with vF = WLM/(2 √ 3ℏ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (We always use perturbative drives that generally fall in the range of ˜E < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=') We expect that ∆R ≈ ℏvF LM ˜E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S50) Such an approximation works well for low-frequency res- onant drives that couple states near the Dirac points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' However, resonant drives with higher frequencies, like those used in the main text, couple states closer to the Γ- points of the TBG energy dispersion where the bands are nonlinear in q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In such a case, higher order (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=', O( ˜E2)) contributions (from O(q2) contributions of the band- structure) to ∆R become dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the present exam- ple, the energy of the tight binding model for graphene is quadratic in momentum near the Γ point, so we write an ansatz ∆R ≈ ℏvF LM (f R 1 ˜E + f R 2 ˜E2), (S51) and fit f R 1 and f R 2 to match ∆R obtained by numerically diagonalizing the Floquet Hamiltonian, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We can estimate the Floquet-induced K-point gap, ∆K, by considering the time-dependent Dirac Hamilto- nian HK(q, t) = ℏvF (ξqxσx + qyσy) + vF eA[ξ cos(Ωt)σx − sin(Ωt)σy].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S52) and performing a Van Vleck expansion [6–8] to obtain an effective Floquet Hamiltonian HK,eff(q) = H(0) K + [H(−1) K , H(1) K ] ℏΩ = HK + ξ e2v2 F A2 ℏΩ σz (S53) with H(n) K (q) = 1 2π/Ω � 2π/Ω 0 HK(q, t)e−inΩtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S54) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S53, we can extract ∆K = 2e2v2 F ℏΩ A2 = 6ℏv2 F L2 MΩ ˜E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S55) IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' FLOQUET BOLTZMANN EQUATION Here, we present the full expression for the Floquet- Boltzmann equation [9], which we copy below for conve- nience: ∂tFkα(t) = Iel-ph kα [{Fkα(t)}] + Iel-el kα [{Fkα(t)}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S56) The electron-phonon collision integral is Iel-ph kα [{Fkα}] = 2π ℏ � k′∈BZ � α′ � j � n |Gk′α′ kα (n, j)|2 × � � Fk′α′(1 − Fkα)N(ℏωj(k′ − k)) − Fkα(1 − Fk′α′)[1 + N(ℏωj(k′ − k))] � × δ(εk′α′ − εkα + ℏωj(q) + nℏΩ) + � Fk′α′(1 − Fkα)[1 + N(ℏωj(k′ − k))] − Fkα(1 − Fk′α′)N(ℏωj(k′ − k)) � × δ(εk′α′ − εkα − ℏωj(q) + nℏΩ) � (S57) Gk′α′ kα (n, j) = � ν 1 √AMoir´e D √2ρcph � ℏωj(k′ − k)e−|k′−k+Gj|2l2 w/4 � m ⟨φn+m k′α′ |νk′⟩⟨νk|φm kα⟩ (S58) where ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='52 × 10−6 kg/m2 is the 2D density of the graphene layers, D is the deformation potential, and the acoustic phonon mode j has frequency ωj(q) = ℏcph|q + Gj| with {Gj} being the set of all possible reciprocal 8 lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The function N(ε) = (e−ε/kBTph − 1)−1 is the Bose-Einstein occupation of the phonon bath at temperature Tph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The electron-electron collision integral is Iel-el kα [{Fkα}] = 4π ℏ 1 N 2 � k2∈BZ � k3∈BZ � α2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α4 � n � G |V(k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='(k1+k2−k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α4) (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='(k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α2) (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' G)|2× × δ(εkα + εk2α2 − εk3α3 − εk+k2−k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α4 + nℏΩ)× × [(1 − Fkα)(1 − Fk2α2)Fk3α3Fk1+k2−k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α4 − FkαFk2α2(1 − Fk3α3)(1 − Fk1+k2−k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α4)] (S59) V(k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='(k1+k2−k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α4) (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='(k2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α2) (n) = � ν1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='ν2 � ν3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='ν4 � n2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='n3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='n4 V (k2 − k3 + G)e−|q+G|2l2 w/2δν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='ν3δν2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='ν4⟨φn−n2+n3+n4 kα |ν1k⟩⟨φn2 k2α2|ν2k2⟩× × ⟨ν3k3|φn3 k3α3⟩⟨ν4k4|φn4 k+k2−k3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='α4⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S60) We solve for ∂tFkα = 0 using the Newton-Raphson al- gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' To ensure charge neutrality, we add a La- grange multiplier term λ(� kα Fkα − N) to the Floquet- Boltzmann equation, choosing some large constant λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' MONKHORST-PACK GRID, NUMERICAL INTEGRATION, AND CONVERGENCE In this section, we describe the methods we use to dis- cretize the momentum Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We perform the Boltzmann equation integrals, introduced in Equations S57 and S59, over an N × N Monkhorst-Pack (MP) set of grid points [10], with k-points km,n = mG1 + nG2 N , (S61) odd N, and m, n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Specifically, we avoid values of N(mod 3) = 0 that generate a k-point ex- actly at the high-symmetry point of K, because such grids converge poorly when the drive strength is weak and Floquet-induced gap ∆K is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Energy and Momentum Conservation Here, we discuss in detail how we impose momentum and energy conservation on this MP grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The space of MP k vectors are closed under addition and subtraction (modulo a reciprocal lattice vector), so conservation of momentum (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=', k+k2−k3 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S59), is simple to im- plement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We impose energy conservation via a smeared Dirac Delta function δ(ε) = � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='04766e−ε2/2σ2/(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='5066283σ), if |ε| < 2σ, 0, otherwise, (S62) (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Convergence of anomalous conductivity with grid size for (a) N(mod 3) = 1 and (b) N(mod 3) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Due to the positioning of grid points near the K point, the results at low grid resolutions show significant disagreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, E0 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='41 × 104 V/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' where we have chosen numerical factors so that � ∞ −∞ δ(ε)dε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' (S63) The smearing parameter σ is one-tenth of the maxi- mum quasienergy spacing between nearest-neighbor MP k-points σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1 max ⟨k,k′⟩,α |ε(ξ) kα − ε(ξ) k′α|, (S64) where ⟨k, k′⟩ restricts k′ to be a nearest-neighbor of k, and we have tuned the prefactor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1 so that upon calculating the steady-state without Floquet-Umklapp processes, we obtain a Fermi-Dirac distribution, F (ξ) kα = (eεkα/kBTph + 1)−1 with temperature Tph of the phonon bath [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='15 θ [◦] 0 5 10 E∗ [105 V/m] 12 15 18 21 [km/s] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The requirement that the laser drive strength E is perturbative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' a fraction of electron bandwidth eELM < W, narrows the range of E values that can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' As a result, the range of cph whose E∗ is visible is limited as well we postulate that they are pushed to higher drive strengths E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Convergence of Conductivities In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S9, we show the convergence of the Hall con- ductivity σxy with grid size, using ℏΩ = 5 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' In the main text, we use a 163×163 MP grid for non-interacting calculations, and a 73 × 73 grid for interacting calcula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' BERRY CURVATURE CALCULATIONS We follow the Berry curvature calculation presented in [12], defining U(1) link variables Uµ(k, t) = ⟨α(k, t)|α(k + ˆµ, t)⟩ |⟨α(k, t)|α(k + ˆµ, t)⟩| (S65) where µ = x, y, ˆµ = Gµ/N, and |α(k, t)⟩ are the Bloch vectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=', |ψkα(t)⟩ = e−ik·r|α(k, t)⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The Berry cur- vature is Bkα(t) = (2π)2 N 2AM arg �Ux(k, t)Uy(k + ˆx, t) Ux(k + ˆy, t)Uy(k, t) � (S66) and we use the time-averaged Berry curvature Bkα ≡ 1 2π/Ω � 2π/Ω 0 Bkα(t)dt (S67) in transport calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' XII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' THE PERTURBATIVE REGIME AT DIFFERENT TWIST ANGLES We have treated the laser drive as a perturbation to the undriven TBG Hamiltonian, which restricts the range of field strengths E we can use to a weak perturbative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' This also narrows the range of phonon speeds cph that will generate a critical field strength E∗ in the perturbative regime, hence the narrow range of cph we have considered in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=', Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' For various twist angles, we estimate the range of drive strengths E that are perturbative in the unshaded region of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S10 and overlap in solid lines the predicted value of E∗ for different speeds of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' The shaded, non-perturbative regime corresponds to drive energy scales vF eE/Ω greater than a fraction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='3, of the bandwidth W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Here, we follow the analysis in [4] to estimate the undriven Fermi velocity vF (θ) = �� (1 − 3α2)/(1 + 3α2(1 + η2)) × vml F �2 + v2 min, (S68) where vmin = 104 m/s is a manually set minimum Fermi velocity of the undriven flat bands, and we use the same parameters as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' We also adjust Ω such that Ω/vF (θ) is constant and equal to those considered in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Rudner and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Lindner, The floquet engineer’s handbook (2020), arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='08252 [cond-mat].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' [2] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Du, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Fiete, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' B 97, 035422 (2018).' 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Journal of Experimental and Theoretical Physics 30 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Fukui, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Hatsugai, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content=' Suzuki, Journal of the Physical Society of Japan 74, 1674 (2005), https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1143/JPSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} +page_content='1674' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE0T4oBgHgl3EQfUQCb/content/2301.02248v1.pdf'} diff --git a/RdE4T4oBgHgl3EQfKwxc/content/tmp_files/load_file.txt b/RdE4T4oBgHgl3EQfKwxc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5363e7c262a4de7e37d0db4f8d93b65b1e65f13 --- /dev/null +++ b/RdE4T4oBgHgl3EQfKwxc/content/tmp_files/load_file.txt @@ -0,0 +1,506 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf,len=505 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='04932v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='AG] 12 Jan 2023 VECTOR BUNDLE CONSTRUCTION VIA MONADS ON MULTIPROJECTIVE SPACES DAMIAN M MAINGI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' In this paper we establish the existence of monads on multiprojective spaces X = P2n+1 × P2n+1 × · · · × P2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We prove stability of the kernel bundle which is a dual of a generalized Schwarzenberger bundle associated to the monads and prove that the cohomology vector bundle is simple, a generalization of instanton bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Next we construct monads on Pa1 × · · · × Pan and prove stability of the kernel bundle and that the cohomology vector bundle is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Introduction The existence of indecomposable low rank vector bundles on algebraic varieties in compar- ison with the ambient space has been a fertile area in algebraic geometry for the last 45 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Some of the remarkable works in this regard are: the famous Horrocks-Mumford bundle of rank 2 over P4[11], the Horrocks vector bundle of rank 3 on P5[9] the Tango bundles[23] of rank n − 1 on Pn for n ≥ 3 and the rank 2 vector bundle on P 5 in charac- teristic 2 by Tango[24] are all obtained as cohomologies of certain monads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' In spite of this fact it remains intriguing and fascinating to construct new examples of indecomposable low rank vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' One of the techniques used to construct these vector bundles is via monads which appear in many contexts within algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' They were first introduced by Horrocks[10] were he proved that all vector bundles E on P3 could be obtained as the cohomology bundle of a monad of a given kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' In vector bundle construction via monads on a given algebraic variety, the first task is to show the existence of monads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Fløystad[5] gave a theorem on the existence of monads over projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Costa and Miro-Roig [3] extended these results to smooth quadric hypersurfaces of dimension at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Marchesi, Marques and Soares[12] generalized Fløystad’s theorem to a larger set of varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Maingi[16, 17, 18] proved the existence of monads on Pn×Pm, P2n+1×P2n+1 and Pa1 ×Pa1 ×Pa2 ×Pa2 ×· · ·×Pan ×Pan respectively and proved simplicity of the cohomology bundles associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The flow of the results is similar to a paper by Ancona and Ottaviani [1] where they proved that special instanton bundles on P2n+1 are simple by first proving stability of Date: January, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Monads, multiprojective spaces, simple vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 1 2 DAMIAN M MAINGI Schwarzenberger bundles and also the results of Maingi[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Here the methods used gen- eralize methods previously used by several authors for odd dimensional projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' A natural and efficient technique to construct monads and hence more examples of vector bundles is to vary the ambient variety and choose a different polarisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We first gener- alize the work of Maingi[17] by construction of monads on P2n+1 × · · · × P2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We then prove stability of the kernel bundle which is a generalization of the dual of Schwarzenberger (steiner) bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Next we prove simplicity of the cohomolgy vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Specifically we establish the existence of monads 0 −−−→ OX(−1, · · · , −1)⊕k −−−→ f ⊕O⊕2n+2k X −−−→ g OX(1, · · · , 1)⊕k −−−→ 0 on X = P2n+1 × · · · × P2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We shall call this monad, Type I in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Next we establish the existence of monads on X = Pa1 × · · · × Pan for the polarisation L = OX(α1, · · · , αt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' This is a generalization of the results of Maingi[16] gave a condi- tional variant theorem on Pa1 × · · · × Pan, here we give a biconditional theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='2) but for all ai = 1, i = 1, · · · , n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Specifically we establish the existence of monads 0 −−−→ OX(−α1, · · · , −αt)⊕α −−−→ f ⊕O⊕β X −−−→ g OX(α1, · · · , αt)⊕γ −−−→ 0 on X = Pa1 × · · · × Pan which we shall call monad Type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We then prove stability of the kernel bundle ker g and finally prove that the cohomology vector bundle, E = ker g/ imf is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Preliminaries In this work we give generalizations for previous results by several authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' To be specific we build upon results by Maingi[16, 17, 18] therefore the definitions, notation, the methods applied are quite similar and the trend follows the paper by Ancona and Ottaviani[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='In this section we define and give notation in order to set up for the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Most of the definitions are from the book by Okonek, Schneider and Spindler[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X be a nonsingular projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' (a) A monad on X is a complex of vector bundles: 0 � M0 α � M1 β � M2 � 0 which is exact at M0 and at M2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' α is injective and β surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' VECTOR BUNDLE CONSTRUCTION VIA MONADS ON MULTIPROJECTIVE SPACES 3 (b) A monad as defined above has a display diagram of short exact sequences as shown below: 0 0 \uf8e6\uf8e6� \uf8e6\uf8e6� 0 −−−→ M0 −−−→ ker β −−−→ E −−−→ 0 || \uf8e6\uf8e6� \uf8e6\uf8e6� 0 −−−→ M0 −−−→ α M1 −−−→ coker α −−−→ 0 β \uf8e6\uf8e6� \uf8e6\uf8e6� M2 M2 \uf8e6\uf8e6� \uf8e6\uf8e6� 0 0 (c) The kernel of the map β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' F = ker β and the cokernel of α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' coker α for the given monad are also vector bundles and the vector bundle E = ker(β)/ im(α) and is called the cohomology bundle of the monad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X be a nonsingular projective variety, let L be a very ample line sheaf, and V, W, U be finite dimensional k-vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' A linear monad on X is a complex of sheaves, M• : 0 � V ⊗ L −1 A � W ⊗ OX B � U ⊗ L � 0 where A ∈ Hom(V, W) ⊗ H0L is injective and B ∈ Hom(W, U) ⊗ H0L is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The existence of the monad M• is equivalent to: A and B being of maximal rank and BA being the zero matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X be a non-singular irreducible projective variety of dimension d and let L be an ample line bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' For a torsion-free sheaf F on X we define (a) the degree of F relative to L as degL F := c1(F) · L d−1, where c1(F) is the first Chern class of F (b) the slope of F as µL (F) := degL F rk(F ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Hoppe’s Criterion over polycyclic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Suppose that the Picard group Pic(X) ≃ Zl where l ≥ 2 is an integer then X is a polycyclic variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Given a divisor B on X we define δL (B) := degL OX(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Then one has the following stability criterion [13], Theorem 3: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='4 (Generalized Hoppe Criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let G → X be a holomorphic vector bundle of rank r ≥ 2 over a polycyclic variety X equiped with a polarisation L if H0(X, (∧sG) ⊗ OX(B)) = 0 for all B ∈ Pic(X) and s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , r − 1} such that δL (B) < −sµL (G) then G is stable and if δL (B) ≤ −sµL (G) then G is semi-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 4 DAMIAN M MAINGI Conversely if then G is (semi-)stable then H0(X, G ⊗ OX(B)) = 0 for all B ∈ Pic(X) and all s ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , r − 1} such that δL (B) < −sµL (G) or δL (B) ≤ −sµL (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Suppose the ambient space is X = Pa1 × · · · × Pan then Pic(X) ≃ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We shall denote by gi for i = 1 · · · , n the generators of the Picard group of X, Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Denote by OX(g1, g2, · · · , gn) := p1∗OPa1(g1) ⊗ · · · ⊗ pn∗OPan(gn), where pi for i = 1, · · · , n are natural projections from X onto Pai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' For any line bundle L = OX(g1, g2, · · · , gn) on X and a vector bundle E, we write E(g1, g2, · · · , gn) = E ⊗ OX(g1, g2, · · · , gn) and (g1, g2, · · · , gn) := 1 · [g1 × Pa1] + · · · + 1 · [Pan × gn] representing its corresponding divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The normalization of E on X with respect to L is defined as follows: Set d = degL (OX(1, 0, · · · , 0)), since degL (E(−kE, 0, · · · , 0)) = degL (E) − nk · rank(E) there is a unique integer kE := ⌈µL (E)/d⌉ such that 1−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' rank(E) ≤ degL (E(−kE, 0, · · · , 0)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The twisted bundle EL −norm := E(−kE, 0, · · · , 0) is called the L -normalization of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Lastly, the linear functional δL on Zn is defined as δL (p1, p2, · · · , pn) := degL OX(p1, p2, · · · , pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' For the q−th cohomology group we use the notation Hq(F) in place of Hq(X, F), for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The following results are generalized versions to a multiprojective space for the purposes of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X be a polycyclic variety with Picard number n, let L be an ample line bundle and let E be a rank r > 1 holomorphic vector bundle over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' If H0(X, (�q E)L −norm(p1, · · · , pn)) = 0 for 1 ≤ q ≤ r − 1 and every (p1, · · · , pn) ∈ Zn such that δL ≤ 0 then E is L -stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let 0 → E → F → G → 0 be an exact sequence of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Then we have the following exact sequence involving exterior and symmetric powers 0 −→ q� E −→ q� F −→ q−1 � F ⊗ G −→ · · · −→ F ⊗ Sq−1G −→ SqG −→ 0 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='8 (K¨unneth formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X and Y be projective varieties over a field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let F and G be coherent sheaves on X and Y respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let F ⊠ G denote p∗ 1(F) ⊗ p∗ 2(G ) then Hm(X × Y, F ⊠ G ) ∼= � p+q=m Hp(X, F) ⊗ Hq(Y, G ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' VECTOR BUNDLE CONSTRUCTION VIA MONADS ON MULTIPROJECTIVE SPACES 5 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X = Pa1 × · · · × Pan then Ht(X, OX(p1, · · · , pn)) ∼= � �t qi=1 Hq1(Pa1, OPa1(p1)) ⊗ Hq2(Pa2, OPa2(p2)) ⊗ · · · ⊗ Hqn(Pan, OPan(pn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='10 ([21], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let n ≥ 1 be an integer and d be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We denote by Sd the space of homogeneous polynomials of degree d in n + 1 variables (conven- tionally if d < 0 then Sd = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Then the following statements are true: (a) H0(Pn, OPn(d)) = Sd for all d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' (b) Hi(Pn, OPn(d)) = 0 for 1 < i < n and for all d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' (c) Hn(Pn, OPn(d)) ∼= H0(Pn, OPn(−d − n − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' If n � i=1 pi >0 then hp(X, OX(−p1, · · · , −pn)⊕k) = 0 where X = Pa1×· · ·×Pan and for 0 ≤ p < dim(X) − 1, for k a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='12 ([12], Lemma 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let A and B be vector bundles canonically pulled back from A′ on Pn and B′ on Pm then Hq( s� (A ⊗ B)) = � k1+···+ks=q � s � i=1 ( s � j=0 ki � m=0 Hm(∧j(A)) ⊗ (Hki−m(∧s−j(B)))) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='13 ([5], Main Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' There exists monads on Pk whose maps are matrices of linear forms, 0 −−−→ OPk(−1)⊕a −−−→ A O⊕b Pk −−−→ B OPk(1)⊕c −−−→ 0 if and only if at least one of the following is fulfilled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' (1)b ≥ 2c + k − 1 , b ≥ a + c and (2)b ≥ a + c + k Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='14 ([17], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let n and k be positive integers and A and B be morphisms of linear forms as in B := Ñ x0 · · · xn y0 · · · yn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' x0 · · · xn y0 · · · yn é and A := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed −y0 · · · −yn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −y0 · · · −yn x0 · · · xn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' x0 · · · xn \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 then there exists a linear monad of the form 0 −−−→ OP2n+1(−1)⊕k −−−→ A O⊕2n+2k P2n+1 −−−→ B OP2n+1(1)⊕k −−−→ 0 6 DAMIAN M MAINGI Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='15 ([16], Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X = Pn × Pm and let L = OX(ρ, σ) be an ample line bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Denote by N = h0(OX(ρ, σ)) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let α, β, γ be positive integers such that at least one of the following conditions holds (1)β ≥ 2γ + N − 1, and β ≥ α + γ, (2)β ≥ α + γ + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Then, there exists a linear monad on X of the form 0 −−−→ OX(−ρ, −σ)⊕α −−−→ A O⊕β X −−−→ B OX(ρ, σ)⊕γ −−−→ 0 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X be a projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' A sheaf S on X is a steiner bundle if has short exact sequence of the form 0 −−−→ OX(−1)⊕a −−−→ O⊕b X −−−→ S −−−→ 0 They were first defined by Dolgachev and Kapranov[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' [22] Let k ≥ 0 the exact sequence of sheaves on P2n+1 0 −−−→ OX(−1)⊕a −−−→ φ O⊕b X −−−→ S −−−→ 0 where φ is given by the matrix \uf8ee \uf8f0 x0 · · · xn y0 · · · yn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' x0 · · · xn y0 · · · yn \uf8f9 \uf8fb defines a 2n + k−bundle S on P2n+1 called a (generalized) Schwarzenberger bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' As we have set up the necessary tools for this work we proceed to the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Monad Type I and associated vector bundles The goal of this section is to construct monads over a multiprojective space of m copies of P2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' More specifically we generalize the results of Maingi [17] by varying the ambient space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We rely on methods similar to those used in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The kernel bundle T is a more generalized version of the dual of a Schwarzenberger vector bundle and we prove that it is stable and consequently we prove that the cohomology vector bundle E associated to the monad on X is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The vector bundle E is a generalized version of an instanton bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' VECTOR BUNDLE CONSTRUCTION VIA MONADS ON MULTIPROJECTIVE SPACES 7 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let n and k be positive integers and X = P2n+1 × · · · × P2n+1 then there exists a monad of the form M• : 0 −−−→ OX(−1, · · · , −1)⊕k −−−→ f O⊕2n⊕2k X −−−→ g OX(1, · · · , 1)⊕k −−−→ 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let a = c = k, b = 2n + 2k and N = 2n + 1 from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='13 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='14 there exists a linear monad M• : 0 −−−→ OP2n+1(−1)⊕k −−−→ f O⊕2n⊕2k P2n+1 −−−→ g OP2n+1(1)⊕k −−−→ 0 and for a line bundle L = OX(1, · · · , 1) we have the Segre embedding i∗ : X = P2n+1 · · · × P2n+1 ֒→ P�H0(X, OX(1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , 1))� ∼= Pm(2n+2)−1 such that i∗(OX(1)) ≃ L and supposing that one of the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='13 is satified then the morphisms A and B in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='14 induce the desired monad whose morpsims are f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' □ The kernel bundle F of the above monad is a generalization of the dual of Schwarzenberger vector bundles [2] which we now proceed to prove that it is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let T be a vector bundle on X = P2n+1 ×· · ·×P2n+1 defined by the sequence 0 −−−→ T −−−→ O⊕2n+2k X −−−→ OX(1, · · · , 1)⊕k −−−→ 0 then T is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We show that H0(X, �q T(−p1, · · · , −pm)) = 0 for all m � i pi > 0 and 1 ≤ q ≤ rank(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Consider the ample line bundle L = OX(1, · · · , 1) = O(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Its class in Pic(X) = ⟨[g1 × P2n+1], · · · , [P2n+1 × gm]⟩ corresponds to the class m � i=1 1 · [gi × P2n+1], where gi, i = 1, · · · , n are hyperplanes of P2n+1 with the intersection product induced by g2n+1 i = 1 and g2n+2 i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Now from the display diagram of the monad we get c1(T) = c1(O2n+2k X ) − c1(OX(1, · · · , 1)⊕k) = (2n + 2k)(0, · · · , 0) − k(1, · · · , 1) = (−k, · · · , −k) 8 DAMIAN M MAINGI Now L(4n+2)m > 0 hence , the degree of T is: degL T = −k([g1 × P2n+1] + · · · + [P2n+1 × gm]) · ( m � i=1 1 · [gi × P2n+1])m(2n+1)−1 = −kLm(2n+1) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Since degL T < 0, then (�q T)L −norm = (�q T) and it suffices by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='6, to prove that h0(�q T(−p1, · · · , −pm)) = 0 with m � i=1 pi ≥ 0 and for all 1 ≤ q ≤ rank(T) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Next we twist the exact sequence 0 −−−→ T −−−→ O⊕2n+2k X −−−→ OX(1, · · · , 1)⊕k −−−→ 0 by OX(−p1, · · · , −pm) we get, 0 −→ T(−p1, · · · , −pm) −→ OX(−p1, · · · , −pm)⊕2n+2k −→ OX(1−p1, · · · , 1−pm)⊕k −→ 0 and taking the exterior powers of the sequence by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='7 we get 0 −→ q� T(−p1, · · · , −pm) −→ q� (OX(−p1, · · · , −pm)⊕2n+2k) −→ q−1 � (OX(1−2p1, · · · , 1−2pm)⊕2n+2k) · · · Taking cohomology we have the injection: 0 −→ H0(X, q� T(−p1, · · · , −pm)) ֒→ H0(X, q� (OX(−p1, · · · , −pm)⊕2n+2k)) Set G = OX(−p1, · · · , −pm)2n+2k = OX(−p1, · · · , −p2) ⊗ O⊕2n+2k X and using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='11 H0(X, �q G ) expands into H0(X, q � j=0 ∧jOX(−p1, · · · , −p2) ⊗ O⊕2n+2k X ) and since m � i pi > 0 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='12 then h0(X, q� (OX(−p1, · · · , −pm)⊕2n+2k)) = h0(X, q� T(−p1, · · · , −pm)) = 0 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' h0(�q T(−p1, · · · , −pm)) = 0 and thus T is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X = P2n+1 × · · · × P2n+1, then the cohomology vector bundle E asso- ciated to the monad 0 −−−→ OX(−1, · · · , −1)⊕k −−−→ A O⊕2n+2k X −−−→ B OX(1, · · · , 1)⊕k −−−→ 0 of rank 2n is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' VECTOR BUNDLE CONSTRUCTION VIA MONADS ON MULTIPROJECTIVE SPACES 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The display of the monad is 0 0 \uf8e6\uf8e6� \uf8e6\uf8e6� 0 −−−→ OX(−1, · · · , −1)⊕k −−−→ T −−−→ E −−−→ 0 || \uf8e6\uf8e6� \uf8e6\uf8e6� 0 −−−→ OX(−1, · · · , −1)⊕k −−−→ α O⊕2n+2k X −−−→ Q −−−→ 0 β \uf8e6\uf8e6� \uf8e6\uf8e6� OX(1, · · · , 1)⊕k OX(1, · · · , 1)⊕k \uf8e6\uf8e6� \uf8e6\uf8e6� 0 0 Since T is stable from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='11 we prove that the cohomology vector bundle E with rank 2n is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The first step is to take the dual short exact sequence 0 −−−→ OX(−1, · · · , −1)⊕k −−−→ T −−−→ E −−−→ 0 to get 0 −−−→ E∗ −−−→ T ∗ −−−→ OX(1, · · · , 1)⊕k −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Tensoring by E we get 0 −−−→ E ⊗ E∗ −−−→ E ⊗ T ∗ −−−→ E(1, · · · , 1)k −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Now taking cohomology gives: 0 −−−→ H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' E ⊗ E∗) −−−→ H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' E ⊗ T ∗) −−−→ H0(E(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 1)k) −−−→ · · · which implies that h0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' E ⊗ E∗) ≤ h0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' E ⊗ T ∗) (1) Now we dualize the short exact sequence 0 −−−→ T −−−→ O⊕2n+2k X −−−→ OX(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 1)⊕k −−−→ 0 to get 0 −−−→ OX(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)⊕k −−−→ O⊕2n+2k X −−−→ T ∗ −−−→ 0 Now twisting by OX(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1) and taking cohomology and get 0 −→ H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OX(−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · − 2)k) −→ H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OX(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)2n+2k) −→ H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' T ∗(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)) −→ −→ H1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OX(−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −2)k) −→ H1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OX(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)2n+2k) −→ H1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' T ∗(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)) −→ −→ H2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OX(−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −2)k) −→ H2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OX(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)2n+2k) −→ H2(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' T ∗(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)) −→ · · · 10 DAMIAN M MAINGI from which we deduce H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' T ∗(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)) = 0 and H1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' T ∗(−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −1)) = 0 from Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Lastly, tensor the short exact sequence 0 −−−→ O(−1, · · · , −1)⊕k −−−→ T −−−→ E −−−→ 0 by T ∗ to get 0 −−−→ T ∗(−1, · · · , −1)k −−−→ T ⊗ T ∗ −−−→ E ⊗ T ∗ −−−→ 0 and taking cohomology we have 0 −−−→ H0(X, T ∗(−1, · · · , −1)k) −−−→ H0(X, T ⊗ T ∗) −−−→ H0(X, E ⊗ T ∗) −−−→ −−−→ H1(X, T ∗(−1, · · · , −1)k) −−−→ · · But H1(X, T ∗(−1, · · · , −1)k = 0 for k > 1 from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' so we have 0 −−−→ H0(X, T ∗(−1, · · · , −1)k) −−−→ H0(X, T ⊗ T ∗) −−−→ H0(X, E ⊗ T ∗) −−−→ 0 This implies that h0(X, T ⊗ T ∗) ≤ h0(X, E ⊗ T ∗) (2) Since T is stable then it follows that it is simple which implies h0(X, T ⊗ T ∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' From (1) and now (2) and putting these together we have;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 1 ≤ h0(X, E ⊗ E∗) ≤ h0(X, E ⊗ T ∗) = h0(X, T ⊗ T ∗) = 1 We have h0(X, E ⊗ E∗) = 1 and therefore E is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Monad Type II and associated vector bundles The goal of this section is to construct monads over a multiprojectivespace Pa1 ×· · ·×Pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' More specifically we generalize the results of Maingi [16] by varying the ambient space and the polarisation L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We prove that the kernel bundle F is stable and thereafter we prove that the cohomology vector bundle E associated to the monad on X is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X = Pa1 · · · × Pan and L = OX(α1, · · · , αt) an ample line bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Denote by N = h0(OX(α1, · · · , αt)) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Then there exists a linear monad M• on X of the form M• : 0 −−−→ OX(−α1, · · · , −αt)⊕α −−−→ f O⊕β X −−−→ g OX(α1, · · · , αt)⊕γ −−−→ 0 if and only if atleast one of the following is satified (a) β ≥ 2γ + N − 1, and β ≥ α + γ, (b) β ≥ α + γ + N, where α, β, γ be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' VECTOR BUNDLE CONSTRUCTION VIA MONADS ON MULTIPROJECTIVE SPACES 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' For the ample line bundle L = OX(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , αt) we have the Segre embedding i∗ : X = Pa1 · · · × Pan ֒→ P � H0(X, OX(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , αt)) � ∼= PN such that i∗(OX(1)) ≃ L and where N = ÇÇ a1 + α1 α1 åÇ a2 + α2 α2 å · · Ç an + αt αt åå − 1 Suppose that one of the conditions of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='13 is satified thus there exists a linear monad 0 −−−→ OPN(−1)⊕α −−−→ A O⊕β PN −−−→ B OPN(1)⊕γ −−−→ 0 on PN whose morphisms are matrices A and B with entries monomials of degree one where A ∈ Hom(OP2n+1(−1)⊕α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' O⊕β P2n+1) ∼= H0(P2n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OP2n+1(1)⊕αβ) B ∈ Hom(O⊕β P2n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OP2n+1(1)⊕γ) ∼= H0(P2n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' OP2n+1(1)⊕βγ) Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' A and B induce a monad on X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 0 −−−→ L −1⊕α ¯ A −−−→ O⊕β X ¯ B −−−→ L ⊕γ −−−→ 0 where whose morphisms are matrices ¯A and ¯B with entries multidegree monomials such that ¯A ∈ Hom(OX(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , −αt)⊕α, O⊕β X ) and ¯B ∈ Hom(O⊕β X , OX(α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , α1)⊕γ) □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let F be a vector bundle on X = Pa1 ×· · ·×Pan defined by the short exact sequence 0 −−−→ F −−−→ O⊕β X −−−→ g OX(α1, · · · , αt)⊕γ −−−→ 0 then F is stable for an ample line bundle L = OX(α1, · · · , αt) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' We are going to show that H0(X, �q F(−p1, · · · , −pn)) = 0 for all n � i=1 pi ≥ 0 and 1 ≤ q ≤ rank(F) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Consider the ample line bundle L = OX(α1, · · · , αt) = O(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Its class in Pic(X) = ⟨[gi × Pai], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , n]⟩ corresponds to n � i=1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' [gi × Pai] where each gi is a hyperplane in Pai with intersection product induced by gai i = 1 and gai+1 i = 0 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' From the display of the monad we get c1(F) = c1(O⊕β X ) − c1(OX(α1, · · · , αt)⊕γ) = (−γα1, · · · , −γαt) 12 DAMIAN M MAINGI Since La1+···+an > 0 the degree of F is degL F = c1(T) · L d−1 that is = −γn t � i=1 αi([g1 × Pa1] + · · · + [Pan × g2n]) � n � i=1 1 · [gi × Pai] ��n i=1 ai−1 = −γn t � i=1 αiL(a1+···+an) < 0 Since degL F < 0, then (�q F)L −norm = (�q F) and it suffices by the generalized Hoppe Criterion (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='6), to prove that h0(�q F(−p1, −p2, · · · , −pn)) = 0 with n � i=1 pi ≥ 0 and for all 1 ≤ q ≤ rank(F) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Next consider the exact sequence 0 −−−→ F −−−→ O⊕β X −−−→ g OX(α1, · · · , αt)⊕γ −−−→ 0 on twisting it by OX(−p1, · · · , −pn) one gets, 0 −−−→ F(−p1, · · · , −pn) −−−→ O⊕β X (−p1, · · · , −pn) −−−→ g OX(α1 − p1, · · · , αt − pn)⊕γ −−−→ 0 and taking the exterior powers of the sequence by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='10 one gets 0 −→ q� F(−p1, · · · , −pn) −→ q� (OX(−p1, · · · , −pn)⊕β) −→ q−1 � (OX(α1−2p1, · · · , αt−2pn)⊕γ) −→ · · · Taking cohomology we have the injection: 0 −→ H0(X, q� F(−p1, · · · , −pn)) ֒→ H0(X, q� (OX(−p1, · · · , −pn)⊕β) From here h0(X, �q F(−p1, · · · , −pn)) = 0 is proved in the same way as Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='2 the last part and thus F is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' □ Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Let X = Pa1 × · · · × Pan, then the cohomology vector bundle E associated to the monad 0 −−−→ OX(−α1, · · · , −αt)⊕α −−−→ f O⊕β X −−−→ g OX(α1, · · · , αt)⊕γ −−−→ 0 of rank β − α − γ is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' VECTOR BUNDLE CONSTRUCTION VIA MONADS ON MULTIPROJECTIVE SPACES 13 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' The display of the monad is 0 0 \uf8e6\uf8e6� \uf8e6\uf8e6� 0 −−−→ OX(−α1, · · · , −αt)⊕α −−−→ F = ker g −−−→ E −−−→ 0 || \uf8e6\uf8e6� \uf8e6\uf8e6� 0 −−−→ OX(−α1, · · · , −αt)⊕α −−−→ f O⊕β X −−−→ Q = coker f −−−→ 0 g \uf8e6\uf8e6� \uf8e6\uf8e6� OX(α1, · · · , αt)⊕γ OX(α1, · · · , αt)⊕γ \uf8e6\uf8e6� \uf8e6\uf8e6� 0 0 Since E is simple if its only endomorphisms are the homotheties then we need to prove that Hom(E, E) = k which is equivalent to h0(E ⊗ E∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' On taking the dual of the short exact sequence on the first row of the display diagram and tensoring by E we obtain 0 −−−→ E ⊗ E∗ −−−→ E ⊗ F ∗ −−−→ E(t, · · · , t)⊕α −−−→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Now taking cohomology gives: 0 −−−→ H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' E ⊗ E∗) −−−→ H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' E ⊗ F ∗) −−−→ H0(E(α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' αt)⊕α) −−−→ · · · which implies that h0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' E ⊗ E∗) ≤ h0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' E ⊗ F ∗) (3) Dualize the short exact sequence on the first column of the display diagram to get 0 −−−→ OX(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)⊕γ −−−→ Oβ X −−−→ F ∗ −−−→ 0 Now twisting the short exact sequence above by OX(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt) one obtains the short exact sequence 0 −−−→ OX(−2α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −2αt)⊕γ −−−→ OX(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)β −−−→ F ∗(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt) −−−→ 0 next on taking cohomology one gets 0 −→ H0(OX(−2α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −2αt)⊕γ) −→ H0(OX(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)β) −→ H0(F ∗(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)) −→ 0 −→ H1(OX(−2α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −2αt)⊕γ) −→ H1(OX(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)β) −→ H1(F ∗(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)) −→ −→ H2(OX(−2α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −2αt)⊕γ) −→ H2(OX(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)β) −→ H2(F ∗(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)) −→ · · · from which we deduce H0(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' T ∗(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)) = 0 and H1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' T ∗(−α1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' −αt)) = 0 14 DAMIAN M MAINGI from Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Lastly, tensor the short exact sequence 0 −−−→ O(−α1, · · · , −αt)⊕k −−−→ T −−−→ E −−−→ 0 by T ∗ to get 0 −−−→ T ∗(−α1, · · · , −αt)k −−−→ T ⊗ T ∗ −−−→ E ⊗ T ∗ −−−→ 0 and taking cohomology we have 0 −−−→ H0(X, T ∗(−α1, · · · , −αt)k) −−−→ H0(X, T ⊗ T ∗) −−−→ H0(X, E ⊗ T ∗) −−−→ −−−→ H1(X, T ∗(−α1, · · · , −αt)k) −−−→ · · But since H0(X, T ∗(−α1, · · · , −αt)) = H1(X, T ∗(−α1, · · · , −αt)) = 0 from above then it follows H1(X, T ∗(−α1, · · · , −αt)k) = 0 for k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' so we have 0 −−−→ H0(X, T ∗(−α1, · · · , −αt)k) −−−→ H0(X, T ⊗ T ∗) −−−→ H0(X, E ⊗ T ∗) −−−→ 0 This implies that h0(X, T ⊗ T ∗) ≤ h0(X, E ⊗ T ∗) (4) Since T is stable then it follows that it is simple which implies h0(X, T ⊗ T ∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' From (3) and (4) and putting these together we have;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 1 ≤ h0(X, E ⊗ E∗) ≤ h0(X, E ⊗ T ∗) = h0(X, T ⊗ T ∗) = 1 We have h0(X, E ⊗ E∗) = 1 and therefore E is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Acknowledgment I wish to express sincere thanks to the Department of Mathematics, College of Science, Sultan Qaboos University staff for providing a conducive enviroment to be able to carry out research despite the overwhelming duties in teaching and community service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' I would also wish to express my sincere thanks to my collegues at the Department of Mathematics at the University of Nairobi for granting me leave in order to pursue my research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Lastly, I am extremely grateful to Melissa, my wife and our 3 kids Amelia, Jerome and Chuksie who are always supportive of my pursuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Data Availability statement My manuscript has no associate data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' VECTOR BUNDLE CONSTRUCTION VIA MONADS ON MULTIPROJECTIVE SPACES 15 Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Construction of rank two vector bundles on P4 in positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Invent math 130, 277–286 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' doi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1007/s002220050185 [15] Kumar N, Peterson C and Rao A P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Construction of low rank vector bundles on P4 and P5, Journal of Algebraic Geometry 11 (2) (2002), 203–217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' doi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1090/S1056-3911-01-00309-5 [16] Maingi D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Vector Bundles of low rank on a multiprojective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Le Matematiche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' LXIX (2014) Fasc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' pp 31-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='4418/2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' [17] Maingi D (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Indecomposable Vector Bundles associated to Monads on Cartesian products of projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Turkish Journal of Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 45: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Article 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Pages 2126-2139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='3906/mat-2101-6 [18] Maingi D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Monads on multiprojective Products of Projective Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' to appear in Manuscripta Math- ematica (2023),doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1007/s00229-022-01449-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' [19] Marchesi S, Marques P M and Soares H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Monads on a Projective Varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Pacific Journal of Math- ematics, vol 296 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 1, 155-180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' doi: 10.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Springer, 1980, doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1007/978-1-4757-1460-9 [21] Perrin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' G´eom´etrie alg´ebrique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Une introduction, (1995), EDP Sciences/CNRS ´edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' ISBN-2-7296-0563-0 [22] Spindler H and Trautmann G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Special Instanton bundles on P2n+1 their geometry and their moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Mathematische Annalen 286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' 1-3 (1990): 559-592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1215/S0012-7094-93-07125-6 [23] Tango, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' An example of indecomposable vector bundle of rank n − 1 on Pn, n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' Journal of Mathematics of Kyoto University, 16, (1976): 137-141, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content='1215/kjm/1250522965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' [24] Tango H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' On morphisms from projective space Pn to the Grassmann variety Gr(n, d), Journal of Mathematics of Kyoto University, 16 (1976), 201-207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RdE4T4oBgHgl3EQfKwxc/content/2301.04932v1.pdf'} +page_content=' doi: 10.' metadata={'source': 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sha256:4a35c6e41baf2129e63114c6d3a66670b0ffca5f23a6e19d1a7903360b7ecd38 +size 7536685 diff --git a/VtE_T4oBgHgl3EQfyBw5/content/tmp_files/2301.08315v1.pdf.txt b/VtE_T4oBgHgl3EQfyBw5/content/tmp_files/2301.08315v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fb31d54b962eb67b5a763d9d0f4fb6fc7f9bd0c5 --- /dev/null +++ b/VtE_T4oBgHgl3EQfyBw5/content/tmp_files/2301.08315v1.pdf.txt @@ -0,0 +1,3415 @@ +arXiv:2301.08315v1 [math.PR] 19 Jan 2023 +NONLINEAR FUNCTIONALS OF HYPERBOLIC +RANDOM WAVES: THE WIENER CHAOS APPROACH +FRANCESCO GROTTO AND GIOVANNI PECCATI +Abstract. We consider Gaussian random waves on hyperbolic spaces and +establish variance asymptotics and central limit theorems for a large class of +their integral functionals, both in the high-frequency and large domain limits. +Our strategy of proof relies on a fine analysis of Wiener chaos expansions, which +in turn requires us to analytically assess the fluctuations of integrals involving +mixed moments of covariance kernels. Our results complement several recent +findings on non-linear transforms of planar and arithmetic random waves, as +well as of random spherical harmonics. In the particular case of 2-dimensional +hyperbolic spaces, our analysis reveals an intriguing discrepancy between the +high-frequency and large domain fluctuations of the so-called fourth polyspectra +— a phenomenon that has no counterpart in the Euclidean setting. We develop +applications of a geometric flavor, most notably to excursion volumes and +occupation densities. +Contents +1. +Introduction +2 +1.1. +Overview +2 +1.2. +First definitions +2 +1.3. +Motivation and background +3 +1.4. +Main contributions +3 +1.5. +Structure +4 +1.6. +Notation +4 +1.7. +Acknowledgments +5 +2. +Geometry of Hyperbolic Space and Random Waves +5 +2.1. +Spectral Theory of Hyperbolic Space +5 +2.2. +Waves on Hyperbolic Spaces +6 +2.3. +Hyperbolic Random Waves +8 +2.4. +Curvature, Large Scale and Local Behavior of Random Waves +10 +3. +Integral Functionals: Wiener Chaos Expansion and CLTs +12 +3.1. +Some elements of Gaussian analysis +12 +3.2. +CLTs for integral functionals +16 +3.3. +First application: excursion volumes at non-zero levels +20 +3.4. +Second application: Leray measures of nodal sets +22 +4. +Covariance Functions and their Moments +25 +4.1. +Approximating Covariance Functions +25 +4.2. +Double Integrals on Hyperbolic Balls +29 +4.3. +Asymptotics for large λ, fixed R +29 +Universit`a di Pisa, Dipartimento di Matematica, 5 Largo Bruno Pontecorvo, 56127 Pisa, +Italia +Universit´e du Luxembourg, Maison du Nombre, 6 Avenue de la Fonte, 4364 Esch-sur- +Alzette, Luxembourg +E-mail addresses: francesco.grotto at unipi.it, giovanni.peccati at uni.lu. +Date: January 23, 2023. +Key words and phrases. Random Waves, Hyperbolic Space, Wiener Chaos. +1 + +2 +F. GROTTO AND G. PECCATI +4.4. +Asymptotics for large R, fixed λ +34 +Appendix A. +Repository of Formulae for Special Functions +37 +A.1. +Bessel Functions +37 +A.2. +Hypergeometric Functions +37 +A.3. +Relations with Legendre Functions +38 +References +38 +1. Introduction +1.1. Overview. The aim of this paper is to initiate the study of non-linear func- +tionals of Gaussian random waves (that is, generalized Gaussian eigenfunctions of +the Laplacian) defined on hyperbolic spaces of arbitrary dimension — with specific +emphasis on variance asymptotics and central limit theorems. As put forward in +the title, our approach is based on a careful analysis of Wiener chaos expansions, +which we implement by using several non-trivial refinements of the general theory +developed in [45, 46], see Section 3. One of the main contributions of our work is +the derivation of new analytic estimates for covariance kernels of hyperbolic waves +(stated in Section 4), which will allow us to deal simultaneously both with the +high-frequency and large domain asymptotic regimes. We will see that our findings +naturally complement several recent studies of Gaussian random waves on mani- +folds, such as Euclidean random waves [8, 9, 17, 16, 47, 51, 43], arithmetic random +waves [12, 18, 32, 49, 53] and random spherical harmonics [39, 38, 34, 40, 36, 37, 52]. +1.2. First definitions. Denote by Hn, n ≥ 2, the n-dimensional hyperbolic space +(that is, the simply connected manifold with constant negative sectional curvature) +and let λ ≥ (n − 1)2/4. The hyperbolic random wave with frequency λ, written +uλ := {uλ(x) : x ∈ Hn}, is defined as the unique (in distribution) centered and +unit variance real Gaussian field on Hn such that (i) the law of uλ is invariant with +respect to the isometries of Hn, (ii) paths of uλ solve a.s. the Laplace-Beltrami +eigenvalue problem ∆Hnuλ + λuλ = 0, where ∆Hn is the hyperbolic Laplacian (see +Proposition 2.7 for details). +The random wave uλ is the exact hyperbolic counterpart of the Euclidean ran- +dom wave vλ := {vλ(x) : x ∈ Rn} (see [8, 9]), that one can similarly characterize +as being the unique centered and unit variance real Gaussian field on Rn veri- +fying properties (i) and (ii) above, with Hn and ∆Hn replaced, respectively, by +Rn and ∆ = − � +i ∂2/∂x2 +i . Further remarkable examples of non-Euclidean ran- +dom waves, to which uλ should be compared, are the already discussed random +spherical harmonics and arithmetic random waves (that are, respectively, Laplace +eigenfunctions on the sphere Sn and on the flat torus Tn), as well as the class of +Gaussian monochromatic random waves on general compact manifolds [14, 19, 60]. +We also recall that hyperbolic random waves appeared in another guise in [15], +in the context of spectral decomposition of stationary Gaussian fields on Hn (the +latter as a particular case of homegeneous space). +Remark 1.1. +(i) For future reference, we recall that (as an application e.g. +of [3, Theorem 5.7.2]) the above characterization of vλ is equivalent to +requiring that, for x, y ∈ Rn, +(1.1) +E [vλ(x)vλ(y)] = Cn,λ(x, y) := +1 +ωn−1 +� +Sn−1 ei +√ +λu·(x−y)du += (2π)n/2 +ωn−1 +�√ +λ|x − y| +�1−n/2 +Jn/2−1( +√ +λ|x − y|), + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +3 +where ωn−1 is the hypersurface volume of Sn−1 and Jν is the Bessel function +of order ν (see e.g. [48, Chapter 10]); we also point out that an analogous +representation of the covariance of uλ will emerge from the statement of +Proposition 2.7 below. +(ii) The central role played by Euclidean random waves in the probabilistic +analysis of Laplace eigenfunctions is amplified by the so-called Berry’s ran- +dom wave conjecture — originally formulated in [8] — according to which +the unit energy random wave v1 is a universal model for the high-frequency +local behavior of deterministic Laplace eigenfunctions on chaotic billiards, +among which negatively curved manifolds are paradigmatic examples. We +refer the reader to [28] for a discussion of the role of random wave models +in the physical literature, and to [1, 27] for mathematically rigorous ap- +proaches toward Berry’s conjecture. See also [14, 19], as well as Section 2.4 +below. +1.3. Motivation and background. As discussed e.g. in the survey [59] (to which +we refer the reader for an exhaustive list of references), in recent years considerable +attention has been devoted to local geometric functionals associated with level sets +of random waves, such as excursion volumes, occupation measures and volumes of +level sets — among which nodal volumes (i.e., the Haussdorff measures of zero loci) +play a pivotal role. +A remarkable phenomenon is that in a number of crucial cases (see [13, 37, +36, 35, 47, 50, 51, 43] for a sample) the study of these geometric functionals can +be fruitfully reduced to the asymptotic analysis of their orthogonal projections on +Wiener chaoses, as formally defined in Section 3.1.2. Such a strategy — which +corresponds to the “Wiener chaos approach” advertised in the title — is described +in detail in the forthcoming Section 3.1.3 and relies pervasively on the abstract +theory of probabilistic approximations presented in [46]; see also [31, 54, 55] for +some earlier use of Wiener chaos in the geometric study of Euclidean Gaussian +fields1. +The main contribution of our work consists in the first explicit application of +Wiener chaos techniques to a class of integral functionals associated with non- +Euclidean random waves on non-compact manifolds, thus setting the bases for the +asymptotic analysis of more general geometric quantities. +Moreover, one could +regard this as a natural first step towards the analysis of Berry’s conjecture for +classical chaotic billiards, such as Artin’s billiard (the geodesic flow on modular +surface H2/ PSL(2, Z)) and more general non-compact hyperbolic surfaces, of which +the hyperbolic plane is the universal cover. +However, in the latter setting the +spectral theory of Laplace operator is much more complicated than on H2: we refer +to [7] for a proper overview on the topic, in particular Chapter 9 concerning the +high-frequency limit and Quantum Unique Ergodicity. It is not clear to us whether +information on random combinations of Laplace eigenfunctions on H2 can actually +give any insight on their analogs on other hyperbolic surfaces, so we leave this as +an open question. +1.4. Main contributions. The principal focus of our paper is on integral func- +tionals of the form +G(uλ) = +� +BR +G(uλ(x))dmn(x), +BR ⊂ Hn, +G : R → R, +1Here, an important caveat is that the covariance structure of random waves typically does not +satisfy the integrability assumptions required in order to directly apply the results from [31, 54, 55], +in such a way that, for random waves, several ad hoc arguments have to be developed on a case- +by-case basis. + +4 +F. GROTTO AND G. PECCATI +where mn is the hyperbolic volume, and BR is a ball of radius R in the hyperbolic +distance. Most of our efforts will be devoted to the study of those functionals G(uλ) +(typically called polyspectra) obtained by taking G to be a Hermite polynomial of a +fixed order, whose behavior is investigated in two different limiting regimes: high- +energy (λ → ∞) and large domain (R → ∞). Our main results, stated in full detail +in Section 3.2, yield variance estimates and Central Limit Theorems (CLTs) for +polyspectra of arbitrary orders, from which one can deduce CLTs for functionals +G(uλ) associated with a generic G. +Remark 1.2. Theorem 3.6 — which is one of the main contributions of the present +work — will reveal an interesting phenomenon, namely: +whereas in the high- +frequency regime the asymptotic behavior of hyperbolic and Euclidean polyspectra +roughly coincide, the same conclusion does not hold in the large-domain limit in the +case n = 2. In the parlance of time-series analysis, such a result seems to indicate +that, unlike Euclidean random waves, hyperbolic random waves on H2 display a +form of short memory, see e.g. [20, 44] for an introduction to this concept. +In Sections 3.3 and 3.4, we will apply our results to study two remarkable func- +tionals associated with the excursions of uλ: +(a) the volume of the excursion set +(1.2) +mn ({x ∈ BR : uλ(x) > t}) = +� +BR +1uλ>t(x)dmn(x); +(b) the Leray measure +(1.3) +LR,λ := lim +ε→0 +1 +2ε |{x ∈ BR : |uλ(x)| ≤ ε}| , +which can be formally understood as the integral of a generalized function, +as follows: +LR,λ = +� +BR +δ0(uλ(x))dmn(x). +Remark 1.3. With probability one, the nodal set of uλ is a submanifold of codimen- +sion 1. As a consequence a – perhaps more natural – local functional to consider is +the induced (n − 1)-dimensional volume. However, a functional such as the nodal +length in dimension n = 2, (formally) given by +length({x ∈ BR : uλ(x) = 0}) = +� +BR +δ0(uλ(x)) +� +⟨duλ, duλ⟩T ∗ +x Hndmn(x), +also involves the differential duλ of the random field, making its study not directly +achievable by the techniques of the present paper. We prefer to regard such an +issue as a separate topic and defer it to future investigations. +1.5. Structure. In Section 2, we recall the necessary preliminaries on geometry +and spectral theory of Hn, and then rigorously introduce the hyperbolic random +wave model. Section 3 contains a discussion of our main results on functionals of +random waves. Finally, Section 4 is devoted to the technical core of the proofs. +1.6. Notation. We write X ∼ Y when random variables X, Y –taking values in +the same space– have the same law. We write N(α, β2) to indicate a Gaussian +random variable with mean α ∈ R and variance β2 ≥ 0. The term distribution will +always refer to an element of the dual space of smooth functions on some manifold, +that is a generalized function, never to the law of a random variable. Landau O’s +and o’s have their usual meaning, subscripts indicating eventual dependence on +parameters. The symbol C will denote a positive constant, possibly differing in any +of its occurrence even in the same formula, depending only on eventual subscripts. + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +5 +The expression A ≃a,b B indicates that B is both an upper and lower bound by A up +to strictly positive multiplicative constants depending only on eventual subscripts +a, b. Expressions A ≲a,b B or A ≳a,b B indicate respectively an upper and lower +bound in the same sense. +1.7. Acknowledgments. Research supported by the Luxembourg National Re- +search Fund (Grant: O21/16236290/HDSA). F. G. acknowledges support of +INdAM through the INdAM-GNAMPA Project CUP E55F22000270001. +2. Geometry of Hyperbolic Space and Random Waves +The hyperbolic space Hn is the simply connected n-dimensional Riemannian +manifold of constant negative curvature −1. +It is modeled by one sheet of the +two-sheeted hyperboloid x2 +0 − x2 +1 − · · · − x2 +n = 1 in Rn+1, say x0 > 0, with the +Riemannian metric being induced by Minkowski metric −dx2 +0 + dx2 +1 + · · · + dx2 +n +on the ambient space Rn+1. +The Riemannian distance in this parametrization +Hn ∋ x = (x0, x1, . . . , xn) is given by +d(x, y) = cosh−1 [x, y] , +[x, y] = x0y0 − x1y1 − · · · − xnyn, +x, y ∈ Hn. +We will denote by dmn the Riemannian volume on Hn, or rather an arbitrarily +fixed positive multiple of it, such choice being completely irrelevant for our goals; +accordingly, we will write for simplicity L2(Hn) = L2(Hn, dmn). +Besides Cartesian coordinates of the hyperboloid model, we will often employ +polar (geodesic) coordinates Hn ∋ x = (r, ϑ), r = d(x, x0) > 0, ϑ ∈ Sn−1, around a +given point x0 ∈ Hn, in terms of which the volume element is given by +dmn(x) = cn sinh(r)n−1drdςn−1(ϑ) +with dςn−1(ϑ) denoting2 the volume form on the sphere Sn−1. We will also employ +the usual notation ωn = +� +Sn dςn, with ω0 = 2. +The content of the forthcoming Sections 2.1 and 2.2 is classical: the reader is +referred to [57, Section 4] and [26, Section 2] for definitions, proofs, and examples. +2.1. Spectral Theory of Hyperbolic Space. We will denote by ∆ = ∆Hn the +Laplace-Beltrami operator on Hn. Since the metric of Hn is induced by the embed- +ding into Minkowski space, we have a convenient representation of the Laplacian +on Hn in terms of the d’Alembert operator □ = −∂2/∂x2 +0 + ∂2/∂x2 +1 + · · · + ∂2/∂x2 +n +on the ambient space Rn+1 ⊃ Hn, that is +(2.1) +∆Hnf = □ f +� +x/ +� +[x, x] +���� +x∈Hn . +We recall the spectral theorem on the hyperbolic space: +Theorem 2.1. [26, Example 2.11, Theorem 2.12] The Laplace-Beltrami operator +∆ on Hn, regarded as an unbounded operator on L2(Hn) densely defined on smooth +functions, is essentially self-adjoint; its spectrum is purely absolutely continuous +and given by +(2.2) +��n − 1 +2 +�2 +, ∞ +� += +� +λ = σ2 + α2, σ = σn = n − 1 +2 +, α ∈ R +� +. +2We prefer the graphic variant ς (‘final sigma’) since the symbol σ is customarily used to param- +etrize the Laplacian’s spectrum, see the subsequent Section. + +6 +F. GROTTO AND G. PECCATI +The projection operator on the eigenspace relative to λ = σ2 + α2 is given by +Pλf(z) = ωn−1ρn(α) +� +Hn Fn,λ(d(x, y))f(y)dmn(y), +f ∈ L2(Hn), +ρn(α) = +1 +(2π)n +���� +Γ(σ + iα) +Γ(α) +���� +2 +, +which is expressed in terms of the so-called spherical function [57, (4.3)] +Fn,λ(d(x, y)) = +1 +ωn−1 +� +Sn−1 [x, (1, ϑ)]−σ+iα [y, (1, ϑ)]−σ−iα dςn−1(ϑ), +(2.3) +Fn,λ(r) = ωn−2 +ωn−1 +� π +0 +(cosh r − sinh r cos θ)−σ+iα (sin θ)n−2dθ. +(2.4) +The projection operators Pλ naturally satisfy f(x) = +� ∞ +0 +Pσ2+α2f(x)dα [57, +(4.2)], and the function ρn(α) is thus the spectral measure (see Proposition 2.5). +Spherical functions take such a name because ψ(x, y) = Fn,λ(d(x, y)) is the unique +(real) radial solution of the eigenvalue problem +∆xψ(x, y) = λψ(x, y), +ψ(x, x) = 1, +x, y ∈ Hn, +where ∆x indicates an application of the Laplacian ∆Hn to the mapping x �→ +ψ(x, y). +Remark 2.2. In what follows, we will use Fn,λ(d(x, y)) as the covariance function of +a Gaussian field. In fact, formulas (2.3) and (2.4) define positive-definite functions +also when α = 0 and σ ∈ (0, (n−1)/2] [15, Sec. 5.3], thus one can consider Gaussian +fields with such covariance for this additional choice of parameters, see [15]. We +leave the study of these fields open for future research. +Remark 2.3 (Notational). Throughout the paper, the parameters λ, n, σ, α will al- +ways be related through the relations put forward in formula (2.2). In particular, +dependence n can be given in terms of σ only and, and given n, a dependence on +λ can be given in terms of α2 only. Equation (2.3) is the prototypical example of +this situation: it is easy to observe that the right-hand side does not depend on the +sign of α, so overall dependence is on λ. +Writing the eigenvalue problem in polar coordinates [15, (4.6)] one readily obtains +the following ODE satisfied by Fn,λ: +d2 +dr2 Fn,λ(r) + n − 1 +tanh r +d +dr Fn,λ(r) + λFn,λ(r) = 0, +r > 0, +(2.5) +Fn,λ(0) = 1, +F ′ +n,λ(0) = 0. +As we recall in Section 4, solutions of such ODE can be represented with hyperge- +ometric functions; together with the integral representation (2.4) this will allow us +to obtain good approximations on Fn,λ on which our arguments heavily rely. +2.2. Waves on Hyperbolic Spaces. As recalled in the Introduction, the class +of Euclidean plane waves is the collection of all exponential functions x �→ eix·k, +k ∈ Rd, and that each of them trivially verifies the Laplace equation +∆Rdeix·k = |k|2eix·k, +x ∈ Rd. +(2.6) +Euclidean plane waves are generalized eigenfunctions of the Laplacian ∆Rd = +− �d ∂2 +j , in the sense that they are smooth functions satisfying (2.6) but they +do not belong to L2(Rd) (in which we set spectral theory). Plane waves as above +are indexed by k ∈ Rd, or equivalently by their wavenumber |k| ∈ R+ (indicating +the relative eigenvalue |k|2) and the direction of the wave k/|k| ∈ Sn−1. + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +7 +On the hyperbolic space Hn, one actually has a perfect analog of plane random +waves, that is obtained (for each n ≥ 2) by considering the smooth functions +x �→ en(x, α, u) derived from the following mappings on the product space Hn × +R∗ × Sn−1: +en : Hn × R∗ × Sn−1 → C, +en(x, α, u) = [x, (1, u)]−σ+iα , +in such a way that the following equation is satisfied: +∆Hn en(x, α, u) = (σ2 + α2) en(x, α, u), +x ∈ Hn. +(2.7) +Note that in (2.7) the operator ∆Hn is applied to the variable x; the formula can +be directly checked by applying the expression (2.1) to en(x, α, u) and carrying +through the tedious but elementary computation. The functions en(·, α, u) are thus +generalized eigenfunctions of the Laplace-Beltrami operator ∆ = ∆Hn, and they are +parametrized by the wavenumber α ∈ R and the “direction” of the wave u ∈ Sn−1. +Remark 2.4. The analogy with the Euclidean case is perhaps more geometrically +intuitive in the case n = 2, once one moves to the disk model of the hyperbolic +plane: in such a setting, e2 is rewritten as an imaginary exponential involving the +distance between the horocycle through x and u ∈ S1 and the origin, the direction +u ∈ S1 being naturally identified with a point of the boundary of the Poincar´e disk +(a point at infinity of the hyperbolic plane). We refer to [24, Introduction] for a +thorough comparison between Euclidean and hyperbolic settings. We also observe +that the analogy with the Euclidean case carries through when considering wave +equations, justifying the “wave” terminology. In particular, solutions of the wave +equation on Hn, +(2.8) +� +∂2 +t + ∆Hn − σ2� +u(x) = 0, +(see [5] for a discussion of this PDE) can be written as superpositions of waves +eitα en(x, α, u). Notice that the wave operator in the previous display takes into +account that the spectrum of ∆Hn begins at σ2. +Just as on Rn, planar waves can be used to set up Fourier analysis on Hn. +Proposition 2.5. [26, Ssec. 2.11.4] Given f ∈ C∞ +c (Hn), define its Fourier trans- +form as +(2.9) +Ff(α, ϑ) = +� +Hn en(x, −α, ϑ)f(x)dmn(x), +α ∈ [0, ∞), ϑ ∈ Sn−1. +It holds (transform inversion) +(2.10) +f(x) = +� ∞ +0 +� +Sn−1 Ff(α, ϑ) en(x, α, ϑ)ρn(α)dαdςn−1(ϑ). +Moreover (Plancherel formula) F extends to an isometry +F : L2(Hn, mn) → L2([0, ∞) × Sn−1, ρn(α)dαdςn−1), +whose inverse is given by (the extension of) (2.10). +Spherical functions can be regarded as spherical averages of waves en, +Fn,λ(d(x, (1, 0, . . . , 0))) = +1 +ωn−1 +� +Sn−1 en(x, α, ϑ)dςn−1(ϑ), +(a special case of Equation 2.3) thus playing the role of Bessel functions in the +Euclidean case — see (1.1). + +8 +F. GROTTO AND G. PECCATI +2.3. Hyperbolic Random Waves. In what follows we will consider both real- +valued and complex-valued random fields; we refer to [25, Chapter 6] for a discussion +of white noise analysis in the complex setting. +Remark 2.6 (Real and complex white noise). Before stating the main result of the +present Section, and for the reader’s convenience, we recall the definition and basic +properties of complex white noises, in the sense of [25]. Fix a finite mesure space +(X, F, µ) and denote by L2(X, µ; R) and L2(X, µ; C), respectively, the associated +L2 spaces of real- and complex-valued functions. A (real) white noise on (X, F, µ) +(often called an isonormal Gaussian process with intensity µ — see e.g. [46, Chapter +2]) is a centred real Gaussian family of the type U = {U(f) : f ∈ L2(X, µ; R)} such +that +E [U(f)U(g)] = +� +X +fg dµ, +f, g ∈ L2(X, µ; R); +the definition of U is customarily extended to all f ∈ L2(X, µ; C) by setting U(f) := +U(Re(f)) + iU(Im(f)). A complex white noise W on a finite measure space (X, µ) +is a complex Gaussian family W = {W(f) : f ∈ L2(X, µ; C)} having the law of +U + iV , where U, V are i.i.d. real white noises on (X, F, µ), as defined above. The +following computational rules can be easily checked: for all f, g ∈ L2(X, µ; C), one +has that +E +� +W(f)W(g) +� += +� +X +fg dµ, +E [W(f)W(g)] = 0, +(2.11) +E [Re[W(f)] Re[W(g)]] = +� +X +[Re(f) Re(g) + Im(f) Im(g)]dµ, +(2.12) +E [Re[W(f)] Im[W(g)]] = +� +X +[Re(f) Im(g) − Im(f) Re(g)]dµ, +(2.13) +and the second and third equalities continue to hold when one switches the symbols +‘Re’ and ‘Im’ on both sides of each equation. +The next proposition singles out a class of stationary random fields that can be +regarded as canonical Gaussian Laplace eigenfunctions on Hn — they will constitute +our main object of study. +Proposition 2.7. Fix α ∈ [0, ∞), and set λ = σ2 + α2, where the constant σ2 is +the same as in (2.2). +(1) There exists a unique (in law) random field uλ : Hn → R such that +(i) uλ(x) is a Gaussian variable N(0, 1) for all x ∈ Hn; +(ii) the law of uλ is invariant under isometries of Hn; +(iii) almost all samples of uλ are generalized λ-eigenfunction of ∆Hn of +class C∞(Hn). +The same conclusion holds for the complex version uC +λ : Hn → C if at +Point (i) one replaces the standard Gaussian random variable N(0, 1) with +a standard complex Gaussian variable NC(0, 1). +(2) The Gaussian random field uλ is equivalently characterized by its mean and +covariance function +(2.14) +E [uλ(x)] = 0, +E [uλ(x)uλ(y)] = Fλ(d(x, y)), +for all x, y ∈ Hn; samples of uλ are of class C∞(Hn). Moreover, +1 +√ +2 Re uC +λ +and +1 +√ +2 Im uC +λ are two independent identically distributed real Gaussian ran- +dom fields with the same law as uλ. + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +9 +(3) We have the following representation: if W is a complex white noise on +(Sn−1, ςn−1), then uC +λ has the same law as the stochastic integral +(2.15) +uC +λ(x) ∼ +� +Sn−1 en(x, α, ϑ)W(dϑ). +Remark 2.8. The random fields uC +λ are discussed – with different notation and from +a slightly different perspective – in [15, Section 5.3], to which the reader is referred +for further background material. For the rest of the paper, we will refer to uλ and +uC +λ, respectively, as the real and complex hyperbolic random wave with eigenvalue +λ. +Proof of Proposition 2.7. It is convenient to start by defining uC +λ(x) by means of +(2.15) and show that it satisfies the properties put forward at Point (2). This will +show in particular that the real-valued function (x, y) �→ Fn,λ(d(x, y)) is positive +definite for all λ ∈ [σ2, ∞), so that (2.14) uniquely identifies the law of a real +Gaussian random field. To prove that the properties at Point (2) are met by the +random field in (2.15), we start by observing that the Gaussianity of the stochastic +integral is trivial, and so is the fact that +E +�� +Sn−1 en(x, α, ϑ)W(dϑ) +� += 0 +for all x ∈ Hn and α ≥ 0. As for the covariance, we deduce from (2.11) that +E +�� +Sn−1 en(x, α, ϑ)W(dϑ) +� +Sn−1 en(y, α, ϑ)W(dϑ) +� += +� +Sn−1 en(x, α, ϑ) en(y, −α, ϑ)dςn−1(ϑ), +which by (2.3) equals Fn,λ(d(z, w)). Moreover, (2.12) and (2.13), combined with +the fact that (by definition) Re[en(x, α, ϑ)] = Re[en(x, −α, ϑ)] and Im[en(x, α, ϑ)] = +− Im[en(x, −α, ϑ)], show that +1 +√ +2 Re uC +λ and +1 +√ +2 Im uC +λ are two i.i.d. centered real +Gaussian random fields with covariance function Fn,λ(d(·, ·)). +This shows that +(2.15) satisfies the properties at Point (2) (note that uλ and uC +λ have paths of class +C∞ because the covariance function of these fields is of class C∞: this implication +is proved e.g. in [41, Subsection A.9] for Gaussian fields on Euclidean spaces, and +it is straightforwardly adapted to the hyperbolic setting after composition with a +(smooth, global) chart of Hn. The proof of the Theorem is concluded if we show +the equivalence of (2.14) and of the properties listed at Point (1). Let us assume +that uλ verifies the properties at Point (1). Then, by invariance under isometries +(and since the isometry group of Hn acts transitively), the covariance function +(2.16) +C(z, w) = E [uλ(z)uλ(w)] = f(d(z, w)) +only depends on the distance d(z, w). Since the samples of uλ are smooth general- +ized eigenfunctions, we then deduce that +(2.17) +∆HnC(z, w) = E [∆Hnuλ(z)uλ(w)] = λE [uλ(z)uλ(w)] = λC(z, w), +and from the discussion in Subsection 2.1, we conclude that C(z, w) = Fλ(d(z, w)). +The proof that (2.14) implies the conditions at Point (1) follows from similar ar- +guments, both in the real- and complex-valued cases. +□ +To further the analogy with Berry’s random waves on Rn, one can also derive +the random field uλ with a Central Limit result for a superposition of finitely many +generalized eigenfunctions of ∆Hn. + +10 +F. GROTTO AND G. PECCATI +Proposition 2.9. Let α ∈ [0, ∞), λ = σ2 + α2 be fixed, and consider two in- +dependent sequences of i.i.d. +uniform random variables ϑ1, ϑ2, . . . on Sn−1 and +φ1, φ2, . . . on [0, 2π]. Define the following finite combination of hyperbolic waves +(2.18) +uN +λ (z) = +1 +√ +N +N +� +j=1 +eiφj en(x, α, ϑj), +to be regarded as a random element of C∞(Hn; C). As N → ∞, finite-dimensional +distributions of uN +λ converge in law to the ones of uC +λ. +A real analogue of the latter can be obtained by taking the real (or imaginary) +part of all involved objects. Notice how, in sight of Subsection 2.1, this result repre- +sents uλ as a stochastic superposition of single waves solving (2.8) with wavenumber +α. +Proof. For fixed x ∈ Hn, uN +λ (z) can be regarded as the duality coupling between +the smooth function en(x, α, ·) ∈ C∞(Sn−1; C) and the generalized function +1 +√ +N +N +� +j=1 +eiφjδbj(·) ∈ C∞(Sn−1; C)∗. +Since generalized functions eiφjδbj can be regarded as i.i.d. random elements of +the Sobolev space Hs(Sn−1; C) for s < −n/2, the Central Limit Theorem for i.i.d. +variables in Hilbert spaces applies (cf. [33, 10.1]), and the sum in display converges +in law to complex white noise W on Sn−1. The thesis then follows by Proposition 2.7 +considering couplings with en(x, α, ·) at finitely many distinct points x. +□ +2.4. Curvature, Large Scale and Local Behavior of Random Waves. As +already discussed, the principal aim of the present paper is to characterize the +fluctuations of integral functionals of the hyperbolic waves {uλ}, as defined in the +previous Subsection 2.3, both as λ → ∞ on a fixed domain (high-frequency limit), +and for fixed λ on expanding domains (large domain limit). Our main achievements +on the matter are discussed in full detail in the forthcoming Section 3: in particular, +our findings will show some remarkable discrepancies between the large domain +behaviours of hyperbolic and Euclidean polyspectra. +In order to develop some basic intuition on the relation between hyperbolic and +Euclidean settings, in Proposition 2.10 we will characterize the local behaviour of +hyperbolic random waves around a fixed point – that we will encode in terms of the +scaling limit of the associated pullback waves on tangent spaces. Some preliminary +considerations are, however, in order. +2.4.1. Remarks on scaling limits. We start by pointing out a fundamental difference +between the hyperbolic and Euclidean settings, that is: in the hyperbolic framework +– and differently from the Euclidean one – there is no direct relation linking high- +frequency and large distance limits. +To see this, fix λ > 0 and recall the definition of the Euclidean random waves +{vλ} introduced in (1.1). Trivially, the fact that Cn,λ(x, y) is a function of +√ +λ|x−y| +makes it so that for Euclidean random waves it is equivalent to consider limits at +high frequency (for a fixed distance) and at large distance (at a fixed frequency). +We will see that this is not the case for (functionals of) hyperbolic waves. +Indeed, the counterpart of scaling lengths on a Euclidean space is to consider +a positive multiple of the metric tensor on a Riemannian manifold. +Namely, if +M = (M, g) is a Riemannian manifold we set MR = (M, R2g), R > 0, a transfor- +mation that amounts to multiply all distances by R: if x, y ∈ M, dM(x, y) = r, + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +11 +then dMR(x, y) = Rr. Under this transformation, eigenvalues of Laplace-Beltrami +operator are scaled by a factor 1/R2. +In the Euclidean case M = Rn, if φR +λ (x, y) = φR +λ (|x − y|) is the unique radial +solution of +∆RφR +λ (x, y) = λφR +λ (x, y), +φR +λ (x, x) = 1, +where ∆R = +1 +R2 ∆ is the Laplace operator on Rn +R, then +(2.19) +φR +λ (|x − y|) = φ1 +R2λ(|x − y|) = CR2λ(x, y) = φ1 +λ(R|x − y|), +where the last equality is a consequence of the particular form of spherical functions +on flat space. +Consider now the hyperbolic case: a crucial difference is that Rn +R, R > 0, are +all isometric, whereas Hn +R has sectional curvature −1/R2. +Looking at spherical +functions, if ψR +λ (x, y) = ψR +λ (d(x, y)) is the unique radial solution of +∆Hn +RψR +λ (x, y) = λψR +λ (x, y), +ψR +λ (x, x) = 1, +then the first equation of (2.19) still holds, +ψR +λ (d(x, y)) = ψ1 +R2λ(d(x, y)) +it being a general fact (notice that d(x, y) is the distance of Hn = Hn +1, not the +rescaled one). However, in sight of the last display and the previous paragraphs, +we can write +ψR +λ (x, y) = Fn,R2λ(d(x, y)) += Cn +� π +0 +(cosh d(x, y) − sinh d(x, y) cos θ)−σ+iR√ +λ−σ2/R2 (sin θ)n−2dθ, +in terms of the function Fn,λ defined in (2.4), which makes it clear that +ψR +λ (d(x, y)) = ψ1 +R2λ(d(x, y)) ̸= ψ1 +λ(Rd(x, y)), +marking the difference with the Euclidean case. +2.4.2. A local scaling limit result. In the light of the above discussion, a natural +question is whether the local behavior of the hyperbolic waves uλ around a given +point resembles that of Berry’s model at high frequencies. This turns out to be the +case — at least from the standpoint of covariance functions. Since the two models +are defined on different manifolds, such a statement is made precise by comparing +the planar random wave on Rn with covariance function as in (1.1) and frequency +λ = 1, and a properly rescaled pullback of uλ to the tangent space (at a given point +x ∈ Hn) given by the exponential map. +Proposition 2.10. Let α ≥ 0 be fixed, set λ = σ2+α2 and fix x ∈ Hn. Consider the +covariance function of the pullback random wave uλ(expx(·/ +√ +λ)) on TxHn ≃ Rn, +CH +n,λ(u, v) = Fλ +� +d +� +expx +v +√ +λ +, expx +v′ +√ +λ +�� +, +v, v′ ∈ Rn. +Let rλ = o( +√ +λ) as λ → ∞. Then, recalling that Cn,λ denotes the covariance of +Berry’s Euclidean model (1.1), one has that +sup +v,v′∈R2:|v|,|v′| 0, +(3.3) +W1(X, Y ) = b · W1(X/b, Y/b). +(c) let Xk, k ≥ 1, be a centered and square-integrable random variable such +that a c2 +k ≤ Var Xk ≤ b c2 +k, for some strictly positive sequence {c2 +k} and +constants a, b > 0, and let Nk denote a centered Gaussian random variable +with variance Var Xk/c2 +k; then if +(3.4) +W1(Xk/ck, Nk) → 0, +for every subsequence k(n) → ∞ there exists a sub-subsequence k(n′) such +that Var Xk(n′)/c2 +k(n′) converges and +(3.5) +Xk(n′) +ck(n′) +Law +=⇒ N(0, c2), +where c2 := limn′ c−2 +k(n′) Var Xk(n′) > 0. +The content of Point (c) amplifies the relevance of the forthcoming Theorem 3.4. +3.1. Some elements of Gaussian analysis. + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +13 +3.1.1. Representation of hyperbolic waves. Let the notation and assumptions of the +previous Sections prevail. Then, for all n ≥ 1 and λ ∈ +�� n−1 +2 +�2 , ∞ +� +(see (2.2)), +one has that there exists a finite measure space (X, F, µ), a real white noise W on +(X, µ) (as defined in Remark 2.6) and an integral kernel Kn,λ : Hn × X → R such +that the random field +(3.6) +Hn ∋ x �→ I1(Kn,λ(x, ·)) := +� +X +Kn,λ(x, y)W(dy), +has the same law as {uλ(x) : x ∈ Hn}. A representation of the type (3.6) can +be deduced from Point (3) of Proposition 2.7 (in which case, X = Sn−1 and µ +is the uniform measure). +In general, we stress that (i) the representation (3.6) +is not unique, (ii) the validity of (3.6) is a standard consequence of the fact that +each random field {uλ} is separable, and (iii) the space (X, F, µ) and the random +measure W can be chosen to be the same for each n. +Without loss of generality, from now on we will assume that representation (3.6) +is in order, and that the real white noise W on (X, µ) is defined on a probability +space (Ω, G, P) in such a way that W is the same for each n and G is the canonical +completion of σ(W). +3.1.2. Wiener chaos. The following basic elements of Gaussian stochastic analysis +will be used for the rest of the paper — see [29, 46] for a full discussion. +For +every f ∈ L2(X, F, µ), the stochastic integral I1(f) := +� +X f dW is well-defined, +and the class {I1(f) : f ∈ L2(X, F, µ)} is a centered Gaussian family (known +as the first Wiener chaos of W) with covariance given by the relation: for all +f, g ∈ L2(X, F, µ) := L2(µ), E [I1(f)I1(g)] = ⟨f, g⟩L2(µ). +It is a standard fact that the space L2(P) := L2(Ω, G, P) admits the Wiener +chaotic decomposition +(3.7) +L2(P) = +∞ +� +q=0 +H:q:, +H:0: := R, H:q: := +� +Iq(f) : f ∈ L2 +sym(µq) +� +, q ≥ 1, +where the symbol L2 +sym(µq) stands for the Hilbert subspace of L2(Xq, F⊗q, µq) := +L2(µq) composed of (the equivalence classes of) those kernels f that are µq-almost +everywhere symmetric, and Iq(f) denotes the q-fold stochastic integral of f with +respect to W. For q ≥ 1, H:q: is called the qth Wiener chaos of W, and one has the +isometric relation: E[Iq(f)Ip(g)] = 1p=q q! ⟨f, g⟩L2(µq), valid for all p, q ≥ 1, and all +f ∈ L2 +sym(µq) and g ∈ L2 +sym(µp). +We will often exploit the fact that, for all f ∈ L2(X, F, µ), one has that Iq(f ⊗q) = +Hq(I1(f)), where Hq is the q-th (probabilistic) Hermite polynomial on the real-line; +see e.g. [46, Theorem 2.7.7]. We recall that the collection {Hq : q = 0, 1, ...} of +Hermite polynomials coincides with the coefficients of the exponential generating +function +est−t2/2 = +∞ +� +q=0 +Hq(s)tq +q!, +t, s ∈ R.3 +It is well-known that +� +Hq/√q! +� +q≥0 is an orthonormal basis of L2(R, φ(s)ds), where +φ(s) = (2π)−1/2e−s2/2 is the standard Gaussian density. +According to the previous conventions and discussion, for all x ∈ Hn, uλ(x) = +I1(Kn,λ(x, ·)) belongs to the first Wiener chaos H:1: and, consequently, the random +3The first few polynomials are: H0(x) = 1, H1(x) = x, H2(x) = x2 − 1, H3(x) = x3 − 3x, +H4(x) = x4 − 6x2 + 3, and so on. + +14 +F. GROTTO AND G. PECCATI +variable +Hq(uλ(x)) = Iq � +Kn,λ(x, ·)⊗q� +is an element of H:q:, as defined in (3.7). Given G ∈ L2(R) we deduce — e.g. by +dominated convergence and Jensen’s inequality — that the chaos expansion of the +integral functional GR(uλ) defined in (3.1) is given by +GR(uλ) = +∞ +� +q=0 +hn,q +R,λ +1 +q! +� +R +G(t)Hq(t)φ(t)dt, +(3.8) +hn,q +R,λ = Iq +�� +BR +Kn,λ(x, ·)⊗qdmn(x) +� +, +(3.9) +where the series (3.8) converges in L2(P) and its qth summand coincides with the +projection of GR(uλ) onto H:q:; in particular, for q ≥ 1, the polyspectrum hn,q +R,λ is +an element of H:q: and (by applying e.g. a stochastic Fubini argument) it is easily +seen to coincide with (3.2) above. The smallest q ≥ 0 for which the coefficient +� +R G(t)Hq(t)φ(t)dt does not vanish is called the Hermite rank of G (and, by ex- +tension, of the functional GR(uλ)). As recalled in the Introduction, the notion of +Hermite rank plays a pivotal role in the asymptotic theory of Gaussian-subordinated +random fields on Euclidean spaces, see e.g. [46, Chapter 7]. +3.1.3. The Wiener chaos approach to CLTs. Mantaining the notation and assump- +tions of the previous Section, we will now state three results allowing one to prove +CLTs by using Wiener chaos expansions: these statements are the core of the +“Wiener chaos approach” advertised in the title of the paper. +Given q ≥ 1 and f, g ∈ L2 +sym(µq), and r = 0, ..., q, the r-contraction of f and g is +defined as f ⊗0 g = f ⊗ g and, for r = 1, ..., q, +(f ⊗r g)(x1, . . . x2q−2r) += +� +Xr f(x1, . . . xq−r, z1, . . . , zr)g(xq−r+1, . . . , x2q−2r, z1, . . . , zr)dµr(z1, . . . , zr), +in such a way that f ⊗r g ∈ L2(µ2q−2r), with the convention L2(µ0) := R. +As made clear in the next statement, which is a direct consequence of [46, The- +orem 5.2.7 and Theorem 6.3.1], contractions can be used in order to quantitatively +assess the distance to Gaussian within a fixed Wiener chaos. +Theorem 3.2. Fix q ≥ 2 and let f ∈ L2 +sym(µq) be such that q!∥f∥2 +L2(µq) = +E +� +Iq(f)2� += σ2 > 0. Then, there exists a combinatorial constant Γ(q) > 0, uniquely +depending on q, such that +(3.10) +W1(Iq(f), N(0, σ2)) ≤ Γ(q) +σ +max +r=1,...,q−1∥f ⊗r f∥L2(µ2(q−r)). +Combining (3.3) with [45, Proposition 3.7], one can use contractions to derive +effective bounds on the normal approximation of random variables living in a finite +sum of Wiener chaoses. +Proposition 3.3. Let Q ≥ 1 and let +F = +Q +� +q=1 +Iq(gq), +gq ∈ L2 +sym(µq), +be such that E +� +F 2� += �Q +q=1 q!∥gq∥2 +L2(µq) = σ2 > 0. Then, there exists a combina- +torial constant Γ1(Q), uniquely depending on Q, such that +(3.11) +W1(F, N(0, σ2)) ≤ Γ1(Q) +σ +· max +q,r ∥gq ⊗ gq∥L2(q−r). + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +15 +where the maximum runs over all q = 1, ..., Q and all r = 1, ..., q − 1. +Finally, in order to derive CLTs in the context of generic nonlinear functionals +of hyperbolic waves, we will need the following general result. +Theorem 3.4. Under the above assumptions and notation, consider a sequence of +square-integrable random variables with the form +Fj = +∞ +� +q=1 +Iq(fj,q), +fj,q ∈ L2 +sym(µq), +j ≥ 1, +and write c2 +j,q := q!∥fj,q∥2 +L2(µq). Consider a sequence c2(j) → ∞ with the following +property: there exist T, T0 ⊂ N such that T ∩ T0 = ∅, T ̸= ∅ and T ∪ T0 = N, as +well as finite constants 0 < b1 < b2 and σ2 +q ≥ 0, verifying +(1) �∞ +q=1 σ2 +q := Σ0 < ∞; +(2) for all q ∈ T , σ2 +q > 0 and +b1 σ2 +q c2(j) ≤ c2 +j,q ≤ b2 σ2 +q c2(j), +j ≥ 1; +(3) for all q ∈ T0, c2 +q,j ≤ b2 c2(j) σ2 +q; +(4) for fixed q ≥ 1 and all r = 1, . . . , q − 1, +lim +j→∞ c−2(j) ∥fj,q ⊗r fj,q∥L2(µ2(q−r)) = 0. +Then, as j → ∞, +(3.12) +Var(Fj) ≃ c2(j) +and +W1 +� Fj +c(j), N(0, γ2(j)) +� +−→ 0, +where γ2(j) := Var(Fj)/c2(j). +Remark 3.5. The discussion around formulae (3.4)–(3.5) above illustrates the sig- +nificance of the second asymptotic relation in (3.12). +Proof of Theorem 3.4. One has that +b1 +� +q∈T +σ2 +q c2(j) ≤ Var(Fj) ≤ b2 Σ0 c2(j), +from which one deduces that Var(Fj) ≃ c2(j). Now write Nj := N(0, γ2(j)), fix Q +such that T ∩ {1, ..., Q} ̸= ∅ (such a Q exists because T is non empty), and observe +that – by applying twice the triangle inequality – +W1(Fj/c(j), Nj) ≤ 2 +� +b2 +� +q≥Q+1 +σ2q + W1 +� +1 +c(j) +Q +� +q=1 +Iq(fj,q), N(0, σ2(j, Q)) +� +where σ2(j, Q) := c(j)−2 �Q +q=1 c2 +j,q ≥ b1σ2 +q0 > 0, and q0 is any element of T ∩ +{1, ..., Q}. It follows from Proposition 3.3 that +W1(Fj/c(j), Nj) ≤ 2 +� +b2 +� +q≥Q+1 +σ2q + Γ1(Q) +b1/2 +1 +σq0 +max +q,r +∥fj,q ⊗r fj,q∥L2(µ2(q−r)) +c2(j) +, +:= +A(Q) + B(Q, j). +where the maximum runs over all q ≤ Q and all r = 1, ..., q − 1. Since A(Q) → 0 +(as Q → ∞) and B(Q, j) → 0 (as j → ∞ for fixed Q), the conclusion follows. +□ + +16 +F. GROTTO AND G. PECCATI +3.2. CLTs for integral functionals. Our aim is now to apply Theorem 3.4 to +the integral functionals GR(uλ) (as defined in (3.1)), both as λ → ∞ and as R → +∞. In view of (3.8)–(3.9), this task requires one to assess both the variances and +the contraction norms associated with the polyspectra hn,q +R,λ, once these random +elements are represented as multiple stochastic integrals — with respect to the +white noise W on (X, µ) featured in (3.6) — of kernels of the type +� +BR +Kn,λ(x, ·)⊗qdmn(x) ∈ L2 +sym(µq). +One fundamental fact (explaining why, for our purposes, the precise choice of +the white noise W and measure space (X, F, µ) is immaterial) is that both the +variances and the contraction norms associated with the polyspectra hn,q +R,λ uniquely +depend on the covariance function Cov(uλ(x), uλ(y)) = Fn,λ(d(x, y)). To see this, +we observe that, by a standard argument based e.g. on [46, Proposition 2.2.1], for +all q ≥ 1 one has that +Var(hn,q +R,λ) += +E +�� +BR +� +BR +Hq(uλ(x))Hq(uλ(y))dmn(x)dmn(y) +� +(3.13) += +q! +� +BR +� +BR +Fn,λ(d(x, y))qdmn(x)dmn(y). +Analogously, a direct computation (based on the use of Fubini theorem as well as +on the representation (3.6) and the isometric properties of real white noises stated +in Remark 2.6) yields the equation +(3.14) +���� +�� +BR +Kn,λ(x, ·)⊗qdmn(x) +� +⊗r +�� +BR +Kn,λ(y, ·)⊗qdmn(y) +����� +2 +L2(µ⊗2(q−r)) += +� +B4 +R +Fn,λ(d(x, y))rFn,λ(d(y, z))q−r +· Fn,λ(d(z, w))rFn,λ(d(w, x))q−rdm⊗4 +n (x, y, z, w). +Appropriate tools for estimating expressions such as (3.13)–(3.14) are developed +in Section 4. As an application of these techniques, we will prove the following +statement, which is one of the main achievements of the paper. +Theorem 3.6. Let n ≥ 2 and q ≥ 1. We have the following asymptotic relations +for q!−1 Var(hn,q +R,λ): +q!−1 Var(hn,q +R,λ) +λ → ∞, R fixed +R → ∞, λ fixed +q = 1 +≲n,R λ−σ−1 +≲n,λ mn(BR) +q = 2 +≃n,R λ−σ +≃n,λ R · mn(BR) +even q ≥ 4, except n = 2, q = 4 +≃n,R λ−σ−1/2 +≃n,λ mn(BR) +n = 2, q = 4 +≃n,R log(λ)/λ +≃n,λ mn(BR) +odd q ≥ 3, except n = q = 3 +≲n,R λ−σ−1/2 +≲n,λ mn(BR) +n = q = 3 +≲n,R λ−3/2 log(λ) +≲n,λ mn(BR) +In both limiting regimes considered above, for all 1 ≤ r ≤ q−1, the squared contrac- +tion norms (3.14) are of order o +� +(q′)!−2 Var(hn,q′ +R,λ)2� +for all even q′ ≤ q, where the +implicit constants depend on n and R (as λ → ∞) and on n and λ (as R → ∞). +As a consequence, one has that, for all n ≥ 1 and all q even, +(3.15) +hn,q +R,λ +Var(hn,q +R,λ)1/2 +Law +=⇒ N(0, 1), +both as λ → ∞, for R fixed, and as R → ∞, for λ fixed. + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +17 +The first part of the statement (concerning variances) is proved in Lemma 4.6 +and Lemma 4.13; the second part (on contraction norms) is proved in Lemma 4.12 +and Lemma 4.15. The CLT (3.15) is a direct consequence of (3.10). +Remark 3.7. By inspection of the proofs of the four Lemmas 4.6, 4.13, 4.12 and 4.15, +one can extrapolate some more precise information about the rate of convergence +of squared contraction norms. For instance, in the case q = 2, one has that, writing +X(R, λ, n) for the ratio +� +B4 +R Fn,λ(d(x, y))Fn,λ(d(y, z)) · Fn,λ(d(z, w))Fn,λ(d(w, x))dm4 +n(x, y, z, w) +Var(hn,2 +R,λ)2 +, +for all n ≥ 1, +(3.16) +X(R, λ, n) +� +≲n,R λ−σ−2, +λ → ∞, +≲n,λ 1 +R, +R → ∞; +Finally, applying Theorem 3.2 yields the following estimate on the speed of conver- +gence in (3.15): +(3.17) +W1 +� +hn,2 +R,λ +Var(hn,2 +R,λ)1/2 , N(0, 1) +� +≤ c · X(R, λ, n)1/2, +where c > 0 is some absolute constant. For general q > 2 even, similar estimates can +be deduced from the proof of Lemma 4.12 and the statement of Lemma 4.15. Again +by virtue of Theorem 3.2, such estimates yield bounds on the speed of convergence +(in the 1-Wasserstein distance) in the CLT (3.15). +Remark 3.8. We recall that, as R → ∞, one has that mn(BR) ≃n e2σR. In the +previous statement, we preferred to use expressions involving mn(BR) in order to +better streamline the comparison with the Euclidean case, as done in the next +remark. +Remark 3.9 (Comparison with the Euclidean case). For every n ≥ 2, consider +the Euclidean random wave with energy λ > 0 featured in (1.1), and define the +Euclidean polyspectrum hn,q +e;R,λ according to (3.2), by replacing uλ with vλ, and +by considering that BR is the Euclidean ball of radius R centered at the origin, +and mn is the Lebesgue measure on Rn. Then, the following table of asymptotic +relations (that we extrapolated from [42, Theorem V.1.1] – see also [47, 51, 43] and +[16], respectively, for the cases n = 2 and n = 3) is valid, with σ defined as in (2.2): +q!−1 Var(hn,q +e;R,λ) +λ → ∞, R fixed +R → ∞, λ fixed +q = 1 +≲n,R λ−σ−1 +≲n,λ mn(BR)/R +q = 2 +≃n,R λ−σ +≃n,λ R · mn(BR) +even q ≥ 2, except n = 2, q = 4 +≃n,R λ−σ−1/2 +≃n,λ mn(BR) +n = 2, q = 4 +≃R log(λ)/λ +≃λ log(R) · mn(BR) +odd q ≥ 3, except n = q = 3 +≲n,R λ−σ−1/2 +≲n,λ mn(BR) +n = q = 3 +≲R λ−3/2 log(λ) +≲λ log(R) · mn(BR) +Moreover, estimates on the contraction norms analogous to the ones in the state- +ment of Theorem 3.6 hold. As discussed in the Introduction, the most remark- +able difference between this table and the one in Theorem 3.6 is that, in the case +n = 2, q = 4 and for R → ∞, the variance of hn,q +e;R,λ does not display any logarithmic +correction. +Remark 3.10 (Comparison with random spherical harmonics). Laplace-Beltrami +operator ∆Sn on the sphere Sn has a discrete spectrum, and an orthonormal basis + +18 +F. GROTTO AND G. PECCATI +of L2(Sn) that diagonalizes ∆Sn is provided by spherical harmonics +∆SnYℓ,m;n = ℓ(ℓ + n − 1)Yℓ,m;n, +ℓ ∈ N, m = 1, 2 . . ., dℓ;n, +dℓ;n = 2ℓ + n − 1 +ℓ +�ℓ + n − 2 +ℓ − 1 +� +, +ℓ playing a role similar to the one of α in hyperbolic waves, that is ℓ is proportional +to the square root of the eigenvalue λ = ℓ(ℓ + n − 1) and ℓ ∈ N parametrizes the +spectrum. The index m on the other hand parametrizes a single eigenspace, dℓ;n +denoting its dimension. Random spherical harmonics are defined by +Tλ(x) = +dℓ;n +� +m=1 +aℓ,mYℓ,m;n, +x ∈ Sn, λ = ℓ(ℓ + n − 1), +with aℓ,m being i.i.d. Gaussian variables with E [aℓ,maℓ,m′] = δm=m′ωn/dℓ;n, and +we can consider polyspectra hn,q +sph,λ = +� +Sn Hq(Tλ(x))dςn(x). We cannot consider a +large-domain limiting regime on Sn: in this case we report variance asymptotics +only in the high-frequency regime, matching the ones of Euclidean and hyperbolic +cases. +q!−1 Var(hn,q +sph,λ) +λ → ∞ +q = 1 += 0 +q = 2 +≃n λ−σ +even q ≥ 2, except n = 2, q = 4 +≃n λ−σ−1/2 +n = 2, q = 4 +≃ log λ/λ +odd q ≥ 3 +≲n λ−σ−1/2 +We refer to [36] for the latter results. Notice that the case n = q = 3 is not included +as an exception, because in fact hn,q +sph,λ identically vanish if n, q are both odd, just +as in the case q = 1 (any n) in the table. This is an artifact of having chosen the +whole Sn as integration domain: symmetries of spherical harmonics come into play +producing cancellations. In dimension n = 2 the study was extended to polyspectra +over spherical caps in [58], obtaining the following: for Ω ⊂ S2 a spherical cap +subtended by a solid angle, +q!−1 Var( +� +Ω Hq(Tλ(x))dς2(x)) +λ → ∞ +q = 1 +≲ λ−3/2 +q = 2 +≃n λ−1/2 +q = 4 +≃ log λ/λ +even q ≥ 6 +≃n λ−1 +odd q ≥ 3 +≲n λ−1 +(constants in the estimates are independent of Ω). We refer to the series of works +[39, 38, 34, 40, 36, 37, 52] for a complete overview of the Wiener chaos approach to +integral functionals of random spherical harmonics. +Remark 3.11. When q is odd, Theorem 3.6 only yields upper bounds for Var(hn,q +R,λ) +(in both regimes): obtaining a lower bound in this case is indeed complicated by +the fact that such a variance displays oscillations that are arbitrarily close to zero. +For this reason, in the forthcoming Proposition 3.13 we are not able establish CLTs +for functionals of odd Hermite rank. This point, together with an explanation of +the fact that the cases n = 2, q = 4 and n = q = 3 of the table feature a logarithmic +correction in the high-frequency regime, will be fully discussed in Section 4 below. +Remark 3.12. In the table appearing in Theorem 3.6, the asymptotic relations for +q ≥ 5 have no explicit dependence on q (the latter is indeed not featured among + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +19 +the subscripts). In fact, for all n ≥ 2, +(3.18) +sup +q≥6 +q!−1 Var(hn,q +R,λ) ≤ sup +q≥6 +� +BR +� +BR +|Fn,λ(d(x, y))|qdmn(x)dmn(y) +≤ +� +BR +� +BR +|Fn,λ(d(x, y))|6dmn(x)dmn(y) += 6!−1 Var(hn,6 +R,λ) +� +≲n,R λ−σ−1/2, +λ → ∞, +≲n,λ mn(BR), +R → ∞, +so that, together with upper bounds for polyspectra with smaller q (if needed), one +can deduce asymptotic upper bounds for variances of polyspectra that are uniform +in q. +Combining Theorem 3.6 and Theorem 3.4, one deduces the following general +result for integral functionals of hyperbolic random waves (see once again the dis- +cussion around (3.4)–(3.5) in order to appreciate the significance of the conclusion +(3.21)). +Proposition 3.13. Consider the random variable GR(uλ), as defined in (3.8) for +some G ∈ L2(R, φ(s)ds) with even Hermite rank q0 ≥ 2. Write v2(q0; R; λ) := +Var(hn,q0 +R,λ ). +(1) Suppose q0 ≥ 4. Then, the following asymptotic relations hold: +Var GR(uλ) ≃n,R,G v2(q0; R; λ), +as λ → ∞, +(3.19) +Var GR(uλ) ≃n,λ,G v2(q0; R; λ), +as R → ∞. +(3.20) +Moreover, denoting by N(R; λ) a centered Gaussian random variable with +the same variance of � +GR(uλ) := GR(uλ)/v(q0; R; λ), one has that, both as +λ → ∞ and as R → ∞, +(3.21) +W1 +� +� +GR(uλ), N(R; λ) +� +−→ 0. +(2) If q0 = 2, then, writing a(G) := 1 +2 +� +R G(t)H2(t)φ(t)dt ̸= 0, one has that +(3.22) +Var GR(uλ) +a2(G) · v2(2; R; λ) −→ 1, both as λ → ∞ and as R → ∞. +Moreover, one has the following bounds: if R → ∞, +(3.23) +W1 +� +GR(uλ) +|a(G)| · v(2; R; λ), N(0, 1) +� +≲n,λ,G +1 +R1/2 ; +if λ → ∞, +(3.24) +W1 +� +GR(uλ) +|a(G)| · v(2; R; λ), N(0, 1) +� + + + + + + + +≲R,G +� +log λ +λ , +n = 2, +≲R,G +� +log λ +λ1/2 , +n = 3, +≲n,R,G +1 +λ1/4 , +n ≥ 4. +(3) If q0 = 4 and n = 2, writing b(G) := +1 +24 +� +R G(t)H4(t)φ(t)dt ̸= 0, it holds +moreover that, as λ → ∞, +Var GR(uλ) +b2(G) · v2(4; R; λ) −→ 1, +(3.25) +W1 +� +GR(uλ) +|b(G)| · v(4; R; λ), N(0, 1) +� +≲R +1 +log λ. +(3.26) + +20 +F. GROTTO AND G. PECCATI +In the case q0 = 2, any n ≥ 2 and both regimes, and q0 = 4, n = 2, large fre- +quency, we are able to obtain quantitative statements since the first non-vanishing +Hermite projection dominates the chaos expansion in the asymptotic regime. In +the other discussed cases this does not happen, and a finer control of the considered +functional is required. +Proof. [Proof of (1)] Since the arguments needed to deal with the case R → ∞ +are analogous, we will only discuss the limiting regime λ → ∞. Let q0 ≥ 2 be +the Hermite rank of GR(uλ). Since we are assuming that q0 is even, the table in +the statement of Theorem 3.6, combined with Remark 3.12, implies the uniform +estimate +sup +q>q0 +q!−1 Var(hn,q +R,λ) = On,R(Var(hn,q0 +R,λ )). +The conclusion now follows by selecting a sequence λj → ∞ and by applying +Theorem 3.4 to the following special case: +(a) c2(j) = v2(q0; R; λj); +(b) T0 = {q0}; +(c) +fj,q = +� +BR +Kn,λj(x, ·)⊗qdmn(x) · q!−1 +� +R +G(s)Hq(s)φ(s)ds; +(d) σ2 +q = 1 +q! +�� +R G(s)Hq(s)φ(s)ds +�2; +(e) Σ0 = E +� +G(N)2� +, where N is a standard Gaussian random variable. +[Proof of (2)] By virtue of (3.8), the projection of GR(uλ) onto the second Wiener +chaos H:2: is given by Z := a(G) · hn,2 +R,λ. Using the triangle inequality, one has that +W1 +� +GR(uλ) +|a(G)| · v(2; R; λ), N +� +≤ +W1 +� +Z +|a(G)| · v(2; R; λ), N +� ++Var(GR(uλ) − Z)1/2 +|a(G)| · v(2; R; λ) +:= A + B. +A direct application of (3.17) yields that A ≤ cX(R, λ, n)1/2, where c is some +absolute combinatorial constant. On the other hand, one can directly bound B by +exploiting the orthogonality of distinct Wiener chaoses together with the asymptotic +relations put forward in Theorem 3.6. Combining these estimates with (3.16) yields +the desired result. The [Proof of (3)] follows along the same lines, the asymptotics +of the ratio between contractions and variance being directly deduced from the ones +collected in the proof of Lemma 4.12. +□ +3.3. First application: excursion volumes at non-zero levels. As before, we +write φ to indicate the standard Gaussian density, and also introduce the notation +Φ(t) := +� t +−∞ φ(s)ds. Our aim in this Section is to use Proposition 3.13 in order to +study the asymptotic behavior of random variables of the type +(3.27) +ΦR,λ(t) := +� +BR +1(−∞,t](uλ(x))dmn(x) = mn{x ∈ BR : uλ(x) ≤ t}, +defined for all t ∈ R and all R > 0, λ ≥ σ2 (where σ2 is defined, as usual, in (2.2)). +We will first derive the chaotic expansion of ΦR,λ(t). A direct application of +Tonelli’s theorem shows immediately that +E [ΦR,λ(t)] = mn(BR)Φ(t). + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +21 +Moreover, for all t ∈ R the Hermite polynomial expansion of the function 1(−∞,t], +in the space L2(R, φ(s)ds), is given by +(3.28) +1(−∞,t](s) = +∞ +� +q=0 +1 +q!ψq(t)Hq(s), +where ψq(t) = −Hq−1(t)φ(t), q ≥ 1, and ψ0(t) = Φ(t).4 +An immediate, easy +consequence of (3.28) is also that, for N a standard Gaussian random variable, +Var(χ(−∞,t](N)) = Φ(t)(1 − Φ(t)) = +∞ +� +q=1 +ψq(t)2 +q! +. +We observe that, since the functions Hq−1 are odd for even q ≥ 2, one has that +ψq(0) = 0 for all q ≥ 2 even: in particular, this implies that the Hermite expansion +of 1(−∞,t] in the critical case t = 0 uniquely involves Hermite polynomials of odd +order (in such a way that this special case cannot be dealt with by using the results +of the present paper). Reasoning as in the previous Section, we also deduce that, +for all t ∈ R, the Wiener chaos expansion of ΦR,λ(t) is given by +(3.29) +ΦR,λ(t) = +∞ +� +q=0 +Aq(t)hn,q +R,λ, +A0(t) = Φ(t), +Aq(t) = ψq(t)/q!, q ≥ 1. +where the polyspectra hn,q +R,λ are defined as in (3.2). +Combining the above discussion Proposition 3.13 we deduce the following state- +ment: +Proposition 3.14. Let the above assumptions and notation prevail and fix n ≥ 1 +and t ̸= 0. For fixed R > 0 and λ → ∞, +Var(ΦR,λ(t)) ≃n,R t2φ(t)2 · λ−σ, +whereas, for fixed λ ≥ σ2 and R → ∞, +Var(ΦR,λ(t)) ≃n,λ t2φ(t)2 · R mn(BR). +Moreover, writing �ΦR,λ(t) := +� +ΦR,λ(t) − mn(BR)Φ(t) +� +/ Var(ΦR,λ(t))1/2, one has +the following explicit estimates: if R → ∞, then +W1 +� +�ΦR,λ(t), N(0, 1) +� +→ 0 +at a speed upper-bounded by the right-hand side of (3.23); if λ → ∞, then +W1 +� +�ΦR,λ(t), N(0, 1) +� +→ 0 +with an upper bound given by the right-hand side of (3.24). +Proof. The projection on H:1: of ΦR,λ(t) is P(t, λ, R) := φ(t)hn,1 +R,λ; according to +Theorem 3.6, one has that Var P(t, λ, R)/ Var(hn,2 +R,λ) ≲n,R λ−1, as λ → ∞ and +Var P(t, λ, R)/ Var(hn,2 +R,λ) ≲n,λ R−2, as R → ∞. The result now follows from an +application of Proposition 3.13 to the function G(x) = 1(−∞,t](x) − Φ(t) − φ(t) x +(which has Hermite rank equal to 2), and from the fact that the projection on H:2: +of ΦR,λ(t) is t +2φ(t)hn,2 +R,λ. +□ +4The functions ψq are the Hermite functions, forming an orthonormal basis of L2(R, dx) that +diagonalizes the Fourier transform operator on the real line. + +22 +F. GROTTO AND G. PECCATI +Remark 3.15. As already recalled, in the nodal case t = 0 all projections on even +Wiener chaoses in (3.29) vanish: since we only have upper asymptotic bounds on +odd polyspectra, in that case we are not able to deduce a CLT for the functional. +In fact, this is the same issue faced in the context of 2d random spherical harmonics +by [39], to which we refer also for an interesting chaining argument produce a CLT +for the normalized functional ΦR,λ(t)−mn(BR)Φ(t) as a stochastic process indexed +by t ∈ R ∖ {0}. Limit theorems at t = 0 were later derived for random waves on +S2 in [38] by means of an ad hoc argument, that we are not yet able to replicate in +our setting. +In the next Section, we present a direct analysis of the so-called “Leray measures” +of nodal sets of hyperbolic waves. +3.4. Second application: Leray measures of nodal sets. According to our +previous discussion, the samples of uλ are smooth functions on Hn, and classical +results on stationary Gaussian random fields ensure moreover that, for all t, the +level set u−1 +λ (t) is a submanifold of codimension 1 almost surely (cf. Bulinskaya’s +Lemma, [6, Proposition 6.12]). One can thus consider generalized functions on Hn +supported on u−1 +λ (t), the fundamental one being defined as +(3.30) +� +ϕ, δu−1 +λ (t) +� += +� +u−1 +λ (t) +ϕ(x)dω(x), +where dω indicates integration with respect to the volume form on u−1 +λ (t) induced +by the metric of Hn, and with brackets denoting duality coupling with smooth +functions ϕ ∈ C∞ +c +(we refer e.g. to [22, III.1] for the classical theory of distributions +in this setting). +The aim of this Section is to study the high-frequency and large domain asymp- +totic behavior of random variables LR,λ that are formally obtained from (3.30) by +taking t = 0 and ϕ(x) = 1BR(x). More precisely, for λ ∈ +�� n−1 +2 +�2 , ∞ +� +and R > 0 +we define the Leray measure of the nodal set of uλ restricted to BR (also called the +occupation density at zero of uλ on BR) to be the quantity +(3.31) +LR,λ := lim +ε→0 +1 +2ε |{x ∈ BR : |uλ(x)| ≤ ε}| =: lim +ε→0 Lε +R,λ +whenever such a limit is well-defined in the sense of convergence in probability. +Heuristically, the Leray measure LR,λ is the rescaled volume of the subset of BR +in which uλ takes values that are infinitesimally close to zero — see the classical +reference [23] for a general discussion of occupation densities, as well as [49, 50] for +similar studies of the Leray measures associated with arithmetic random waves. +Should the limit (3.31) exist in L2(P), one could derive the chaos expansion of +LR,λ from that of Lε +R,λ, which is easily seen to be the following: +Lε +R,λ = 1 +2ε +� +BR +1[−ε,ε](uλ(x))dmn = ΦR,λ(ε) − ΦR,λ(−ε) +2ε += +∞ +� +q=0 +Bε +qhn,q +R,λ, +where hn,q +R,λ is defined in (3.2) and, in the notation of Proposition 3.14, +Bε +q = Aq(ε) − Aq(−ε) +2ε +, +with Bq := lim +ε→0 Bε +q = A′ +q(0) = Hq(0)φ(0) +q! +. +The last asymptotic relation yields that, if (3.31) holds in L2(P), then the chaos +expansion of LR,λ is obtained from that of Lε +R,λ by replacing each Bε +q with Bq. +Here are some additional (useful) remarks on the coefficients Bq: + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +23 +– one can regard the sequence (Bq)q≥1 as given by the coefficients of a formal +Hermite decomposition +δ0(Z) = +� +q≥0 +BqHq(Z), +Z ∼ N(0, 1), +Bq = ⟨δ0, Hq/q!⟩L2(R,φ(s)ds) , +where Dirac’s delta is regarded as a generalized function on R; +– Bq = 0 for all odd q ≥ 1, whereas for ℓ ∈ N one has that +(3.32) +B2ℓ = +H2ℓ(0) +� +2π(2ℓ)! +, +H2ℓ(0) = (−1)ℓ(2ℓ − 1)!!, +the latter being a consequence of the definition of Hq; +– the following asymptotic relation is derived from Stirling’s formula and +(3.32): +B2 +2ℓ ≃ 1 +√ +ℓ +, +ℓ → ∞. +In order to establish the existence in L2(P) of the limit (3.31), we will rely on +the following result, which we state and prove in a general setting because of its +independent interest. +Proposition 3.16. Let (Z, Z , µ) be a finite measure space, and consider a real- +valued centered Gaussian field X(z), defined on a probability space (Ω, F, P), in- +dexed by z ∈ Z with X(z) ∼ N(0, 1) for all z ∈ Z and covariance function +R(z, z′) = Cov(X(z), X(z′)) (which therefore takes values in [−1, 1]). For every +measurable C ⊂ Z × Z, one has that +(3.33) +∞ +� +ℓ=0 +B2 +2ℓ +� +C +R(z, z′)2ℓdµ2(z, z′) = 1 +2π +� +C +1 +� +1 − R(z, z′)2 dµ2(z, z′). +Moreover, the limit +(3.34) +R = +� +Z +δ0(X(z))dµ(z) := lim +ε→0 +1 +2ε +� +Z +1[−ε,ε](X(z))dµ(z), +exists in L2(P) if and only if either side of (3.33) is finite when C = Z × Z. In +this case, one has that +(3.35) +E [R] = µ(Z) +√ +2π +, +E +� +R2� += 1 +2π +� +Z2 +1 +� +1 − R(z, z′)2 dµ2(z, z′). +Remark 3.17. A by-product of Proposition 3.16 is that, if R is well-defined as a +limit in L2(P), the set of those (z, y) ∈ Z2 such that R(z, y) = ±1 is necessarily +µ2-negligible. +Proof of Proposition 3.16. From (3.32), it follows that +(3.36) +∞ +� +ℓ=0 +B2 +2ℓx2ℓ = 1 +2π +∞ +� +ℓ=0 +�2ℓ +ℓ +�x2ℓ +4ℓ = 1 +2π +1 +√ +1 − x2 , +x ∈ (−1, 1). +This directly implies (3.33) by monotone convergence. To prove the second part +of the statement, for every ε > 0 denote by R(ε) the argument of the limit on +the right-hand side of (3.34): R(ε) is a square-integrable variable and its chaos +decomposition is given as before by +(3.37) +R(ε) = +∞ +� +q=0 +Bε +q +� +Z +Hq(X(z))dµ(z). +We recall that Bε +q = Bq = 0 for all ε > 0 and odd q. If either side of (3.33) is finite +for C = Z × Z, recalling the elementary relation +E [Hq(X(z))Hq′(X(z′))] = q!R(z, z′)q · 1q=q′, + +24 +F. GROTTO AND G. PECCATI +we deduce that the series +X = +∞ +� +ℓ=0 +B2 +2ℓ +� +Z +H2ℓ(X(z))µ(dz) +converges in L2(P), and +E +� +(R(ε) − X)2� += +∞ +� +ℓ=0 +(Bε +2ℓ − B2ℓ)2 +� +Z +� +Z +R(z, y)2lµ(dz)µ(dy). +Since Bε +q +ε→0 +−−−→ Bq one can apply the well-known inequality for Hermite polynomials +(see [2, 22.14.16]), +|H2ℓ−1(x)| ≤ xex2/4(2ℓ)! +2ℓℓ! +, +ℓ ≥ 1, +from which one infers the estimates +|Bε +2ℓ| ≤ φ(ε)eε2/4 +2ℓℓ! +≤ φ(0) +2ℓℓ! = φ(0)(2ℓ − 1)!! +(2ℓ)! += |B2ℓ| , +l ≥ 1, +so that +(Bε +2ℓ − B2ℓ)2 ≤ 2B2 +2ℓ. +By dominated convergence, this yields that E +� +(R(ε) − X)2� +→ 0. This last relation +implies that R is well-defined in L2(P) and that R = X. Conversely, if R is well- +defined in L2(P), then its projection on the q-th Wiener chaos is obtained from the +one of R(ε), as given in (3.37), by taking the limit ε → 0 (because of the continuity +of the projection map from L2(P) to a closed subspace). The conclusions in (3.35) +now follow immediately by combining the relation B0 = +1 +√ +2π with (3.33). +□ +The next statement contains the main results of the Section. +Proposition 3.18. The functional LR,λ is well-defined as the L2(P)-limit of Lε +R,λ +(as ε → 0; see (3.31)) and its chaos expansion is given by +LR,λ = +∞ +� +ℓ=1 +B2ℓhn,2ℓ +R,λ = mn(BR) +√ +2π +− +1 +2 +√ +2πhn,2 +R,λ + . . . , +where the series converges in L2(P). The following asymptotic relations hold: +Var(LR,λ) +� +≃n,R λ−σ, +λ → ∞, +≃n,λ R · mn(BR), +R → ∞. +Moreover, setting �LR,λ := +� +LR,λ − mn(BR) +√ +2π +� +Var(LR,λ)−1/2, one has that �LR,λ con- +verges in distribution towards a standard Gaussian random variable, both as λ → ∞, +with R > 0 fixed, and as R → ∞, with λ ≥ σ2 fixed. +Proof. We start by recalling that, according to the table in the statement of The- +orem 3.6 one has that +Var(hn,2 +R,λ) +� +≃n,R λ−σ, +λ → ∞, +≃n,λ R · mn(BR), +R → ∞. +Also, Corollary 4.11 and Lemma 4.14 below imply the following bounds: +����� +� +B2 +R +dm⊗2 +n (x, y) +� +1 − Fn,λ(d(x, y))2 − mn(BR)2 − 1 +2 Var hn,2 +R,λ +����� +� +≲n,R λ−σ−1/2, +λ → ∞, +≲n,λ mn(BR), +R → ∞. +Combining these asymptotic relations, one infers that: +– the random variable LR,λ is well-defined in L2(P) (as a direct application +of Proposition 3.16 in the case Z = BR, µ = mn and X = uλ); + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +25 +– denoting P[R, λ ; ≥ 3] the projection of LR,λ onto the direct sum � +q≥3 H:q:, +one has that +Var(P[R, λ ; ≥ 3]) = o(Var(hn,2 +R,λ)), +both as λ → ∞, with R > 0 fixed, and as R → ∞, with λ ≥ σ2 fixed. +The conclusion now follows immediately from the second part of the statement of +Theorem 3.6. +□ +4. Covariance Functions and their Moments +The arguments in the previous Section are based on asymptotic estimates of +multiple integrals involving the covariance function Fn,λ(d(x, y)) of uλ, such as the +moments +(4.1) +Cn,q +R,λ = Var(hn,q +R,λ) = +� +BR +� +BR +Fn,λ(d(x, y))qdmn(x)dmn(y), +q ≥ 0, +and 4-fold integrals appearing in (3.14) as representations of kernel contractions in +Theorem 3.4. In order to obtain bounds on these integrals we need precise estimates +on Fn,λ itself, which we derive in the next paragraph. The proof of Proposition 2.10 +also requires such estimates, so we report it at the end of Subsection 4.1, before +moving to the main technical arguments of the paper in the remainder of the Sec- +tion. +4.1. Approximating Covariance Functions. The following statement collects +approximations of Fn,λ we will employ to derive estimates on the double integrals +Cn,q +R,λ defined in (4.1). +Lemma 4.1. Let n ≥ 2, α ∈ R∗, λ = σ2 + α2, r ≥ 0. It holds +(1) (uniform bound) +(4.2) +|Fn,λ(r)| ≲n e−σr, +uniformly in r ≥ 0; moreover, for any r0 > 0, uniformly in r ≥ r0, +(4.3) +��sinh(r)σFn,λ(r) − Re +� +cn(α) sinh(r)iα��� ≲n |α|−2 sinh(r)−2, +where +(4.4) +cn(α) = 22σ−1Γ(iα)Γ(σ + 1/2) +√πΓ(σ + iα) +, +is bounded for α ∈ R∗ away from zero, and cn(α) ≃n |α|−σ as |α| → ∞; +(2) (decay in α at fixed r) for all r > 0 it holds, as |α| → ∞, +(4.5) +Fn,λ(r) = Cn +Re[cosh(r)iα] +|α|σ sinh(r)σ + on,r(|α|−σ); +(3) (approximation with Bessel functions) as |α| → ∞, uniformly in r > 0, +(4.6) +Fn,λ(r) = (2π)n/2 +ωn−1 +� +r +sinh r +· +� +α − 1 +2i +�1−n/2 +(sinh r)1−n/2Jn/2−1(αr) (1 + On(1/|α|)) . +Remark 4.2. The Harish-Chandra function cn(α) (cf. [24, I.4]) is closely related to +the spectral density (introduced in Theorem 2.1) by +ρn(α) = +2n−2 +ω2 +n−1|cn(α)|2 . + +26 +F. GROTTO AND G. PECCATI +As a function of α, cn(α) can be extended to a meromorphic function on C with +poles on iN. +In what follows we regard Fλ,n(r) as a hypergeometric function and make use of +a number of known facts on that kind of special functions. Section A collects the +formulae we are using, together with precise bibliographic references. +We begin by recalling that Fλ,n(r) is the unique (smooth) solution of the ODE +(4.7) +Fn,λ(r)′′ + 2σ coth(r)Fn,λ(r)′ + λFn,λ(r) = 0, +Fn,λ(0) = 1, F ′ +n,λ(0) = 0, +on the positive real axis r > 0. +This is the hyperbolic analogue of the ODE +characterizing radial solutions of Helmholtz equation on Rn, the Euclidean case +being recovered by replacing coth(r) with r. +Remark 4.3. A relevant difference with Euclidean setting: the function z−νJν(z) +appearing in the covariance of Berry’s model (1.1) is an entire function on C co- +inciding with its power series expansion at 0, whereas a power series of Fn,λ at 0 +that one can derive from (4.7) has a finite radius of convergence of order O(λ−2σ) +as λ → ∞ (cf. [48, 15.2]). +When rewritten in terms of the variable sinh2(r), (4.7) becomes a special case +of the hypergeometric equation (A.5): it is thus possible to represent Fn,λ with the +principal branch of the hypergeometric function (A.2), +(4.8) +Fn,λ(r) = 2F1 +�σ + iα +2 +, σ − iα +2 +, n +2 , − sinh2(r) +� +(the branch cut of 2F1, with respect to its last variable, is [1, ∞]). We refer to +[11, Section 4.1-4.2] for more details on the representation with hypergeometric +functions and asymptotics at the boundary on the half-plane model for n = 2. +Remark 4.4. The arguments in the remainder of the paper rely on asymptotic +properties of Fn,λ(r), which we often deduce comparing (4.8) with other special +functions, namely Legendre and Bessel functions. It is of course possible to obtain +those asymptotics directly from the definition and basic properties of hypergeo- +metric functions: we refer to [30] for a proper discussion. We choose to proceed +through comparison with other special functions appearing in the study of random +waves on different geometries in order to emphasize analogies for readers who are +familiar with those other models. +Proof of Lemma 4.1, item (2). Rewrite the representation (2.4) as an oscillatory +integral: denoting t(r) = tanh(r) to lighten notation, +Fn,λ(r) = ωn−2 +2ωn−1 +cosh(r)iα−σ +� π +−π +eiβφ(r,θ)ψn(r, θ)dθ, +β = αt(r), +φ(r, θ) = log(1 + t(r) cos θ) +t(r) +, +ψn(r, θ) = +sin(θ)n−2 +(1 + t(r) cos θ)σ . +Since |t(r)| < 1 for all r ∈ R and t(r) ∼ r as r → 0, the phase φ is uniformly +bounded in both variables. As a function of θ, it has two simple stationary points +at θ = 0 and π, in which +d +dθφ(r, θ) +���� +θ=0,π += 0, +d2 +d2θφ(r, θ) +���� +θ=0,π += +t(r) ± 1 +(1 ± t(r))2 ̸= 0, +r ∈ R. +For all r ∈ R, the amplitude ψn(r, θ) is a smooth function of θ, vanishing at critical +points of the phase, θ = 0, π, with order n − 2 and leading coefficient (1 ± t(r))−σ. + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +27 +By the standard stationary phase method, see [10, Section 6.1], we deduce the +following asymptotic +� π +−π +eiβφ(r,θ)ψ(r, θ)dθ = Cnβ−σ + on(β−σ), +β → ∞ +(here Cn ∈ C ∖ {0}, depending on n only). +□ +Proof of Lemma 4.1, item (1). A power series expansion at large r for 2F1 can be +deduced from (A.2) and (A.3), yielding an asymptotic for r → ∞ at any fixed +α ∈ R ∖ {0}, +(4.9) +Fn,λ(r) = Re +� +cn(α)(sinh r)iα−σ� ++ Oλ(sinh(r)1−σ), +where cn(α) is the Harish-Chandra function defined in (4.4). Since sinh(r) ≃ er as +r → ∞, we can rewrite the asymptotic expression (4.9) as +Fλ(r) = sinh(r)−σ (c1(α) cos(αr) + c2(α) sin(αr)) + Oλ(e−(σ+1)r), +with c1 and c2 real functions of α. By standard properties of Gamma function (see +[48, 5.3]) we have that |cn(α)| is uniformly bounded5 in α away from 0, so functions +c1, c2 are uniformly bounded away from 0. +Since Fn,λ is a smooth solution of the ODE (4.7) with Fn,λ(0) = 1, it is bounded +in a neighborhood of 0: combined with the last displayed equation this proves (4.2). +We will now bound the deviation from the main asymptotic by means of the ODE. +Equation (4.7) is put into canonical form by the substitution +H(r) = sinh(r)σFλ(r), +H′′(r) + b(r)H(r) = 0, +b(r) = α2 − σ(σ − 1) +sinh(r)2 . +This is the reason why we expressed the exponential behavior of Fλ in r in terms +of sinh(r) above. Since b(r) converges to α2 for large r, it is easy to compare H +with the harmonic oscillator of frequency α. We thus define the remainder +(4.10) +Rλ(r) = sinh(r)σFλ(r) − (c1(α) cos(αr) + c2(α) sin(αr)) , +which we know to be Oλ(e−r) by the asymptotics above, and that satisfies +R′′ +λ(r) + α2Rλ(r) = σ(σ − 1) sinh(r)σ−2Fλ(r), +as an immediate consequence of the ODE for H. The general solution for Rλ(r) is +given by +Rλ(r) = A1 cos(αr) + A2 sin(αr) − 1 +α +� ∞ +r +sin(α(t − r)) +sinh(t)2−σ Fλ(t)dt, +in which we can immediately tell that A1 = A2 = 0 because of the known asymp- +totics of Fλ and R as r → ∞. Substituting the definition (4.10) of Rλ, we obtain +the integral equation +Rλ(r) = 1 +α +� ∞ +r +sin(α(t − r)) (c1(α) cos(αt) + c2(α) sin(αt)) +sinh(t)2 +dt +− 1 +α +� ∞ +r +sin(α(t − r))Rλ(t) +sinh(t)2 +dt. +The first summand on the right-hand side can be controlled by means of van der +Corput Lemma (see [56, Chap. VIII, Sec. 1.2], we omit the elementary computa- +tion), leading to the estimate +|Rλ(r)| ≲ +1 +α2 sinh(r)2 + 1 +α +� ∞ +r +|Rλ(r)| +sinh(t)2 dt, +5We refer to [57, Eqs. 4.4’,4.4”] for representations of |cn(α)|, for instance in terms of infinite +products. + +28 +F. GROTTO AND G. PECCATI +to which we apply Gr¨onwall inequality (we omit again an elementary computation) +concluding +|Rλ(r)| ≲ +1 +α2 sinh(r)2 . +We thus obtained, for r ≥ 0, +□ +(4.11) +|sinh(r)σFλ(r) − (c1(α) cos(αr) + c2(α) sin(αr))| ≲ +1 +α2 sinh(r)2 . +Proof of Lemma 4.1, item (3). For any n ≥ 2, the integral expression (2.4) of Fn,λ +coincides with a Legendre function of complex degree: from (A.7) we derive +(4.12) +Fn,λ(r) = +� +2 +sinh r +�n/2−1 +Γ +�n +2 +� +P 1−n/2 +iα−1/2(cosh r). +Expressing the latter in terms of hypergeometric functions can be done combining +(A.6) and (A.7), thus deriving the hypergeometric representation (4.8) in a different +way. From (4.12), the thesis follows by a straightforward application of (A.8). +□ +Item (3) of Lemma 4.1 conveniently relates the hyperbolic spherical function +with Bessel’s function appearing in (1.1), in the high-frequency limit. In fact, (4.6) +plays a role akin to the one of Hilb’s asymptotic for Legendre polynomials in the +context of random wave models on the sphere Sn, see [36], and we can immediately +deduce the result on local behavior of uλ we stated in Proposition 2.10. +Proof of Proposition 2.10. Denote +dr,λ = d +� +expx(v/ +√ +λ), expx(v′/ +√ +λ) +� +. +By invariance under rotations we reduce ourselves to the case in which v, v′ ∈ Rn +lie on the plane R2×{0} ⊂ Rn and have polar coordinates respectively (r, 0), (cr, θ), +with r > 0, 0 < c < 1 and θ ∈ [0, π]. The exponential map becomes identity when +expressed in polar coordinates both on its domain and on the image Hn (i.e. with +hyperbolic polar coordinates centered at x). By the first hyperbolic law of cosines +cosh dr,λ = cosh r +√ +λ +cosh cr +√ +λ +− sinh r +√ +λ +sinh cr +√ +λ +cos θ += 1 + r2 +2λ +� +1 + c2 − 2c cosθ +� ++ O +� r4 +λ2 +� +, +the second step coming from first order expansion in r/ +√ +λ = o(1). Notice that +1 + c2 − 2c cosθ = |1 − ceiθ|2 is uniformly bounded by 2. We are thus expressing +the fact that the local behavior of hyperbolic and Euclidean distance is the same: +dr,λ = +r +√ +λ +|1 − ceiθ| + O +� r3 +λ3/2 +� += |v − v′| +√ +λ ++ O +� r3 +λ3/2 +� +, +λ → ∞. +We conclude the proof combining (4.6) with the asymptotic we obtained for dr,λ: +Fλ(dr,λ) = (2π)n/2 +ωn−1 +�α|v − v′| +√ +λ +�1−n/2 +Jn/2−1 +�α|v − v′| +√ +λ +� +(1 + O(r/|α|)) += Cn,1(v, v′) (1 + O(r/|α|)) , +where the last step simply uses the definition of Berry’s covariance function (1.1) +and +√ +λ = +√ +α2 + σ2 ≃ α. +□ + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +29 +4.2. Double Integrals on Hyperbolic Balls. Given a measurable function f : +[0, ∞) → [0, ∞), we are interested in the integral of f(d(z, w)) over x, y ∈ BR, the +ball of Hn of radius R (to fix ideas one can consider the one centered at the origin) +with respect to the hyperbolic volume, that is +I(f, R) = +� +BR +� +BR +f(d(x, y))dmn(x)dmn(y). +To deal with this kind of integrals, we use once again hyperbolic polar coordinates. +Let (r, ϑ) be polar coordinates of x with respect to the center of BR, and (s, ϑ′) ∈ +[0, ∞) × Sn−1 be coordinates of y with respect to x; we can write +I(f, R) = Cn +� +Sn−1 dςn−1(ϑ) +� R +0 +sinh(r)n−1dr +� +BR +dmn(y)f(d(x, y)), +in which the innermost integral does not depend on ϑ ∈ Sn−1, thus neither does +the integral in dr. We can thus fix a particular (arbitrary) ϑ, replace the outer +integral with ωn−1 and write +I(f, R) = Cn +� R +0 +sinh(r)n−1dr +� +Sn−1 dςn−1(ϑ′) +� R(r,ϑ′) +0 +sinh(s)n−1dsf(s), +where R(r, ϑ′) is the hyperbolic length of the geodesic arc leaving x = (r, ϑ) in +direction ϑ′ and ending and the boundary ∂BR. The function R(r, ϑ′) of course +varies between its minimum R(r, ϑ) = R − r and maximum R(r, −ϑ) = R + r. If +f is non-negative, replacing R(r, ϑ′) with one of these two values (and integrating +out the now muted variable ϑ′) provides respectively a lower and an upper bound +for the integral. +Remark 4.5. By means of hyperbolic trigonometry (we refer to [4, Section 5.6]) one +can explicitly represent the function R(r, ϑ′); for instance in the case n = 2, fixing +ϑ = 0 ∈ [0, 2π] ≃ S1, it holds +cosh (R(r, ϑ′)) = +cosh(R) cosh(r) + cos(ϑ′) sinh(r) +� +sinh2(R) − sin2(ϑ′) sinh2(r) +1 + sin2(ϑ′) sinh2(r) +, +r > 0, ϑ′ ∈ [0, 2π] ≃ S1. +4.3. Asymptotics for large λ, fixed R. When the domain of integration is fixed, +for moments of order q ≥ 2 we can determine the asymptotic of Cn,q +R,λ as λ ↑ ∞ +by separating the contributions of regions in which integrating variables x, y ∈ BR +are respectively closer or farther than 1/α with respect to each other. To this end, +the above change of variables reduces our task to control one-dimensional integrals +close to and far from the lower extremum of integration. We can proceed this way +since for q ≥ 2 we can derive good controls simply from the exponential decay of +Fn,λ. +For odd q ≥ 3, oscillations provide additional cancellations and we will thus +establish only an upper bound that suffices to our needs. However, in the special +case of q = 1 the decay of Fn,λ is outweighed by the volume element, so the +oscillations play a determinant role and can not be neglected. We thus need a more +careful analysis in that case, so we will discuss it separately by means of Fourier +analysis on Hn. +Lemma 4.6. As λ → ∞, for a fixed R > 0, we have the following asymptotics: +• (q = 1, any n ≥ 2) Cn,1 +R,λ ≲R |α|−2σ−2; +• (q = 2, any n ≥ 2) Cn,2 +R,λ ≃R |α|−2σ; +• (even q ≥ 2, any n ≥ 2) Cn,q +R,λ ≃R |α|−2σ−1 except for the single case +(n = 2, q = 4) for which C2,4 +R,λ ≃R |α|−2 log(|α|); + +30 +F. GROTTO AND G. PECCATI +• (odd q ≥ 2, any n ≥ 2) Cn,q +R,λ ≲R |α|−2σ−1 except for the single case +(q = n = 3) for which C3,3 +R,λ ≲R |α|−3 log(|α|). +Remark 4.7. The two exceptional cases (n = 2, q = 4) and (q = n = 3) correspond, +as we show below, to critical choices of the parameters. However, for (n = 2, q = 4) +a logarithmic correction appears in the true asymptotic, whereas in the other case +the factor log α appearing in the upper bound we state is not present in the real +asymptotic, as it is canceled by oscillations that persist due to q = 3 being odd. +This last fact is irrelevant in our scope, a careful (and lengthy) estimate would be +required to prove it, so we do not discuss it any further. +We will assume α > 0, the negative case being identical. +Proof of case q = 1. Assume that BR is centered at the origin of Hn; since the +indicator function 1BR(x) = 1[0,R](d(x, o)) is radial, so is its Fourier transform +fR(α) = F1BR(α, ϑ) = +� +BR +en(x, −α, ϑ)dmn(x) = cn +� R +0 +Fn,λ(r)(sinh r)n−1dr += cn +� R +0 +� π +0 +(cosh r − sinh r cos θ)−σ+iα (sin θ)n−2(sinh r)n−1dθdr, +(the latter descending directly from definitions in Section 2 and change of variables). +Standard stationary phase analysis (see [10, 8.4]) allows to obtain asymptotics of +the oscillatory integral in display: we have fR(α) = OR(α−σ−1) for α → ∞ (see +Subsection 4.1 for completely analogous computations with details). +We recall (2.3) in the form +Fn,λ(d(x, y)) = +1 +ωn−1 +� +Sn−1 en(x, α, ϑ) en(y, −α, ϑ)dςn−1(ϑ). +Finally, we will use the (formal) orthogonality relation +ρn(α) +� +Hn en(x, α, ϑ) en(x, −α′, ϑ′)dmn(x) = δα=α′δϑ=ϑ′ +in our computation: it can be derived by applying FF−1 to the right-hand side, +regarded as a Dirac delta on R+ ×Sn−1. A standard mollification argument –or the +extension of Fourier transform to distributions– allows to make it rigorous, we omit +details for the sake of brevity. Combining the formulas above and Fourier inversion +(Proposition 2.5), the thesis follows from: +Cn,1 +R,λ = +� +(Hn)2 1BR(x)1BR(y)Fn,λ(d(x, y))dmn(x)dmn(y) += Cn +� +(Hn)2 dm2 +n(x, y) +� +(Sn−1)3 dς3 +n−1(ϑ, ϑ′, ϑ′′) +� +(R+)2 ρn(α′)ρn(α′′)dα′dα′′ +× fR(α′)fR(α′′) en(x, α′, ϑ′) en(y, α′′, ϑ′′) en(x, α, ϑ) en(y, −α, ϑ) += Cn +� +(Sn−1)3 dς3 +n−1(ϑ, ϑ′, ϑ′′) +� +(R+)2 ρn(α′)ρn(α′′)dα′dα′′ +× fR(α′)fR(α′′)δα=−α′δϑ=ϑ′δα=α′′δϑ=ϑ′′ρn(α)−2 = Cn|fR(α)|2. +□ +Remark 4.8. The Fourier transform fR(α) of 1BR is in fact an oscillating function, +and it has a discrete, countable set of zeros on R+. In sight of the discussion in +Subsection 3.1, this means that the variance of hn,1 +R,λ vanishes for infinite values of +α, at which one can not divide by the variance in order to study a Central Limit +Theorem. The same might happen for all other odd orders q ≥ 3, and this is an open +question even for random wave models on other geometries: it was conjectured in +[39] that variances of odd polyspectra do not vanish in the high of random spherical + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +31 +Substantiating this claim would require a much more careful control of Cn,q +R,λ for odd +orders q, which we are not yet able to produce. +Proof of case q ≥ 2, upper bound. By Subsection 4.2 and Lemma 4.1, +Cn,q +R,λ ≲n,R +� 2R +0 +|Fn,λ(s)|q sinh(s)2σds +(4.13) +≲n +� 1/α +0 +sinh(s)2σds + +1 +αqσ +� 2R +1/α +sinh(s)σ(2−q)ds =: (A) + (B), +where the second step uses the fact that Fn,λ is uniformly bounded to control the +integral over small s < 1/α, and (4.5) for large values of s, neglecting trigonometric +oscillations. The asymptotic behavior as α → ∞ is then determined by recalling +that sinh(x) ∼ x as x → 0: summand (A) on the right-hand side is of order α−2σ−1, +so we need to discuss whether (B) provides a relevant correction. +The integrand of (B) is integrable at 0 when σ(2 − q) > −1, which is the case +only when: +• (q = 2, any n ≥ 2) in this case it actually is σ(2−q) = 0, the second integral +can be bounded with the one over [0, 2R], thus it is of order |α|−2σ, and it +prevails in the asymptotic; +• (q = 3, n = 2) that is σ(2 − q) = −1/2 < 0, so summand (B) does not +provide a correction to (A). +The case σ(2 − q) = −1 is realized in two cases: +• (q = 4, n = 2) +C2,4 +R,λ ≲R +1 +α2 + 1 +α2 +� 2R +1/α +sinh(s)−1ds ≃R +log α +α2 ; +• (q = n = 3) +C3,3 +R,λ ≲R +1 +α3 + + 1 +α3 +� 2R +1/α +sinh(s)−1ds ≃R +log α +α3 +≲R +1 +α3 . +Finally, if σ(2 − q) < −1, +(4.14) +(B) = +1 +αqσ +� 2R +1/α +sinh(s)σ(2−q)ds ≃R +1 +α2σ+1 +has the same asymptotic behavior of (A). +□ +Proof of case q ≥ 2 and even, lower bound. Once again we apply the argument of +Subsection 4.2: this time we replace R(r, θ) with its minimum R − r, +(4.15) +Cn,q +R,λ ≳ +� R +0 +� R−r +0 +Fn,λ(s)q sinh(s)2σ sinh(r)2σdsdr += +� R +0 +gR(s) sinh(s)2σFn,λ(s)qds, +gn,R(s) = +� R−s +0 +sinh(r)2σdr. +Once again we divide the integral in ds over intervals [0, 1/α] and [1/α, R]. In sight +of the upper bound, in the cases q = 2, all n ≥ 2, and q = 4, n = 2 we can actually + +32 +F. GROTTO AND G. PECCATI +neglect the contribution of [0, 1/α]: applying (4.3) for s > 1/α we obtain +(q = 2) +Cn,2 +R,λ ≳R +1 +α2σ +� R +1/α +gn,R(s) Re +� +cosh(s)iα�2 ds ≃R +1 +α2σ , +(q = 4, n = 2) +C2,4 +R,λ ≳R +1 +α2 +� R +1/α +gn,R(s) +sinh(s) Re +� +cosh(s)iα�4 ds ≃R +log α +α2 . +Derivation of asymptotics on the right-hand side is as follows: since gn,R(s) vanishes +at s = R, once again the relevant contribution comes from the lower integration +extremum, close to which gn,R is bounded from below (in terms of n, R), while +cosh(s)iα ∼ 1 for small s. We omit the tedious, but elementary computation. +When q ≥ 6, the contribution relative to s ∈ [0, 1/α] is the relevant one, matching +the asymptotic obtained in the upper bound. Indeed, for all q ≥ 2 and ε > 0, +|Fn,λ(s)|q ≥ 1 − ε on s ∈ [0, 1/α] if α is large enough, since Fn,λ(0) = 1 and Fn,λ is +continuous, so we can bound +Cn,q +R,λ ≳ +� R +0 +gR(s) sinh(s)2σFn,λ(s)qds ≳R (1 − ε) +� 1/α +0 +sinh(s)2σds ≃ +1 +α2σ+1 . +Therefore, there is actually no need to estimate a lower bound on the contribution +of s ∈ [1/α, R]. +□ +Lemma 4.9. Let n ≥ 2, R > 0, q ∈ 2N and +gq(x) = +∞ +� +ℓ=q/2 +�2ℓ +ℓ +�x2ℓ +4ℓ , +x ∈ (−1, 1). +As λ → ∞ it holds +(4.16) +� +B2 +R +gq(Fn,λ(d(x, y)))dm⊗2 +n (x, y) ≲n,R Cn,q +R,λ. +Remark 4.10. It is worth observing that Fn,λ(r) = 0 if and only if r = 0, for +any n, λ; we report here a direct argument to prove it. Take points x ∈ Hn and +y = (1, 0, . . . , 0) ∈ Hn with x0 = r, so that Fnλ(d(x, y)) = Fnλ(r), and assume the +latter to equal 1. Then, by (2.3), we have +1 = Fnλ(r) = +1 +ωn−1 +� +Sn−1[x, (1, ϑ)]−σ+iαdςn−1(ϑ), +and since [x, y] ≥ 1 for any x, y ∈ Hn, thus |[x, (1, ϑ)]−σ+iα| ≤ 1, we deduce that +actually it must hold |[x, (1, ϑ)]−σ+iα| = 1, thus |[x, (1, ϑ)]| = 1, for almost all +ϑ ∈ Sn−1, which implies that r = x0 = 0. +Proof. The power series defining gq has radius of convergence 1, hence the integrand +in (4.16) is well-defined for all distinct x, y, that is m⊗2 +n -almost everywhere, thanks +to Remark 4.10. Denote by S = S(n, R, λ, q) the left-hand side of (4.16). As in +(4.13) we split the integration domain, +S ≲n,R +� 2R +0 +gq(Fn,λ(s)) sinh(s)2σds +≲n,R +� 1/α +0 +sinh(s)2σds +� +1 − Fn,λ(s)2 + +� 2R +1/α +Fn,λ(s)q sinh(s)2σds := (A) + (B), +in which: +• we applied the inequality |gq(x)| ≤ 1/ +√ +1 − x2 on the interval [0, 1/α] (gq +is the tail of the series (3.36) having only positive terms); + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +33 +• we replaced gq(Fn,λ) with F q +n,λ in the integral over [1/α, R] thanks to the +fact that |Fn,λ(s)| ≤ Cn,R < 1 uniformly on s ∈ [1/α, R], as it directly +descends from either (4.3) or (4.5), and observing that x−qgq(x) is uniformly +bounded on compacts of [0, 1). +It holds (B) ≃n,R Cn,q +R,λ (see the proof of Lemma 4.6), so we are left to control (A). +We apply the asymptotic (4.6) to control Fn,λ close to s = 0. From (4.6) and +the definition of sinh(r) we derive +Fn,λ(s) = (2π)n/2 +ωn−1 +(αs)1−n/2Jn/2−1(αs)(1 + O(1/α)) +=: f(αs)(1 + O(1/α)), +s ∈ [0, 1/α], +where we stress that Landau’s O does not depend on parameters. From the defini- +tion of Bessel functions in (A.1) we know that f(αs) = 1 − α2s2 + . . . is (a power +series in αs) analytic on the whole R. This, once again recalling that sinh(r) ≃ r, +r → 0, implies that the integral (A) converges and that +(A) ≲n,R +� 1/α +0 +sinh(s)2σds +αs +≃n,R +1 +α2σ+1 = O(Cn,q +R,λ) +for all q, the last step following from Lemma 4.6. +□ +Recalling the power series expansion (3.36), Lemma 4.9 implies: +Corollary 4.11. Let n ≥ 2, R > 0; as λ → ∞ it holds +(4.17) +����� +� +B2 +R +dm⊗2 +n (x, y) +� +1 − Fn,λ(d(x, y))2 − mn(BR)2 − 1 +2 Var hn,2 +R,λ +����� ≲n,R Var hn,4 +R,λ, +in particular, the first summand of the left-hand side is uniformly bounded in λ. +Moving to contractions in Equation 3.14, we have the following: +Lemma 4.12. Let n ≥ 2, R > 0, p, p′ ≥ 1 and even 2 ≤ q ≤ p + p′. As λ → ∞ it +holds +(4.18) +� +B4 +R +Fn,λ(d(x, y))pFn,λ(d(y, z))p′ +· Fn,λ(d(z, w))pFn,λ(d(w, x))p′dm⊗4 +n (x, y, z, w) = oR +� +Var(hn,q +R,λ)2� +. +Proof. Denote by I(λ) = In,R,p,p′(λ) the left-hand side of (4.18), assume without +loss of generality p ≤ p′ and drop dependences of constants on n, R, p, p′ to lighten +notation. Applying GM-QM inequality to the last two factors of the integrand we +obtain +I(λ) ≲ +� +B4 +R +|Fn,λ(d(x, y))|p|Fn,λ(d(y, z))|p′|Fn,λ(d(z, w))|2pdm⊗4 +n (x, y, z, w) + . . . , +where dots stand for an analogous term with higher exponents (and thus lower +asymptotic order). Integration variables can now be decoupled: moving to polar +coordinates we can write +I(λ) ≲ Jp(λ)Jp′(λ)J2p(λ), +Jp(λ) = +� R +0 +|Fn,λ(r)|p sinh(r)2σdr, + +34 +F. GROTTO AND G. PECCATI +where we can apply the asymptotic estimates obtained in the proof of the previous +Lemma, +Jp(λ) + + + + + + + + + +≲ λ−σ−1 +p = 1, +≃ λ−σ +p = 2, +≲ λ−σ−1/2 +p ≥ 3 excluding n = 2, p = 4 and n = p = 3, +≲ log(λ)λ−σ−1/2 +n = 2, p = 4 or n = p = 3. +The thesis now follows from a case-by-case check, which we summarize in the fol- +lowing table (the right-most column coming once again from Lemma 4.6). +q, n +possible p, p′ +JpJp′J2p ≲ +Var(hn,q +R,λ)2 ≃ +q = 2, all n +1,1 +λ−3σ−2 +λ−2σ +q = 4, n = 2 +1,3 +λ−3 +λ−2 log(λ)2 +2,2 +λ−2 +q = 4, n = 3 +1,3 +λ−9/2 log(λ)2 +λ−3 +2,2 +λ−7/2 +q = 4, n ≥ 4, +1,3 +λ−3σ−3/2 +λ−2σ−1 +thus σ ≥ 3/2 +2,2 +λ−3σ−1/2 +q ≥ 6, all n +1, ≥ 5 +λ−3σ−3/2 +λ−2σ−1 +2, ≥ 4 +λ−3σ−1 log(λ)2 +both ≥ 3 +λ−3σ−3/2 log(λ)4 +The thesis follows since asymptotics in the third column vanish faster then the ones +in the fourth one; notice that in the case q ≥ 6 we included in the third column +possible logarithmic corrections occurring in low dimension since Var(hn,q +R,λ)2 in this +case vanishes much slower for all n. On the other hand, one should be aware that +in the case q = 4, n = 2 a careful account of the logarithmic correction in the +asymptotic of the variance is crucial. +□ +4.4. Asymptotics for large R, fixed λ. The main difference between the analysis +in high-frequency regimes outlined above, and the one on large domains we carry +through in this paragraph, is that in the latter case the oscillations of Fn,λ (more +pronounced close to r = 0) actually play no role, and the determinant factor is the +exponential decay at r → ∞. +Lemma 4.13. As R → ∞, for a fixed λ > σ2 ( i.e. for fixed α ∈ R∗), we have the +following asymptotics: +• (q = 1, any n ≥ 2) Cn,1 +R,λ ≲n,λ e2σR; +• (q = 2, any n ≥ 2) Cn,2 +R,λ ≃n,λ Re2σR; +• (even q ≥ 2, any n ≥ 2) Cn,q +R,λ ≃n,λ e2σR; +• (odd q ≥ 2, any n ≥ 2) Cn,q +R,λ ≲n,λ e2σR. +Proof. Starting from the case q = 1, we established in the previous Section that +Cn,1 +R,λ = Cn +�� R +0 +Fn,λ(r)(sinh r)2σdr +�2 +, +so we simply discuss the limiting behavior of the right-hand side as R → ∞ instead +of λ → ∞. In the large R case, we are not dealing with an oscillatory integral, and + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +35 +the estimate is actually easier: by (4.3) it holds +� R +0 +Fn,λ(r) sinh(r)2σdr += Cn,λ +� R +0 +Re[cn(α) sinh(r)iα] sinh(r)σdr + On,λ +�� R +1 +sinh(r)σ−2dr +� +≲n,λ +� R +0 +eσr cos(αr + φn)dr +for some phase φn ∈ [0, 2π], from which the statement for q = 1 directly follows. +Moving to q ≥ 2, let us establish upper bounds by neglecting oscillations as +above. From the discussion in Subsection 4.2 we deduce +Cn,q +R,λ ≲n +� R +0 +dr sinh(r)2σ +� R+r +0 +ds sinh(s)2σ|Fn,λ(s)|q, +At this point we can just apply a rough consequence of (4.3), |Fn,λ(s)| ≲n,λ e−σs, +to deduce +Cn,q +R,λ ≲n,λ +� R +0 +dre2σr +� R+r +0 +dseσ(2−q)s, +from which upper bounds coherent with the statement directly follow. Just as in +the high-frequency case, we only claimed exact asymptotics only for even q ≥ 2, +since lower bounds for odd q are once again made harder to derive by oscillations +of Fn,λ. +As for lower bounds for even q ≥ 2, starting from the change of variables of +Subsection 4.2, this time we replace R(r, θ) with its minimum R − r, +Cn,q +R,λ ≳ +� R +0 +� R−r +0 +Fn,λ(s)q sinh(s)2σ sinh(r)2σdsdr = +� R +0 +gR(s) sinh(s)2σFn,λ(s)qds, +gn,R(s) = +� R−s +0 +sinh(r)2σdr ≃n mn(BR−s) ≃n e2σ(R−s). +The case q ≥ 4 is the easier one: combining (4.3) and the last displayed formulae, +Cn,q +R,λ ≳n,λ e2σR +� R +0 +Re[cn(α) sinh(s)iα]q sinh(s)σqds ≃n,λ e2σR, +since the integrand consists in a positive power of a trigonometric function (as +opposed to the case q = 1 above) and a rapidly decreasing function, thus the +integral is overall O(1) as R → ∞. +The last estimate clearly holds true also for q = 2, but in that case it does not +capture the correct asymptotic behavior, due to the rough control from below in +replacing R(r, ϑ) �→ R−r. In this case we proceed more carefully: in the notation of +Subsection 4.2, an easy geometric argument reveals that there exists a solid angle6 +6Recall that, given a point x whose distance from the origin is r, R(r, ϑ) is the geodesic distance +of the boundary of BR from x in direction ϑ, so the solid angle Σ is centered around the geodesic +arc joining x and the origin. As R increases, one can actually take larger and larger Σ. + +36 +F. GROTTO AND G. PECCATI +Σ ⊂ Sn−1 such that R(r, θ) ≥ R for all r ∈ [0, R] and ϑ ∈ Σ, so that we can control +� +BR +dmn(x) +� +BR +dmn(y)Fn,λ(d(x, y))2 +≃n +� R +0 +sinh(r)2σdr +� +Sn−1 dςn−1(ϑ) +� R(r,ϑ) +0 +Fn,λ(s)2 sinh(s)2σds +≥ ςn−1(Σ) +� R +0 +sinh(r)2σdr +� R +0 +Fn,λ(s)2 sinh(s)2σds +≃n,λ mn(BR) +� R +0 +Re[cn(α) sinh(s)iα]2ds ≃n,λ R · mn(BR). +□ +Lemma 4.14. Let n ≥ 2, λ > σ2; as R → ∞ it holds +(4.19) +� +B2 +R +dm⊗2 +n (x, y) +� +1 − Fn,λ(d(x, y))2 − mn(BR)2 − 1 +2 Var hn,2 +R,λ ≲n,λ mn(BR). +Proof. We only provide a sketch of the proof: we can argue in the case R → ∞ just +like we did for obtaining Corollary 4.11, that is using the arguments of Lemma 4.6 +on single polyspectra applied to power series in Lemma 4.9. The thesis can be +rephrased as +� +B2 +R +g(Fn,λ(d(x, y)))dm⊗2 +n (x, y) ≲n,λ Cn,4 +R,λ, +g(x) = +∞ +� +ℓ=2 +�2ℓ +ℓ +�x2ℓ +4ℓ , +x ∈ (−1, 1). +Close to d(x, y) = 0 we can apply an expansion deduced from (A.2) and (4.8), +Fn,λ(r) = 1 − λ +2n sinh(r)2 + On,λ(sinh(r)4) +(keep in mind that λ here is fixed), from which convergence of the integral follows +(thanks to Remark 4.10). We can then restrict the integration domain to DR = +B2 +R ∖ {d(x, y) > 1}, since the latter provides the relevant contribution as R → +∞. By Lemma 4.1, 0 < Cn,λ ≤ Fn,λ(d(x, y)) ≤ C′ +n,λ < 1 for all (x, y) ∈ DR, +and moreover Fn,λ(r) ≲n,λ e−σr as r → ∞; this, together with the fact that +x−4g(x) is bounded on compacts of [0, 1), leads to the desired estimate with a +direct computation. +□ +Lemma 4.15. Let n ≥ 2, R > 0, p, q ∈ N0. As R → ∞ it holds +(4.20) +� +B4 +R +Fn,λ(d(x, y))pFn,λ(d(y, z))q +· Fn,λ(d(z, w))pFn,λ(d(w, x))qdm⊗4 +n (x, y, z, w) += +� +oλ(R2mn(BR)2), if p = q = 1, +oλ(mn(BR)2), +if p + q > 2. +Proof. As in the proof of Lemma 4.12, we denote by I(R) = In,λ,p,q(R) the left-hand +side of (4.20), we assume without loss of generality p ≤ q and drop dependencies of +constants on n, λ, p, q to lighten notation, and we apply GM-QM inequality to the +last two factors of the integrand: +I(R) ≲ +� +B4 +R +|Fn,λ(d(x, y))|p|Fn,λ(d(y, z))|q|Fn,λ(d(z, w))|2pdm⊗4 +n (x, y, z, w) + . . . , +where dots stand for an analogous term with higher exponents. Integration variables +can now be decoupled: moving to polar coordinates and recalling the uniform bound + +NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES +37 +|Fn,λ(r)| ≤ e−σr, r ≥ 0, we control +I(R) ≲ mn(BR) +�� R +0 +e−pσr sinh(r)σdr +� +· +�� R +0 +e−qσr sinh(r)σdr +� �� R +0 +e−2pσr sinh(r)σdr +� +. +Since p ≥ 1, the last factor is always O(R) as R → ∞. Similarly, the other integrals +into parentheses are O(R) or O(eσR) respectively when the exponent p, q equals 1 +or not. Thus, recalling that mn(BR) ≃ e2σR as R → ∞, our estimate suffices to +conclude the thesis, since it implies: +I(R) ≲ + + + + + +Re4σR +if p = q = 1 +R2e3σR +if 1 = p < 2 ≤ q +e2σR +if p, q ≥ 2 += +� +o(R2e4σR) +if p = q = 1 +o(e4σR) +otherwise +. +□ +Appendix A. Repository of Formulae for Special Functions +A.1. Bessel Functions. For ν ∈ [0, ∞), the Bessel function of first kind Jν(z) is +defined as, [48, 10.2.2], +(A.1) +Jν(z) = +�z +2 +�ν +∞ +� +k=0 +� +−z2/4 +�k +k!Γ(ν + k + 1), +where the power series is convergent and analytic on the whole C, and when ν is +not an integer the principal branch of zν (with branch cut (−∞, 0]) is customarily +chosen for the prefactor. The analytic function z−νJν(z) provides a representation +for the Fourier transform of the sphere as reported in Equation 1.1. +The modified Bessel function of first kind Iν (often appearing instead of Jν in the +references below) is simply obtained from Jν by the relation Iν(z) = i±νJν(i∓1z). +A.2. Hypergeometric Functions. For a, b, c ∈ C, the hypergeometric function +2F1(a, b, c, z) is defined as the analytic continuation of +2F1(a, b, c, z) = +∞ +� +n=0 +(a)n(bn) +(c)nn! zn, +|z| < 1, +(A.2) +(a)0 = 1, +(a)n = a(a + 1) · · · (a + n − 1), +on C ∖ [1, ∞], [48, 15.2.1]. The following linear transformation properties hold: for +z /∈ [0, ∞], [48, 15.8.2], +(A.3) +sin(π(b − a)) +πΓ(c) +2F1(a, b, c, z) += +(−z)−a +Γ(b)Γ(c − a)Γ(a − b + 1) 2F1(a, a − c + 1, a − b + 1, 1/z) +− +(−z)−b +Γ(a)Γ(c − b)Γ(b − a + 1) 2F1(b, b − c + 1, b − a + 1, 1/z), +and for z /∈ [1, ∞], [48, 15.8.1], +(A.4) +2F1(a, b, c, z) = (1 − z)−a2F1(a, c − b, c, z/(z − 1)) += (1 − z)−b2F1(c − a, b, c, z/(z − 1)) = (1 − z)c−a−b2F1(c − a, c − b, c, z). +The hypergeometric equation +(A.5) +z(1 − z)d2f +dz2 + (c − (a + b + 1)z) df +dz − abf = 0 + +38 +F. GROTTO AND G. PECCATI +is a complex-valued ODE with regular singularities at z = 0, 1, ∞. When c, a − +b, c−a−b are not integers, the above power series provides two linearly independent +solutions close to z = 0, +f1(z) = 2F1(a, b, c, z), +f2(z) = z1−c2F1(a − c + 1, b − c + 1, 2 − c, z), +and the aforementioned analytic extension of 2F1(a, b, c, z) solves A.5 on (−∞, 0]. +A.3. Relations with Legendre Functions. For a particular choice of parame- +ters, the hypergeometric function provides a representation for Legendre functions +(we are interested in particular to those of the first kind). For x ∈ (0, ∞), µ, ν ∈ C, +[48, 14.3.6] and [21, pag. 122, (3)], +(A.6) +P µ +ν (x) = +1 +Γ(1 − µ) +�x + 1 +x − 1 +�µ/2 +2F1(ν + 1, −ν, 1 − µ, (1 − x)/2). +We have the following integral representation for the Legendre function P µ +ν when +Re µ < 1/2, r > 0, [21, pag. 156, (7)], +(A.7) +P µ +ν (cosh(r)) = +2µ +√π sinh(r)µΓ(1/2 − µ) +� π +0 +(cosh r + sinh r cos t)µ+ν +(sin t)−2µ +dt. +For fixed µ ∈ R+, as ν → ∞, [48, 14.15.13] +(A.8) +P −µ +ν +(cosh r) = +1 +(iν)µ +� +r +sinh r +�1/2 +Jµ +�� +ν + 1 +2 +� +ir +� � +1 + O +� 1 +ν +�� +uniformly in r ∈ (0, ∞). +References +[1] Miklos Abert, Nicolas Bergeron, and Etienne Le Masson. Eigenfunctions and random waves +in the benjamini-schramm limit. October 2018. +[2] Milton Abramowitz, editor. 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Math. +Soc., Providence, RI, 2009. + diff --git a/VtE_T4oBgHgl3EQfyBw5/content/tmp_files/load_file.txt b/VtE_T4oBgHgl3EQfyBw5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cfe814928abfc5af125323140bc204a5a688edb7 --- /dev/null +++ b/VtE_T4oBgHgl3EQfyBw5/content/tmp_files/load_file.txt @@ -0,0 +1,1696 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf,len=1695 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='08315v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='PR] 19 Jan 2023 NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES: THE WIENER CHAOS APPROACH FRANCESCO GROTTO AND GIOVANNI PECCATI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We consider Gaussian random waves on hyperbolic spaces and establish variance asymptotics and central limit theorems for a large class of their integral functionals, both in the high-frequency and large domain limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Our strategy of proof relies on a fine analysis of Wiener chaos expansions, which in turn requires us to analytically assess the fluctuations of integrals involving mixed moments of covariance kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Our results complement several recent findings on non-linear transforms of planar and arithmetic random waves, as well as of random spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In the particular case of 2-dimensional hyperbolic spaces, our analysis reveals an intriguing discrepancy between the high-frequency and large domain fluctuations of the so-called fourth polyspectra — a phenomenon that has no counterpart in the Euclidean setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We develop applications of a geometric flavor, most notably to excursion volumes and occupation densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Overview 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' First definitions 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Motivation and background 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Main contributions 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Structure 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Notation 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Acknowledgments 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Geometry of Hyperbolic Space and Random Waves 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Spectral Theory of Hyperbolic Space 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Waves on Hyperbolic Spaces 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Hyperbolic Random Waves 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Curvature, Large Scale and Local Behavior of Random Waves 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Integral Functionals: Wiener Chaos Expansion and CLTs 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Some elements of Gaussian analysis 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' CLTs for integral functionals 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' First application: excursion volumes at non-zero levels 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Second application: Leray measures of nodal sets 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Covariance Functions and their Moments 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Approximating Covariance Functions 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Double Integrals on Hyperbolic Balls 29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Asymptotics for large λ, fixed R 29 Universit`a di Pisa, Dipartimento di Matematica, 5 Largo Bruno Pontecorvo, 56127 Pisa, Italia Universit´e du Luxembourg, Maison du Nombre, 6 Avenue de la Fonte, 4364 Esch-sur- Alzette, Luxembourg E-mail addresses: francesco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='grotto at unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='it, giovanni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='peccati at uni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Date: January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Random Waves, Hyperbolic Space, Wiener Chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 1 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' GROTTO AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' PECCATI 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Asymptotics for large R, fixed λ 34 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Repository of Formulae for Special Functions 37 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Bessel Functions 37 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Hypergeometric Functions 37 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Relations with Legendre Functions 38 References 38 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The aim of this paper is to initiate the study of non-linear func- tionals of Gaussian random waves (that is, generalized Gaussian eigenfunctions of the Laplacian) defined on hyperbolic spaces of arbitrary dimension — with specific emphasis on variance asymptotics and central limit theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' As put forward in the title, our approach is based on a careful analysis of Wiener chaos expansions, which we implement by using several non-trivial refinements of the general theory developed in [45, 46], see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' One of the main contributions of our work is the derivation of new analytic estimates for covariance kernels of hyperbolic waves (stated in Section 4), which will allow us to deal simultaneously both with the high-frequency and large domain asymptotic regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We will see that our findings naturally complement several recent studies of Gaussian random waves on mani- folds, such as Euclidean random waves [8, 9, 17, 16, 47, 51, 43], arithmetic random waves [12, 18, 32, 49, 53] and random spherical harmonics [39, 38, 34, 40, 36, 37, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' First definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Denote by Hn, n ≥ 2, the n-dimensional hyperbolic space (that is, the simply connected manifold with constant negative sectional curvature) and let λ ≥ (n − 1)2/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The hyperbolic random wave with frequency λ, written uλ := {uλ(x) : x ∈ Hn}, is defined as the unique (in distribution) centered and unit variance real Gaussian field on Hn such that (i) the law of uλ is invariant with respect to the isometries of Hn, (ii) paths of uλ solve a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' the Laplace-Beltrami eigenvalue problem ∆Hnuλ + λuλ = 0, where ∆Hn is the hyperbolic Laplacian (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The random wave uλ is the exact hyperbolic counterpart of the Euclidean ran- dom wave vλ := {vλ(x) : x ∈ Rn} (see [8, 9]), that one can similarly characterize as being the unique centered and unit variance real Gaussian field on Rn veri- fying properties (i) and (ii) above, with Hn and ∆Hn replaced, respectively, by Rn and ∆ = − � i ∂2/∂x2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Further remarkable examples of non-Euclidean ran- dom waves, to which uλ should be compared, are the already discussed random spherical harmonics and arithmetic random waves (that are, respectively, Laplace eigenfunctions on the sphere Sn and on the flat torus Tn), as well as the class of Gaussian monochromatic random waves on general compact manifolds [14, 19, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We also recall that hyperbolic random waves appeared in another guise in [15], in the context of spectral decomposition of stationary Gaussian fields on Hn (the latter as a particular case of homegeneous space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (i) For future reference, we recall that (as an application e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' of [3, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2]) the above characterization of vλ is equivalent to requiring that, for x, y ∈ Rn, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1) E [vλ(x)vλ(y)] = Cn,λ(x, y) := 1 ωn−1 � Sn−1 ei √ λu·(x−y)du = (2π)n/2 ωn−1 �√ λ|x − y| �1−n/2 Jn/2−1( √ λ|x − y|), NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES 3 where ωn−1 is the hypersurface volume of Sn−1 and Jν is the Bessel function of order ν (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' [48, Chapter 10]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' we also point out that an analogous representation of the covariance of uλ will emerge from the statement of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (ii) The central role played by Euclidean random waves in the probabilistic analysis of Laplace eigenfunctions is amplified by the so-called Berry’s ran- dom wave conjecture — originally formulated in [8] — according to which the unit energy random wave v1 is a universal model for the high-frequency local behavior of deterministic Laplace eigenfunctions on chaotic billiards, among which negatively curved manifolds are paradigmatic examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We refer the reader to [28] for a discussion of the role of random wave models in the physical literature, and to [1, 27] for mathematically rigorous ap- proaches toward Berry’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' See also [14, 19], as well as Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Motivation and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' As discussed e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' in the survey [59] (to which we refer the reader for an exhaustive list of references), in recent years considerable attention has been devoted to local geometric functionals associated with level sets of random waves, such as excursion volumes, occupation measures and volumes of level sets — among which nodal volumes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=', the Haussdorff measures of zero loci) play a pivotal role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' A remarkable phenomenon is that in a number of crucial cases (see [13, 37, 36, 35, 47, 50, 51, 43] for a sample) the study of these geometric functionals can be fruitfully reduced to the asymptotic analysis of their orthogonal projections on Wiener chaoses, as formally defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Such a strategy — which corresponds to the “Wiener chaos approach” advertised in the title — is described in detail in the forthcoming Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3 and relies pervasively on the abstract theory of probabilistic approximations presented in [46];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' see also [31, 54, 55] for some earlier use of Wiener chaos in the geometric study of Euclidean Gaussian fields1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The main contribution of our work consists in the first explicit application of Wiener chaos techniques to a class of integral functionals associated with non- Euclidean random waves on non-compact manifolds, thus setting the bases for the asymptotic analysis of more general geometric quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Moreover, one could regard this as a natural first step towards the analysis of Berry’s conjecture for classical chaotic billiards, such as Artin’s billiard (the geodesic flow on modular surface H2/ PSL(2, Z)) and more general non-compact hyperbolic surfaces, of which the hyperbolic plane is the universal cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' However, in the latter setting the spectral theory of Laplace operator is much more complicated than on H2: we refer to [7] for a proper overview on the topic, in particular Chapter 9 concerning the high-frequency limit and Quantum Unique Ergodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' It is not clear to us whether information on random combinations of Laplace eigenfunctions on H2 can actually give any insight on their analogs on other hyperbolic surfaces, so we leave this as an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The principal focus of our paper is on integral func- tionals of the form G(uλ) = � BR G(uλ(x))dmn(x), BR ⊂ Hn, G : R → R, 1Here, an important caveat is that the covariance structure of random waves typically does not satisfy the integrability assumptions required in order to directly apply the results from [31, 54, 55], in such a way that, for random waves, several ad hoc arguments have to be developed on a case- by-case basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' GROTTO AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' PECCATI where mn is the hyperbolic volume, and BR is a ball of radius R in the hyperbolic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Most of our efforts will be devoted to the study of those functionals G(uλ) (typically called polyspectra) obtained by taking G to be a Hermite polynomial of a fixed order, whose behavior is investigated in two different limiting regimes: high- energy (λ → ∞) and large domain (R → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Our main results, stated in full detail in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2, yield variance estimates and Central Limit Theorems (CLTs) for polyspectra of arbitrary orders, from which one can deduce CLTs for functionals G(uλ) associated with a generic G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='6 — which is one of the main contributions of the present work — will reveal an interesting phenomenon, namely: whereas in the high- frequency regime the asymptotic behavior of hyperbolic and Euclidean polyspectra roughly coincide, the same conclusion does not hold in the large-domain limit in the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In the parlance of time-series analysis, such a result seems to indicate that, unlike Euclidean random waves, hyperbolic random waves on H2 display a form of short memory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' [20, 44] for an introduction to this concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4, we will apply our results to study two remarkable func- tionals associated with the excursions of uλ: (a) the volume of the excursion set (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2) mn ({x ∈ BR : uλ(x) > t}) = � BR 1uλ>t(x)dmn(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (b) the Leray measure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3) LR,λ := lim ε→0 1 2ε |{x ∈ BR : |uλ(x)| ≤ ε}| , which can be formally understood as the integral of a generalized function, as follows: LR,λ = � BR δ0(uλ(x))dmn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' With probability one, the nodal set of uλ is a submanifold of codimen- sion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' As a consequence a – perhaps more natural – local functional to consider is the induced (n − 1)-dimensional volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' However, a functional such as the nodal length in dimension n = 2, (formally) given by length({x ∈ BR : uλ(x) = 0}) = � BR δ0(uλ(x)) � ⟨duλ, duλ⟩T ∗ x Hndmn(x), also involves the differential duλ of the random field, making its study not directly achievable by the techniques of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We prefer to regard such an issue as a separate topic and defer it to future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In Section 2, we recall the necessary preliminaries on geometry and spectral theory of Hn, and then rigorously introduce the hyperbolic random wave model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Section 3 contains a discussion of our main results on functionals of random waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Finally, Section 4 is devoted to the technical core of the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We write X ∼ Y when random variables X, Y –taking values in the same space– have the same law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We write N(α, β2) to indicate a Gaussian random variable with mean α ∈ R and variance β2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The term distribution will always refer to an element of the dual space of smooth functions on some manifold, that is a generalized function, never to the law of a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Landau O’s and o’s have their usual meaning, subscripts indicating eventual dependence on parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The symbol C will denote a positive constant, possibly differing in any of its occurrence even in the same formula, depending only on eventual subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES 5 The expression A ≃a,b B indicates that B is both an upper and lower bound by A up to strictly positive multiplicative constants depending only on eventual subscripts a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Expressions A ≲a,b B or A ≳a,b B indicate respectively an upper and lower bound in the same sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Research supported by the Luxembourg National Re- search Fund (Grant: O21/16236290/HDSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' acknowledges support of INdAM through the INdAM-GNAMPA Project CUP E55F22000270001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Geometry of Hyperbolic Space and Random Waves The hyperbolic space Hn is the simply connected n-dimensional Riemannian manifold of constant negative curvature −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' It is modeled by one sheet of the two-sheeted hyperboloid x2 0 − x2 1 − · · · − x2 n = 1 in Rn+1, say x0 > 0, with the Riemannian metric being induced by Minkowski metric −dx2 0 + dx2 1 + · · · + dx2 n on the ambient space Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The Riemannian distance in this parametrization Hn ∋ x = (x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' , xn) is given by d(x, y) = cosh−1 [x, y] , [x, y] = x0y0 − x1y1 − · · · − xnyn, x, y ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We will denote by dmn the Riemannian volume on Hn, or rather an arbitrarily fixed positive multiple of it, such choice being completely irrelevant for our goals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' accordingly, we will write for simplicity L2(Hn) = L2(Hn, dmn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Besides Cartesian coordinates of the hyperboloid model, we will often employ polar (geodesic) coordinates Hn ∋ x = (r, ϑ), r = d(x, x0) > 0, ϑ ∈ Sn−1, around a given point x0 ∈ Hn, in terms of which the volume element is given by dmn(x) = cn sinh(r)n−1drdςn−1(ϑ) with dςn−1(ϑ) denoting2 the volume form on the sphere Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We will also employ the usual notation ωn = � Sn dςn, with ω0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The content of the forthcoming Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2 is classical: the reader is referred to [57, Section 4] and [26, Section 2] for definitions, proofs, and examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Spectral Theory of Hyperbolic Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We will denote by ∆ = ∆Hn the Laplace-Beltrami operator on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Since the metric of Hn is induced by the embed- ding into Minkowski space, we have a convenient representation of the Laplacian on Hn in terms of the d’Alembert operator □ = −∂2/∂x2 0 + ∂2/∂x2 1 + · · · + ∂2/∂x2 n on the ambient space Rn+1 ⊃ Hn, that is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1) ∆Hnf = □ f � x/ � [x, x] ���� x∈Hn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We recall the spectral theorem on the hyperbolic space: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' [26, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='11, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='12] The Laplace-Beltrami operator ∆ on Hn, regarded as an unbounded operator on L2(Hn) densely defined on smooth functions, is essentially self-adjoint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' its spectrum is purely absolutely continuous and given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2) ��n − 1 2 �2 , ∞ � = � λ = σ2 + α2, σ = σn = n − 1 2 , α ∈ R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 2We prefer the graphic variant ς (‘final sigma’) since the symbol σ is customarily used to param- etrize the Laplacian’s spectrum, see the subsequent Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' GROTTO AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' PECCATI The projection operator on the eigenspace relative to λ = σ2 + α2 is given by Pλf(z) = ωn−1ρn(α) � Hn Fn,λ(d(x, y))f(y)dmn(y), f ∈ L2(Hn), ρn(α) = 1 (2π)n ���� Γ(σ + iα) Γ(α) ���� 2 , which is expressed in terms of the so-called spherical function [57, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3)] Fn,λ(d(x, y)) = 1 ωn−1 � Sn−1 [x, (1, ϑ)]−σ+iα [y, (1, ϑ)]−σ−iα dςn−1(ϑ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3) Fn,λ(r) = ωn−2 ωn−1 � π 0 (cosh r − sinh r cos θ)−σ+iα (sin θ)n−2dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4) The projection operators Pλ naturally satisfy f(x) = � ∞ 0 Pσ2+α2f(x)dα [57, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2)], and the function ρn(α) is thus the spectral measure (see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Spherical functions take such a name because ψ(x, y) = Fn,λ(d(x, y)) is the unique (real) radial solution of the eigenvalue problem ∆xψ(x, y) = λψ(x, y), ψ(x, x) = 1, x, y ∈ Hn, where ∆x indicates an application of the Laplacian ∆Hn to the mapping x �→ ψ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In what follows, we will use Fn,λ(d(x, y)) as the covariance function of a Gaussian field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In fact, formulas (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4) define positive-definite functions also when α = 0 and σ ∈ (0, (n−1)/2] [15, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3], thus one can consider Gaussian fields with such covariance for this additional choice of parameters, see [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We leave the study of these fields open for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3 (Notational).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Throughout the paper, the parameters λ, n, σ, α will al- ways be related through the relations put forward in formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In particular, dependence n can be given in terms of σ only and, and given n, a dependence on λ can be given in terms of α2 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3) is the prototypical example of this situation: it is easy to observe that the right-hand side does not depend on the sign of α, so overall dependence is on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Writing the eigenvalue problem in polar coordinates [15, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='6)] one readily obtains the following ODE satisfied by Fn,λ: d2 dr2 Fn,λ(r) + n − 1 tanh r d dr Fn,λ(r) + λFn,λ(r) = 0, r > 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='5) Fn,λ(0) = 1, F ′ n,λ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' As we recall in Section 4, solutions of such ODE can be represented with hyperge- ometric functions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' together with the integral representation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4) this will allow us to obtain good approximations on Fn,λ on which our arguments heavily rely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Waves on Hyperbolic Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' As recalled in the Introduction, the class of Euclidean plane waves is the collection of all exponential functions x �→ eix·k, k ∈ Rd, and that each of them trivially verifies the Laplace equation ∆Rdeix·k = |k|2eix·k, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='6) Euclidean plane waves are generalized eigenfunctions of the Laplacian ∆Rd = − �d ∂2 j , in the sense that they are smooth functions satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='6) but they do not belong to L2(Rd) (in which we set spectral theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Plane waves as above are indexed by k ∈ Rd, or equivalently by their wavenumber |k| ∈ R+ (indicating the relative eigenvalue |k|2) and the direction of the wave k/|k| ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES 7 On the hyperbolic space Hn, one actually has a perfect analog of plane random waves, that is obtained (for each n ≥ 2) by considering the smooth functions x �→ en(x, α, u) derived from the following mappings on the product space Hn × R∗ × Sn−1: en : Hn × R∗ × Sn−1 → C, en(x, α, u) = [x, (1, u)]−σ+iα , in such a way that the following equation is satisfied: ∆Hn en(x, α, u) = (σ2 + α2) en(x, α, u), x ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7) Note that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7) the operator ∆Hn is applied to the variable x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' the formula can be directly checked by applying the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1) to en(x, α, u) and carrying through the tedious but elementary computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The functions en(·, α, u) are thus generalized eigenfunctions of the Laplace-Beltrami operator ∆ = ∆Hn, and they are parametrized by the wavenumber α ∈ R and the “direction” of the wave u ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The analogy with the Euclidean case is perhaps more geometrically intuitive in the case n = 2, once one moves to the disk model of the hyperbolic plane: in such a setting, e2 is rewritten as an imaginary exponential involving the distance between the horocycle through x and u ∈ S1 and the origin, the direction u ∈ S1 being naturally identified with a point of the boundary of the Poincar´e disk (a point at infinity of the hyperbolic plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We refer to [24, Introduction] for a thorough comparison between Euclidean and hyperbolic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We also observe that the analogy with the Euclidean case carries through when considering wave equations, justifying the “wave” terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In particular, solutions of the wave equation on Hn, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='8) � ∂2 t + ∆Hn − σ2� u(x) = 0, (see [5] for a discussion of this PDE) can be written as superpositions of waves eitα en(x, α, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Notice that the wave operator in the previous display takes into account that the spectrum of ∆Hn begins at σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Just as on Rn, planar waves can be used to set up Fourier analysis on Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' [26, Ssec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4] Given f ∈ C∞ c (Hn), define its Fourier trans- form as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='9) Ff(α, ϑ) = � Hn en(x, −α, ϑ)f(x)dmn(x), α ∈ [0, ∞), ϑ ∈ Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' It holds (transform inversion) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='10) f(x) = � ∞ 0 � Sn−1 Ff(α, ϑ) en(x, α, ϑ)ρn(α)dαdςn−1(ϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Moreover (Plancherel formula) F extends to an isometry F : L2(Hn, mn) → L2([0, ∞) × Sn−1, ρn(α)dαdςn−1), whose inverse is given by (the extension of) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Spherical functions can be regarded as spherical averages of waves en, Fn,λ(d(x, (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' , 0))) = 1 ωn−1 � Sn−1 en(x, α, ϑ)dςn−1(ϑ), (a special case of Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3) thus playing the role of Bessel functions in the Euclidean case — see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' GROTTO AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' PECCATI 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Hyperbolic Random Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In what follows we will consider both real- valued and complex-valued random fields;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' we refer to [25, Chapter 6] for a discussion of white noise analysis in the complex setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='6 (Real and complex white noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Before stating the main result of the present Section, and for the reader’s convenience, we recall the definition and basic properties of complex white noises, in the sense of [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Fix a finite mesure space (X, F, µ) and denote by L2(X, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' R) and L2(X, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' C), respectively, the associated L2 spaces of real- and complex-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' A (real) white noise on (X, F, µ) (often called an isonormal Gaussian process with intensity µ — see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' [46, Chapter 2]) is a centred real Gaussian family of the type U = {U(f) : f ∈ L2(X, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' R)} such that E [U(f)U(g)] = � X fg dµ, f, g ∈ L2(X, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' the definition of U is customarily extended to all f ∈ L2(X, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' C) by setting U(f) := U(Re(f)) + iU(Im(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' A complex white noise W on a finite measure space (X, µ) is a complex Gaussian family W = {W(f) : f ∈ L2(X, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' C)} having the law of U + iV , where U, V are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' real white noises on (X, F, µ), as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The following computational rules can be easily checked: for all f, g ∈ L2(X, µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' C), one has that E � W(f)W(g) � = � X fg dµ, E [W(f)W(g)] = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='11) E [Re[W(f)] Re[W(g)]] = � X [Re(f) Re(g) + Im(f) Im(g)]dµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='12) E [Re[W(f)] Im[W(g)]] = � X [Re(f) Im(g) − Im(f) Re(g)]dµ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='13) and the second and third equalities continue to hold when one switches the symbols ‘Re’ and ‘Im’ on both sides of each equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The next proposition singles out a class of stationary random fields that can be regarded as canonical Gaussian Laplace eigenfunctions on Hn — they will constitute our main object of study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Fix α ∈ [0, ∞), and set λ = σ2 + α2, where the constant σ2 is the same as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (1) There exists a unique (in law) random field uλ : Hn → R such that (i) uλ(x) is a Gaussian variable N(0, 1) for all x ∈ Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (ii) the law of uλ is invariant under isometries of Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (iii) almost all samples of uλ are generalized λ-eigenfunction of ∆Hn of class C∞(Hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The same conclusion holds for the complex version uC λ : Hn → C if at Point (i) one replaces the standard Gaussian random variable N(0, 1) with a standard complex Gaussian variable NC(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' (2) The Gaussian random field uλ is equivalently characterized by its mean and covariance function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='14) E [uλ(x)] = 0, E [uλ(x)uλ(y)] = Fλ(d(x, y)), for all x, y ∈ Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' samples of uλ are of class C∞(Hn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Moreover, 1 √ 2 Re uC λ and 1 √ 2 Im uC λ are two independent identically distributed real Gaussian ran- dom fields with the same law as uλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES 9 (3) We have the following representation: if W is a complex white noise on (Sn−1, ςn−1), then uC λ has the same law as the stochastic integral (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='15) uC λ(x) ∼ � Sn−1 en(x, α, ϑ)W(dϑ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The random fields uC λ are discussed – with different notation and from a slightly different perspective – in [15, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3], to which the reader is referred for further background material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' For the rest of the paper, we will refer to uλ and uC λ, respectively, as the real and complex hyperbolic random wave with eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' It is convenient to start by defining uC λ(x) by means of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='15) and show that it satisfies the properties put forward at Point (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' This will show in particular that the real-valued function (x, y) �→ Fn,λ(d(x, y)) is positive definite for all λ ∈ [σ2, ∞), so that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='14) uniquely identifies the law of a real Gaussian random field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' To prove that the properties at Point (2) are met by the random field in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='15), we start by observing that the Gaussianity of the stochastic integral is trivial, and so is the fact that E �� Sn−1 en(x, α, ϑ)W(dϑ) � = 0 for all x ∈ Hn and α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' As for the covariance, we deduce from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='11) that E �� Sn−1 en(x, α, ϑ)W(dϑ) � Sn−1 en(y, α, ϑ)W(dϑ) � = � Sn−1 en(x, α, ϑ) en(y, −α, ϑ)dςn−1(ϑ), which by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3) equals Fn,λ(d(z, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Moreover, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='13), combined with the fact that (by definition) Re[en(x, α, ϑ)] = Re[en(x, −α, ϑ)] and Im[en(x, α, ϑ)] = − Im[en(x, −α, ϑ)], show that 1 √ 2 Re uC λ and 1 √ 2 Im uC λ are two i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' centered real Gaussian random fields with covariance function Fn,λ(d(·, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' This shows that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='15) satisfies the properties at Point (2) (note that uλ and uC λ have paths of class C∞ because the covariance function of these fields is of class C∞: this implication is proved e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' in [41, Subsection A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='9] for Gaussian fields on Euclidean spaces, and it is straightforwardly adapted to the hyperbolic setting after composition with a (smooth, global) chart of Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The proof of the Theorem is concluded if we show the equivalence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='14) and of the properties listed at Point (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Let us assume that uλ verifies the properties at Point (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Then, by invariance under isometries (and since the isometry group of Hn acts transitively), the covariance function (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='16) C(z, w) = E [uλ(z)uλ(w)] = f(d(z, w)) only depends on the distance d(z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Since the samples of uλ are smooth general- ized eigenfunctions, we then deduce that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='17) ∆HnC(z, w) = E [∆Hnuλ(z)uλ(w)] = λE [uλ(z)uλ(w)] = λC(z, w), and from the discussion in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1, we conclude that C(z, w) = Fλ(d(z, w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The proof that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='14) implies the conditions at Point (1) follows from similar ar- guments, both in the real- and complex-valued cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' □ To further the analogy with Berry’s random waves on Rn, one can also derive the random field uλ with a Central Limit result for a superposition of finitely many generalized eigenfunctions of ∆Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' GROTTO AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' PECCATI Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Let α ∈ [0, ∞), λ = σ2 + α2 be fixed, and consider two in- dependent sequences of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' uniform random variables ϑ1, ϑ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' on Sn−1 and φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' on [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Define the following finite combination of hyperbolic waves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='18) uN λ (z) = 1 √ N N � j=1 eiφj en(x, α, ϑj), to be regarded as a random element of C∞(Hn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' As N → ∞, finite-dimensional distributions of uN λ converge in law to the ones of uC λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' A real analogue of the latter can be obtained by taking the real (or imaginary) part of all involved objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Notice how, in sight of Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1, this result repre- sents uλ as a stochastic superposition of single waves solving (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='8) with wavenumber α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' For fixed x ∈ Hn, uN λ (z) can be regarded as the duality coupling between the smooth function en(x, α, ·) ∈ C∞(Sn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' C) and the generalized function 1 √ N N � j=1 eiφjδbj(·) ∈ C∞(Sn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' C)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Since generalized functions eiφjδbj can be regarded as i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' random elements of the Sobolev space Hs(Sn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' C) for s < −n/2, the Central Limit Theorem for i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' variables in Hilbert spaces applies (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' [33, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1]), and the sum in display converges in law to complex white noise W on Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' The thesis then follows by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='7 considering couplings with en(x, α, ·) at finitely many distinct points x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Curvature, Large Scale and Local Behavior of Random Waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' As already discussed, the principal aim of the present paper is to characterize the fluctuations of integral functionals of the hyperbolic waves {uλ}, as defined in the previous Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='3, both as λ → ∞ on a fixed domain (high-frequency limit), and for fixed λ on expanding domains (large domain limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Our main achievements on the matter are discussed in full detail in the forthcoming Section 3: in particular, our findings will show some remarkable discrepancies between the large domain behaviours of hyperbolic and Euclidean polyspectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In order to develop some basic intuition on the relation between hyperbolic and Euclidean settings, in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='10 we will characterize the local behaviour of hyperbolic random waves around a fixed point – that we will encode in terms of the scaling limit of the associated pullback waves on tangent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Some preliminary considerations are, however, in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Remarks on scaling limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We start by pointing out a fundamental difference between the hyperbolic and Euclidean settings, that is: in the hyperbolic framework – and differently from the Euclidean one – there is no direct relation linking high- frequency and large distance limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' To see this, fix λ > 0 and recall the definition of the Euclidean random waves {vλ} introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Trivially, the fact that Cn,λ(x, y) is a function of √ λ|x−y| makes it so that for Euclidean random waves it is equivalent to consider limits at high frequency (for a fixed distance) and at large distance (at a fixed frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' We will see that this is not the case for (functionals of) hyperbolic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Indeed, the counterpart of scaling lengths on a Euclidean space is to consider a positive multiple of the metric tensor on a Riemannian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Namely, if M = (M, g) is a Riemannian manifold we set MR = (M, R2g), R > 0, a transfor- mation that amounts to multiply all distances by R: if x, y ∈ M, dM(x, y) = r, NONLINEAR FUNCTIONALS OF HYPERBOLIC RANDOM WAVES 11 then dMR(x, y) = Rr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Under this transformation, eigenvalues of Laplace-Beltrami operator are scaled by a factor 1/R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In the Euclidean case M = Rn, if φR λ (x, y) = φR λ (|x − y|) is the unique radial solution of ∆RφR λ (x, y) = λφR λ (x, y), φR λ (x, x) = 1, where ∆R = 1 R2 ∆ is the Laplace operator on Rn R, then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='19) φR λ (|x − y|) = φ1 R2λ(|x − y|) = CR2λ(x, y) = φ1 λ(R|x − y|), where the last equality is a consequence of the particular form of spherical functions on flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Consider now the hyperbolic case: a crucial difference is that Rn R, R > 0, are all isometric, whereas Hn R has sectional curvature −1/R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Looking at spherical functions, if ψR λ (x, y) = ψR λ (d(x, y)) is the unique radial solution of ∆Hn RψR λ (x, y) = λψR λ (x, y), ψR λ (x, x) = 1, then the first equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='19) still holds, ψR λ (d(x, y)) = ψ1 R2λ(d(x, y)) it being a general fact (notice that d(x, y) is the distance of Hn = Hn 1, not the rescaled one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' However, in sight of the last display and the previous paragraphs, we can write ψR λ (x, y) = Fn,R2λ(d(x, y)) = Cn � π 0 (cosh d(x, y) − sinh d(x, y) cos θ)−σ+iR√ λ−σ2/R2 (sin θ)n−2dθ, in terms of the function Fn,λ defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4), which makes it clear that ψR λ (d(x, y)) = ψ1 R2λ(d(x, y)) ̸= ψ1 λ(Rd(x, y)), marking the difference with the Euclidean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' A local scaling limit result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' In the light of the above discussion, a natural question is whether the local behavior of the hyperbolic waves uλ around a given point resembles that of Berry’s model at high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' This turns out to be the case — at least from the standpoint of covariance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Since the two models are defined on different manifolds, such a statement is made precise by comparing the planar random wave on Rn with covariance function as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1) and frequency λ = 1, and a properly rescaled pullback of uλ to the tangent space (at a given point x ∈ Hn) given by the exponential map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Let α ≥ 0 be fixed, set λ = σ2+α2 and fix x ∈ Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Consider the covariance function of the pullback random wave uλ(expx(·/ √ λ)) on TxHn ≃ Rn, CH n,λ(u, v) = Fλ � d � expx v √ λ , expx v′ √ λ �� , v, v′ ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Let rλ = o( √ λ) as λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content=' Then, recalling that Cn,λ denotes the covariance of Berry’s Euclidean model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE_T4oBgHgl3EQfyBw5/content/2301.08315v1.pdf'} +page_content='1), one has that sup v,v′∈R2:|v|,|v′|